The following pages are from the solutions manual for Business Finance. The chapters are out of order. Chapters 4 to 12 are in order, then the syllabus, then chapter one.
Chapter 4
Analysis of Financial Statements
Learning Objectives
After reading this chapter, students should be able to:
Explain why ratio analysis is usually the first step in the analysis of a company’s financial statements.
List the five groups of ratios, specify which ratios belong in each group, and explain what information each group gives us about the firm’s financial position.
State what trend analysis is, and why it is important.
Describe how the basic Du Pont equation is used, and how it may be modified to form the extended Du Pont equation, which includes the effect of financial leverage.
Explain "benchmarking" and its purpose.
List several limitations of ratio analysis.
Identify some of the problems with ROE that can arise when firms use it as a sole measure of performance.
Identify some of the qualitative factors that must be considered when evaluating a company’s financial performance.
Lecture Suggestions
Chapter 4 shows how financial statements are analyzed to determine firms’ strengths and weaknesses. On the basis of this information, management can take actions to exploit strengths and correct weaknesses.
At Florida, we find a significant difference in preparation between our accounting and nonaccounting students. The accountants are relatively familiar with financial statements, and they have covered in depth in their financial accounting course many of the ratios discussed in Chapter 4. We pitch our lectures to the nonaccountants, which means concentrating on the use of statements and ratios, and the "big picture," rather than on details such as seasonal adjustments and the effects of different accounting procedures. Details are important, but so are general principles, and there are courses other than the introductory finance course where details can be addressed.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 4, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 3 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
41 The emphasis of the various types of analysts is by no means uniform nor should it be. Management is interested in all types of ratios for two reasons. First, the ratios point out weaknesses that should be strengthened; second, management recognizes that the other parties are interested in all the ratios and that financial appearances must be kept up if the firm is to be regarded highly by creditors and equity investors. Equity investors (stockholders) are interested primarily in profitability, but they examine the other ratios to get information on the riskiness of equity commitments. Longterm creditors are more interested in the debt, TIE, and EBITDA coverage ratios, as well as the profitability ratios. Shortterm creditors emphasize liquidity and look most carefully at the current ratio.
42 The inventory turnover ratio is important to a grocery store because of the much larger inventory required and because some of that inventory is perishable. An insurance company would have no inventory to speak of since its line of business is selling insurance policies or other similar financial products—contracts written on paper and entered into between the company and the insured. This question demonstrates that the student should not take a routine approach to financial analysis but rather should examine the business that he or she is analyzing.
43 Given that sales have not changed, a decrease in the total assets turnover means that the company’s assets have increased. Also, the fact that the fixed assets turnover ratio remained constant implies that the company increased its current assets. Since the company’s current ratio increased, and yet, its cash and equivalents and DSO are unchanged means that the company has increased its inventories.
44 Differences in the amounts of assets necessary to generate a dollar of sales cause asset turnover ratios to vary among industries. For example, a steel company needs a greater number of dollars in assets to produce a dollar in sales than does a grocery store chain. Also, profit margins and turnover ratios may vary due to differences in the amount of expenses incurred to produce sales. For example, one would expect a grocery store chain to spend more per dollar of sales than does a steel company. Often, a large turnover will be associated with a low profit margin, and vice versa.
45 Inflation will cause earnings to increase, even if there is no increase in sales volume. Yet, the book value of the assets that produced the sales and the annual depreciation expense remain at historic values and do not reflect the actual cost of replacing those assets. Thus, ratios that compare current flows with historic values become distorted over time. For example, ROA will increase even though those assets are generating the same sales volume.
When comparing different companies, the age of the assets will greatly affect the ratios. Companies with assets that were purchased earlier will reflect lower asset values than those that purchased assets later at inflated prices. Two firms with similar physical assets and sales could have significantly different ROAs. Under inflation, ratios will also reflect differences in the way firms treat inventories. As can be seen, inflation affects both income statement and balance sheet items.
46 ROE, using the extended Du Pont equation, is the return on assets multiplied by the equity multiplier. The equity multiplier, defined as total assets divided by common equity, is a measure of debt utilization; the more debt a firm uses, the lower its equity, and the higher the equity multiplier. Thus, using more debt will increase the equity multiplier, resulting in a higher ROE.
47 a. Cash, receivables, and inventories, as well as current liabilities, vary over the year for firms with seasonal sales patterns. Therefore, those ratios that examine balance sheet figures will vary unless averages (monthly ones are best) are used.
b. Common equity is determined at a point in time, say December 31, 2005. Profits are earned over time, say during 2005. If a firm is growing rapidly, yearend equity will be much larger than beginningofyear equity, so the calculated rate of return on equity will be different depending on whether endofyear, beginningofyear, or average common equity is used as the denominator. Average common equity is conceptually the best figure to use. In public utility rate cases, people are reported to have deliberately used endofyear or beginningofyear equity to make returns on equity appear excessive or inadequate. Similar problems can arise when a firm is being evaluated.
48 Firms within the same industry may employ different accounting techniques that make it difficult to compare financial ratios. More fundamentally, comparisons may be misleading if firms in the same industry differ in their other investments. For example, comparing Pepsico and CocaCola may be misleading because apart from their soft drink business, Pepsi also owns other businesses, such as FritoLay.
49 The three components of the extended Du Pont equation are profit margin, assets turnover, and the equity multiplier. One would not expect the three components of the discount merchandiser and highend merchandiser to be the same even though their ROEs are identical. The discount merchandiser’s profit margin would be lower than the highend merchandiser, while the assets turnover would be higher for the discount merchandiser than for the highend merchandiser.
410 Total Current Effect on
Current Assets Ratio Net Income
a. Cash is acquired through issuance of additional
common stock. + + 0
b. Merchandise is sold for cash. + + +
c. Federal income tax due for the previous year is paid. – + 0
d. A fixed asset is sold for less than book value. + + –
e. A fixed asset is sold for more than book value. + + +
f. Merchandise is sold on credit. + + +
g. Payment is made to trade creditors for previous purchases. – + 0
h. A cash dividend is declared and paid. – – 0
i. Cash is obtained through shortterm bank loans. + – 0
j. Shortterm notes receivable are sold at a discount. – – –
k. Marketable securities are sold below cost. – – –
l. Advances are made to employees. 0 0 0
m. Current operating expenses are paid. – – –
n. Shortterm promissory notes are issued to trade creditors
in exchange for past due accounts payable. 0 0 0
o. 10year notes are issued to pay off accounts payable. 0 + 0
Total Current Effect on
Current Assets Ratio Net Income
p. A fully depreciated asset is retired. 0 0 0
q. Accounts receivable are collected. 0 0 0
r. Equipment is purchased with shortterm notes. 0 – 0
s. Merchandise is purchased on credit. + – 0
t. The estimated taxes payable are increased. 0 – –
Solutions of EndofChapter Problems
41 DSO = 40 days; S = $7,300,000; AR = ?
DSO =
40 =
40 = AR/$20,000
AR = $800,000.
42 A/E = 2.4; D/A = ?
43 ROA = 10%; PM = 2%; ROE = 15%; S/TA = ?; TA/E = ?
ROA = NI/A; PM = NI/S; ROE = NI/E.
ROA = PM ´ S/TA
NI/A = NI/S ´ S/TA
10% = 2% ´ S/TA
S/TA = TATO = 5.
ROE = PM ´ S/TA ´ TA/E
NI/E = NI/S ´ S/TA ´ TA/E
15% = 2% ´ 5 ´ TA/E
15% = 10% ´ TA/E
TA/E = EM = 1.5.
44 TA = $10,000,000,000; CL = $1,000,000,000; LT debt = $3,000,000,000; CE = $6,000,000,000; Shares outstanding = 800,000,000; P_{0} = $32; M/B = ?
Book value = = $7.50.
M/B = = 4.2667.
45 EPS = $2.00; CFPS = $300; P/CF = 8.0´ ; P/E = ?
P/CF = 8.0
P/$3.00 = 8.0
P = $24.00.
P/E = $24.00/$2.00 = 12.0´ .
46 PM = 2%; EM = 2.0; Sales = $100,000,000; Assets = $50,000,000; ROE = ?
ROE = PM ´ TATO ´ EM
= NI/S ´ S/TA ´ A/E
= 2% ´ $100,000,00/$50,000,000 ´ 2
= 8%.
47 Step 1: Calculate total assets from information given.
Sales = $6 million.
3.2´ = Sales/TA
3.2´ =
Assets = $1,875,000.
Step 2: Calculate net income.
There is 50% debt and 50% equity, so Equity = $1,875,000 ´ 0.5 = $937,500.
ROE = NI/S ´ S/TA ´ TA/E
0.12 = NI/$6,000,000 ´ 3.2 ´ $1,875,000/$937,500
0.12 =
$720,000 = 6.4NI
$112,500 = NI.
48 ROA = 8%; net income = $600,000; TA = ?
ROA =
8% =
TA = $7,500,000.
To calculate BEP, we still need EBIT. To calculate EBIT construct a partial income statement:
EBIT $1,148,077 ($225,000 + $923,077)
Interest 225,000 (Given)
EBT $ 923,077 $600,000/0.65
Taxes (35%) 323,077
NI $ 600,000
BEP =
=
= 0.1531 = 15.31%.
49 Stockholders’ equity = $3,750,000,000; M/B = 1.9; P = ?
Total market value = $3,750,000,000(1.9) = $7,125,000,000.
Market value per share = $7,125,000,000/50,000,000 = $142.50.
Alternative solution:
Stockholders’ equity = $3,750,000,000; Shares outstanding = 50,000,000; P = ?
Book value per share = $3,750,000,000/50,000,000 = $75.
Market value per share = $75(1.9) = $142.50.
410 We are given ROA = 3% and Sales/Total assets = 1.5´ .
From the basic Du Pont equation: ROA = Profit margin ´ Total assets turnover
3% = Profit margin(1.5)
Profit margin = 3%/1.5 = 2%.
We can also calculate the company’s debt ratio in a similar manner, given the facts of the problem. We are given ROA(NI/A) and ROE(NI/E); if we use the reciprocal of ROE we have the following equation:
Alternatively, using the extended Du Pont equation:
ROE = ROA ´ EM
5% = 3% ´ EM
EM = 5%/3% = 5/3 = TA/E.
Take reciprocal: E/TA = 3/5 = 60%; therefore, D/A = 1 – 0.60 = 0.40 = 40%.
Thus, the firm’s profit margin = 2% and its debt ratio = 40%.
411 TA = $30,000,000,000; EBIT/TA = 20%; TIE = 8; DA = $3,200,000,000; Lease payments = $2,000,000,000; Principal payments = $1,000,000,000; EBITDA coverage = ?
EBIT/$30,000,000,000 = 0.2
EBIT = $6,000,000,000.
8 = EBIT/INT
8 = $6,000,000,000/INT
INT = $750,000,000.
EBITDA = EBIT + DA
= $6,000,000,000 + $3,200,000,000
= $9,200,000,000.
EBITDA coverage ratio =
=
= = 2.9867.
412 TA = $12,000,000,000; T = 40%; EBIT/TA = 15%; ROA = 5%; TIE = ?
= 0.15
EBIT = $1,800,000,000.
= 0.05
NI = $600,000,000.
Now use the income statement format to determine interest so you can calculate the firm’s TIE ratio.
EBIT $1,800,000,000 See above.
INT 800,000,000
EBT $1,000,000,000 EBT = $600,000,000/0.6
Taxes (40%) 400,000,000
NI $ 600,000,000 See above.
TIE = EBIT/INT
= $1,800,000,000/$800,000,000
= 2.25.
413 TIE = EBIT/INT, so find EBIT and INT.
Interest = $500,000 ´ 0.1 = $50,000.
Net income = $2,000,000 ´ 0.05 = $100,000.
Pretax income (EBT) = $100,000/(1 – T) = $100,000/0.7 = $142,857.
EBIT = EBT + Interest = $142,857 + $50,000 = $192,857.
TIE = $192,857/$50,000 = 3.86´ .
414 ROE = Profit margin ´ TA turnover ´ Equity multiplier
= NI/Sales ´ Sales/TA ´ TA/Equity.
Now we need to determine the inputs for the extended Du Pont equation from the data that were given. On the left we set up an income statement, and we put numbers in it on the right:
Sales (given) $10,000,000
– Cost na
EBIT (given) $ 1,000,000
– INT (given) 300,000
EBT $ 700,000
– Taxes (34%) 238,000
NI $ 462,000
Now we can use some ratios to get some more data:
Total assets turnover = 2 = S/TA; TA = S/2 = $10,000,000/2 = $5,000,000.
D/A = 60%; so E/A = 40%; and, therefore,
Equity multiplier = TA/E = 1/(E/A) = 1/0.4 = 2.5.
Now we can complete the extended Du Pont equation to determine ROE:
ROE = $462,000/$10,000,000 ´ $10,000,000/$5,000,000 ´ 2.5 = 0.231 = 23.1%.
415 Currently, ROE is ROE_{1} = $15,000/$200,000 = 7.5%.
The current ratio will be set such that 2.5 = CA/CL. CL is $50,000, and it will not change, so we can solve to find the new level of current assets: CA = 2.5(CL) = 2.5($50,000) = $125,000. This is the level of current assets that will produce a current ratio of 2.5´ .
At present, current assets amount to $210,000, so they can be reduced by $210,000 – $125,000 = $85,000. If the $85,000 generated is used to retire common equity, then the new common equity balance will be $200,000 – $85,000 = $115,000.
Assuming that net income is unchanged, the new ROE will be ROE_{2} = $15,000/$115,000 = 13.04%. Therefore, ROE will increase by 13.04% – 7.50% = 5.54%.
The new CA level is $125,000; CL remain at $50,000; and the new Inventory level = $150,000 – $85,000 = $65,000. Thus, the new quick ratio is calculated as follows:
New quick ratio =
=
= 1.2´ .
416 Known data:
TA = $1,000,000; Int. rate = 8%; T = 40%; BEP = 0.2 = EBIT/Total assets, so EBIT = 0.2($1,000,000) = $200,000; D/A = 0.5 = 50%, so Equity = $500,000.
D/A = 0% D/A = 50%
EBIT $200,000 $200,000
Interest 0 40,000*
EBT $200,000 $160,000
Tax (40%) 80,000 64,000
NI $120,000 $ 96,000
ROE = = = 12% = 19.2%
Difference in ROE = 19.2% – 12.0% = 7.2%.
*If D/A = 50%, then half of the assets are financed by debt, so Debt = $500,000. At an 8% interest rate, INT = $40,000.
417 Statement a is correct. Refer to the solution setup for Problem 416 and think about it this way: (1) Adding assets will not affect common equity if the assets are financed with debt. (2) Adding assets will cause expected EBIT to increase by the amount EBIT = BEP(added assets). (3) Interest expense will increase by the amount Int. rate(added assets). (4) Pretax income will rise by the amount (added assets)(BEP – Int. rate). Assuming BEP > Int. rate, if pretax income increases so will net income. (5) If expected net income increases but common equity is held constant, then the expected ROE will also increase. Note that if Int. rate > BEP, then adding assets financed by debt would lower net income and thus the ROE. Therefore, Statement a is true—if assets financed by debt are added, and if the expected BEP on those assets exceeds the interest rate on debt, then the firm’s ROE will increase.
Statements b, c, and d are false, because the BEP ratio uses EBIT, which is calculated before the effects of taxes or interest charges are felt. Of course, Statement e is also false.
418 TA = $5,000,000,000; T = 40%; EBIT/TA = 10%; ROA = 5%; TIE ?
Now use the income statement format to determine interest so you can calculate the firm’s TIE ratio.
EBIT $500,000,000 See above.
INT 83,333,333
EBT $416,666,667 EBT = $250,000,000/0.6
Taxes (40%) 166,666,667
NI $250,000,000 See above.
TIE = EBIT/INT
= $500,000,000/$83,333,333
= 6.0.
419 Present current ratio = = 2.5.
Minimum current ratio = = 2.0.
$1,312,500 + D NP = $1,050,000 + 2D NP
D NP = $262,500.
Shortterm debt can increase by a maximum of $262,500 without violating a 2 to 1 current ratio, assuming that the entire increase in notes payable is used to increase current assets. Since we assumed that the additional funds would be used to increase inventory, the inventory account will increase to $637,500 and current assets will total $1,575,000, and current liabilities will total $787,500.
420 Step 1: Solve for current annual sales using the DSO equation:
55 = $750,000/(Sales/365)
55Sales = $273,750,000
Sales = $4,977,272.73.
Step 2: If sales fall by 15%, the new sales level will be $4,977,272.73(0.85) = $4,230,681.82. Again, using the DSO equation, solve for the new accounts receivable figure as follows:
35 = AR/($4,230,681.82/365)
35 = AR/$11,590.91
AR = $405,681.82 » $405,682.
421 The current EPS is $2,000,000/500,000 shares or $4.00. The current P/E ratio is then $40/$4 = 10.00. The new number of shares outstanding will be 650,000. Thus, the new EPS = $3,250,000/650,000 = $5.00. If the shares are selling for 10 times EPS, then they must be selling for $5.00(10) = $50.
422 1. Total debt = (0.50)(Total assets) = (0.50)($300,000) = $150,000.
2. Accounts payable = Total debt – Longterm debt = $150,000 – $60,000
= $90,000.
3. Common stock = – Debt – Retained earnings
= $300,000 – $150,000 – $97,500 = $52,500.
4. Sales = (1.5)(Total assets) = (1.5)($300,000) = $450,000.
5. Inventories = Sales/5 = $450,000/5 = $90,000.
6. Accounts receivable = (Sales/365)(DSO) = ($450,000/365)(36.5) = $45,000.
7. Cash + Accounts receivable + Inventories = (1.8)(Accounts payable)
Cash + $45,000 + $90,000 = (1.8)($90,000)
Cash + $135,000 = $162,000
Cash = $27,000.
8. Fixed assets = Total assets – (Cash + Accts rec. + Inventories)
= $300,000 – ($27,000 + $45,000 + $90,000)
= $138,000.
9. Cost of goods sold = (Sales)(1 – 0.25) = ($450,000)(0.75) = $337,500.
423 a. (Dollar amounts in thousands.)
Industry
Firm Average

= 

= 
1.98´ 
2.0´ 


= 

= 
1.25´ 
1.3´ 

DSO 
= 

= 

= 
76.3 days 
35 

= 

= 
6.66´ 
6.7´ 


= 

= 
1.70´ 
3.0´ 


= 

= 
1.7% 
1.2% 


= 

= 
2.9% 
3.6% 


= 

= 
7.6% 
9.0% 


= 

= 
61.9% 
60.0% 
b. For the firm,
ROE = PM ´ T.A. turnover ´ EM = 1.7% ´ 1.7 ´ = 7.6%.
For the industry, ROE = 1.2% ´ 3 ´ 2.5 = 9%.
Note: To find the industry ratio of assets to common equity, recognize that 1 – (Total debt/Total assets) = Common equity/Total assets. So, Common equity/Total assets = 40%, and 1/0.40 = 2.5 = Total assets/Common equity.
c. The firm’s days sales outstanding is more than twice as long as the industry average, indicating that the firm should tighten credit or enforce a more stringent collection policy. The total assets turnover ratio is well below the industry average so sales should be increased, assets decreased, or both. While the company’s profit margin is higher than the industry average, its other profitability ratios are low compared to the industry—net income should be higher given the amount of equity and assets. However, the company seems to be in an average liquidity position and financial leverage is similar to others in the industry.
d. If 2005 represents a period of supernormal growth for the firm, ratios based on this year will be distorted and a comparison between them and industry averages will have little meaning. Potential investors who look only at 2005 ratios will be misled, and a return to normal conditions in 2006 could hurt the firm’s stock price.
424 a. Industry
Firm Average
Current ratio 
= 

= 

= 
2.73´ 
2´ 

= 

= 

= 
30.00% 
30.00% 

= 

= 

= 
11´ 
7´ 

= 

= 

= 
9.46´ 
9´ 

= 

= 

= 
5´ 
10´ 
DSO 
= 

= 

= 
30.3 days 
24 

= 

= 

= 
5.41´ 
6´ 

= 

= 

= 
1.77´ 
3´ 
Profit margin 
= 

= 

= 
3.40% 
3.00% 

= 

= 

= 
6.00% 
9.00% 

= 
ROA ´ EM 
= 
6% ´ 1.4286 
= 
8.57% 
12.90% 
Alternatively, ROE = = = 8.57% » 8.6%.
b. ROE = Profit margin ´ Total assets turnover ´ Equity multiplier
= ´ ´
= ´ ´ = 3.4% ´ 1.77 ´ 1.4286 = 8.6%.
Firm Industry Comment
Profit margin 3.4% 3.0% Good
Total assets turnover 1.77´ 3.0´ Poor
Equity multiplier 1.4286 1.43* O.K.
* 1 – =
1 – 0.30 = 0.7
EM = = = 1.43.
Alternatively, EM = ROE/ROA = 12.9%/9% = 1.43.
c. Analysis of the extended Du Pont equation and the set of ratios shows that the turnover ratio of sales to assets is quite low. Either sales should be higher given the present level of assets, or the firm is carrying more assets than it needs to support its sales.
d. The comparison of inventory turnover ratios shows that other firms in the industry seem to be getting along with about half as much inventory per unit of sales as the firm. If the company’s inventory could be reduced, this would generate funds that could be used to retire debt, thus reducing interest charges and improving profits, and strengthening the debt position. There might also be some excess investment in fixed assets, perhaps indicative of excess capacity, as shown by a slightly lowerthanaverage fixed assets turnover ratio. However, this is not nearly as clearcut as the overinvestment in inventory.
e. If the firm had a sharp seasonal sales pattern, or if it grew rapidly during the year, many ratios might be distorted. Ratios involving cash, receivables, inventories, and current liabilities, as well as those based on sales, profits, and common equity, could be biased. It is possible to correct for such problems by using average rather than endofperiod figures.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
425
a. Corrigan's liquidity position has improved from 2004 to 2005; however, its current ratio is still below the industry average of 2.7.
b. Corrigan's inventory turnover, fixed assets turnover, and total assets turnover have improved from 2004 to 2005; however, they are still below industry averages. The firm's days sales outstanding has increased from 2004 to 2005—which is bad. In 2004, its DSO was close to the industry average. In 2005, its DSO is somewhat higher. If the firm's credit policy has not changed, it needs to look at its receivables and determine whether it has any uncollectibles. If it does have uncollectible receivables, this will make its current ratio look worse than what was calculated above.
c. Corrigan's debt ratio has increased from 2004 to 2005, which is bad. In 2004, its debt ratio was right at the industry average, but in 2005 it is higher than the industry average. Given its weak current and asset management ratios, the firm should strengthen its balance sheet by paying down liabilities.
d. Corrigan's profitability ratios have declined substantially from 2004 to 2005, and they are substantially below the industry averages. Corrigan needs to reduce its costs, increase sales, or both.
e. Corrigan's P/E ratio has increased from 2004 to 2005, but only because its net income has declined significantly from the prior year. Its P/CF ratio has declined from the prior year and is well below the industry average. These ratios reflect the same information as Corrigan's profitability ratios. Corrigan needs to reduce costs to increase profit, lower its debt ratio, increase sales, and improve its asset management.
f.
Looking at the extended Du Pont equation, Corrigan's profit margin is significantly lower than the industry average and it has declined substantially from 2004 to 2005. The firm's total assets turnover has improved slightly from 2004 to 2005, but it's still below the industry average. The firm's equity multiplier has increased from 2004 to 2005 and is higher than the industry average. This indicates that the firm's debt ratio is increasing and it is higher than the industry average.
Corrigan should increase its net income by reducing costs, lower its debt ratio, and improve its asset management by either using less assets for the same amount of sales or increase sales.
g. If Corrigan initiated costcutting measures, this would increase its net income. This would improve its profitability ratios and market value ratios. If Corrigan also reduced its levels of inventory, this would improve its current ratio—as this would reduce liabilities as well. This would also improve its inventory turnover and total assets turnover ratio. Reducing costs and lowering inventory would also improve its debt ratio.
Chapter 5
Financial Markets and Institutions
Learning Objectives
After reading this chapter, students should be able to:
Describe three ways in which the transfer of capital takes place.
List some of the many different types of financial markets, and identify several recent trends taking place in the financial markets.
Identify some of the most important money and capital market instruments, and list the characteristics of each.
Compare and contrast major financial institutions.
Distinguish between the two basic types of stock markets.
Identify the three classifications of stock market transactions.
Read stock quotations from a variety of sources/publications.
Briefly explain the Efficient Markets Hypothesis (EMH), identify the three levels of efficiency, and discuss the implications of market efficiency.
Briefly discuss behavioral finance and its impact on the support for EMH.
Lecture Suggestions
Chapter 5 presents an overview of financial markets and institutions and it leads right into the next chapter. Additionally, students have an interest in financial markets and institutions.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 5, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 2 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
51 The prices of goods and services must cover their costs. Costs include labor, materials, and capital. Capital costs to a borrower include a return to the saver who supplied the capital, plus a markup (called a "spread") for the financial intermediary that brings the saver and the borrower together. The more efficient the financial system, the lower the costs of intermediation, the lower the costs to the borrower, and, hence, the lower the prices of goods and services to consumers.
52 In a wellfunctioning economy, capital will flow efficiently from those who supply capital to those who demand it. This transfer of capital can take place in three different ways:
1. Direct transfers of money and securities occur when a business sells its stocks or bonds directly to savers, without going through any type of financial institution. The business delivers its securities to savers, who in turn give the firm the money it needs.
2. Transfers may also go through an investment banking house which underwrites the issue. An underwriter serves as a middleman and facilitates the issuance of securities. The company sells its stocks or bonds to the investment bank, which in turn sells these same securities to savers. The businesses’ securities and the savers’ money merely "pass through" the investment banking house.
3. Transfers can also be made through a financial intermediary. Here the intermediary obtains funds from savers in exchange for its own securities. The intermediary uses this money to buy and hold businesses’ securities. Intermediaries literally create new forms of capital. The existence of intermediaries greatly increases the efficiency of money and capital markets.
53 A primary market is the market in which corporations raise capital by issuing new securities. An initial public offering is a stock issue in which privately held firms go public. Therefore, an IPO would be an example of a primary market transaction.
54 A money market transaction occurs in the financial market in which funds are borrowed or loaned for short periods (less than one year). A capital market transaction occurs in the financial market in which stocks and intermediate—or longterm debt (one year or longer)—are issued.
a. A U.S. Treasury bill is an example of a money market transaction.
b. Longterm corporate bonds are examples of capital market transactions.
c. Common stocks are examples of capital market transactions.
d. Preferred stocks are examples of capital market transactions.
e. Dealer commercial paper is an example of a money market transaction.
55 It would be difficult for firms to raise capital. Thus, capital investment would slow down, unemployment would rise, the output of goods and services would fall, and, in general, our standard of living would decline.
56 Financial markets have experienced many changes during the last two decades. Technological advances in computers and telecommunications, along with the globalization of banking and commerce, have led to deregulation, and this has increased competition throughout the world. The result is a much more efficient, internationally linked market, but one that is far more complex than existed a few years ago. While these developments have been largely positive, they have also created problems for policy makers. Large amounts of capital move quickly around the world in response to changes in interest and exchange rates, and these movements can disrupt local institutions and economies.
Globalization has exposed the need for greater cooperation among regulators at the international level. Factors that complicate coordination include (1) the differing structures among nations’ banking and securities industries, (2) the trend in Europe toward financial services conglomerates, and (3) reluctance on the part of individual countries to give up control over their national monetary policies.
Another important trend in recent years has been the increased use of derivatives. The market for derivatives has grown faster than any other market in recent years, providing corporations with new opportunities but also exposing them to new risks. Derivatives can be used either to reduce risks or to speculate. It’s not clear whether recent innovations have "increased or decreased the inherent stability of the financial system."
57 The physical location exchanges are tangible physical entities. Each of the larger ones occupies its own building, has a limited number of members, and has an elected governing body. A dealer market is defined to include all facilities that are needed to conduct security transactions not made on the physical location exchanges. These facilities include (1) the relatively few dealers who hold inventories of these securities and who are said to "make a market" in these securities; (2) the thousands of brokers who act as agents in bringing the dealers together with investors; and (3) the computers, terminals, and electronic networks that provide a communication link between dealers and brokers.
58 The two leading stock markets today are the New York Stock Exchange (NYSE) and the Nasdaq stock market. The NYSE is a physical location exchange, while the Nasdaq is an electronic dealerbased market.
59 The three forms, or levels, of market efficiency are: weakform efficiency, semistrongform efficiency, and strongform efficiency. The weak form of the EMH states that all information contained in past stock price movements is fully reflected in current market prices. The semistrong form of the EMH states that current market prices reflect all publicly available information. The strong form of the EMH states that current market prices reflect all pertinent information, whether publicly available or privately held.
510 If the market is semistrongform efficient and the company announces a 1% increase when investors had expected it to announce a 10% earnings increase, you would expect the stock’s price to fall because the earnings increase was less than expected, which is Statement b. In fact, if the assumption were made that weakform efficiency existed then you would expect the stock’s price to increase slightly because the company had a slight increase in earnings, which is Statement a. If the assumption were made that strongform efficiency existed then you would expect the stock’s price to remain the same because earnings announcements have no effect because all information, whether publicly available or privately held, is already reflected in the stock price.
511 a. False; derivatives can be used either to reduce risks or to speculate.
b. True; hedge funds generally charge large fees, often a fixed amount plus 15% to 20% of the fund’s capital gains.
c. False; hedge funds are largely unregulated.
d. True; the NYSE is a physical location exchange with a tangible physical location that conducts auction markets in designated securities.
e. False; a larger bidask spread means the dealer will realize a higher profit.
f. False; the EMH does not assume that all investors are rational.
Chapter 6
Interest Rates
Learning Objectives
After reading this chapter, students should be able to:
Explain how capital is allocated in a supply/demand framework, and list the fundamental factors that affect the cost of money.
Write out two equations for the nominal, or quoted, interest rate, and briefly discuss each component.
Define what is meant by the term structure of interest rates, and graph a yield curve for a given set of data.
Explain what factors determine the shape of the yield curve.
Use the yield curve and the information embedded in it to estimate the market’s expectations regarding future inflation and risk.
List four additional factors that influence the level of interest rates and the slope of the yield curve.
Discuss country risk.
Briefly explain how interest rate levels affect business decisions.
Lecture Suggestions
Chapter 6 is important because it lays the groundwork for the following chapters. Additionally, students have a curiosity about interest rates, so this chapter stimulates their interest in the course.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 6, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 2 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
61 Regional mortgage rate differentials do exist, depending on supply/demand conditions in the different regions. However, relatively high rates in one region would attract capital from other regions, and the end result would be a differential that was just sufficient to cover the costs of effecting the transfer (perhaps ½ of one percentage point). Differentials are more likely in the residential mortgage market than the business loan market, and not at all likely for the large, nationwide firms, which will do their borrowing in the lowestcost money centers and thereby quickly equalize rates for large corporate loans. Interest rates are more competitive, making it easier for small borrowers, and borrowers in rural areas, to obtain lower cost loans.
62 Shortterm interest rates are more volatile because (1) the Fed operates mainly in the shortterm sector, hence Federal Reserve intervention has its major effect here, and (2) longterm interest rates reflect the average expected inflation rate over the next 20 to 30 years, and this average does not change as radically as yeartoyear expectations.
63 Interest rates will fall as the recession takes hold because (1) business borrowings will decrease and (2) the Fed will increase the money supply to stimulate the economy. Thus, it would be better to borrow shortterm now, and then to convert to longterm when rates have reached a cyclical low. Note, though, that this answer requires interest rate forecasting, which is extremely difficult to do with better than 50% accuracy.
64 a. If transfers between the two markets are costly, interest rates would be different in the two areas. Area Y, with the relatively young population, would have less in savings accumulation and stronger loan demand. Area O, with the relatively old population, would have more savings accumulation and weaker loan demand as the members of the older population have already purchased their houses and are less consumption oriented. Thus, supply/demand equilibrium would be at a higher rate of interest in Area Y.
b. Yes. Nationwide branching, and so forth, would reduce the cost of financial transfers between the areas. Thus, funds would flow from Area O with excess relative supply to Area Y with excess relative demand. This flow would increase the interest rate in Area O and decrease the interest rate in Y until the rates were roughly equal, the difference being the transfer cost.
65 A significant increase in productivity would raise the rate of return on producers’ investment, thus causing the investment curve (see Figure 61 in the textbook) to shift to the right. This would increase the amount of savings and investment in the economy, thus causing all interest rates to rise.
66 a. The immediate effect on the yield curve would be to lower interest rates in the shortterm end of the market, since the Fed deals primarily in that market segment. However, people would expect higher future inflation, which would raise longterm rates. The result would be a much steeper yield curve.
b. If the policy is maintained, the expanded money supply will result in increased rates of inflation and increased inflationary expectations. This will cause investors to increase the inflation premium on all debt securities, and the entire yield curve would rise; that is, all rates would be higher.
67 a. S&Ls would have a higher level of net income with a "normal" yield curve. In this situation their liabilities (deposits), which are shortterm, would have a lower cost than the returns being generated by their assets (mortgages), which are longterm. Thus, they would have a positive "spread."
b. It depends on the situation. A sharp increase in inflation would increase interest rates along the entire yield curve. If the increase were large, shortterm interest rates might be boosted above the longterm interest rates that prevailed prior to the inflation increase. Then, since the bulk of the fixedrate mortgages were initiated when interest rates were lower, the deposits (liabilities) of the S&Ls would cost more than the returns being provided on the assets. If this situation continued for any length of time, the equity (reserves) of the S&Ls would be drained to the point that only a "bailout" would prevent bankruptcy. This has indeed happened in the United States. Thus, in this situation the S&L industry would be better off selling their mortgages to federal agencies and collecting servicing fees rather than holding the mortgages they originated.
68 Treasury bonds, along with all other bonds, are available to investors as an alternative investment to common stocks. An increase in the return on Treasury bonds would increase the appeal of these bonds relative to common stocks, and some investors would sell their stocks to buy Tbonds. This would cause stock prices, in general, to fall. Another way to view this is that a relatively riskless investment (Tbonds) has increased its return by 4 percentage points. The return demanded on riskier investments (stocks) would also increase, thus driving down stock prices. The exact relationship will be discussed in Chapter 8 (with respect to risk) and Chapters 7 and 9 (with respect to price).
69 A trade deficit occurs when the U.S. buys more than it sells. In other words, a trade deficit occurs when the U.S. imports more than it exports. When trade deficits occur, they must be financed, and the main source of financing is debt. Therefore, the larger the U.S. trade deficit, the more the U.S. must borrow, and as the U.S. increases its borrowing, this drives up interest rates.
Solutions to EndofChapter Problems
61 a. Term Rate
6 months 5.1%
1 year 5.5
2 years 5.6
3 years 5.7
4 years 5.8
5 years 6.0
10 years 6.1
20 years 6.5
30 years 6.3
b. The yield curve shown is an upward sloping yield curve.
c. This yield curve tells us generally that either inflation is expected to increase or there is an increasing maturity risk premium.
d. It would make sense to borrow long term because each year the loan is renewed interest rates are higher. This exposes you to rollover risk. If you borrow for 30 years outright you have locked in a 6.3% interest rate each year.
62 Tbill rate = r* + IP
5.5% = r* + 3.25%
r* = 2.25%.
63 r* = 3%; I_{1} = 2%; I_{2} = 4%; I_{3} = 4%; MRP = 0; r_{T2} = ?; r_{T3} = ?
r = r* + IP + DRP + LP + MRP.
Since these are Treasury securities, DRP = LP = 0.
r_{T2} = r* + IP_{2}.
IP_{2} = (2% + 4%)/2 = 3%.
r_{T2} = 3% + 3% = 6%.
r_{T3} = r* + IP_{3}.
IP_{3} = (2% + 4% + 4%)/3 = 3.33%.
r_{T3} = 3% + 3.33% = 6.33%.
64 r_{T10} = 6%; r_{C10} = 8%; LP = 0.5%; DRP = ?
r = r* + IP + DRP + LP + MRP.
r_{T10} = 6% = r* + IP_{10} + MRP_{10}; DRP = LP = 0.
r_{C10} = 8% = r* + IP_{10} + DRP + 0.5% + MRP_{10}.
Because both bonds are 10year bonds the inflation premium and maturity risk premium on both bonds are equal. The only difference between them is the liquidity and default risk premiums.
r_{C10} = 8% = r* + IP + MRP + 0.5% + DRP. But we know from above that r* + IP_{10} + MRP_{10} = 6%; therefore,
r_{C10} = 8% = 6% + 0.5% + DRP
1.5% = DRP.
65 r* = 3%; IP_{2} = 3%; r_{T2} = 6.2%; MRP_{2} = ?
r_{T2} = r* + IP_{2} + MRP_{2} = 6.2%
r_{T2} = 3% + 3% + MRP_{2} = 6.2%
MRP_{2} = 0.2%.
66 r* = 5%; I_{14} = 16%; MRP = DRP = LP = 0; r_{4} = ?
r_{4} = r_{RF}.
r_{RF} = (1 + r*)(1 + I) – 1
= (1.05)(1.16) – 1
= 0.218 = 21.8%.
67 r_{T1} = 5%; _{1}r_{T1} = 6%; r_{T2} = ?
(1 + r_{T2})^{2} = (1.05)(1.06)
(1 + r_{T2})^{2} = 1.113
1 + r_{T2} = 1.055
r_{T2} = 5.5%.
68 Let X equal the yield on 2year securities 4 years from now:
(1.07)^{4}(1 + X)^{2} = (1.075)^{6}
(1.3108)(1 + X)^{2} = 1.5433
1 + X =
X = 8.5%.
69 r = r* + IP + MRP + DRP + LP.
r* = 0.03.
IP = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035.
MRP = 0.0005(6) = 0.003.
DRP = 0.
LP = 0.
r_{T7} = 0.03 + 0.035 + 0.003 = 0.068 = 6.8%.
610 Basic relevant equations:
r_{t} = r* + IP_{t} + DRP_{t} + MRP_{t} + IP_{t}.
But here IP_{t} is the only premium, so r_{t} = r* + IP_{t}.
IP_{t} = Avg. inflation = (I_{1} + I_{2} + . . .)/N.
We know that I_{1} = IP_{1} = 3% and r* = 2%. Therefore,
r_{T1} = 2% + 3% = 5%. r_{T3} = r_{T1} + 2% = 5% + 2% = 7%. But,
r_{T3} = r* + IP_{3} = 2% + IP_{3} = 7%, so
IP_{3} = 7% – 2% = 5%.
We also know that I_{t} = Constant after t = 1.
We can set up this table:
r* I Avg. I = IP_{t} r = r* + IP_{t}
1 2% 3% 3%/1 = 3% 5%
2 2% I (3% + I)/2 = IP_{2}
3 2% I (3% + I + I)/3 = IP_{3} r_{3} = 7%, so IP_{3} = 7% – 2% = 5%.
IP_{3} = (3% + 2I)/3 = 5%
2I = 12%
I = 6%.
611 We’re given all the components to determine the yield on the bonds except the default risk premium (DRP) and MRP. Calculate the MRP as 0.1%(5 – 1) = 0.4%. Now, we can solve for the DRP as follows:
7.75% = 2.3% + 2.5% + 0.4% + 1.0% + DRP, or DRP = 1.55%.
612 First, calculate the inflation premiums for the next three and five years, respectively. They are IP_{3} = (2.5% + 3.2% + 3.6%)/3 = 3.1% and IP_{5} = (2.5% + 3.2% + 3.6% + 3.6% + 3.6%)/5 = 3.3%. The real riskfree rate is given as 2.75%. Since the default and liquidity premiums are zero on Treasury bonds, we can now solve for the maturity risk premium. Thus, 6.25% = 2.75% + 3.1% + MRP_{3}, or MRP_{3} = 0.4%. Similarly, 6.8% = 2.75% + 3.3% + MRP_{5}, or MRP_{5} = 0.75%. Thus, MRP_{5} – MRP_{3} = 0.75% – 0.40% = 0.35%.
613 r_{C8} = r* + IP_{8} + MRP_{8} + DRP_{8} + LP_{8}
8.3% = 2.5% + (2.8% ´ 4 + 3.75% ´ 4)/8 + 0.0% + DRP_{8} + 0.75%
8.3% = 2.5% + 3.275% + 0.0% + DRP_{8} + 0.75%
8.3% = 6.525% + DRP_{8}
DRP_{8} = 1.775%.
614 a. (1.045)^{2} = (1.03)(1 + X)
1.092/1.03 = 1 + X
X = 6%.
b. For riskless bonds under the expectations
theory, the interest rate for a bond of any maturity is
r_{N} = r* + average inflation over N years. If r* = 1%, we can
solve for IP_{N}:
Year 1: r_{1} = 1% + I_{1} = 3%;
I_{1} = expected inflation = 3% – 1% = 2%.
Year 2: r_{1} = 1% + I_{2} = 6%;
I_{2} = expected inflation = 6% – 1% = 5%.
Note also that the average inflation rate is (2% + 5%)/2 = 3.5%, which, when added to r* = 1%, produces the yield on a 2year bond, 4.5%. Therefore, all of our results are consistent.
615 r* = 2%; MRP = 0%; r_{1} = 5%; r_{2} = 7%; X = ?
X represents the oneyear rate on a bond one year from now (Year 2).
(1.07)^{2} = (1.05)(1 + X)
= 1 + X
X = 9%.
9% = r* + I_{2}
9% = 2% + I_{2}
7% = I_{2}.
The average interest rate during the 2year period differs from the 1year interest rate expected for Year 2 because of the inflation rate reflected in the two interest rates. The inflation rate reflected in the interest rate on any security is the average rate of inflation expected over the security’s life.
616 r_{RF} = r_{6} = 20.84%; MRP = DRP = LP = 0; r* = 6%; I = ?
20.84% = (1.06)(1 + I) – 1
1.2084 = (1.06)(1 + I)
1.14 = 1 + I
0.14 = I.
617 r_{T5} = 5.2%; r_{T10} = 6.4%; r_{C10} = 8.4%; IP_{10} = 2.5%; MRP = 0. For Treasury securities, DRP = LP = 0.
DRP_{5} + LP_{5} = DRP_{10} + LP_{10}. r_{C5} = ?
r_{T10} = r* + IP_{10}
6.4% = r* + 2.5%
r* = 3.9%.
r_{T5} = r* + IP_{5}
5.2% = 3.9% + IP_{5}
1.3% = IP_{5}.
r_{C10} = r* + IP_{10} + DRP_{10} + LP_{10}
8.4% = 3.9% + 2.5% + DRP_{10} + LP_{10}
2% = DRP_{10} + LP_{10}.
r_{C5} = 3.9% + 1.3% + DRP_{5} + LP_{5}, but DRP_{5} + LP_{5} = DRP_{10} + LP_{10} = 2%. So,
r_{C5} = 3.9% + 1.3% + 2%
= 7.2%.
618 a. Years to Real RiskFree
Maturity Rate (r*) IP** MRP r_{T} = r* + IP + MRP
1 2% 7.00% 0.2% 9.20%
2 2 6.00 0.4 8.40
3 2 5.00 0.6 7.60
4 2 4.50 0.8 7.30
5 2 4.20 1.0 7.20
10 2 3.60 1.0 6.60
20 2 3.30 1.0 6.30
**The computation of the inflation premium is as follows:
Expected Average
Year Inflation Expected Inflation
1 7% 7.00%
2 5 6.00
3 3 5.00
4 3 4.50
5 3 4.20
10 3 3.60
20 3 3.30
For example, the calculation for 3 years is as follows:
= 5.00%.
Thus, the yield curve would be as follows:
b. The interest rate on the ExxonMobil bonds has the same components as the Treasury securities, except that the ExxonMobil bonds have default risk, so a default risk premium must be included. Therefore,
r_{ExxonMobil} = r* + IP + MRP + DRP.
For a strong company such as ExxonMobil, the default risk premium is virtually zero for shortterm bonds. However, as time to maturity increases, the probability of default, although still small, is sufficient to warrant a default premium. Thus, the yield risk curve for the ExxonMobil bonds will rise above the yield curve for the Treasury securities. In the graph, the default risk premium was assumed to be 1.0 percentage point on the 20year ExxonMobil bonds. The return should equal 6.3% + 1% = 7.3%.
c. Exelon bonds would have significantly more default risk than either Treasury securities or ExxonMobil bonds, and the risk of default would increase over time due to possible financial deterioration. In this example, the default risk premium was assumed to be 1.0 percentage point on the 1year Exelon bonds and 2.0 percentage points on the 20year bonds. The 20year return should equal 6.3% + 2% = 8.3%.
619 a. The average rate of inflation for the 5year period is calculated as:
= (0.13 + 0.09 + 0.07 + 0.06 + 0.06)/5 = 8.20%.
b. r = r* + IP_{Avg.} = 2% + 8.2% = 10.20%.
c. Here is the general situation:
Year 
1Year Expected Inflation 
Arithmetic Average Expected Inflation 
r* 
Maturity Risk Premium 
Estimated Interest Rates 
1 
13% 
13.0% 
2% 
0.1% 
15.1% 
2 
9 
11.0 
2 
0.2 
13.2 
3 
7 
9.7 
2 
0.3 
12.0 
5 
6 
8.2 
2 
0.5 
10.7 
. . . 
. . . 
. . . 
. . . 
. . . 
. . . 
10 
6 
7.1 
2 
1.0 
10.1 
20 
6 
6.6 
2 
2.0 
10.6 
d. The "normal" yield curve is upward sloping because, in "normal" times, inflation is not expected to trend either up or down, so IP is the same for debt of all maturities, but the MRP increases with years, so the yield curve slopes up. During a recession, the yield curve typically slopes up especially steeply, because inflation and consequently shortterm interest rates are currently low, yet people expect inflation and interest rates to rise as the economy comes out of the recession.
e. If inflation rates are expected to be constant, then the expectations theory holds that the yield curve should be horizontal. However, in this event it is likely that maturity risk premiums would be applied to longterm bonds because of the greater risks of holding longterm rather than shortterm bonds:
If maturity risk premiums were added to the yield curve in Part e above, then the yield curve would be more nearly normal; that is, the longterm end of the curve would be raised. (The yield curve shown in this answer is upward sloping; the yield curve shown in part c is downward sloping.)
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
620 a. 1. This action will increase the supply of money; therefore, interest rates will decline.
2. This action will cause interest rates to increase.
3. The larger the federal deficit, other things held constant, the higher the level of interest rates.
4. This expectation will cause interest rates to increase.
b.
c.
d. The real riskfree rate would be the same for the corporate and treasury bonds. Similarly, without information to the contrary, we would assume that the maturity and inflation premiums would be the same for bonds with the same maturities. However, the corporate bond would have a liquidity premium and a default premium. If we assume that these premiums are constant across maturities, then we can use the LP and DRP premiums as determined above and add them to the Tbond yields to find the corporate yields. This procedure was used in the table below.
Now we can graph the data in the first 3 columns of the above table to get the Treasury and corporate (Arated) yield curves:
Note that if we constructed yield curves for corporate bonds with other ratings, the higher the rating, the lower the curves would be. Note too that the DRP for different ratings can change over time as investors' (1) risk aversion and (2) perceptions of risk change, and this can lead to different yield spreads and curve positions. Expectations for inflation can also change, and this will lead to upward or downward shifts in all the yield curves.
e. Shortterm rates are more volatile than longerterm rates; therefore, the left side of the yield curve would be most volatile over time.
f. 1. The 1year rate, one year from now.
2. The 5year rate, five years from now.
3. The 10year rate, ten years from now.
4. The 10year rate, twenty years from now.
Chapter 7
Bonds and Their Valuation
Learning Objectives
After reading this chapter, students should be able to:
List the four main classifications of bonds and differentiate among them.
Identify the key characteristics common to all bonds.
Calculate the value of a bond with annual or semiannual interest payments.
Calculate the yield to maturity, the yield to call, and the current yield on a bond.
Explain why the market value of an outstanding fixedrate bond will fall when interest rates rise on new bonds of equal risk, or vice versa.
Differentiate between interest rate risk, reinvestment rate risk, and default risk.
List major types of corporate bonds and distinguish among them.
Explain the importance of bond ratings and list some of the criteria used to rate bonds.
Differentiate among the following terms: Insolvent, liquidation, and reorganization.
Read and understand the information provided on the bond market page of your newspaper.
Lecture Suggestions
This chapter serves two purposes. First, it provides important and useful information on bonds per se. Second, it provides a good example of the use of time value concepts, so it reinforces the topics covered in Chapter 2.
We begin our lecture with a discussion of the different types of bonds and their characteristics. Then we move on to how bond values are established, how yields are determined, the effects of changing interest rates on bond prices, and the riskiness inherent in different types of bonds.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 7, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 4 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
71 From the corporation’s viewpoint, one important factor in establishing a sinking fund is that its own bonds generally have a higher yield than do government bonds; hence, the company saves more interest by retiring its own bonds than it could earn by buying government bonds. This factor causes firms to favor the second procedure. Investors also would prefer the annual retirement procedure if they thought that interest rates were more likely to rise than to fall, but they would prefer the government bond purchase program if they thought rates were likely to fall. In addition, bondholders recognize that, under the government bond purchase scheme, each bondholder would be entitled to a given amount of cash from the liquidation of the sinking fund if the firm should go into default, whereas under the annual retirement plan, some of the holders would receive a cash benefit while others would benefit only indirectly from the fact that there would be fewer bonds outstanding.
On balance, investors seem to have little reason for choosing one method over the other, while the annual retirement method is clearly more beneficial to the firm. The consequence has been a pronounced trend toward annual retirement and away from the accumulation scheme.
72 Yes, the statement is true.
73 False. Shortterm bond prices are less sensitive than longterm bond prices to interest rate changes because funds invested in shortterm bonds can be reinvested at the new interest rate sooner than funds tied up in longterm bonds.
For example, consider two bonds, both with a 10% annual coupon and a $1,000 par value. The only difference between them is their maturity. One bond is a 1year bond, while the other is a 20year bond. Consider the values of each at 5%, 10%, 15%, and 20% interest rates.
1year 20year
5% $1,047.62 $1,623.11
10% 1,000.00 1,000.00
15% 956.52 687.03
20% 916.67 513.04
As you can see, the price of the 20year bond is much more volatile than the price of the 1year bond.
74 The price of the bond will fall and its YTM will rise if interest rates rise. If the bond still has a long term to maturity, its YTM will reflect longterm rates. Of course, the bond’s price will be less affected by a change in interest rates if it has been outstanding a long time and matures shortly. While this is true, it should be noted that the YTM will increase only for buyers who purchase the bond after the change in interest rates and not for buyers who purchased previous to the change. If the bond is purchased and held to maturity, the bondholder’s YTM will not change, regardless of what happens to interest rates. For example, consider two bonds with an 8% annual coupon and a $1,000 par value. One bond has a 5year maturity, while the other has a 20year maturity. If interest rates rise to 15% immediately after issue the value of the 5year bond would be $765.35, while the value of the 20year bond would be $561.85.
75 If interest rates decline significantly, the values of callable bonds will not rise by as much as those of bonds without the call provision. It is likely that the bonds would be called by the issuer before maturity, so that the issuer can take advantage of the new, lower rates.
76 As an investor with a short investment horizon, I would view the 20year Treasury security as being more risky than the 1year Treasury security. If I bought the 20year security, I would bear a considerable amount of interest rate risk. Since my investment horizon is only one year, I would have to sell the 20year security one year from now, and the price I would receive for it would depend on what happened to interest rates during that year. However, if I purchased the 1year security I would be assured of receiving my principal at the end of that one year, which is the 1year Treasury’s maturity date.
77 a. If a bond’s price increases, its YTM decreases.
b. If a company’s bonds are downgraded by the rating agencies, its YTM increases.
c. If a change in the bankruptcy code made it more difficult for bondholders to receive payments in the event a firm declared bankruptcy, then the bond’s YTM would increase.
d. If the economy entered a recession, then the possibility of a firm defaulting on its bond would increase; consequently, its YTM would increase.
e. If a bond were to become subordinated to another debt issue, then the bond’s YTM would increase.
78 If a company sold bonds when interest rates were relatively high and the issue is callable, then the company could sell a new issue of lowyielding securities if and when interest rates drop. The proceeds of the new issue would be used to retire the highrate issue, and thus reduce its interest expense. The call privilege is valuable to the firm but detrimental to longterm investors, who will be forced to reinvest the amount they receive at the new and lower rates.
79 A sinking fund provision facilitates the orderly retirement of the bond issue. Although sinking funds are designed to protect investors by ensuring that the bonds are retired in an orderly fashion, sinking funds can work to the detriment of bond holders. On balance, however, bonds that have a sinking fund are regarded as being safer than those without such a provision, so at the time they are issued sinking fund bonds have lower coupon rates than otherwise similar bonds without sinking funds.
710 A call for sinking fund purposes is quite different from a refunding call a sinking fund call requires no call premium, but only a small percentage of the issue is normally callable in a given year. A refunding call gives the issuer the right to call all the bond issue for redemption. The call provision generally states that the issuer must pay the bondholders an amount greater than the par value if they are called.
711 Convertibles and bonds with warrants are offered with lower coupons than similarlyrated straight bonds because both offer investors the chance for capital gains as compensation for the lower coupon rate. Convertible bonds are exchangeable into shares of common stock, at a fixed price, at the option of the bondholder. On the other hand, bonds issued with warrants are options that permit the holder to buy stock for a stated price, thereby providing a capital gain if the stock’s price rises.
712 This statement is false. Extremely strong companies can use debentures because they simply do not need to put up property as security for their debt. Debentures are also issued by weak companies that have already pledged most of their assets as collateral for mortgage loans. In this latter case, the debentures are quite risky, and that risk will be reflected in their interest rates.
713 The yield spread between a corporate bond over a Treasury bond with the same maturity reflects both investors’ risk aversion and their optimism or pessimism regarding the economy and corporate profits. If the economy appeared to be heading into a recession, the spread should widen. The change in spread would be even wider if a firm’s credit strength weakened.
714 Assuming a bond issue is callable, the YTC is a better estimate of a bond’s expected return when interest rates are below an outstanding bond’s coupon rate. The YTM is a better estimate of a bond’s expected return when interest rates are equal or above an outstanding bond’s coupon rate.
Solutions to EndofChapter Problems
71 With your financial calculator, enter the following:
N = 10; I/YR = YTM = 9%; PMT = 0.08 ´ 1,000 = 80; FV = 1000; PV = V_{B} = ?
PV = $935.82.
72 V_{B} = $985; M = $1,000; Int = 0.07 ´ $1,000 = $70.
a. Current yield = Annual interest/Current price of bond
= $70/$985.00
= 7.11%.
b. N = 10; PV = 985; PMT = 70; FV = 1000; YTM = ?
Solve for I/YR = YTM = 7.2157% » 7.22%.
c. N = 7; I/YR = 7.2157; PMT = 70; FV = 1000; PV = ?
Solve for V_{B} = PV = $988.46.
73 The problem asks you to find the price of a bond, given the following facts: N = 2 ´ 8 = 16; I/YR = 8.5/2 = 4.25; PMT = 45; FV = 1000.
With a financial calculator, solve for PV = $1,028.60.
74 With your financial calculator, enter the following to find YTM:
N = 10 ´ 2 = 20; PV = 1100; PMT = 0.08/2 ´ 1,000 = 40; FV = 1000; I/YR = YTM = ?
YTM = 3.31% ´ 2 = 6.62%.
With your financial calculator, enter the following to find YTC:
N = 5 ´ 2 = 10; PV = 1100; PMT = 0.08/2 ´ 1,000 = 40; FV = 1050; I/YR = YTC = ?
YTC = 3.24% ´ 2 = 6.49%.
Since the YTC is less than the YTM, investors would expect the bonds to be called and to earn the YTC.
75 a. 1. 5%: Bond L: Input N = 15, I/YR = 5, PMT = 100, FV = 1000, PV = ?, PV = $1,518.98.
Bond S: Change N = 1, PV = ? PV = $1,047.62.
2. 8%: Bond L: From Bond S inputs, change N = 15 and I/YR = 8, PV = ?, PV = $1,171.19.
Bond S: Change N = 1, PV = ? PV = $1,018.52.
3. 12%: Bond L: From Bond S inputs, change N = 15 and I/YR = 12, PV = ?, PV = $863.78.
Bond S: Change N = 1, PV = ? PV = $982.14.
b. Think about a bond that matures in one month. Its present value is influenced primarily by the maturity value, which will be received in only one month. Even if interest rates double, the price of the bond will still be close to $1,000. A 1year bond’s value would fluctuate more than the onemonth bond’s value because of the difference in the timing of receipts. However, its value would still be fairly close to $1,000 even if interest rates doubled. A longterm bond paying semiannual coupons, on the other hand, will be dominated by distant receipts, receipts that are multiplied by 1/(1 + r_{d}/2)^{t}, and if r_{d} increases, these multipliers will decrease significantly. Another way to view this problem is from an opportunity point of view. A 1month bond can be reinvested at the new rate very quickly, and hence the opportunity to invest at this new rate is not lost; however, the longterm bond locks in subnormal returns for a long period of time.
76 a. Years to Maturity Price of Bond C Price of Bond Z
4 $1,012.79 $ 693.04
3 1,010.02 759.57
2 1,006.98 832.49
1 1,003.65 912.41
0 1,000.00 1,000.00
b.
77 Percentage
Price at 8% Price at 7% Change
10year, 10% annual coupon $1,134.20 $1,210.71 6.75%
10year zero 463.19 508.35 9.75
5year zero 680.58 712.99 4.76
30year zero 99.38 131.37 32.19
$100 perpetuity 1,250.00 1,428.57 14.29
78 The rate of return is approximately 15.03%, found with a calculator using the following inputs:
N = 6; PV = 1000; PMT = 140; FV = 1090; I/YR = ? Solve for I/YR = 15.03%.
Despite a 15% return on the bonds, investors are not likely to be happy that they were called. Because if the bonds have been called, this indicates that interest rates have fallen sufficiently that the YTC is less than the YTM. (Since they were originally sold at par, the YTM at issuance= 14%.) Rates are sufficiently low to justify the call. Now investors must reinvest their funds in a much lower interest rate environment.
79 a. V_{B} =
M = $1,000. PMT = 0.09($1,000) = $90.
1. V_{B} = $829: Input N = 4, PV = 829, PMT = 90, FV = 1000, YTM = I/YR = ? I/YR = 14.99%.
2. V_{B} = $1,104: Change PV = 1104, YTM = I/YR = ? I/YR = 6.00%.
b. Yes. At a price of $829, the yield to maturity, 15%, is greater than your required rate of return of 12%. If your required rate of return were 12%, you should be willing to buy the bond at any price below $908.88.
710 a. Solving for YTM:
N = 9, PV = 901.40, PMT = 80, FV = 1000
I/YR = YTM = 9.6911%.
b. The current yield is defined as the annual coupon payment divided by the current price.
CY = $80/$901.40 = 8.875%.
Expected capital gains yield can be found as the difference between YTM and the current yield.
CGY = YTM – CY = 9.691% – 8.875% = 0.816%.
Alternatively, you can solve for the capital gains yield by first finding the expected price next year.
N = 8, I/YR = 9.6911, PMT = 80, FV = 1000
PV = $908.76. V_{B} = $908.76.
Hence, the capital gains yield is the percent price appreciation over the next year.
CGY = (P_{1} – P_{0})/P_{0} = ($908.76 – $901.40)/$901.40 = 0.816%.
c. As long as promised coupon payments are made, the current yield will not change as a result of changing interest rates. However, as rates change they will cause the endofyear price to change and thus the realized capital gains yield to change. As a result, the realized return to investors will differ from the YTM.
711 a. Using a financial calculator, input the following to solve for YTM:
N = 20, PV = 1100, PMT = 60, FV = 1000, and solve for YTM = I/YR = 5.1849%.
However, this is a periodic rate. The nominal YTM = 5.1849%(2) = 10.3699% » 10.37%.
For the YTC, input the following:
N = 8, PV = 1100, PMT = 60, FV = 1060, and solve for YTC = I/YR = 5.0748%.
However, this is a periodic rate. The nominal YTC = 5.0748%(2) = 10.1495% » 10.15%.
So the bond is likely to be called, and investors are most likely to earn a 10.15% yield.
b. The current yield = $120/$1,100 = 10.91%. The current yield will remain the same; however, if the bond is called the YTC reflects the total return (rather than the YTM) so the capital gains yield will be different.
c. YTM = Current yield + Capital gains (loss) yield
10.37% = 10.91% + Capital loss yield
_{ 0.54% = Capital loss yield. This is the capital loss yield if the YTM is expected. However, based on our calculations in part a the total return expected would actually be the YTC = 10.15%. So, the expected capital loss yield = 10.15% – 10.91% = 0.76%. 712 a. Yield to maturity (YTM): With a financial calculator, input N = 28, PV = 1165.75, PMT = 95, FV = 1000, I/YR = ? I/YR = YTM = 8.00%. Yield to call (YTC): With a calculator, input N = 3, PV = 1165.75, PMT = 95, FV = 1090, I/YR = ? I/YR = YTC = 6.11%. b. Knowledgeable investors would expect the return to be closer to 6.1% than to 8%. If interest rates remain substantially lower than 9.5%, the company can be expected to call the issue at the call date and to refund it with an issue having a coupon rate lower than 9.5%. c. If the bond had sold at a discount, this would imply that current interest rates are above the coupon rate (i.e., interest rates have risen). Therefore, the company would not call the bonds, so the YTM would be more relevant than the YTC. 713 The problem asks you to solve for the YTM and Price, given the following facts: N = 5 ´ 2 = 10, PMT = 80/2 = 40, and FV = 1000. In order to solve for I/YR we need PV. However, you are also given that the current yield is equal to 8.21%. Given this information, we can find PV (Price). Current yield = Annual interest/Current price 0.0821 = $80/PV PV = $80/0.0821 = $974.42. Now, solve for the YTM with a financial calculator: N = 10, PV = 974.42, PMT = 40, and FV = 1000. Solve for I/YR = YTM = 4.32%. However, this is a periodic rate so the nominal YTM = 4.32%(2) = 8.64%. 714 The problem asks you to solve for the current yield, given the following facts: N = 14, I/YR = 10.5883/2 = 5.29415, PV = 1020, and FV = 1000. In order to solve for the current yield we need to find PMT. With a financial calculator, we find PMT = $55.00. However, because the bond is a semiannual coupon bond this amount needs to be multiplied by 2 to obtain the annual interest payment: $55.00(2) = $110.00. Finally, find the current yield as follows: Current yield = Annual interest/Current price = $110/$1,020 = 10.78%. 715 a. The bond is selling at a large premium, which means that its coupon rate is much higher than the going rate of interest. Therefore, the bond is likely to be called—it is more likely to be called than to remain outstanding until it matures. Therefore, the likely life remaining on these bonds is 5 years (the time to call). b. Since the bonds are likely to be called, they will probably provide a return equal to the YTC rather than the YTM. So, there is no point in calculating the YTM—just calculate the YTC. Enter these values: N = 2 ´ 5 = 10, PV = 1353.54, PMT = 0.14/2 ´ 1,000 = 70, FV = 1050, and then solve for YTC = I/YR. The periodic rate is 3.2366%, so the nominal YTC is 2 ´ 3.2366% = 6.4733% » 6.47%. This would be close to the going rate, and it is about what the firm would have to pay on new bonds. 716 First, we must find the amount of money we can expect to sell this bond for in 5 years. This is found using the fact that in five years, there will be 15 years remaining until the bond matures and that the expected YTM for this bond at that time will be 8.5%. N = 15, I/YR = 8.5, PMT = 90, FV = 1000 PV = $1,041.52. VB = $1,041.52. This is the value of the bond in 5 years. Therefore, we can solve for the maximum price we would be willing to pay for this bond today, subject to our required rate of return of 10%. N = 5, I/YR = 10, PMT = 90, FV = 1041.52 PV = $987.87. VB = $987.87. You would be willing to pay up to $987.87 for this bond today. 717 Before you can solve for the price, we must find the appropriate semiannual rate at which to evaluate this bond. EAR = (1 + INOM/2)2 – 1 0.0816 = (1 + INOM/2)2 – 1 INOM = 0.08. Semiannual interest rate = 0.08/2 = 0.04 = 4%. Solving for price: N = 2 ´ 10 = 20, I/YR = 4, PMT = 0.09/2 ´ 1,000 = 45, FV = 1000 PV = $1,067.95. VB = $1,067.95. 718 First, we must find the price Joan paid for this bond. N = 10, I/YR = 9.79, PMT = 110, FV = 1000 PV = $1,075.02. VB = $1,075.02. Then to find the oneperiod return, we must find the sum of the change in price and the coupon received divided by the starting price. Oneperiod return = Oneperiod return = ($1,060.49 – $1,075.02 + $110)/$1,075.02 Oneperiod return = 8.88%. 719 a. According to Table 74, the yield to maturities for Albertson’s and Ford Motor Co. bonds are 6.303% and 8.017%, respectively. So, Albertson’s would need to set a coupon of 6.3% to sell its bonds at par, while Ford would need to set a coupon of 8%. b. Current investments in Albertson’s and Ford Motor Co. would be expected to earn returns equal to their expected present yields. The return is safer for Albertson’s. Looking at the table, we see that the Ford Motor Co. bonds were originally issued with a lower coupon but their yields have increased greatly (resulting in a spread of 320 basis points, compared to Albertson’s spread of 149 basis points). 720 a. Find the YTM as follows: N = 10, PV = 1175, PMT = 110, FV = 1000 I/YR = YTM = 8.35%. b. Find the YTC, if called in Year 5 as follows: N = 5, PV = 1175, PMT = 110, FV = 1090 I/YR = YTC = 8.13%. c. The bonds are selling at a premium which indicates that interest rates have fallen since the bonds were originally issued. Assuming that interest rates do not change from the present level, investors would expect to earn the yield to call. (Note that the YTC is less than the YTM.) d. Similarly from above, YTC can be found, if called in each subsequent year. If called in Year 6: N = 6, PV = 1175, PMT = 110, FV = 1080 I/YR = YTC = 8.27%. If called in Year 7: N = 7, PV = 1175, PMT = 110, FV = 1070 I/YR = YTC = 8.37%. If called in Year 8: N = 8, PV = 1175, PMT = 110, FV = 1060 I/YR = YTC = 8.46%. If called in Year 9: N = 9, PV = 1175, PMT = 110, FV = 1050 I/YR = YTC = 8.53%. According to these calculations, the latest investors might expect a call of the bonds is in Year 6. This is the last year that the expected YTC will be less than the expected YTM. At this time, the firm still finds an advantage to calling the bonds, rather than seeing them to maturity. Comprehensive/Spreadsheet Problem Note to Instructors: The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD. 721 a. Bond A is selling at a discount because its coupon rate (7%) is less than the going interest rate (YTM = 9%). Bond B is selling at par because its coupon rate (9%) is equal to the going interest rate (YTM = 9%). Bond C is selling at a premium because its coupon rate (11%) is greater than the going interest rate (YTM = 9%). b. c. d. e. 1. 5.83% 2. 5.26% 3. The bond is selling at a premium, which means that interest rates have declined. If interest rates remain at current levels, then Mr. Clark should expect the bond to be called. Consequently, he would earn the YTC not the YTM. f. Interest rate (price) risk is the risk of a decline in a bond's price due to an increase in interest rates. Reinvestment rate risk is the risk that a decline in interest rates will lead to a decline in income from a bond portfolio. Ranking the bonds above in order from the most interest rate risk to the least interest rate risk: 10year bond with a zero coupon, 10year bond with a 9 percent annual coupon, 5year bond with a zero coupon, and 5year bond with a 9 percent annual coupon. You can double check this ranking by calculating the prices of each bond at 2 different interest rates, and then determining the percentage change in value. (See calculations above.) g. 1. 2. 3. Chapter 8 Risk and Rates of Return Learning Objectives After reading this chapter, students should be able to: Define risk and calculate the expected rate of return, standard deviation, and coefficient of variation for a probability distribution. Specify how risk aversion influences required rates of return. Graph diversifiable risk and market risk; explain which of these is relevant to a welldiversified investor. State the basic proposition of the Capital Asset Pricing Model (CAPM) and explain how and why a portfolio’s risk may be reduced. Explain the significance of a stock’s beta coefficient, and use the Security Market Line to calculate a stock’s required rate of return. List changes in the market or within a firm that would cause the required rate of return on a firm’s stock to change. Identify concerns about beta and the CAPM. Explain the implications of risk and return for corporate managers and investors. Lecture Suggestions Risk analysis is an important topic, but it is difficult to teach at the introductory level. We just try to give students an intuitive overview of how risk can be defined and measured, and leave a technical treatment to advanced courses. Our primary goals are to be sure students understand (1) that investment risk is the uncertainty about returns on an asset, (2) the concept of portfolio risk, and (3) the effects of risk on required rates of return. What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 8, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes. DAYS ON CHAPTER: 3 OF 58 DAYS (50minute periods) Answers to EndofChapter Questions 81 a. No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but inflation could erode the portfolio’s purchasing power. If the actual inflation rate is greater than that expected, interest rates in general will rise to incorporate a larger inflation premium (IP) and—as we saw in Chapter 7—the value of the portfolio would decline. b. No, you would be subject to reinvestment rate risk. You might expect to "roll over" the Treasury bills at a constant (or even increasing) rate of interest, but if interest rates fall, your investment income will decrease. c. A U.S. governmentbacked bond that provided interest with constant purchasing power (that is, an indexed bond) would be close to riskless. The U.S. Treasury currently issues indexed bonds. 82 a. The probability distribution for complete certainty is a vertical line. b. The probability distribution for total uncertainty is the Xaxis from ¥ to +¥ . 83 a. The expected return on a life insurance policy is calculated just as for a common stock. Each outcome is multiplied by its probability of occurrence, and then these products are summed. For example, suppose a 1year term policy pays $10,000 at death, and the probability of the policyholder’s death in that year is 2%. Then, there is a 98% probability of zero return and a 2% probability of $10,000: Expected return = 0.98($0) + 0.02($10,000) = $200. This expected return could be compared to the premium paid. Generally, the premium will be larger because of sales and administrative costs, and insurance company profits, indicating a negative expected rate of return on the investment in the policy. b. There is a perfect negative correlation between the returns on the life insurance policy and the returns on the policyholder’s human capital. In fact, these events (death and future lifetime earnings capacity) are mutually exclusive. The prices of goods and services must cover their costs. Costs include labor, materials, and capital. Capital costs to a borrower include a return to the saver who supplied the capital, plus a markup (called a "spread") for the financial intermediary that brings the saver and the borrower together. The more efficient the financial system, the lower the costs of intermediation, the lower the costs to the borrower, and, hence, the lower the prices of goods and services to consumers. c. People are generally risk averse. Therefore, they are willing to pay a premium to decrease the uncertainty of their future cash flows. A life insurance policy guarantees an income (the face value of the policy) to the policyholder’s beneficiaries when the policyholder’s future earnings capacity drops to zero. 84 Yes, if the portfolio’s beta is equal to zero. In practice, however, it may be impossible to find individual stocks that have a nonpositive beta. In this case it would also be impossible to have a stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably be better off just purchasing Treasury bills, or other zero beta investments. 85 Security A is less risky if held in a diversified portfolio because of its negative correlation with other stocks. In a singleasset portfolio, Security A would be more risky because s A > s B and CVA > CVB. 86 No. For a stock to have a negative beta, its returns would have to logically be expected to go up in the future when other stocks’ returns were falling. Just because in one year the stock’s return increases when the market declined doesn’t mean the stock has a negative beta. A stock in a given year may move counter to the overall market, even though the stock’s beta is positive. 87 The risk premium on a highbeta stock would increase more than that on a lowbeta stock. RPj = Risk Premium for Stock j = (rM – rRF)bj. If risk aversion increases, the slope of the SML will increase, and so will the market risk premium (rM – rRF). The product (rM – rRF)bj is the risk premium of the jth stock. If bj is low (say, 0.5), then the product will be small; RPj will increase by only half the increase in RPM. However, if bj is large (say, 2.0), then its risk premium will rise by twice the increase in RPM. 88 According to the Security Market Line (SML) equation, an increase in beta will increase a company’s expected return by an amount equal to the market risk premium times the change in beta. For example, assume that the riskfree rate is 6%, and the market risk premium is 5%. If the company’s beta doubles from 0.8 to 1.6 its expected return increases from 10% to 14%. Therefore, in general, a company’s expected return will not double when its beta doubles. 89 a. A decrease in risk aversion will decrease the return an investor will require on stocks. Thus, prices on stocks will increase because the cost of equity will decline. b. With a decline in risk aversion, the risk premium will decline as compared to the historical difference between returns on stocks and bonds. c. The implication of using the SML equation with historical risk premiums (which would be higher than the "current" risk premium) is that the CAPM estimated required return would actually be higher than what would be reflected if the more current risk premium were used. Solutions to EndofChapter Problems 81 = (0.1)(50%) + (0.2)(5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%) = 11.40%. s 2 = (50% – 11.40%)2(0.1) + (5% – 11.40%)2(0.2) + (16% – 11.40%)2(0.4) + (25% – 11.40%)2(0.2) + (60% – 11.40%)2(0.1) s 2 = 712.44; s = 26.69%. CV = = 2.34. 82 Investment Beta $35,000 0.8 40,000 1.4 Total $75,000 bp = ($35,000/$75,000)(0.8) + ($40,000/$75,000)(1.4) = 1.12. 83 rRF = 6%; rM = 13%; b = 0.7; r = ? r = rRF + (rM – rRF)b = 6% + (13% – 6%)0.7 = 10.9%. 84 rRF = 5%; RPM = 6%; rM = ? rM = 5% + (6%)1 = 11%. r when b = 1.2 = ? r = 5% + 6%(1.2) = 12.2%. 85 a. r = 11%; rRF = 7%; RPM = 4%. r = rRF + (rM – rRF)b 11% = 7% + 4%b 4% = 4%b b = 1. b. rRF = 7%; RPM = 6%; b = 1. r = rRF + (rM – rRF)b = 7% + (6%)1 = 13%. 86 a. . = 0.1(35%) + 0.2(0%) + 0.4(20%) + 0.2(25%) + 0.1(45%) = 14% versus 12% for X. b. s = . = (10% – 12%)2(0.1) + (2% – 12%)2(0.2) + (12% – 12%)2(0.4) + (20% – 12%)2(0.2) + (38% – 12%)2(0.1) = 148.8%. s X = 12.20% versus 20.35% for Y. CVX = s X/X = 12.20%/12% = 1.02, while CVY = 20.35%/14% = 1.45. If Stock Y is less highly correlated with the market than X, then it might have a lower beta than Stock X, and hence be less risky in a portfolio sense. 87 Portfolio beta = (1.50) + (0.50) + (1.25) + (0.75) bp = (0.1)(1.5) + (0.15)(0.50) + (0.25)(1.25) + (0.5)(0.75) = 0.15 – 0.075 + 0.3125 + 0.375 = 0.7625. rp = rRF + (rM – rRF)(bp) = 6% + (14% – 6%)(0.7625) = 12.1%. Alternative solution: First, calculate the return for each stock using the CAPM equation [rRF + (rM – rRF)b], and then calculate the weighted average of these returns. rRF = 6% and (rM – rRF) = 8%. Stock Investment Beta r = rRF + (rM – rRF)b Weight A $ 400,000 1.50 18% 0.10 B 600,000 (0.50) 2 0.15 C 1,000,000 1.25 16 0.25 D 2,000,000 0.75 12 0.50 Total $4,000,000 1.00 rp = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%. 88 In equilibrium: rJ = = 12.5%. rJ = rRF + (rM – rRF)b 12.5% = 4.5% + (10.5% – 4.5%)b b = 1.33. 89 We know that bR = 1.50, bS = 0.75, rM = 13%, rRF = 7%. ri = rRF + (rM – rRF)bi = 7% + (13% – 7%)bi. rR = 7% + 6%(1.50) = 16.0% rS = 7% + 6%(0.75) = 11.5 4.5% 810 An index fund will have a beta of 1.0. If rM is 12.0% (given in the problem) and the riskfree rate is 5%, you can calculate the market risk premium (RPM) calculated as rM – rRF as follows: r = rRF + (RPM)b 12.0% = 5% + (RPM)1.0 7.0% = RPM. Now, you can use the RPM, the rRF, and the two stocks’ betas to calculate their required returns. Bradford: rB = rRF + (RPM)b = 5% + (7.0%)1.45 = 5% + 10.15% = 15.15%. Farley: rF = rRF + (RPM)b = 5% + (7.0%)0.85 = 5% + 5.95% = 10.95%. The difference in their required returns is: 15.15% – 10.95% = 4.2%. 811 rRF = r* + IP = 2.5% + 3.5% = 6%. rs = 6% + (6.5%)1.7 = 17.05%. 812 Using Stock X (or any stock): 9% = rRF + (rM – rRF)bX 9% = 5.5% + (rM – rRF)0.8 (rM – rRF) = 4.375%. 813 a. ri = rRF + (rM – rRF)bi = 9% + (14% – 9%)1.3 = 15.5%. b. 1. rRF increases to 10%: rM increases by 1 percentage point, from 14% to 15%. ri = rRF + (rM – rRF)bi = 10% + (15% – 10%)1.3 = 16.5%. 2. rRF decreases to 8%: rM decreases by 1%, from 14% to 13%. ri = rRF + (rM – rRF)bi = 8% + (13% – 8%)1.3 = 14.5%. c. 1. rM increases to 16%: ri = rRF + (rM – rRF)bi = 9% + (16% – 9%)1.3 = 18.1%. 2. rM decreases to 13%: ri = rRF + (rM – rRF)bi = 9% + (13% – 9%)1.3 = 14.2%. 814 Old portfolio beta = (b) + (1.00) 1.12 = 0.95b + 0.05 1.07 = 0.95b 1.1263 = b. New portfolio beta = 0.95(1.1263) + 0.05(1.75) = 1.1575 » 1.16. Alternative solutions: 1. Old portfolio beta = 1.12 = (0.05)b1 + (0.05)b2 + ... + (0.05)b20 }1.12 = (0.05)
= 1.12/0.05 = 22.4.
New portfolio beta = (22.4 – 1.0 + 1.75)(0.05) = 1.1575 » 1.16.
2. excluding the stock with the beta equal to 1.0 is 22.4 – 1.0 = 21.4, so the beta of the portfolio excluding this stock is b = 21.4/19 = 1.1263. The beta of the new portfolio is:
1.1263(0.95) + 1.75(0.05) = 1.1575 » 1.16.
815 b_{HRI} = 1.8; b_{LRI} = 0.6. No changes occur.
r_{RF} = 6%. Decreases by 1.5% to 4.5%.
r_{M} = 13%. Falls to 10.5%.
Now SML: r_{i} = r_{RF} + (r_{M} – r_{RF})b_{i}.
r_{HRI} = 4.5% + (10.5% – 4.5%)1.8 = 4.5% + 6%(1.8) = 15.3%
r_{LRI} = 4.5% + (10.5% – 4.5%)0.6 = 4.5% + 6%(0.6) = 8.1%
Difference 7.2%
816 Step 1: Determine the market risk premium from the CAPM:
0.12 = 0.0525 + (r_{M} – r_{RF})1.25
(r_{M} – r_{RF}) = 0.054.
Step 2: Calculate the beta of the new portfolio:
($500,000/$5,500,000)(0.75) + ($5,000,000/$5,500,000)(1.25) = 1.2045.
Step 3: Calculate the required return on the new portfolio:
5.25% + (5.4%)(1.2045) = 11.75%.
817 After additional investments are made, for the entire fund to have an expected return of 13%, the portfolio must have a beta of 1.5455 as shown below:
13% = 4.5% + (5.5%)b
b = 1.5455.
Since the fund’s beta is a weighted average of the betas of all the individual investments, we can calculate the required beta on the additional investment as follows:
1.5455 = +
1.5455 = 1.2 + 0.2X
0.3455 = 0.2X
X = 1.7275.
818 a. ($1 million)(0.5) + ($0)(0.5) = $0.5 million.
b. You would probably take the sure $0.5 million.
c. Risk averter.
d. 1. ($1.15 million)(0.5) + ($0)(0.5) = $575,000, or an expected profit of $75,000.
2. $75,000/$500,000 = 15%.
3. This depends on the individual’s degree of risk aversion.
4. Again, this depends on the individual.
5. The situation would be unchanged if the stocks’ returns were perfectly positively correlated. Otherwise, the stock portfolio would have the same expected return as the single stock (15%) but a lower standard deviation. If the correlation coefficient between each pair of stocks was a negative one, the portfolio would be virtually riskless. Since for stocks is generally in the range of +0.35, investing in a portfolio of stocks would definitely be an improvement over investing in the single stock.
819 = 10%; b_{X} = 0.9; s _{X} = 35%.
= 12.5%; b_{Y} = 1.2; s _{Y} = 25%.
r_{RF} = 6%; RP_{M} = 5%.
a. CV_{X} = 35%/10% = 3.5. CV_{Y} = 25%/12.5% = 2.0.
b. For diversified investors the relevant risk is measured by beta. Therefore, the stock with the higher beta is more risky. Stock Y has the higher beta so it is more risky than Stock X.
c. r_{X} = 6% + 5%(0.9)
= 10.5%.
r_{Y} = 6% + 5%(1.2)
= 12%.
d. r_{X} = 10.5%; = 10%.
r_{Y} = 12%; = 12.5%.
Stock Y would be most attractive to a diversified investor since its expected return of 12.5% is greater than its required return of 12%.
e. b_{p} = ($7,500/$10,000)0.9 + ($2,500/$10,000)1.2
= 0.6750 + 0.30
= 0.9750.
r_{p} = 6% + 5%(0.975)
= 10.875%.
f. If RP_{M} increases from 5% to 6%, the stock with the highest beta will have the largest increase in its required return. Therefore, Stock Y will have the greatest increase.
Check:
r_{X} = 6% + 6%(0.9)
= 11.4%. Increase 10.5% to 11.4%.
r_{Y} = 6% + 6%(1.2)
= 13.2%. Increase 12% to 13.2%.
820 The answers to a, b, c, and d are given below:
r_{A} r_{B} Portfolio
2001 (18.00%) (14.50%) (16.25%)
2002 33.00 21.80 27.40
2003 15.00 30.50 22.75
2004 (0.50) (7.60) (4.05)
2005 27.00 26.30 26.65
Mean 11.30 11.30 11.30
Std. Dev. 20.79 20.78 20.13
Coef. Var. 1.84 1.84 1.78
e. A riskaverse investor would choose the portfolio over either Stock A or Stock B alone, since the portfolio offers the same expected return but with less risk. This result occurs because returns on A and B are not perfectly positively correlated (r_{AB} = 0.88).
821 a. = 0.1(7%) + 0.2(9%) + 0.4(11%) + 0.2(13%) + 0.1(15%) = 11%.
r_{RF} = 6%. (given)
Therefore, the SML equation is:
r_{i} = r_{RF} + (r_{M} – r_{RF})b_{i} = 6% + (11% – 6%)b_{i} = 6% + (5%)b_{i}.
b. First, determine the fund’s beta, b_{F}. The weights are the percentage of funds invested in each stock:
A = $160/$500 = 0.32.
B = $120/$500 = 0.24.
C = $80/$500 = 0.16.
D = $80/$500 = 0.16.
E = $60/$500 = 0.12.
b_{F} = 0.32(0.5) + 0.24(2.0) + 0.16(4.0) + 0.16(1.0) + 0.12(3.0)
= 0.16 + 0.48 + 0.64 + 0.16 + 0.36 = 1.8.
Next, use b_{F} = 1.8 in the SML determined in Part a:
= 6% + (11% – 6%)1.8 = 6% + 9% = 15%.
c. r_{N} = Required rate of return on new stock = 6% + (5%)2.0 = 16%.
An expected return of 15% on the new stock is below the 16% required rate of return on an investment with a risk of b = 2.0. Since r_{N} = 16% > = 15%, the new stock should not be purchased. The expected rate of return that would make the fund indifferent to purchasing the stock is 16%.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
822 a.
b.
On a standalone basis, it would appear that Bartman is the most risky, Reynolds the least risky.
c.
Reynolds now looks most risky, because its risk (SD) per unit of return is highest.
d.
It is clear that Bartman moves with the market and Reynolds moves counter to the market. So, Bartman has a positive beta and Reynolds a negative one.
e. Bartman’s calculations:
Reynolds’ calculations:
Note that these betas are consistent with the scatter diagrams we constructed earlier. Reynolds' beta suggests that it is less risky than average in a CAPM sense, whereas Bartman is more risky than average.
f.
This suggests that Reynolds' stock is like an insurance policy that has a low expected return, but it will pay off in the event of a market decline. Actually, it is hard to find negativebeta stocks, so we would not be inclined to believe the Reynolds' data.
g. The beta of a portfolio is simply a weighted average of the betas of the stocks in the portfolio, so this portfolio's beta would be:
h.
Chapter 9
Stocks and Their Valuation
Learning Objectives
After reading this chapter, students should be able to:
Identify some of the more important rights that come with stock ownership and define the following terms: proxy, proxy fight, takeover, and preemptive right.
Briefly explain why classified stock might be used by a corporation and what founders’ shares are.
Determine the value of a share of common stock when: (1) dividends are expected to grow at some constant rate, (2) dividends are expected to remain constant (zero growth), and (3) dividends are expected to grow at some supernormal, or nonconstant, growth rate.
Calculate the expected rate of return on a constant growth stock.
Apply the total company (corporate valuation) model to value a firm in situations where future dividends are not easily predictable.
Explain why a stock’s intrinsic value might differ between the total company model and the dividend growth model.
Explain the following terms: equilibrium and marginal investor. Identify the two related conditions that must hold in equilibrium.
Explain how changes in the riskfree rate, the market risk premium, the stock’s beta, and the expected growth rate impact equilibrium stock price.
Explain the reasons for investing in international stocks and identify the "bets" an investor is making when he does invest overseas.
Define preferred stock, determine the value of a share of preferred stock, or given its value, calculate its expected return.
Lecture Suggestions
This chapter provides important and useful information on common and preferred stocks. Moreover, the valuation of stocks reinforces the concepts covered in Chapters 2, 7, and 8, so Chapter 9 extends and reinforces concepts discussed in those chapters.
We begin our lecture with a discussion of the characteristics of common stocks and how stocks are valued in the market. Models are presented for valuing constant growth stocks, zero growth stocks, and nonconstant growth stocks. We conclude the lecture with a discussion of preferred stocks.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 9, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 3 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
91 a. The average investor of a firm traded on the NYSE is not really interested in maintaining his or her proportionate share of ownership and control. If the investor wanted to increase his or her ownership, the investor could simply buy more stock on the open market. Consequently, most investors are not concerned with whether new shares are sold directly (at about market prices) or through rights offerings. However, if a rights offering is being used to effect a stock split, or if it is being used to reduce the underwriting cost of an issue (by substantial underpricing), the preemptive right may well be beneficial to the firm and to its stockholders.
b. The preemptive right is clearly important to the stockholders of closely held (private) firms whose owners are interested in maintaining their relative control positions.
92 No. The correct equation has D_{1} in the numerator and a minus sign in the denominator.
93 Yes. If a company decides to increase its payout ratio, then the dividend yield component will rise, but the expected longterm capital gains yield will decline.
94 Yes. The value of a share of stock is the PV of its expected future dividends. If the two investors expect the same future dividend stream, and they agree on the stock’s riskiness, then they should reach similar conclusions as to the stock’s value.
95 A perpetual bond is similar to a nogrowth stock and to a share of perpetual preferred stock in the following ways:
1. All three derive their values from a series of cash inflows—coupon payments from the perpetual bond, and dividends from both types of stock.
All three are assumed to have indefinite lives with no maturity value (M) for the perpetual bond and no capital gains yield for the stocks.
However, there are preferreds that have a stated maturity. In this situation, the preferred would be valued much like a bond with a stated maturity. Both derive their values from a series of cash inflows—coupon payments and a maturity value for the bond and dividends and a stock price for the preferred.
Solutions to EndofChapter Problems
91 D_{0} = $1.50; g_{13} = 7%; g_{n} = 5%; D_{1} through D_{5} = ?
D_{1} = D_{0}(1 + g_{1}) = $1.50(1.07) = $1.6050.
D_{2} = D_{0}(1 + g_{1})(1 + g_{2}) = $1.50(1.07)^{2} = $1.7174.
D_{3} = D_{0}(1 + g_{1})(1 + g_{2})(1 + g_{3}) = $1.50(1.07)^{3} = $1.8376.
D_{4} = D_{0}(1 + g_{1})(1 + g_{2})(1 + g_{3})(1 + g_{n}) = $1.50(1.07)^{3}(1.05) = $1.9294.
D_{5} = D_{0}(1 + g_{1})(1 + g_{2})(1 + g_{3})(1 + g_{n})^{2} = $1.50(1.07)^{3}(1.05)^{2} = $2.0259.
92 D_{1} = $0.50; g = 7%; r_{s} = 15%; = ?
93 P_{0} = $20; D_{0} = $1.00; g = 6%; = ?; r_{s} = ?
= P_{0}(1 + g) = $20(1.06) = $21.20.
= + g = + 0.06
= + 0.06 = 11.30%. r_{s} = 11.30%.
94 a. The terminal, or horizon, date is the date when the growth rate becomes constant. This occurs at the end of Year 2.
b. 0 1 2 3
   
1.25 1.50 1.80 1.89
37.80 =
The horizon, or terminal, value is the value at the horizon date of all dividends expected thereafter. In this problem it is calculated as follows:
c. The firm’s intrinsic value is calculated as the sum of the present value of all dividends during the supernormal growth period plus the present value of the terminal value. Using your financial calculator, enter the following inputs: CF_{0} = 0, CF_{1} = 1.50, CF_{2} = 1.80 + 37.80 = 39.60, I/YR = 10, and then solve for NPV = $34.09.
95 The firm’s free cash flow is expected to grow at a constant rate, hence we can apply a constant growth formula to determine the total value of the firm.
Firm value = FCF_{1}/(WACC – g)
= $150,000,000/(0.10 – 0.05)
= $3,000,000,000.
To find the value of an equity claim upon the company (share of stock), we must subtract out the market value of debt and preferred stock. This firm happens to be entirely equity funded, and this step is unnecessary. Hence, to find the value of a share of stock, we divide equity value (or in this case, firm value) by the number of shares outstanding.
Equity value per share = Equity value/Shares outstanding
= $3,000,000,000/50,000,000
= $60.
Each share of common stock is worth $60, according to the corporate valuation model.
96 D_{p} = $5.00; V_{p} = $60; r_{p} = ?
r_{p} = = = 8.33%.
97 V_{p} = D_{p}/r_{p}; therefore, r_{p} = D_{p}/V_{p}.
a. r_{p} = $8/$60 = 13.33%.
b. r_{p} = $8/$80 = 10.0%.
c. r_{p} = $8/$100 = 8.0%.
d. r_{p} = $8/$140 = 5.71%.
98 a.
b.
99 a. The preferred stock pays $8 annually in dividends. Therefore, its nominal rate of return would be:
Nominal rate of return = $8/$80 = 10%.
Or alternatively, you could determine the security’s periodic return and multiply by 4.
Periodic rate of return = $2/$80 = 2.5%.
Nominal rate of return = 2.5% ´ 4 = 10%.
b. EAR = (1 + r_{NOM}/4)^{4} – 1
= (1 + 0.10/4)^{4} – 1
= 0.103813 = 10.3813%.
910
911 First, solve for the current price.
= D_{1}/(r_{s} – g)
= $0.50/(0.12 – 0.07)
_{ = $10.00. If the stock is in a constant growth state, the constant dividend growth rate is also the capital gains yield for the stock and the stock price growth rate. Hence, to find the price of the stock four years from today: = P0(1 + g)4 = $10.00(1.07)4 = $13.10796 ≈ $13.11. }912 a. 1.
2. = $2/0.15 = $13.33.
3.
4.
b. 1. = $2.30/0 = Undefined.
2. = $2.40/(0.05) = $48, which is nonsense.
These results show that the formula does not make sense if the required rate of return is equal to or less than the expected growth rate.
c. No, the results of part b show this. It is not reasonable for a firm to grow indefinitely at a rate higher than its required return. Such a stock, in theory, would become so large that it would eventually overtake the whole economy.
913 a. r_{i} = r_{RF} + (r_{M} – r_{RF})b_{i}.
r_{C} = 7% + (11% – 7%)0.4 = 8.6%.
r_{D} = 7% + (11% – 7%)(0.5) = 5%.
Note that r_{D} is below the riskfree rate. But since this stock is like an insurance policy because it "pays off" when something bad happens (the market falls), the low return is not unreasonable.
b. In this situation, the expected rate of return is as follows:
= D_{1}/P_{0} + g = $1.50/$25 + 4% = 10%.
However, the required rate of return is 8.6%. Investors will seek to buy the stock, raising its price to the following:
At this point, , and the stock will be in equilibrium.
914 The problem asks you to determine the value of , given the following facts: D_{1} = $2, b = 0.9, r_{RF} = 5.6%, RP_{M} = 6%, and P_{0} = $25. Proceed as follows:
Step 1: Calculate the required rate of return:
r_{s} = r_{RF} + (r_{M} – r_{RF})b = 5.6% + (6%)0.9 = 11%.
Step 2: Use the constant growth rate formula to calculate g:
Step 3: Calculate :
= P_{0}(1 + g)^{3} = $25(1.03)^{3} = $27.3182 » $27.32.
Alternatively, you could calculate D_{4} and then use the constant growth rate formula to solve for :
D_{4} = D_{1}(1 + g)^{3} = $2.00(1.03)^{3} = $2.1855.
= $2.1855/(0.11 – 0.03) = $27.3182 » $27.32.
915 a. r_{s} = r_{RF} + (r_{M} – r_{RF})b = 6% + (10% – 6%)1.5 = 12.0%.
= D_{1}/(r_{s} – g) = $2.25/(0.12 – 0.05) = $32.14.
b. r_{s} = 5% + (9% – 5%)1.5 = 11.0%. = $2.25/(0.110 – 0.05) = $37.50.
c. r_{s} = 5% + (8% – 5%)1.5 = 9.5%. = $2.25/(0.095 – 0.05) = $50.00.
d. New data given: r_{RF} = 5%; r_{M} = 8%; g = 6%, b = 1.3.
r_{s} = r_{RF} + (r_{M} – r_{RF})b = 5% + (8% – 5%)1.3 = 8.9%.
= D_{1}/(r_{s} – g) = $2.27/(0.089 – 0.06) = $78.28.
916 Calculate the dividend cash flows and place them on a time line. Also, calculate the stock price at the end of the supernormal growth period, and include it, along with the dividend to be paid at t = 5, as CF_{5}. Then, enter the cash flows as shown on the time line into the cash flow register, enter the required rate of return as I/YR = 15, and then find the value of the stock using the NPV calculation. Be sure to enter CF_{0} = 0, or else your answer will be incorrect.
D_{0} = 0; D_{1} = 0; D_{2} = 0; D_{3} = 1.00; D_{4} = 1.00(1.5) = 1.5; D_{5} = 1.00(1.5)^{2} = 2.25; D_{6} = 1.00(1.5)^{2}(1.08) = $2.43. = ?
0 1 2 3 4 5 6
      
1.00 1.50 2.25 2.43
0.658 +34.714 =
0.858
18.378 36.964
$19.894 =
= D_{6}/(r_{s} – g) = $2.43/(0.15 – 0.08) = $34.714. This is the stock price at the end of Year 5.
CF_{0} = 0; CF_{12} = 0; CF_{3} = 1.0; CF_{4} = 1.5; CF_{5} = 36.964; I/YR = 15%.
With these cash flows in the CFLO register, press NPV to get the value of the stock today: NPV = $19.89.
917 a. Terminal value = = = $713.33 million.
b. 0 1 2 3 4
    
20 30 40 42.80
($ 17.70)
23.49 = 713.33
522.10 753.33
$527.89
Using a financial calculator, enter the following inputs: CF_{0} = 0; CF_{1} = 20; CF_{2} = 30; CF_{3} = 753.33; I/YR = 13; and then solve for NPV = $527.89 million.
c. Total value_{t=0} = $527.89 million.
Value of common equity = $527.89 – $100 = $427.89 million.
Price per share = = $42.79.
918 The value of any asset is the present value of all future cash flows expected to be generated from the asset. Hence, if we can find the present value of the dividends during the period preceding longrun constant growth and subtract that total from the current stock price, the remaining value would be the present value of the cash flows to be received during the period of longrun constant growth.
D_{1} = $2.00 ´ (1.25)^{1} = $2.50 PV(D_{1}) = $2.50/(1.12)^{1} = $2.2321
D_{2} = $2.00 ´ (1.25)^{2} = $3.125 PV(D_{2}) = $3.125/(1.12)^{2} = $2.4913
D_{3} = $2.00 ´ (1.25)^{3} = $3.90625 PV(D_{3}) = $3.90625/(1.12)^{3} = $2.7804
S PV(D_{1} to D_{3}) = $7.5038
Therefore, the PV of the remaining dividends is: $58.8800 – $7.5038 = $51.3762. Compounding this value forward to Year 3, we find that the value of all dividends received during constant growth is $72.18. [$51.3762(1.12)^{3} = $72.1799 » $72.18.] Applying the constant growth formula, we can solve for the constant growth rate:
= D_{3}(1 + g)/(r_{s} – g)
$72.18 = $3.90625(1 + g)/(0.12 – g)
$8.6616 – $72.18g = $3.90625 + $3.90625g
$4.7554 = $76.08625g
0.0625 = g
6.25% = g.
919 0 1 2 3 4
    
D_{0} = 2.00 D_{1} D_{2} D_{3} D_{4}
a. D_{1} = $2(1.05) = $2.10; D_{2} = $2(1.05)^{2} = $2.2050; D_{3} = $2(1.05)^{3} = $2.31525.
b. Financial calculator solution: Input 0, 2.10, 2.2050, and 2.31525 into the cash flow register, input I/YR = 12, PV = ? PV = $5.28.
c. Financial calculator solution: Input 0, 0, 0, and 34.73 into the cash flow register, I/YR = 12, PV = ? PV = $24.72.
d. $24.72 + $5.28 = $30.00 = Maximum price you should pay for the stock.
e.
f. No. The value of the stock is not dependent upon the holding period. The value calculated in Parts a through d is the value for a 3year holding period. It is equal to the value calculated in Part e. Any other holding period would produce the same value of ; that is, = $30.00.
920 a. Part 1: Graphical representation of the problem:
Supernormal Normal
growth growth
0 1 2 3 ¥
    • • • 
D_{0} D_{1} (D_{2} + ) D_{3} D¥
PVD_{1}
PVD_{2}
P_{0}
D_{1} = D_{0}(1 + g_{s}) = $1.6(1.20) = $1.92.
D_{2} = D_{0}(1 + g_{s})^{2} = $1.60(1.20)^{2} = $2.304.
= PV(D_{1}) + PV(D_{2}) + PV()
=
= $1.92/1.10 + $2.304/(1.10)^{2} + $61.06/(1.10)^{2} = $54.11.
Financial calculator solution: Input 0, 1.92, 63.364(2.304 + 61.06) into the cash flow register, input I/YR = 10, PV = ? PV = $54.11.
Part 2: Expected dividend yield:
D_{1}/P_{0} = $1.92/$54.11 = 3.55%.
Capital gains yield: First, find , which equals the sum of the present values of D_{2} and discounted for one year.
Financial calculator solution: Input 0, 63.364(2.304 + 61.06) into the cash flow register, input I/YR = 10, PV = ? PV = $57.60.
Second, find the capital gains yield:
Dividend yield = 3.55%
Capital gains yield = 6.45
10.00% = r_{s}.
b. Due to the longer period of supernormal growth, the value of the stock will be higher for each year. Although the total return will remain the same, r_{s} = 10%, the distribution between dividend yield and capital gains yield will differ: The dividend yield will start off lower and the capital gains yield will start off higher for the 5year supernormal growth condition, relative to the 2year supernormal growth state. The dividend yield will increase and the capital gains yield will decline over the 5year period until dividend yield = 4% and capital gains yield = 6%.
c. Throughout the supernormal growth period, the total yield, r_{s}, will be 10%, but the dividend yield is relatively low during the early years of the supernormal growth period and the capital gains yield is relatively high. As we near the end of the supernormal growth period, the capital gains yield declines and the dividend yield rises. After the supernormal growth period has ended, the capital gains yield will equal g_{n} = 6%. The total yield must equal r_{s} = 10%, so the dividend yield must equal 10% – 6% = 4%.
d. Some investors need cash dividends (retired people), while others would prefer growth. Also, investors must pay taxes each year on the dividends received during the year, while taxes on the capital gain can be delayed until the gain is actually realized. Currently (2005), dividends to individuals are now taxed at the lower capital gains rate of 15%.
921 a. 0 1 2 3 4
    
3,000,000 6,000,000 10,000,000 15,000,000
Using a financial calculator, enter the following inputs: CF_{0} = 0; CF_{1} = 3000000; CF_{2} = 6000000; CF_{3} = 10000000; CF_{4} = 15000000; I/YR = 12; and then solve for NPV = $24,112,308.
b. The firm’s terminal value is calculated as follows:
c. The firm’s total value is calculated as follows:
0 1 2 3 4 5
     
3,000,000 6,000,000 10,000,000 15,000,000 16,050,000
PV = ? 321,000,000 =
Using your financial calculator, enter the following inputs: CF_{0} = 0; CF_{1} = 3000000; CF_{2} = 6000000; CF_{3} = 10000000; CF_{4} = 15000000 + 321000000 = 336000000; I/YR = 12; and then solve for NPV = $228,113,612.
d. To find Barrett’s stock price, you need to first find the value of its equity. The value of Barrett’s equity is equal to the value of the total firm less the market value of its debt and preferred stock.
Total firm value $228,113,612
Market value, debt + preferred 60,000,000 (given in problem)
Market value of equity $168,113,612
Barrett’s price per share is calculated as:
922 FCF = EBIT(1 – T) + Depreciation – – D
= $500,000,000 + $100,000,000 – $200,000,000 – $0
= $400,000,000.
Firm value =
=
=
= $10,000,000,000.
This is the total firm value. Now find the market value of its equity.
MV_{Total} = MV_{Equity} + MV_{Debt}
$10,000,000,000 = MV_{Equity} + $3,000,000,000
MV_{Equity} = $7,000,000,000.
This is the market value of all the equity. Divide by the number of shares to find the price per share. $7,000,000,000/200,000,000 = $35.00.
923 a. Old r_{s} = r_{RF} + (r_{M} – r_{RF})b = 6% + (3%)1.2 = 9.6%.
New r_{s} = 6% + (3%)0.9 = 8.7%.
Old price:
New price:
Since the new price is lower than the old price, the expansion in consumer products should be rejected. The decrease in risk is not sufficient to offset the decline in profitability and the reduced growth rate.
b. P_{Old} = $58.89. P_{New} = .
Solving for r_{s} we have the following:
$58.89 =
$2.08 = $58.89(r_{s}) – $2.3556
$4.4356 = $58.89(r_{s})
r_{s} = 0.07532.
Solving for b:
7.532% = 6% + 3%(b)
1.532% = 3%(b)
b = 0.5107.
Check: r_{s} = 6% + (3%)0.5107 = 7.532%.
= = $58.89.
Therefore, only if management’s analysis concludes that risk can be lowered to b = 0.5107, should the new policy be put into effect.
924 a. End of Year: 05 06 07 08 09 10 11
      
D_{0} = 1.75 D_{1} D_{2} D_{3} D_{4} D_{5} D_{6}
D_{t} = D_{0}(1 + g)^{t}.
D_{2006} = $1.75(1.15)^{1} = $2.01.
D_{2007} = $1.75(1.15)^{2} = $1.75(1.3225) = $2.31.
D_{2008} = $1.75(1.15)^{3} = $1.75(1.5209) = $2.66.
D_{2009} = $1.75(1.15)^{4} = $1.75(1.7490) = $3.06.
D_{2010} = $1.75(1.15)^{5} = $1.75(2.0114) = $3.52.
b. Step 1:
PV of dividends = .
PV D_{2006} = $2.01/(1.12) = $1.79
PV D_{2007} = $2.31/(1.12)^{2} = $1.84
PV D_{2008} = $2.66/(1.12)^{3} = $1.89
PV D_{2009} = $3.06/(1.12)^{4} = $1.94
PV D_{2010} = $3.52/(1.12)^{5} = $2.00
PV of dividends = $9.46
Step 2:
.
This is the price of the stock 5 years from now. The PV of this price, discounted back 5 years, is as follows:
PV of = $52.80/(1.12)^{5} = $29.96
Step 3:
The price of the stock today is as follows:
= PV dividends Years 20062010 + PV of
= $9.46 + $29.96 = $39.42.
This problem could also be solved by substituting the proper values into the following equation:
.
Calculator solution: Input 0, 2.01, 2.31, 2.66, 3.06, 56.32 (3.52 + 52.80) into the cash flow register, input I/YR = 12, PV = ? PV = $39.43.
c. 2006
D_{1}/P_{0} = $2.01/$39.43 = 5.10%
Capital gains yield = 6.90*
Expected total return = 12.00%
2011
D_{6}/P_{5} = $3.70/$52.80 = 7.00%
Capital gains yield = 5.00
Expected total return = 12.00%
*We know that r_{s} is 12%, and the dividend yield is 5.10%; therefore, the capital gains yield must be 6.90%.
The main points to note here are as follows:
The total yield is always 12% (except for rounding errors).
The capital gains yield starts relatively high, then declines as the supernormal growth period approaches its end. The dividend yield rises.
After 12/31/10, the stock will grow at a 5% rate. The dividend yield will equal 7%, the capital gains yield will equal 5%, and the total return will be 12%.
d. People in highincome tax brackets will be more inclined to purchase "growth" stocks to take the capital gains and thus delay the payment of taxes until a later date. The firm’s stock is "mature" at the end of 2010.
e. Since the firm’s supernormal and normal growth rates are lower, the dividends and, hence, the present value of the stock price will be lower. The total return from the stock will still be 12%, but the dividend yield will be larger and the capital gains yield will be smaller than they were with the original growth rates. This result occurs because we assume the same last dividend but a much lower current stock price.
f. As the required return increases, the price of the stock goes down, but both the capital gains and dividend yields increase initially. Of course, the longterm capital gains yield is still 4%, so the longterm dividend yield is 10%.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solutions for parts a through c of this problem are provided at the back of the text; however, the solution to part d is not. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
925 a. 1. Find the price today.
2. Find the expected dividend yield.
Recall that the expected dividend yield is equal to the next expected annual dividend divided by the price at the beginning of the period.
3. Find the expected capital gains yield.
The capital gains yield can be calculated by simply subtracting the dividend yield from the total expected return.
Alternatively, we can recognize that the capital gains yield measures capital appreciation, hence solve for the price in one year, then divide the change in price from today to one year from now by the current price. To find the price one year from now, we will have to find the present values of the terminal value and second year dividend to time period one.
b. 1. Find the price today.
2. Find the expected dividend yield.
3. Find the expected capital gains yield.
c. We used the 5 year supernormal growth scenario for this calculation, but ultimately it does not matter which example you use, as they both yield the same result.
Upon reflection, we see that these calculations were unnecessary because the constant growth assumption holds that the longterm growth rate is the dividend growth rate and the capital gains yield, hence we could have simply subtracted the longrun growth rate from the required return to find the dividend yield.
d.
The price as estimated by the free cash flow method differs from the discounted dividends method because different assumptions are built into the two situations. If we had projected financial statements, found both dividends and free cash flow from those projected statements, and then applied the two methods, then the prices produced would have been identical. As it stands, though, the two prices were based on somewhat different assumptions, hence different prices were obtained. Note especially that in the FCF model we assumed a WACC of 9% versus 10% for the discounted dividend model. That would obviously tend to raise the price.
Chapter 10
The Cost of Capital
Learning Objectives
After reading this chapter, students should be able to:
Explain what is meant by a firm’s weighted average cost of capital.
Define and calculate the component costs of debt and preferred stock. Explain why the cost of debt is tax adjusted and the cost of preferred is not.
Explain why retained earnings are not free and use three approaches to estimate the component cost of retained earnings.
Briefly explain the two alternative approaches that can be used to account for flotation costs.
Briefly explain why the cost of new common equity is higher than the cost of retained earnings, calculate the cost of new common equity, and calculate the retained earnings breakpoint—which is the point where new common equity would have to be issued.
Calculate the firm’s composite, or weighted average, cost of capital.
Identify some of the factors that affect the WACC—dividing them into factors the firm cannot control and those they can.
Briefly explain how firms should evaluate projects with different risks, and the problems encountered when divisions within the same firm all use the firm’s composite WACC when considering capital budgeting projects.
List some problems with cost of capital estimates.
Lecture Suggestions
Chapter 10 uses the rate of return concepts covered in previous chapters, along with the concept of the weighted average cost of capital (WACC), to develop a corporate cost of capital for use in capital budgeting.
We begin by describing the logic of the WACC, and why it should be used in capital budgeting. We next explain how to estimate the cost of each component of capital, and how to put the components together to determine the WACC. We go on to discuss factors that affect the WACC and how to adjust the cost of capital for risk. We conclude the chapter with a discussion on some problems with cost of capital estimates.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 10, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 3 OF 58 DAYS (50minute periods)
Answers to EndOfChapter Questions
101 Probable Effect on
r_{d}(1 – T) r_{s} WACC
a. The corporate tax rate is lowered. + 0 +
b. The Federal Reserve tightens credit. + + +
c. The firm uses more debt; that is, it increases
its debt/assets ratio. + + 0
d. The dividend payout ratio is increased. 0 0 0
e. The firm doubles the amount of capital it raises
during the year. 0 or + 0 or + 0 or +
f. The firm expands into a risky new area. + + +
g. The firm merges with another firm whose earnings
are countercyclical both to those of the first firm and
to the stock market. – – –
h. The stock market falls drastically, and the firm’s stock
falls along with the rest. 0 + +
i. Investors become more risk averse. + + +
j. The firm is an electric utility with a large investment in
nuclear plants. Several states propose a ban on
nuclear power generation. + + +
102 An increase in the riskfree rate will increase the cost of debt. Remember from Chapter 6, r = r_{RF} + DRP + LP + MRP. Thus, if r_{RF} increases so does r (the cost of debt). Similarly, if the riskfree rate increases so does the cost of equity. From the CAPM equation, r_{s} = r_{RF} + (r_{M} – r_{RF})b. Consequently, if r_{RF} increases r_{s} will increase too.
103 Each firm has an optimal capital structure, defined as that mix of debt, preferred, and common equity that causes its stock price to be maximized. A valuemaximizing firm will determine its optimal capital structure, use it as a target, and then raise new capital in a manner designed to keep the actual capital structure on target over time. The target proportions of debt, preferred stock, and common equity, along with the costs of those components, are used to calculate the firm’s weighted average cost of capital, WACC.
The weights could be based either on the accounting values shown on the firm’s balance sheet (book values) or on the market values of the different securities. Theoretically, the weights should be based on market values, but if a firm’s book value weights are reasonably close to its market value weights, book value weights can be used as a proxy for market value weights. Consequently, target market value weights should be used in the WACC equation.
104 In general, failing to adjust for differences in risk would lead the firm to accept too many risky projects and reject too many safe ones. Over time, the firm would become more risky, its WACC would increase, and its shareholder value would suffer.
The cost of capital for averagerisk projects would be the firm’s cost of capital, 10%. A somewhat higher cost would be used for more risky projects, and a lower cost would be used for less risky ones. For example, we might use 12% for more risky projects and 9% for less risky projects. These choices are arbitrary.
105 The cost of retained earnings is lower than the cost of new common equity; therefore, if new common stock had to be issued then the firm’s WACC would increase.
The calculated WACC does depend on the size of the capital budget. A firm calculates its retained earnings breakpoint (and any other capital breakpoints for additional debt and preferred). This R/E breakpoint represents the amount of capital raised beyond which new common stock must be issued. Thus, a capital budget smaller than this breakpoint would use the lower cost retained earnings and thus a lower WACC. A capital budget greater than this breakpoint would use the higher cost of new equity and thus a higher WACC.
Dividend policy has a significant impact on the WACC. The R/E breakpoint is calculated as the addition to retained earnings divided by the equity fraction. The higher the firm’s dividend payout, the smaller the addition to retained earnings and the lower the R/E breakpoint. (That is, the firm’s WACC will increase at a smaller capital budget.)
Solutions to EndOfChapter Problems
101 r_{d}(1 – T) = 0.12(0.65) = 7.80%.
102 P_{p} = $47.50; D_{p} = $3.80; r_{p} = ?
r_{p} = = = 8%.
103 40% Debt; 60% Common equity; r_{d} = 9%; T = 40%; WACC = 9.96%; r_{s} = ?
WACC = (w_{d})(r_{d})(1 – T) + (w_{c})(r_{s})
0.0996 = (0.4)(0.09)(1 – 0.4) + (0.6)r_{s}
0.0996 = 0.0216 + 0.6r_{s}
0.078 = 0.6r_{s}
r_{s} = 13%.
104 P_{0} = $30; D_{1} = $3.00; g = 5%; r_{s} = ?
a. r_{s} = + g = + 0.05 = 15%.
b. F = 10%; r_{e} = ?
r_{e} = + g = + 0.05
= + 0.05 = 16.11%.
105 Projects A, B, C, D, and E would be accepted since each project’s return is greater than the firm’s WACC.
106 a. r_{s} = + g = + 7% = 9.3% + 7% = 16.3%.
b. r_{s} = r_{RF} + (r_{M} – r_{RF})b
= 9% + (13% – 9%)1.6 = 9% + (4%)1.6 = 9% + 6.4% = 15.4%.
c. r_{s} = Bond rate + Risk premium = 12% + 4% = 16%.
d. Since you have equal confidence in the inputs used for the three approaches, an average of the three methodologies probably would be warranted.
r_{s} = = 15.9%.
107 a. r_{s} = + g
= + 0.06
= 14.83%.
b. F = ($36.00 – $32.40)/$36.00 = $3.60/$36.00 = 10%.
c. r_{e} = D_{1}/[P_{0}(1 – F)] + g = $3.18/$32.40 + 6% = 9.81% + 6% = 15.81%.
108 Debt = 40%, Common equity = 60%.
P_{0} = $22.50, D_{0} = $2.00, D_{1} = $2.00(1.07) = $2.14, g = 7%.
r_{s} = + g = + 7% = 16.51%.
WACC = (0.4)(0.12)(1 – 0.4) + (0.6)(0.1651)
= 0.0288 + 0.0991 = 12.79%.
109 Capital Sources Amount Capital Structure Weight
Longterm debt $1,152 40.0%
Common Equity 1,728 60.0
$2,880 100.0%
WACC = w_{d}r_{d}(1 – T) + w_{c}r_{s} = 0.4(0.13)(0.6) + 0.6(0.16)
= 0.0312 + 0.0960 = 12.72%.
1010 If the investment requires $5.9 million, that means that it requires $3.54 million (60%) of common equity and $2.36 million (40%) of debt. In this scenario, the firm would exhaust its $2 million of retained earnings and be forced to raise new stock at a cost of 15%. Needing $2.36 million in debt, the firm could get by raising debt at only 10%. Therefore, its weighted average cost of capital is: WACC = 0.4(10%)(1 – 0.4) + 0.6(15%) = 11.4%.
1011 r_{s} = D_{1}/P_{0} + g = $2(1.07)/$24.75 + 7%
= 8.65% + 7% = 15.65%.
WACC = w_{d}(r_{d})(1 – T) + w_{c}(r_{s}); w_{c} = 1 – w_{d}.
13.95% = w_{d}(11%)(1 – 0.35) + (1 – w_{d})(15.65%)
0.1395 = 0.0715w_{d} + 0.1565 – 0.1565w_{d}
0.017 = 0.085w_{d}
w_{d} = 0.20 = 20%.
1012 a. r_{d} = 10%, r_{d}(1 – T) = 10%(0.6) = 6%.
D/A = 45%; D_{0} = $2; g = 4%; P_{0} = $20; T = 40%.
Project A: Rate of return = 13%.
Project B: Rate of return = 10%.
r_{s} = $2(1.04)/$20 + 4% = 14.40%.
b. WACC = 0.45(6%) + 0.55(14.40%) = 10.62%.
c. Since the firm’s WACC is 10.62% and each of the projects is equally risky and as risky as the firm’s other assets, MEC should accept Project A. Its rate of return is greater than the firm’s WACC. Project B should not be accepted, since its rate of return is less than MEC’s WACC.
1013 If the firm's dividend yield is 5% and its stock price is $46.75, the next expected annual dividend can be calculated.
Dividend yield = D_{1}/P_{0}
5% = D_{1}/$46.75
D_{1} = $2.3375.
Next, the firm's cost of new common stock can be determined from the DCF approach for the cost of equity.
r_{e} = D_{1}/[P_{0}(1 – F)] + g
= $2.3375/[$46.75(1 – 0.05)] + 0.12
= 17.26%.
1014 r_{p} = = = 11.94%.
1015 a. Examining the DCF approach to the cost of retained earnings, the expected growth rate can be determined from the cost of common equity, price, and expected dividend. However, first, this problem requires that the formula for WACC be used to determine the cost of common equity.
WACC = w_{d}(r_{d})(1 – T) + w_{c}(r_{s})
13.0% = 0.4(10%)(1 – 0.4) + 0.6(r_{s})
10.6% = 0.6r_{s}
r_{s} = 0.17667 or 17.67%.
From the cost of common equity, the expected growth rate can now be determined.
r_{s} = D_{1}/P_{0} + g
0.17667 = $3/$35 + g
g = 0.090952 or 9.10%.
b. From the formula for the longrun growth rate:
g = (1 – Div. payout ratio) ´ ROE = (1 – Div. payout ratio) ´ (NI/Equity)
0.090952 = (1 – Div. payout ratio) ´ ($1,100 million/$6,000 million)
0.090952 = (1 – Div. payout ratio) ´ 0.1833333
0.496104 = (1 – Div. payout ratio)
Div. payout ratio = 0.503896 or 50.39%.
1016 a. With a financial calculator, input N = 5, PV = 4.42, PMT = 0, FV = 6.50, and then solve for I/YR = g = 8.02% » 8%.
b. D_{1} = D_{0}(1 + g) = $2.60(1.08) = $2.81.
c. r_{s} = D_{1}/P_{0} + g = $2.81/$36.00 + 8% = 15.81%.
1017 a. r_{s} = + g
0.09 = + g
0.09 = 0.06 + g
g = 3%.
b. Current EPS $5.400
Less: Dividends per share 3.600
Retained earnings per share $1.800
Rate of return ´ 0.090
Increase in EPS $0.162
Plus: Current EPS 5.400
Next year’s EPS $5.562
Alternatively, EPS_{1} = EPS_{0}(1 + g) = $5.40(1.03) = $5.562.
1018 a. r_{d}(1 – T) = 0.10(1 – 0.3) = 7%.
r_{p} = $5/$49 = 10.2%.
r_{s} = $3.50/$36 + 6% = 15.72%.
b. WACC:
Aftertax Weighted
Component Weight ´ Cost = Cost
Debt [0.10(1 – T)] 0.15 7.00% 1.05%
Preferred stock 0.10 10.20 1.02
Common stock 0.75 15.72 11.79
WACC = 13.86%
c. Projects 1 and 2 will be accepted since their rates of return exceed the WACC.
1019 a. If all project decisions are independent, the firm should accept all projects whose returns exceed their riskadjusted costs of capital. The appropriate costs of capital are summarized below:
Required Rate of Cost of
Project Investment Return Capital
A $4 million 14.0% 12%
B 5 million 1.5 12
C 3 million 9.5 8
D 2 million 9.0 10
E 6 million 12.5 12
F 5 million 12.5 10
G 6 million 7.0 8
H 3 million 11.5 8
Therefore, Ziege should accept projects A, C, E, F, and H.
b. With only $13 million to invest in its capital budget, Ziege must choose the best combination of Projects A, C, E, F, and H. Collectively, the projects would account for an investment of $21 million, so naturally not all these projects may be accepted. Looking at the excess return created by the projects (rate of return minus the cost of capital), we see that the excess returns for Projects A, C, E, F, and H are 2%, 1.5%, 0.5%, 2.5%, and 3.5%. The firm should accept the projects which provide the greatest excess returns. By that rationale, the first project to be eliminated from consideration is Project E. This brings the total investment required down to $15 million, therefore one more project must be eliminated. The next lowest excess return is Project C. Therefore, Ziege's optimal capital budget consists of Projects A, F, and H, and it amounts to $12 million.
c. Since Projects A, F, and H are already accepted projects, we must adjust the costs of capital for the other two value producing projects (C and E).
Required Rate of Cost of
Project Investment Return Capital
C $3 million 9.5% 8% + 1% = 9%
E 6 million 12.5 12% + 1% = 13%
If new capital must be issued, Project E ceases to be an acceptable project. On the other hand, Project C's expected rate of return still exceeds the riskadjusted cost of capital even after raising additional capital. Hence, Ziege's new capital budget should consist of Projects A, C, F, and H and requires $15 million of capital, so $3 million of additional capital must be raised.
1020 a. Aftertax cost of new debt: r_{d}(1 – T) = 0.09(1 – 0.4) = 5.4%.
Cost of common equity: Calculate g as follows:
With a financial calculator, input N = 9, PV = 3.90, PMT = 0, FV = 7.80, and then solve for I/YR = g = 8.01% » 8%.
r_{s} = + g = + 0.08 = + 0.08 = 0.146 = 14.6%.
b. WACC calculation:
Aftertax Weighted
Component Weight ´ Cost = Cost
Debt [0.09(1 – T)] 0.40 5.4% 2.16%
Common equity (RE) 0.60 14.6 8.76
WACC = 10.92%
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
1021 a.
b.
c.
d.
Skye's WACC will be 11.22% so long as it finances with debt, preferred stock, and common equity raised as retained earnings. If it expands so rapidly that it uses up all of its retained earnings and must issue new common stock with a cost of 13.83% (average of DCF and CAPM estimates), then its WACC will increase to 11.51%.
Chapter 11
The Basics of Capital Budgeting
Learning Objectives
After reading this chapter, students should be able to:
Define capital budgeting, explain why it is important, differentiate between security valuation and capital budgeting, and state how project proposals are generally classified.
Calculate net present value (NPV) and internal rate of return (IRR) for a given project and evaluate each method.
Define NPV profiles, the crossover rate, and explain the rationale behind the NPV and IRR methods, their reinvestment rate assumptions, and which method is better when evaluating independent versus mutually exclusive projects.
Briefly explain the problem of multiple IRRs and when this situation could occur.
Calculate the modified internal rate of return (MIRR) for a given project and evaluate this method.
Calculate both the payback and discounted payback periods for a given project and evaluate each method.
Identify at least one relevant piece of information provided to decision makers for each capital budgeting decision method discussed in the chapter.
Identify a number of different types of decisions that use the capital budgeting techniques developed in this chapter.
Identify and explain the purposes of the postaudit in the capital budgeting process.
Lecture Suggestions
This is a relatively straightforward chapter, and, for the most part, it is a direct application of the time value concepts first discussed in Chapter 2. We point out that capital budgeting is to a company what buying stocks or bonds is to an individual—an investment decision, when the company wants to know if the expected value of the cash flows is greater than the cost of the project, and whether or not the expected rate of return on the project exceeds the cost of the funds required to do the project. We cover the standard capital budgeting procedures—NPV, IRR, MIRR, payback and discounted payback.
At this point, students who have not yet mastered time value concepts and how to use their calculator efficiently get another chance to catch on. Students who have mastered those tools and concepts have fun, because they can see what is happening and the usefulness of what they are learning.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 11, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 4 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
111 Project classification schemes can be used to indicate how much analysis is required to evaluate a given project, the level of the executive who must approve the project, and the cost of capital that should be used to calculate the project’s NPV. Thus, classification schemes can increase the efficiency of the capital budgeting process.
112 The regular payback method has three main flaws: (1) Dollars received in different years are all given the same weight. (2) Cash flows beyond the payback year are given no consideration whatever, regardless of how large they might be. (3) Unlike the NPV, which tells us by how much the project should increase shareholder wealth, and the IRR, which tells us how much a project yields over the cost of capital, the payback merely tells us when we get our investment back. The discounted payback corrects the first flaw, but the other two flaws still remain.
113 The NPV is obtained by discounting future cash flows, and the discounting process actually compounds the interest rate over time. Thus, an increase in the discount rate has a much greater impact on a cash flow in Year 5 than on a cash flow in Year 1.
114 Mutually exclusive projects are a set of projects in which only one of the projects can be accepted. For example, the installation of a conveyorbelt system in a warehouse and the purchase of a fleet of forklifts for the same warehouse would be mutually exclusive projects—accepting one implies rejection of the other. When choosing between mutually exclusive projects, managers should rank the projects based on the NPV decision rule. The mutually exclusive project with the highest positive NPV should be chosen. The NPV decision rule properly ranks the projects because it assumes the appropriate reinvestment rate is the cost of capital.
115 The first question is related to Question 113 and the same rationale applies. A high cost of capital favors a shorterterm project. If the cost of capital declined, it would lead firms to invest more in longterm projects. With regard to the last question, the answer is no; the IRR rankings are constant and independent of the firm’s cost of capital.
116 The statement is true. The NPV and IRR methods result in conflicts only if mutually exclusive projects are being considered since the NPV is positive if and only if the IRR is greater than the cost of capital. If the assumptions were changed so that the firm had mutually exclusive projects, then the IRR and NPV methods could lead to different conclusions. A change in the cost of capital or in the cash flow streams would not lead to conflicts if the projects were independent. Therefore, the IRR method can be used in lieu of the NPV if the projects being considered are independent.
117 Payback provides information on how long funds will be tied up in a project. The shorter the payback, other things held constant, the greater the project’s liquidity. This factor is often important for smaller firms that don’t have ready access to the capital markets. Also, cash flows expected in the distant future are generally riskier than nearterm cash flows, so the payback can be used as a risk indicator.
118 Project X should be chosen over Project Y. Since the two projects are mutually exclusive, only one project can be accepted. The decision rule that should be used is NPV. Since Project X has the higher NPV, it should be chosen. The cost of capital used in the NPV analysis appropriately includes risk.
119 The NPV method assumes reinvestment at the cost of capital, while the IRR method assumes reinvestment at the IRR. MIRR is a modified version of IRR that assumes reinvestment at the cost of capital.
The NPV method assumes that the rate of return that the firm can invest differential cash flows it would receive if it chose a smaller project is the cost of capital. With NPV we are calculating present values and the interest rate or discount rate is the cost of capital. When we find the IRR we are discounting at the rate that causes NPV to equal zero, which means that the IRR method assumes that cash flows can be reinvested at the IRR (the project’s rate of return). With MIRR, since positive cash flows are compounded at the cost of capital and negative cash flows are discounted at the cost of capital, the MIRR assumes that the cash flows are reinvested at the cost of capital.
1110 a. In general, the answer is no. The objective of management should be to maximize value, and as we point out in subsequent chapters, stock values are determined by both earnings and growth. The NPV calculation automatically takes this into account, and if the NPV of a longterm project exceeds that of a shortterm project, the higher future growth from the longterm project must be more than enough to compensate for the lower earnings in early years.
b. If the same $100 million had been spent on a shortterm project—one with a faster payback—reported profits would have been higher for a period of years. This is, of course, another reason why firms sometimes use the payback method.
Solutions to EndofChapter Problems
111 Financial calculator solution: Input CF_{0} = 52125, CF_{18} = 12000, I/YR = 12, and then solve for NPV = $7,486.68.
112 Financial calculator solution: Input CF_{0} = 52125, CF_{18} = 12000, and then solve for IRR = 16%.
113 MIRR: PV costs = $52,125.
FV inflows:
PV FV
0 1 2 3 4 5 6 7 8
        
12,000 12,000 12,000 12,000 12,000 12,000 12,000 12,000
13,440
15,053
16,859
18,882
21,148
23,686
26,528
52,125 MIRR = 13.89% 147,596
Financial calculator solution: Obtain the FVA by inputting N = 8, I/YR = 12, PV = 0, PMT = 12000, and then solve for FV = $147,596. The MIRR can be obtained by inputting N = 8, PV = 52125, PMT = 0, FV = 147596, and then solving for I/YR = 13.89%.
114 Since the cash flows are a constant $12,000, calculate the payback period as: $52,125/$12,000 = 4.3438, so the payback is about 4 years.
115 Project K’s discounted payback period is calculated as follows:
Annual Discounted @12%
Period Cash Flows Cash Flows Cumulative
0 ($52,125) ($52,125.00) ($52,125.00)
1 12,000 10,714.29 (41,410.71)
2 12,000 9,566.33 (31,844.38)
3 12,000 8,541.36 (23,303.02)
4 12,000 7,626.22 (15,676.80)
5 12,000 6,809.12 (8,867.68)
6 12,000 6,079.57 (2,788.11)
7 12,000 5,428.19 2,640.08
8 12,000 4,846.60 7,486.68
The discounted payback period is 6 + years, or 6.51 years.
116 a. Project A: Using a financial calculator, enter the following:
CF_{0} = 25, CF_{1} = 5, CF_{2} = 10, CF_{3} = 17, I/YR = 5; NPV = $3.52.
Change I/YR = 5 to I/YR = 10; NPV = $0.58.
Change I/YR = 10 to I/YR = 15; NPV = $1.91.
Project B: Using a financial calculator, enter the following:
CF_{0} = 20, CF_{1} = 10, CF_{2} = 9, CF_{3} = 6, I/YR = 5; NPV = $2.87.
Change I/YR = 5 to I/YR = 10; NPV = $1.04.
Change I/YR = 10 to I/YR = 15; NPV = $0.55.
b. Using the data for Project A, enter the cash flows into a financial calculator and solve for IRR_{A} = 11.10%. The IRR is independent of the WACC, so it doesn’t change when the WACC changes.
Using the data for Project B, enter the cash flows into a financial calculator and solve for IRR_{B} = 13.18%. Again, the IRR is independent of the WACC, so it doesn’t change when the WACC changes.
c. At a WACC = 5%, NPV_{A }> NPV_{B} so choose Project A.
At a WACC = 10%, NPV_{B} > NPV_{A} so choose Project B.
At a WACC = 15%, both NPVs are less than zero, so neither project would be chosen.
117 a. Project A:
CF_{0} = 6000; CF_{15} = 2000; I/YR = 14.
Solve for NPV_{A} = $866.16. IRR_{A} = 19.86%.
MIRR calculation:
0 1 2 3 4 5
     
6,000 2,000 2,000 2,000 2,000 2,000
2,280.00
2,599.20
2,963.09
3,377.92
13,220.21
Using a financial calculator, enter N = 5; PV = 6000; PMT = 0; FV = 13220.21; and solve for MIRR_{A} = I/YR = 17.12%.
Payback calculation:
0 1 2 3 4 5
     
6,000 2,000 2,000 2,000 2,000 2,000
Cumulative CF: 6,000 4,000 2,000 0 2,000 4,000
Regular Payback_{A} = 3 years.
Discounted payback calculation:
0 1 2 3 4 5
     
6,000 2,000 2,000 2,000 2,000 2,000
Discounted CF: 6,000 1,754.39 1,538.94 1,349.94 1,184.16 1,038.74
Cumulative CF: 6,000 4,245.61 2,706.67 1,356.73 172.57 866.17
Discounted Payback_{A} = 4 + $172.57/$1,038.74 = 4.17 years.
Project B:
CF_{0} = 18000; CF_{15} = 5600; I/YR = 14.
Solve for NPV_{B} = $1,255.25. IRR_{B} = 16.80%.
MIRR calculation:
0 1 2 3 4 5
     
18,000 5,600 5,600 5,600 5,600 5,600
6,384.00
7,277.76
8,296.65
9,458.18
37,016.59
Using a financial calculator, enter N = 5; PV = 18000; PMT = 0; FV = 37016.59; and solve for MIRR_{B} = I/YR = 15.51%.
Payback calculation:
0 1 2 3 4 5
     
18,000 5,600 5,600 5,600 5,600 5,600
Cumulative CF: 18,000 12,400 6,800 1,200 4,400 10,000
Regular Payback_{B} = 3 + $1,200/$5,600 = 3.21 years.
Discounted payback calculation:
0 1 2 3 4 5
     
18,000 5,600 5,600 5,600 5,600 5,600
Discounted CF: 18,000 4,912.28 4,309.02 3,779.84 3,315.65 2,908.46
Cumulative CF: 18,000 13,087.72 8,778.70 4,998.86 1,683.21 1,225.25
Discounted Payback_{B} = 4 + $1,683.21/$2,908.46 = 4.58 years.
Summary of capital budgeting rules results:
Project A Project B
NPV $866.16 $1,225.25
IRR 19.86% 16.80%
MIRR 17.12% 15.51%
Payback 3.0 years 3.21 years
Discounted payback 4.17 years 4.58 years
b. If the projects are independent, both projects would be accepted since both of their NPVs are positive.
c. If the projects are mutually exclusive then only one project can be accepted, so the project with the highest positive NPV is chosen. Accept Project B.
d. The conflict between NPV and IRR occurs due to the difference in the size of the projects. Project B is 3 times larger than Project A.
118 a. No mitigation analysis (in millions of dollars):
0 1 2 3 4 5
     
60 20 20 20 20 20
Using a financial calculator, enter the data as follows: CF_{0} = 60; CF_{15} = 20; I/YR = 12. Solve for NPV = $12.10 million and IRR = 19.86%.
With mitigation analysis (in millions of dollars):
0 1 2 3 4 5
     
70 21 21 21 21 21
Using a financial calculator, enter the data as follows: CF_{0} = 70; CF_{15} = 21; I/YR = 12. Solve for NPV = $5.70 million and IRR = 15.24%.
b. The environmental effects if not mitigated could result in additional loss of cash flows and/or fines and penalties due to ill will among customers, community, etc. Therefore, even though the mine is legal without mitigation, the company needs to make sure that they have anticipated all costs in the "no mitigation" analysis from not doing the environmental mitigation.
c. Even when mitigation is considered the project has a positive NPV, so it should be undertaken. The question becomes whether you mitigate or don’t mitigate for environmental problems. Under the assumption that all costs have been considered, the company would not mitigate for the environmental impact of the project since its NPV is $12.10 million vs. $5.70 million when mitigation costs are included in the analysis.
119 a. No mitigation analysis (in millions of dollars):
0 1 2 3 4 5
     
240 80 80 80 80 80
Using a financial calculator, enter the data as follows: CF_{0} = 240; CF_{15 }= 80; I/YR = 17. Solve for NPV = $15.95 million and IRR = 19.86%.
With mitigation analysis (in millions of dollars):
0 1 2 3 4 5
     
280 84 84 84 84 84
Using a financial calculator, enter the data as follows: CF_{0} = 280; CF_{15 }= 84; I/YR = 17. Solve for NPV = $11.25 million and IRR = 15.24%.
b. If the utility mitigates for the environmental effects, the project is not acceptable. However, before the company chooses to do the project without mitigation, it needs to make sure that any costs of "ill will" for not mitigating for the environmental effects have been considered in that analysis.
c. Again, the project should be undertaken only if they do not mitigate for the environmental effects. However, they want to make sure that they’ve done the analysis properly due to any "ill will" and additional "costs" that might result from undertaking the project without concern for the environmental impacts.
1110 Project A: Using a financial calculator, enter the following data: CF_{0} = 400; CF_{13} = 55; CF_{45} = 225; I/YR = 10. Solve for NPV = $30.16.
Project B: Using a financial calculator, enter the following data: CF_{0} = 600; CF_{12} = 300; CF_{34} = 50; CF_{5} = 49; I/YR = 10. Solve for NPV = $22.80.
The decision rule for mutually exclusive projects is to accept the project with the highest positive NPV. In this situation, the firm would accept Project A since NPV_{A} = $30.16 compared to NPV_{B} = $22.80.
1111 Project S: Using a financial calculator, enter the following data: CF_{0} = 15000; CF_{15} = 4500; I/YR = 14. NPV_{S} = $448.86.
Project L: Using a financial calculator, enter the following data: CF_{0} = 37500; CF_{15} = 11100; I/YR = 14. NPV_{L} = $607.20.
The decision rule for mutually exclusive projects is to accept the project with the highest positive NPV. In this situation, the firm would accept Project L since NPV_{L} = $607.20 compared to NPV_{S} = $448.86.
1112 Input the appropriate cash flows into the cash flow register, and then calculate NPV at 10% and the IRR of each of the projects:
Project S: CF_{0} = 1000; CF_{1} = 900; CF_{2} = 250; CF_{34} = 10; I/YR = 10. Solve for NPV_{S} = $39.14; IRR_{S} = 13.49%.
Project L: CF_{0} = 1000; CF_{1} = 0; CF_{2} = 250; CF_{3} = 400; CF_{4} = 800; I/YR = 10. Solve for NPV_{L} = $53.55; IRR_{L} = 11.74%.
Since Project L has the higher NPV, it is the better project, even though its IRR is less than Project S’s IRR. The IRR of the better project is IRR_{L} = 11.74%.
1113 Because both projects are the same size you can just calculate each project’s MIRR and choose the project with the higher MIRR.
Project X: 0 1 2 3 4
    
1,000 100 300 400 700.00
448.00
376.32
140.49
1,000 13.59% = MIRR_{X} 1,664.81
$1,000 = $1,664.81/(1 + MIRR_{X})^{4}.
Project Y: 0 1 2 3 4
    
1,000 1,000 100 50 50.00
56.00
125.44
1,404.93
1,000 13.10% = MIRR_{Y} 1,636.37
$1,000 = $1,636.37/(1 + MIRR_{Y})^{4}.
Thus, since MIRR_{X} > MIRR_{Y}, Project X should be chosen.
Alternate step: You could calculate the NPVs, see that Project X has the higher NPV, and just calculate MIRR_{X}.
NPV_{X} = $58.02 and NPV_{Y} = $39.94.
1114 a. HCC: Using a financial calculator, enter the following data: CF_{0} = 600000; CF_{15} = 50000; I/YR = 7. Solve for NPV = $805,009.87.
LCC: Using a financial calculator, enter the following data: CF_{0} = 100000; CF_{15} = 175000; I/YR = 7. Solve for NPV = $817,534.55.
Since we are examining costs, the unit chosen would be the one that has the lower PV of costs. Since HCC’s PV of costs is lower than LCC’s, HCC would be chosen.
b. The IRR cannot be calculated because the cash flows are all one sign. A change of sign would be needed in order to calculate the IRR.
c. HCC: I/YR = 15; solve for NPV = $767,607.75.
LCC: I/YR = 15; solve for NPV = $686,627.14.
When the WACC increases from 7% to 15%, the PV of costs are now lower for LCC than HCC. The reason is that when you discount at a higher rate you are making negative CFs smaller and thus improving the results, unknowingly. Thus, if you were trying to risk adjust for a riskier project that consisted just of negative CFs then you would use a lower cost of capital rather than a higher cost of capital and this would properly adjust for the risk of a project with only negative CFs.
1115 a. Using a financial calculator, calculate NPVs for each plan (as shown in the table below) and graph each plan’s NPV profile.
Discount Rate NPV Plan A NPV Plan B
0% $2,400,000 $30,000,000
5 1,714,286 14,170,642
10 1,090,909 5,878,484
12 857,143 3,685,832
15 521,739 1,144,596
16.7 339,332 0
20 0 1,773,883
The crossover rate is approximately 16%. If the cost of capital is less than the crossover rate, then Plan B should be accepted; if the cost of capital is greater than the crossover rate, then Plan A is preferred. At the crossover rate, the two projects’ NPVs are equal.
b. Yes. Assuming (1) equal risk among projects, and (2) that the cost of capital is a constant and does not vary with the amount of capital raised, the firm would take on all available projects with returns greater than its 12% WACC. If the firm had invested in all available projects with returns greater than 12%, then its best alternative would be to repay capital. Thus, the WACC is the correct reinvestment rate for evaluating a project’s cash flows.
1116 a. Using a financial calculator, we get:
NPV_{A} = $14,486,808. NPV_{B} = $11,156,893.
IRR_{A} = 15.03%. IRR_{B} = 22.26%.
b. Using a financial calculator, calculate each plan’s NPVs at different discount rates (as shown in the table below) and graph the NPV profiles.
Discount Rate NPV Plan A NPV Plan B
0% $88,000,000 $42,400,000
5 39,758,146 21,897,212
10 14,486,808 11,156,893
15.03 0 4,997,152
20 8,834,690 1,245,257
22.26 11,765,254 0
The crossover rate is somewhere between 11% and 12%.
c. The NPV method implicitly assumes that the opportunity exists to reinvest the cash flows generated by a project at the WACC, while use of the IRR method implies the opportunity to reinvest at the IRR. The firm will invest in all independent projects with an NPV > $0. As cash flows come in from these projects, the firm will either pay them out to investors, or use them as a substitute for outside capital which, in this case, costs 10%. Thus, since these cash flows are expected to save the firm 10%, this is their opportunity cost reinvestment rate.
The IRR method assumes reinvestment at the internal rate of return itself, which is an incorrect assumption, given a constant expected future cost of capital, and ready access to capital markets.
1117 a. Using a financial calculator and entering each project’s cash flows into the cash flow registers and entering I/YR = 12, you would calculate each project’s NPV. At WACC = 12%, Project A has the greater NPV, specifically $200.41 as compared to Project B’s NPV of $145.93.
b. Using a financial calculator and entering each project’s cash flows into the cash flow registers, you would calculate each project’s IRR. IRR_{A} = 18.1%; IRR_{B} = 24.0%.
c. Here is the MIRR for Project A when WACC = 12%:
PV costs = $300 + $387/(1.12)^{1} + $193/(1.12)^{2} + $100/(1.12)^{3} + $180/(1.12)^{7} = $952.00.
TV inflows = $600(1.12)^{3} + $600(1.12)^{2} + $850(1.12)^{1} = $2,547.60.
MIRR is the discount rate that forces the TV of $2,547.60 in 7 years to equal $952.00.
Using a financial calculator enter the following inputs: N = 7, PV = 952, PMT = 0, and FV = 2547.60. Then, solve for I/YR = MIRR_{A} = 15.10%.
Here is the MIRR for Project B when WACC = 12%:
PV costs = $405.
TV inflows = $134(1.12)^{6} + $134(1.12)^{5} + $134(1.12)^{4} + $134(1.12)^{3} + $134(1.12)^{2} + $134(1.12)
= $1,217.93.
MIRR is the discount rate that forces the TV of $1,217.93 in 7 years to equal $405.
Using a financial calculator enter the following inputs: N = 7; PV = 405; PMT = 0; and FV = 1217.93. Then, solve for I/YR = MIRR_{B} = 17.03%.
d. WACC = 12% criteria:
Project A Project B
NPV $200.41 $145.93
IRR 18.1% 24.0%
MIRR 15.1% 17.03%
The correct decision is that Project A should be chosen because NPV_{A} > NPV_{B}.
At WACC = 18%, using your financial calculator enter the cash flows for each project, enter I/YR = WACC = 18, and then solve for each Project’s NPV.
NPV_{A} = $2.66; NPV_{B} = $63.68.
At WACC = 18%, NPV_{B} > NPV_{A} so Project B would be chosen.
e.
Discount Rate NPV_{A} NPV_{B}
0.0% $890 $399
10.0 283 179
12.0 200 146
18.1 0 62
20.0 (49) 41
24.0 (138) 0
30.0 (238) (51)
f. Here is the MIRR for Project A when WACC = 18%:
PV costs = $300 + $387/(1.18)^{1} + $193/(1.18)^{2} + $100/(1.18)^{3} + $180/(1.18)^{7} = $883.95.
TV inflows = $600(1.18)^{3} + $600(1.18)^{2} + $850(1.18)^{1} = $2,824.26.
MIRR is the discount rate that forces the TV of $2,824.26 in 7 years to equal $883.95.
Using a financial calculator enter the following inputs: N = 7; PV = 883.95; PMT = 0; and FV = 2824.26. Then, solve for I/YR = MIRR_{A} = 18.05%.
Here is the MIRR for Project B when WACC = 18%:
PV costs = $405.
TV inflows = $134(1.18)^{6} + $134(1.18)^{5} + $134(1.18)^{4} + $134(1.18)^{3} + $134(1.18)^{2} + $134(1.18)
= $1,492.96.
MIRR is the discount rate that forces the TV of $1,492.26 in 7 years to equal $405.
Using a financial calculator enter the following inputs: N = 7; PV = 405; PMT = 0; and FV = 1492.26. Then, solve for I/YR = MIRR_{B} = 20.48%.
1118 Facts: 5 years remaining on lease; rent = $2,000/month; 60 payments left, payment at end of month.
New lease terms: $0/month for 9 months; $2,600/month for 51 months.
WACC = 12% annual (1% per month).
a. 0 1 2 59 60
   · · ·  
2,000 2,000 2,000 2,000
PV cost of old lease: N = 60; I/YR = 1; PMT = 2000; FV = 0; PV = ? PV = $89,910.08.
0 1 9 10 59 60
  · · ·   · · ·  
0 0 2,600 2,600 2,600
PV cost of new lease: CF_{0} = 0, CF_{19} = 0; CF_{1060} = 2600; I/YR = 1. NPV = $94,611.45.
Sharon should not accept the new lease because the present value of its cost is $94,611.45 – $89,910.08 = $4,701.37 greater than the old lease.
b. At t = 9 the FV of the original lease’s cost = $89,910.08(1.01)^{9} = $98,333.33. Since lease payments for months 09 would be zero, we can calculate the lease payments during the remaining 51 months as follows: N = 51; I/YR = 1; PV = 98333.33; and FV = 0. Solve for PMT = $2,470.80.
Check:
0 1 9 10 59 60
  · · ·   · · ·  
0 0 2,470.80 2,470.80 2,470.80
PV cost of new lease: CF_{0} = 0; CF_{19} = 0; CF_{1060} = 2470.80; I/YR = 1. NPV = $89,909.99.
Except for rounding; the PV cost of this lease equals the PV cost of the old lease.
c. Period Old Lease New Lease D Lease
0 0 0 0
19 2,000 0 2,000
1060 2,000 2,600 600
CF_{0} = 0; CF_{19} = 2000; CF_{1060} = 600; IRR = ? IRR = 1.9113%. This is the periodic rate. To obtain the nominal cost of capital, multiply by 12: 12(0.019113) = 22.94%.
Check: Old lease terms:
N = 60; I/YR = 1.9113; PMT = 2000; FV = 0; PV = ? PV = $71,039.17.
New lease terms:
CF_{0} = 0; CF_{19} = 0; CF_{1060} = 2600; I/YR = 1.9113; NPV = ? NPV = $71,038.98.
Except for rounding differences; the costs are the same.
1119 a. The project’s expected cash flows are as follows (in millions of dollars):
Time Net Cash Flow
0 ($ 2.0)
1 13.0
2 (12.0)
We can construct the following NPV profile:
WACC NPV
0% ($1,000,000)
10 (99,174)
50 1,333,333
80 1,518,519
100 1,500,000
200 1,000,000
300 500,000
400 120,000
410 87,659
420 56,213
430 25,632
450 (33,058)
b. If WACC = 10%, reject the project since NPV < $0. Its NPV at WACC = 10% is equal to $99,174. But if WACC = 20%, accept the project because NPV > $0. Its NPV at WACC = 20% is $500,000.
c. Other possible projects with multiple rates of return could be nuclear power plants where disposal of radioactive wastes is required at the end of the project’s life.
d. MIRR @ WACC = 10%:
PV costs = $2,000,000 + $12,000,000/(1.10)^{2} = $11,917,355.
FV inflows = $13,000,000 ´ 1.10 = $14,300,000.
Using a financial calculator enter the following data: N = 2; PV = 11917355; PMT = 0; and FV = 14300000. Then solve for I/YR = MIRR = 9.54%. (Reject the project since MIRR < WACC.)
MIRR @ WACC = 20%:
PV costs = $2,000,000 + $12,000,000/(1.20)^{2} = $10,333,333.
FV inflows = $13,000,000 ´ 1.20 = $15,600,000.
Using a financial calculator enter the following data: N = 2; PV = 10333333; PMT = 0; and FV = 15600000. Then solve for I/YR = MIRR = 22.87%. (Accept the project since MIRR > WACC.)
Looking at the results, this project’s MIRR calculations lead to the same decisions as the NPV calculations. However, the MIRR method will not always lead to the same accept/reject decision as the NPV method. Decisions involving two mutually exclusive projects that differ in scale (size) may have MIRRs that conflict with NPV. In those situations, the NPV method should be used.
1120 Since the IRR is the discount rate at which the NPV of a project equals zero, the project’s inflows can be evaluated at the IRR and the present value of these inflows must equal the initial investment.
Using a financial calculator enter the following: CF_{0} = 0; CF_{1} = 7500; N_{j} = 10; CF_{1} = 10000; N_{j} = 10; I/YR = 10.98. NPV = $65,002.11.
Therefore, the initial investment for this project is $65,002.11. Using a calculator, the project's NPV at the firm’s WACC can now be solved.
CF_{0} = 65002.11; CF_{1} = 7500; N_{j} = 10; CF_{1} = 10000; N_{j} = 10; I/YR = 9. NPV = $10,239.20.
1121 Step 1: Determine the PMT:
0 1 10
  · · · 
1,000 PMT PMT
The IRR is the discount rate at which the NPV of a project equals zero. Since we know the project’s initial investment, its IRR, the length of time that the cash flows occur, and that each cash flow is the same, then we can determine the project’s cash flows by setting it up as a 10year annuity. With a financial calculator, input N = 10, I/YR = 12, PV = 1000, and FV = 0 to obtain PMT = $176.98.
Step 2: Since we’ve been given the WACC, once we have the project’s cash flows we can now determine the project’s MIRR.
Calculate the project’s MIRR:
0 1 2 9 10
   · · ·  
1,000 176.98 176.98 176.98 176.98
194.68
.
.
.
379.37
417.31
1,000 10.93% = MIRR TV = 2,820.61
FV of inflows: With a financial calculator, input N = 10, I/YR = 10, PV = 0, and PMT = 176.98 to obtain FV = $2,820.61. Then input N = 10, PV = 1000, PMT = 0, and FV = 2820.61 to obtain I/YR = MIRR = 10.93%.
1122 The MIRR can be solved with a financial calculator by finding the terminal future value of the cash inflows and the initial present value of cash outflows, and solving for the discount rate that equates these two values. In this instance, the MIRR is given, but a cash outflow is missing and must be solved for. Therefore, if the terminal future value of the cash inflows is found, it can be entered into a financial calculator, along with the number of years the project lasts and the MIRR, to solve for the initial present value of the cash outflows. One of these cash outflows occurs in Year 0 and the remaining value must be the present value of the missing cash outflow in Year 2.
Cash Inflows Compounding Rate FV in Year 5 @ 10%
CF_{1} = $202 ´ (1.10)^{4} $ 295.75
CF_{3} = 196 ´ (1.10)^{2} 237.16
CF_{4} = 350 ´ 1.10 385.00
CF_{5} = 451 ´ 1.00 451.00
$1,368.91
Using the financial calculator to solve for the present value of cash outflows: N = 5; I/YR = 14.14; PV = ?; PMT = 0; FV = 1368.91
The total present value of cash outflows is $706.62, and since the outflow for Year 0 is $500, the present value of the Year 2 cash outflow is $206.62. Therefore, the missing cash outflow for Year 2 is $206.62 ×(1.1)^{2 }= $250.01.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
1123 a. Project A:
Using a financial calculator, enter the following data:
CF_{0} = 30; CF_{1} = 5; CF_{2} = 10; CF_{3} = 15; CF_{4} = 20; I/YR = 10; and solve for NPV_{A} = $7.74; IRR_{A} = 19.19%.
Calculate MIRR_{A} at WACC = 10%:
Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated. With a financial calculator, enter the cash flow stream into the cash flow registers, then enter I/YR = 10, and solve for NPV = $37.739.
Step 2: Calculate the FV of the cash flow stream as follows:
Enter N = 4, I/YR = 10, PV = 37.739, and PMT = 0 to solve for FV = $55.255.
Step 3: Calculate MIRR_{A} as follows:
Enter N = 4, PV = 30, PMT = 0, and FV = 55.255 to solve for I/YR = 16.50%.
Payback A (cash flows in millions):
Annual
Period Cash Flows Cumulative
0 ($30) ($30)
1 5 (25)
2 10 (15)
3 15 0
4 20 20
Payback_{A} = 3 years.
Discounted Payback A (cash flows in millions):
Annual Discounted @10% Cumulative
Period Cash Flows Cash Flows Cash Flows
0 ($30) ($30.00) ($30.00)
1 5 4.55 (25.45)
2 10 8.26 (17.19)
3 15 11.27 (5.92)
4 20 13.66 7.74
Discounted Payback_{A} = 3 + $5.92/$13.66 = 3.43 years.
Project B:
Using a financial calculator, enter the following data:
CF_{0} = 30; CF_{1} = 20; CF_{2} = 10; CF_{3} = 8; CF_{4} = 6; I/YR = 10; and solve for NPV_{B} = $6.55; IRR_{B} = 22.52%.
Calculate MIRR_{B} at WACC = 10%:
Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated. With a financial calculator, enter the cash flow stream into the cash flow registers, then enter I/YR = 10, and solve for NPV = $36.55.
Step 2: Calculate the FV of the cash flow stream as follows:
Enter N = 4, I/YR = 10, PV = 36.55, and PMT = 0 to solve for FV = $53.52.
Step 3: Calculate MIRR_{B} as follows:
Enter N = 4, PV = 30, PMT = 0, and FV = 53.52 to solve for I/YR = 15.57%.
Payback B (cash flows in millions):
Annual
Period Cash Flows Cumulative
0 ($30) ($30)
1 20 (10)
2 10 0
3 8 8
4 6 14
Payback_{B} = 2 years.
Discounted Payback B (cash flows in millions):
Annual Discounted @10% Cumulative
Period Cash Flows Cash Flows Cash Flows
0 ($30) ($30.00) ($30.00)
1 20 18.18 (11.82)
2 10 8.26 (3.56)
3 8 6.01 (2.45)
4 6 4.10 6.55
Discounted Payback_{B} = 2 + $3.56/$6.01 = 2.59 years.
Summary:
Project A Project B
NPV $7.74 $6.55
IRR 19.19% 22.52%
MIRR 16.50% 15.57%
Payback 3 years 2 years
Discounted Payback 3.43 years 2.59 years
b. If the two projects are independent, both projects will be accepted because their NPVs are greater than zero.
c. If the two projects are mutually exclusive, at WACC = 10% Project A should be chosen since NPV_{A} > NPV_{B}.
d. WACC NPV_{A} NPV_{B}
0% $20.00 $14.00
5 13.24 9.96
10 7.74 6.55
15 3.21 3.64
19.19 0 1.52
20 (0.56) 1.13
22.52 (2.23) 0
e. At WACC = 5% and the two projects are mutually exclusive, NPV_{A} > NPV_{B} so choose Project A. This doesn’t change our recommendation. At WACC = 15% and the two projects are mutually exclusive, NPV_{B} > NPV_{A} so choose Project B. This does change our recommendation. Both of these decisions can be made from looking at the NPV profile in part d.
f. The crossover rate is the cost of capital at which the NPV profiles of two projects cross and, thus, at which the projects’ NPVs are equal. At a cost of capital less than the crossover rate there is a conflict between NPV and IRR but at a cost of capital greater than the crossover rate there is no conflict between NPV and IRR.
g. It is not possible for conflicts between NPV and IRR when independent projects are being evaluated. NPV is greater than zero at all WACCs < IRR, so the NPV rule would accept these projects. At IRR > WACC, all projects meeting this criterion would be accepted by the IRR rule.
h. Looking at both the payback and discounted payback methods, Project B looks better than A. The faster the payback, the more liquid and less risky the project.
i. The cutoff chosen for both payback periods is arbitrary—but usually based on specific information the firm has on past projects. However, the criteria for the NPV and the IRR methods are not arbitrary.
j. The MIRR is the discount rate at which the present value of a project’s cost is equal to the present value of its terminal value, where the terminal value is found as the sum of the future values of the cash inflows, compounded at the firm’s cost of capital. The difference between the IRR and MIRR is the reinvestment rate assumption. The reinvestment rate of the IRR is the project’s return, while the reinvestment rate of the MIRR is the firm’s cost of capital. Consequently, MIRR gives a better idea of the rate of return on the project.
k. Academics prefer NPV to IRR because NPV gives an estimate (a dollar value) of how much a potential project will contribute to shareholder wealth. However, executives tend to like IRR because it gives a measure of the project’s "bang for the buck" and gives information concerning a project’s safety margin.
Chapter 12
Cash Flow Estimation and Risk Analysis
Learning Objectives
After reading this chapter, students should be able to:
Analyze an expansion project and make a decision whether the project should be accepted on the basis of standard capital budgeting techniques.
Discuss difficulties and relevant considerations in estimating net cash flows, and explain how project cash flow differs from accounting income.
Define the following terms: incremental cash flow, replacement analysis, sunk cost, opportunity cost, externalities, and cannibalization effect.
Identify and briefly explain three separate and distinct types of risk.
Demonstrate sensitivity and scenario analyses and explain Monte Carlo simulation.
Explain why conventional DCF techniques do not always lead to proper capital budgeting decisions.
Explain the following terms: real option, abandonment/shutdown option, and option value.
List the steps a firm goes through when establishing its optimal capital budget in practice, and explain what capital rationing is.
Lecture Suggestions
This chapter covers some important but relatively technical topics. Note too that this chapter is more modular than most, i.e., the major sections are discrete, hence they can be omitted without loss of continuity. Therefore, if you are experiencing a time crunch, you could skip sections of the chapter.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 12, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 4 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
121 Only cash can be spent or reinvested, and since accounting profits do not represent cash, they are of less fundamental importance than cash flows for investment analysis. Recall that in the stock valuation chapter we focused on dividends, which represent cash flows, rather than on earnings per share.
122 Capital budgeting analysis should only include those cash flows that will be affected by the decision. Sunk costs are unrecoverable and cannot be changed, so they have no bearing on the capital budgeting decision. Opportunity costs represent the cash flows the firm gives up by investing in this project rather than its next best alternative, and externalities are the cash flows (both positive and negative) to other projects that result from the firm taking on this project. These cash flows occur only because the firm took on the capital budgeting project; therefore, they must be included in the analysis.
123 When a firm takes on a new capital budgeting project, it typically must increase its investment in receivables and inventories, over and above the increase in payables and accruals, thus increasing its net operating working capital (NOWC). Since this increase must be financed, it is included as an outflow in Year 0 of the analysis. At the end of the project’s life, inventories are depleted and receivables are collected. Thus, there is a decrease in NOWC, which is treated as an inflow in the final year of the project’s life.
124 The costs associated with financing are reflected in the weighted average cost of capital. To include interest expense in the capital budgeting analysis would "double count" the cost of debt financing.
125 Daily cash flows would be theoretically best, but they would be costly to estimate and probably no more accurate than annual estimates because we simply cannot forecast accurately at a daily level. Therefore, in most cases we simply assume that all cash flows occur at the end of the year. However, for some projects it might be useful to assume that cash flows occur at midyear, or even quarterly or monthly. There is no clear upward or downward bias on NPV since both revenues and costs are being recognized at the end of the year. Unless revenues and costs are distributed radically different throughout the year, there should be no bias.
126 In replacement projects, the benefits are generally cost savings, although the new machinery may also permit additional output. The data for replacement analysis are generally easier to obtain than for new products, but the analysis itself is somewhat more complicated because almost all of the cash flows are incremental, found by whether the project is a new expansion or a replacement project. A new expansion project is defined as one where subtracting the firm invests in new assets to increase sales. Here the incremental cash flows are simply the cash inflows and outflows. In effect, the company is comparing what its value looks like with and without the proposed project. By contrast, a replacement project occurs when the firm replaces an existing asset with a new one in order to reduce operating costs, to increase output, or to improve product quality. In this case, the incremental cash flows are the additional inflows and outflows that result from replacing the old asset. In a replacement analysis, the company is comparing its value if it makes the replacement versus its value if it continues to use the existing asset.new cost numbers from the old numbers. Similarly, differences in depreciation and any other factor that affects cash flows must also be determined.
127 Standalone risk is the project’s risk if it is held as a lone asset. It disregards the fact that it is but one asset within the firm’s portfolio of assets and that the firm is but one stock in a typical investor’s portfolio of stocks. Standalone risk is measured by the variability of the project’s expected returns. Corporate, or withinfirm, risk is the project’s risk to the corporation, giving consideration to the fact that the project represents only one in the firm’s portfolio of assets, hence some of its risk will be eliminated by diversification within the firm. Corporate risk is measured by the project’s impact on uncertainty about the firm’s future earnings. Market, or beta, risk is the riskiness of the project as seen by welldiversified stockholders who recognize that the project is only one of the firm’s assets and that the firm’s stock is but one small part of their total portfolios. Market risk is measured by the project’s effect on the firm’s beta coefficient.
128 It is often difficult to quantify market risk. On the other hand, we can usually get a good idea of a project’s standalone risk, and that risk is normally correlated with market risk: The higher the standalone risk, the higher the market risk is likely to be. Therefore, firms tend to focus on standalone risk, then deal with corporate and market risk by making subjective, judgmental modifications to the calculated standalone risk.
129 Simulation analysis involves working with continuous probability distributions, and the output of a simulation analysis is a distribution of net present values or rates of return. Scenario analysis involves picking several points on the various probability distributions and determining cash flows or rates of return for these points. Sensitivity analysis involves determining the extent to which cash flows change, given a change in one particular input variable. Simulation analysis is expensive. Therefore, it would more than likely be employed in the decision for the $500 million investment in a satellite system than in the decision for the $30,000 truck.
1210 Scenario analyses, and especially simulation analyses, would probably be reserved for very important "makeorbreak" decisions. They would not be used for every project decision because it is costly (in terms of money and time) to perform the necessary calculations and it is challenging to gather all the required information for a full analysis. Simulation analysis, in particular, requires data from many different departments about costs and projections, including the probability distributions corresponding to those estimates and the correlation coefficients between various variables.
Solutions to EndofChapter Problems
121 a. Equipment $ 9,000,000
NOWC Investment 3,000,000
Initial investment outlay $12,000,000
b. No, last year’s $50,000 expenditure is considered a sunk cost and does not represent an incremental cash flow. Hence, it should not be included in the analysis.
c. The potential sale of the building represents an opportunity cost of conducting the project in that building. Therefore, the possible aftertax sale price must be charged against the project as a cost.
122 a. Operating cash flows: t = 1
Sales revenues $10,000,000
Operating costs 7,000,000
Depreciation 2,000,000
Operating income before taxes $ 1,000,000
Taxes (40%) 400,000
Operating income after taxes $ 600,000
Add back depreciation 2,000,000
Operating cash flow $ 2,600,000
b. The cannibalization of existing sales needs to be considered in this analysis on an aftertax basis, because the cannibalized sales represent sales revenue the firm would realize without the new project but would lose if the new project is accepted. Thus, the aftertax effect would be to reduce the firm’s operating cash flow by $1,000,000(1 – T) = $1,000,000(0.6) = $600,000. Thus, the firm’s OCF would now be $2,000,000 rather than $2,600,000.
c. If the tax rate fell to 30%, the operating cash flow would change to:
Operating income before taxes $1,000,000
Taxes (30%) 300,000
Operating income after taxes $ 700,000
Add back depreciation 2,000,000
Operating cash flow $2,700,000
Thus, the firm’s operating cash flow would increase by $100,000.
123 Equipment’s original cost $20,000,000
Depreciation (80%) 16,000,000
Book value $ 4,000,000
Gain on sale = $5,000,000 – $4,000,000 = $1,000,000.
Tax on gain = $1,000,000(0.4) = $400,000.
AT net salvage value = $5,000,000 – $400,000 = $4,600,000.
124 a. The applicable depreciation values are as follows for the two scenarios:
Scenario 1 Scenario 2
Year (StraightLine) (MACRS)
1 $200,000 $264,000
2 200,000 360,000
3 200,000 120,000
4 200,000 56,000
b. To find the difference in net present values under these two methods, we must determine the difference in incremental cash flows each method provides. The depreciation expenses cannot simply be subtracted from each other, as there are tax ramifications due to depreciation expense. The full depreciation expense is subtracted from Revenues to get operating income, and then taxes due are computed Then, depreciation is added to aftertax operating income to get the project’s operating cash flow. Therefore, if the tax rate is 40%, only 60% of the depreciation expense is actually subtracted out during the aftertax operating income calculation and the full depreciation expense is added back to calculate operating income. So, there is a tax benefit associated with the depreciation expense that amounts to 40% of the depreciation expense. Therefore, the differences between depreciation expenses under each scenario should be computed and multiplied by 0.4 to determine the benefit provided by the depreciation expense.
Depreciation Expense Depreciation Expense
Year Difference (2 – 1) Diff. ´ 0.4 (MACRS)
1 $ 64,000 $25,600
2 160,000 64,000
3 80,000 32,000
4 144,000 57,600
Now to find the difference in NPV to be generated under these scenarios, just enter the cash flows that represent the benefit from depreciation expense and solve for net present value based upon a WACC of 10%.
CF_{0} = 0; CF_{1} = 25600; CF_{2} = 64000; CF_{3} = 32000; CF_{4} = 57600; and I/YR = 10. Solve for NPV = $12,781.64
So, all else equal the use of the accelerated depreciation method will result in a higher NPV (by $12,781.64) than would the use of a straightline depreciation method.
126 a. The net cost is $178,000:
Cost of investment at t = 0:
Base price ($140,000)
Modification (30,000)
Increase in NOWC (8,000)
Cash outlay for new machine ($178,000)
b. The operating cash flows follow:
Year 1 Year 2 Year 3
Aftertax savings $30,000 $30,000 $30,000
Depreciation tax savings 22,440 30,600 10,200
Net operating cash flow $52,440 $60,600 $40,200
Notes:
1. The aftertax cost savings is $50,000(1 – T) = $50,000(0.6) = $30,000.
2. The depreciation expense in each year is the depreciable basis, $170,000, times the MACRS allowance percentages of 0.33, 0.45, and 0.15 for Years 1, 2, and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $56,100, $76,500, and $25,500. The depreciation tax savings is calculated as the tax rate (40%) times the depreciation expense in each year.
c. The terminal cash flow is $48,760:
Salvage value $60,000
Tax on SV* (19,240)
Return of NOWC 8,000
$48,760
*Tax on SV = ($60,000 – $11,900)(0.4) = $19,240.
Remaining BV in Year 4 = $170,000(0.07) = $11,900.
d. The project has an NPV of ($19,549). Thus, it should not be accepted.
Year Net Cash Flow PV @ 12%
0 ($178,000) ($178,000)
1 52,440 46,821
2 60,600 48,310
3 88,960 63,320
NPV = ($ 19,549)
Alternatively, place the cash flows on a time line:
0 1 2 3
   
178,000 52,440 60,600 40,200
48,760
88,960
With a financial calculator, input the cash flows into the cash flow register, I/YR = 12, and then solve for NPV = $19,548.65 » $19,549.
127 a. The $5,000 spent last year on exploring the feasibility of the project is a sunk cost and should not be included in the analysis.
b. The net cost is $126,000:
Price ($108,000)
Modification (12,500)
Increase in NOWC (5,500)
Cash outlay for new machine ($126,000)
c. The operating cash flows follow:
Year 1 Year 2 Year 3
1. Aftertax savings $28,600 $28,600 $28,600
2. Depreciation tax savings 13,918 18,979 6,326
Net cash flow $42,518 $47,579 $34,926
Notes:
1. The aftertax cost savings is $44,000(1 – T) = $44,000(0.65) = $28,600.
2. The depreciation expense in each year is the depreciable basis, $120,500, times the MACRS allowance percentages of 0.33, 0.45, and 0.15 for Years 1, 2, and 3, respectively. Depreciation expense in Years 1, 2, and 3 is $39,765, $54,225, and $18,075. The depreciation tax savings is calculated as the tax rate (35%) times the depreciation expense in each year.
d. The terminal cash flow is $50,702:
Salvage value $65,000
Tax on SV* (19,798)
Return of NOWC 5,500
$50,702
*Tax on SV = ($65,000 – $8,435)(0.35) = $19,798.
BV in Year 4 = $120,500(0.07) = $8,435.
e. The project has an NPV of $10,841; thus, it should be accepted.
Year Net Cash Flow PV @ 12%
0 ($126,000) ($126,000)
1 42,518 37,963
2 47,579 37,930
3 85,628 60,948
NPV = $ 10,841
Alternatively, place the cash flows on a time line:
0 1 2 3
   
126,000 42,518 47,579 34,926
50,702
85,628
With a financial calculator, input the appropriate cash flows into the cash flow register, input I/YR = 12, and then solve for NPV = $10,840.51 » $10,841.
0 1 2 3 4 5
0 1 2 3 4
0 1 5 6 7 8
If the life is as low as 4 years (an unlikely event), the investment will not be desirable. But, if the investment life is longer than 4 years, the investment will be a good one. Therefore, the decision will depend on the managers' confidence in the life of the tractor. Given the low probability of the tractor's life being only 4 years, it is likely that the managers will decide to purchase the tractor.
Notes:
^{ a Depreciation Schedule, Basis = $250,000 MACRS Rate ´ Basis = Year Beg. Bk. Value MACRS Rate Depreciation Ending BV 1 $250,000 0.33 $ 82,500 $167,500 2 167,500 0.45 112,500 55,000 3 55,000 0.15 37,500 17,500 4 17,500 0.07 17,500 0 $250,000 }b. If savings increase by 20%, then savings will be (1.2)($90,000) = $108,000.
If savings decrease by 20%, then savings will be (0.8)($90,000) = $72,000.
(1) Savings increase by 20%:
(2) Savings decrease by 20%:
c. Worstcase scenario:
Basecase scenario:
This was worked out in part a. NPV = $37,035.13.
Bestcase scenario:
Worstcase 0.35 ($ 7,663.52) ($ 2,682.23)
Basecase 0.35 37,035.13 12,962.30
Bestcase 0.30 81,733.79 24,520.14
E(NPV) $34,800.21
s _{NPV} = [(0.35)($7,663.52 – $34,800.21)^{2} + (0.35)($37,035.13 – $34,800.21)^{2} + (0.30)($81,733.79 – $34,800.21)^{2}]^{½}
= [$631,108,927.93 + $1,748,203.59 + $660,828,279.49]^{½}
= $35,967. 84.
CV = $35,967.84/$34,800.21 = 1.03.
1211 a. NPV of abandonment after Year t:
Using a financial calculator, input the following: CF_{0} = 22500, CF_{1} = 23750, and I/YR = 10 to solve for NPV_{1} = $909.09 » $909.
Using a financial calculator, input the following: CF_{0} = 22500, CF_{1} = 6250, CF_{2} = 20250, and I/YR = 10 to solve for NPV_{2} = $82.64 » $83.
Using a financial calculator, input the following: CF_{0} = 22500, CF_{1} = 6250, N_{j} = 2, CF_{3} = 17250, and I/YR = 10 to solve for NPV_{3} = $1,307.29 » $1,307.
Using a financial calculator, input the following: CF_{0} = 22500, CF_{1} = 6250, N_{j} = 3, CF_{4} = 11250, and I/YR = 10 to solve for NPV_{4} = $726.73 » $727.
Using a financial calculator, input the following: CF_{0} = 22500, CF_{1} = 6250, N_{j} = 5, and I/YR = 10 to solve for NPV_{5} = $1,192.42 » $1,192.
The firm should operate the truck for 3 years, NPV_{3} = $1,307.
b. No. Abandonment possibilities could only raise NPV and IRR. The firm’s value is maximized by abandoning the project after Year 3.
1212 a. WACC_{1} = 12%; WACC_{2} = 12.5% after $3,250,000 of new capital is raised.
Since each project is independent and of average risk, all projects whose IRR > WACC_{2} will be accepted. Consequently, Projects A, B, C, D, and E will be accepted and the optimal capital budget is $5,250,000.
b. If Projects C and D are mutually exclusive, then only one of these projects can be accepted. The project that should be accepted is the one whose NPV is greater. Thus, Project D should be chosen because its NPV is greater than NPV_{C}. Projects A, B, D, and E should be chosen and the optimal capital budget = $4,000,000.
c. Project Size IRR Risk RiskAdjusted WACC
A $ 750,000 14.0% High 12.5% + 2% = 14.5%
B 1,250,000 13.5 Average 12.5
C 1,250,000 13.2 Average 12.5
D 1,250,000 13.0 Average 12.5
E 750,000 12.7 Average 12.5
F 750,000 12.3 Low 12.5% – 2% = 10.5
G 750,000 12.2 Low 12.5% – 2% = 10.5
Projects B, C, D, E, F, and G would be accepted because each has IRR > riskadjusted WACC. The optimal capital budget is $6,000,000.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
1213 a.
b. The $30,000 R&D costs are sunk costs. Therefore, these costs will have no effect on NPV and other profitability measures.
c. If the new project will reduce cash flows from the firm's other projects, then this is a negative externality and must be considered in the analysis. Consequently, these should be considered costs of the new project and would reduce the project's NPV. If the project can be housed in an empty building that the firm owns and could sell if it were not used for the project, then this is an opportunity cost which should also be considered as a "cost" of this project. The aftertax sales amount for this building will reduce the project's NPV.
d. The project's cash flows are likely to be positively correlated with returns on the firm's other projects and with the economy. The firm is involved with materials and caulking compound is a building material, so it is a similar product to the firm's other products. In addition, when the economy is booming, housing starts increase—which would mean an increase in sales of the caulking compound. Whether a project is positively or negatively correlated with the firm's other projects impacts the risk of the project and the relevant cost of capital at which it should be evaluated.
e.
f.
g. Note that "bestcase" values for variable costs, fixed costs, WACC, and equipment cost are 20% less than basecase values, while the "worstcase" values for variable costs, fixed costs, WACC, and equipment cost are 20% higher than basecase values.
The scenario analysis suggests that the project could be highly profitable, but also that it is quite risky. There is a 25% probability that the project would result in a loss of $227,902. There is also a 25% probability that it could produce an NPV of $324,244. The standard deviation is high, at $196,458, and the coefficient of variation is a high 7.53.
Eastern University  Syllabus
College of Arts and Sciences
Department of Business
Fin 360 – Business Finance
Section 1 (11:00) & 2 (12:00)
Prof. Jack E. Bower Phone: 610.341.5835
Andrews 310 Fax: 610.341.1466
email: jebower@eastern.edu Home 610.265.5676
Office Hours, MWF 10:00 to 10:50, 3:00 to 5:00, T & R by appointment only
Course Overview: An introduction to the finance functions of business organizations. The course begins with a study of financial statements, ratios, taxes, and cash flows. Next, the course will cover the financial environment of markets and institutions. The course finishes with capital budgeting.
Prerequisite: Two semesters of Principles of Financial Accounting (Acct 107 + 108).
Required Textbooks: Fundamentals of Financial Management, Concise 5th Edition. Eugene F. Brigham and Joel F. Houston, Thomson – Southwest Publishing.
ISBN10: 0324319835 An online version is also available from the publisher’s website.
Accounting Through the Eyes of Faith,(ATEF) 3rd edition, compiled by Jack E. Bower, Thornton Publishing, 2006. ISBN 0977996069
Required Calculator: You must have a calculator that can do TimeValue of Money problems or a laptop with Excel for this course. You are responsible for learning how to operate the calculator you bring to class. Professor Bower will be using the BAII PLUS in class. The Texas Instruments BAII PLUS sells for about $35.
Learning Objectives: Students are expected to become familiar with financial terms, analysis, and the potential use of managerial finance skills. An understanding of the role of financial management and the environment in which business organizations operate is a primary learning objective. Financial decisions must be analyzed in terms of their effect on the value of the company stock; therefore, a knowledge of how alternative decisions affect the company is also a learning objective. By exposing students to the fundamentals of financial concepts and techniques, they will be better able to perform proper financial reasoning and make proper decisions. Students learn finance to the degree to which they can actively manipulate financial terms and concepts within a general economics framework. Students must actively participate in the construction of knowledge.
Grading and Examinations: Grades will be assigned based on the following proportions:
First Exam Ch. 1, 2, 3 & 4 + ATEF essay 23%
Second Exam Ch. 5, 6, 7 & 8 + ATEF essay 23%
Integrated Case 1* 10%
Integrated Case #2 10%
Final Exam Ch. 9, 10 & 11 + ATEF essay 23%
Class Participation** 11%
100%
*Integrated Cases (complete two at 10% each) are found at the end of each chapter starting with chapter two. You may choose cases from any two chapters in the textbook. You may also do these as a team of two students. A team needs to submit only one paper for a grade. The due date for the case is three class meetings after we complete the respective chapter. Extensions will be given upon a reasonable request. Here is a sample of the Integrated Cases presented in the textbook:
Case 242, First National Bank, page 62
Case 316, D’Leon Inc., Part I, page 95
Case 426, D’Leon Inc., Part II, page 136
(The cases follow the problems for each chapter, pick any two chapters)
**Class participation (11%) is based on the effort to interact with other students and to integrate the material. Specifically, this means the effort to:
#1 Reason logically
#2 Think creatively
#3 Appreciate ethical standards and conduct
#4 Communicate effectively
The Undergraduate School uses the following scale:
A+..97100 B+…8789 C+…7779 D....…6069
A…9396 B…..8386 C…..7376 F...below 60
A..9092 B….8092 C......7072
Instructional Method: The course will place more value on insight than on ability to memorize. Students are encouraged to use financial calculators, present value tables and selected equations found in the textbook. The successful students will train themselves to use financial knowledge to attack problems for homework. Success on homework will lead to success on exams.
Accommodations for Students with Disabilities: A request for accommodations on the basis of a disability should be directed to the Cushing Center for Counseling and Academic Support (CCAS) at the beginning of the semester. The request should be submitted well in advance of examinations so that appropriate plans and arrangements can be properly implemented. Essential components of the course will not be changed or eliminated, but modifications in the way you the student meets these essential requirements will be made when this is possible, such as the length of time to take examinations.
Academic dishonesty, add/drop/withdrawal and grade appeal policy: Please consult the Eastern University undergraduate catalog for additional information. All University policies apply to this class.
Tentative Class Schedule
Fall 2007
8/29 Luke 19, Scriptural Admonition to Study Accounting, Lecture Chapter 1
8/31 Questions, Chapter 1, page 19
9/03 No Class
9/05 Lecture Chapter 2, Bring your TVM calculator to class
9/07 Problems Chapter 2
9/10 Problems Chapter 2
9/12 Lecture Chapter 3, Financial Statements
9/14 Lecture Chapter 3, & Problems
9/17 Problems Chapter 3
9/19 Lecture Chapter 4, Analysis of Financial Statements
9/21 Lecture Chapter 4
9/24 Problems Chapter 4
9/26 Review for examination
9/28 Examination Chapters 1, 2, 3 & 4  Brigham and Houston,
Chapter 2, The Financial Accounting Environment, Bower ATEF
10/01 Lecture Chapter 5, Financial Markets and Institutions
10/03 Lecture Chapter 5
10/05 Problems Chapter 5
10/08 Lecture Chapter 6, Interest Rates
10/10 Lecture Chapter 6,
10/12 Problems Chapter 6
10/15 Lecture Chapter 7, Bonds & their Valuation
10/17 Lecture Chapter 7
10/19 No Class
10/22 Problems Chapter 7
10/24 No Class – Professor Bower will be at a tax conference
10/26 No Class
10/29 Lecture Chapter 8, Risk & Rates of Return
10/31 Lecture Chapter 8
11/02 Problems Chapter 8
11/05 Review for Examination
11/07 Examination Chapters 5, 6, 7 & 8  Brigham and Houston,
Chapter 10, The Manna Principle, Daniel Fletcher, ATEF
11/09 Lecture Chapter 9, Stocks and their Valuation
11/12 Lecture Chapter 9
11/14 Problems Chapter 9
11/16 Lecture Chapter 10, Cost of Capital
11/19 Lecture Chapter 10
11/21 to 11/25, No Class, time to praise God for His Blessings
11/26 Problems Chapter 10
11/28 Lecture Chapter 11, Basics of Capital Budgeting
11/30 Lecture Chapter 11
12/03 Problems Chapter 11
12/05 Review for Final Exam
Chapters 9, 10 & 11, Brigham and Houston,
ATEF  Chapter 11, Christian Stewardship, Robert Horst
12/07 No Class, Prof. Bower will be traveling to Minneapolis
The following Final Exam Schedule is posted by the Office of the Registrar:
MWF 
11:0011:50 AM  Monday  December 10  9:0011:00 AM 
MWF 
12:0012:50 PM  Friday  December 14  12:302:30 PM 
Chapter 1
An Overview of Financial Management
Learning Objectives
After reading this chapter, students should be able to:
Identify the three main forms of business organization and describe the advantages and disadvantages of each one.
Identify the primary goal of the management of a publicly held corporation, and understand the relationship between stock prices and shareholder value.
Differentiate between what is meant by a stock’s intrinsic value and its market value and understand the concept of equilibrium in the market.
Briefly explain three important trends that have been occurring in business that have implications for managers.
Define business ethics and briefly explain what companies are doing in response to a renewed interest in ethics, the consequences of unethical behavior, and how employees should deal with unethical behavior.
Briefly explain the conflicts between managers and stockholders, and explain useful motivational tools that can help to prevent these conflicts.
Identify the key officers in the organization and briefly explain their responsibilities.
Lecture Suggestions
Chapter 1 covers some important concepts, and discussing them in class can be interesting. However, students can read the chapter on their own, so it can be assigned but not covered in class.
We spend the first day going over the syllabus and discussing grading and other mechanics relating to the course. To the extent that time permits, we talk about the topics that will be covered in the course and the structure of the book. We also discuss briefly the fact that it is assumed that managers try to maximize stock prices, but that they may have other goals, hence that it is useful to tie executive compensation to stockholderoriented performance measures. If time permits, we think it’s worthwhile to spend at least a full day on the chapter. If not, we ask students to read it on their own, and to keep them honest, we ask one or two questions about the material on the first midterm exam.
One point we emphasize in the first class is that students should print a copy of the PowerPoint slides for each chapter covered and purchase a financial calculator immediately, and bring both to class regularly. We also put copies of the various versions of our "Brief Calculator Manual," which in about 12 pages explains how to use the most popular calculators, in the copy center. Students will need to learn how to use their calculators immediately as time value of money concepts are covered in Chapter 2. It is important for students to grasp these concepts early as many of the remaining chapters build on the TVM concepts.
We are often asked what calculator students should buy. If
they already have a financial calculator that can find IRRs, we tell them that
it will do, but if they do not have one, we recommend either the
HP10BII or 17BII. Please see the "Lecture Suggestions" for Chapter 2 for more
on calculators.
DAYS ON CHAPTER: 1 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
11 When you purchase a stock, you expect to receive dividends plus capital gains. Not all stocks pay dividends immediately, but those corporations that do, typically pay dividends quarterly. Capital gains (losses) are received when the stock is sold. Stocks are risky, so you would not be certain that your expectations would be met—as you would if you had purchased a U.S. Treasury security, which offers a guaranteed payment every 6 months plus repayment of the purchase price when the security matures.
12 No, the stocks of different companies are not equally risky. A company might operate in an industry that is viewed as relatively risky, such as biotechnology—where millions of dollars are spent on R&D that may never result in profit. A company might also be heavily regulated and this could be perceived as increasing its risk. Other factors that could cause a company’s stock to be viewed as relatively risky include: heavy use of debt financing vs. equity financing, stock price volatility, and so on.
13 If investors are more confident that Company A’s cash flows will be closer to their expected value than Company B’s cash flows, then investors will drive the stock price up for Company A. Consequently, Company A will have a higher stock price than Company B.
14 No, all corporate projects are not equally risky. A firm’s investment decisions have a significant impact on the riskiness of the stock. For example, the types of assets a company chooses to invest in can impact the stock’s risk—such as capital intensive vs. labor intensive, specialized assets vs. general (multipurpose) assets—and how they choose to finance those assets can also impact risk.
15 A firm’s intrinsic value is an estimate of a stock’s "true" value based on accurate risk and return data. It can be estimated but not measured precisely. A stock’s current price is its market price—the value based on perceived but possibly incorrect information as seen by the marginal investor. From these definitions, you can see that a stock’s "true longrun value" is more closely related to its intrinsic value rather than its current price.
16 Equilibrium is the situation where the actual market price equals the intrinsic value, so investors are indifferent between buying or selling a stock. If a stock is in equilibrium then there is no fundamental imbalance, hence no pressure for a change in the stock’s price. At any given time, most stocks are reasonably close to their intrinsic values and thus are at or close to equilibrium. However, at times stock prices and equilibrium values are different, so stocks can be temporarily undervalued or overvalued.
17 If the three intrinsic value estimates for Stock X were different, I would have the most confidence in Company X’s CFO’s estimate. Intrinsic values are strictly estimates, and different analysts with different data and different views of the future will form different estimates of the intrinsic value for any given stock. However, a firm’s managers have the best information about the company’s future prospects, so managers’ estimates of intrinsic value are generally better than the estimates of outside investors.
18 If a stock’s market price and intrinsic value are equal, then the stock is in equilibrium and there is no pressure (buying/selling) to change the stock’s price. So, theoretically, it is better that the two be equal; however, intrinsic value is a longrun concept. Management’s goal should be to maximize the firm’s intrinsic value, not its current price. So, maximizing the intrinsic value will maximize the average price over the long run but not necessarily the current price at each point in time. So, stockholders in general would probably expect the firm’s market price to be under the intrinsic value—realizing that if management is doing its job that current price at any point in time would not necessarily be maximized. However, the CEO would prefer that the market price be high—since it is the current price that he will receive when exercising his stock options. In addition, he will be retiring after exercising those options, so there will be no repercussions to him (with respect to his job) if the market price drops—unless he did something illegal during his tenure as CEO.
19 The board of directors should set CEO compensation dependent on how well the firm performs. The compensation package should be sufficient to attract and retain the CEO but not go beyond what is needed. Compensation should be structured so that the CEO is rewarded on the basis of the stock’s performance over the long run, not the stock’s price on an option exercise date. This means that options (or direct stock awards) should be phased in over a number of years so the CEO will have an incentive to keep the stock price high over time. If the intrinsic value could be measured in an objective and verifiable manner, then performance pay could be based on changes in intrinsic value. However, it is easier to measure the growth rate in reported profits than the intrinsic value, although reported profits can be manipulated through aggressive accounting procedures and intrinsic value cannot be manipulated. Since intrinsic value is not observable, compensation must be based on the stock’s market price—but the price used should be an average over time rather than on a spot date.
110 The three principal forms of business organization are sole proprietorship, partnership, and corporation. The advantages of the first two include the ease and low cost of formation. The advantages of the corporation include limited liability, indefinite life, ease of ownership transfer, and access to capital markets.
The disadvantages of a sole proprietorship are (1) difficulty in obtaining large sums of capital; (2) unlimited personal liability for business debts; and (3) limited life. The disadvantages of a partnership are (1) unlimited liability, (2) limited life, (3) difficulty of transferring ownership, and (4) difficulty of raising large amounts of capital. The disadvantages of a corporation are (1) double taxation of earnings and (2) setting up a corporation and filing required state and federal reports, which are complex and timeconsuming.
111 Stockholder wealth maximization is a longrun goal. Companies, and consequently the stockholders, prosper by management making decisions that will produce longterm earnings increases. Actions that are continually shortsighted often "catch up" with a firm and, as a result, it may find itself unable to compete effectively against its competitors. There has been much criticism in recent years that U.S. firms are too shortrun profitoriented. A prime example is the U.S. auto industry, which has been accused of continuing to build large "gas guzzler" automobiles because they had higher profit margins rather than retooling for smaller, more fuelefficient models.
112 Useful motivational tools that will aid in aligning stockholders’ and management’s interests include: (1) reasonable compensation packages, (2) direct intervention by shareholders, including firing managers who don’t perform well, and (3) the threat of takeover.
The compensation package should be sufficient to attract and retain able managers but not go beyond what is needed. Also, compensation packages should be structured so that managers are rewarded on the basis of the stock’s performance over the long run, not the stock’s price on an option exercise date. This means that options (or direct stock awards) should be phased in over a number of years so managers will have an incentive to keep the stock price high over time. Since intrinsic value is not observable, compensation must be based on the stock’s market price—but the price used should be an average over time rather than on a spot date.
Stockholders can intervene directly with managers. Today, the majority of stock is owned by institutional investors and these institutional money managers have the clout to exercise considerable influence over firms’ operations. First, they can talk with managers and make suggestions about how the business should be run. In effect, these institutional investors act as lobbyists for the body of stockholders. Second, any shareholder who has owned $2,000 of a company’s stock for one year can sponsor a proposal that must be voted on at the annual stockholders’ meeting, even if management opposes the proposal. Although shareholdersponsored proposals are nonbinding, the results of such votes are clearly heard by top management.
If a firm’s stock is undervalued, then corporate raiders will see it to be a bargain and will attempt to capture the firm in a hostile takeover. If the raid is successful, the target’s executives will almost certainly be fired. This situation gives managers a strong incentive to take actions to maximize their stock’s price.
113 a. Corporate philanthropy is always a sticky issue, but it can be justified in terms of helping to create a more attractive community that will make it easier to hire a productive work force. This corporate philanthropy could be received by stockholders negatively, especially those stockholders not living in its headquarters city. Stockholders are interested in actions that maximize share price, and if competing firms are not making similar contributions, the "cost" of this philanthropy has to be borne by someonethe stockholders. Thus, stock price could decrease.
b. Companies must make investments in the current period in order to generate future cash flows. Stockholders should be aware of this, and assuming a correct analysis has been performed, they should react positively to the decision. The Mexican plant is in this category. Capital budgeting is covered in depth in Part 4 of the text. Assuming that the correct capital budgeting analysis has been made, the stock price should increase in the future.
c. U.S. Treasury bonds are considered safe investments, while common stock are far more risky. If the company were to switch the emergency funds from Treasury bonds to stocks, stockholders should see this as increasing the firm’s risk because stock returns are not guaranteed—sometimes they go up and sometimes they go down. The firm might need the funds when the prices of their investments were low and not have the needed emergency funds. Consequently, the firm’s stock price would probably fall.
114 a. No, TIAACREF is not an ordinary shareholder. Because it is one of the largest institutional shareholders in the United States and it controls nearly $280 billion in pension funds, its voice carries a lot of weight. This "shareholder" in effect consists of many individual shareholders whose pensions are invested with this group.
b. The owners of TIAACREF are the individual teachers whose pensions are invested with this group.
c. For TIAACREF to be effective in wielding its weight, it must act as a coordinated unit. In order to do this, the fund’s managers should solicit from the individual shareholders their "votes" on the fund’s practices, and from those "votes" act on the majority’s wishes. In so doing, the individual teachers whose pensions are invested in the fund have in effect determined the fund’s voting practices.
115 Earnings per share in the current year will decline due to the cost of the investment made in the current year and no significant performance impact in the short run. However, the company’s stock price should increase due to the significant cost savings expected in the future.
116 The board of directors should set CEO compensation dependent on how well the firm performs. The compensation package should be sufficient to attract and retain the CEO but not go beyond what is needed. Compensation should be structured so that the CEO is rewarded on the basis of the stock’s performance over the long run, not the stock’s price on an option exercise date. This means that options (or direct stock awards) should be phased in over a number of years so the CEO will have an incentive to keep the stock price high over time. If the intrinsic value could be measured in an objective and verifiable manner, then performance pay could be based on changes in intrinsic value. Since intrinsic value is not observable, compensation must be based on the stock’s market price—but the price used should be an average over time rather than on a spot date. The board should probably set the CEO’s compensation as a mix between a fixed salary and stock options. The vice president of Company X’s actions would be different than if he were CEO of some other company.
117 Setting the compensation policy for three division managers would be different than setting the compensation policy for a CEO because performance of each of these managers could be more easily observed. For a CEO an award based on stock price performance makes sense, while in this situation it probably doesn’t make sense. Each of the managers could still be given stock awards; however, rather than the award being based on stock price it could be determined from some observable measure like increased gas output, oil output, etc.
Chapter 2
Time Value of Money
Learning Objectives
After reading this chapter, students should be able to:
Convert time value of money (TVM) problems from words to time lines.
Explain the relationship between compounding and discounting, between future and present value.
Calculate the future value of some beginning amount, and find the present value of a single payment to be received in the future.
Solve for interest rate or time, given the other three variables in the TVM equation.
Find the future value of a series of equal, periodic payments (an annuity) and the present value of such an annuity.
Explain the difference between an ordinary annuity and an annuity due, and calculate the difference in their values—both on a present value and future value basis.
Solve for annuity payments, periods, and interest rates, given the other four variables in the TVM equation.
Calculate the value of a perpetuity.
Demonstrate how to find the present and future values of an uneven series of cash flows and how to solve for the interest rate of an uneven series of cash flows.
Solve TVM problems for nonannual compounding.
Distinguish among the following interest rates: Nominal (or Quoted) rate, Periodic rate, Annual Percentage Rate (APR), and Effective (or Equivalent) Annual Rate; and properly choose among securities with different compounding periods.
Solve time value of money problems that involve fractional time periods.
Construct loan amortization schedules for fullyamortized loans.
Lecture Suggestions
We regard Chapter 2 as the most important chapter in the book, so we spend a good bit of time on it. We approach time value in three ways. First, we try to get students to understand the basic concepts by use of time lines and simple logic. Second, we explain how the basic formulas follow the logic set forth in the time lines. Third, we show how financial calculators and spreadsheets can be used to solve various time value problems in an efficient manner. Once we have been through the basics, we have students work problems and become proficient with the calculations and also get an idea about the sensitivity of output, such as present or future value, to changes in input variables, such as the interest rate or number of payments.
Some instructors prefer to take a strictly analytical approach and have students focus on the formulas themselves. The argument is made that students treat their calculators as "black boxes," and that they do not understand where their answers are coming from or what they mean. We disagree. We think that our approach shows students the logic behind the calculations as well as alternative approaches, and because calculators are so efficient, students can actually see the significance of what they are doing better if they use a calculator. We also think it is important to teach students how to use the type of technology (calculators and spreadsheets) they must use when they venture out into the real world.
In the past, the biggest stumbling block to many of our students has been time value, and the biggest problem was that they did not know how to use their calculator. Since time value is the foundation for many of the concepts that follow, we have moved this chapter to near the beginning of the text. This should give students more time to become comfortable with the concepts and the tools (formulas, calculators, and spreadsheets) covered in this chapter. Therefore, we strongly encourage students to get a calculator, learn to use it, and bring it to class so they can work problems with us as we go through the lectures. Our urging, plus the fact that we can now provide relatively brief, coursespecific manuals for the leading calculators, has reduced if not eliminated the problem.
Our research suggests that the best calculator for the money for most students is the HP10BII. Finance and accounting majors might be better off with a more powerful calculator, such as the HP17BII. We recommend these two for people who do not already have a calculator, but we tell them that any financial calculator that has an IRR function will do.
We also tell students that it is essential that they work lots of problems, including the endofchapter problems. We emphasize that this chapter is critical, so they should invest the time now to get the material down. We stress that they simply cannot do well with the material that follows without having this material down cold. Bond and stock valuation, cost of capital, and capital budgeting make little sense, and one certainly cannot work problems in these areas, without understanding time value of money first.
We base our lecture on the integrated case. The case goes systematically through the key points in the chapter, and within a context that helps students see the real world relevance of the material in the chapter. We ask the students to read the chapter, and also to "look over" the case before class. However, our class consists of about 1,000 students, many of whom view the lecture on TV, so we cannot count on them to prepare for class. For this reason, we designed our lectures to be useful to both prepared and unprepared students.
Since we have easy access to computer projection equipment, we generally use the electronic slide show as the core of our lectures. We strongly suggest to our students that they print a copy of the PowerPoint slides for the chapter from the Web site and bring it to class. This will provide them with a hard copy of our lecture, and they can take notes in the space provided. Students can then concentrate on the lecture rather than on taking notes.
We do not stick strictly to the slide show—we go to the board frequently to present somewhat different examples, to help answer questions, and the like. We like the spontaneity and change of pace trips to the board provide, and, of course, use of the board provides needed flexibility. Also, if we feel that we have covered a topic adequately at the board, we then click quickly through one or more slides.
The lecture notes we take to class consist of our own markedup copy of the PowerPoint slides, with notes on the comments we want to say about each slide. If we want to bring up some current event, provide an additional example, or the like, we use postit notes attached at the proper spot. The advantages of this system are (1) that we have a carefully structured lecture that is easy for us to prepare (now that we have it done) and for students to follow, and (2) that both we and the students always know exactly where we are. The students also appreciate the fact that our lectures are closely coordinated with both the text and our exams.
The slides contain the essence of the solution to each part of the integrated case, but we also provide more indepth solutions in this Instructor’s Manual. It is not essential, but you might find it useful to read through the detailed solution. Also, we put a copy of the solution on reserve in the library for interested students, but most find that they do not need it.
Finally, we remind students again, at the start of the lecture on Chapter 2, that they should bring a printout of the PowerPoint slides to class, for otherwise they will find it difficult to take notes. We also repeat our request that they get a financial calculator and our brief manual for it that can be found on the Web site, and bring it to class so they can work through calculations as we cover them in the lecture.
DAYS ON CHAPTER: 4 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
21 The opportunity cost is the rate of interest one could earn on an alternative investment with a risk equal to the risk of the investment in question. This is the value of I in the TVM equations, and it is shown on the top of a time line, between the first and second tick marks. It is not a single rate—the opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also varies from year to year depending on inflationary expectations (see Chapter 6).
22 True. The second series is an uneven cash flow stream, but it contains an annuity of $400 for 8 years. The series could also be thought of as a $100 annuity for 10 years plus an additional payment of $100 in Year 2, plus additional payments of $300 in Years 3 through 10.
23 True, because of compounding effects—growth on growth. The following example demonstrates the point. The annual growth rate is I in the following equation:
$1(1 + I)^{10} = $2.
We can find I in the equation above as follows:
Using a financial calculator input N = 10, PV = 1, PMT = 0, FV = 2, and I/YR = ? Solving for I/YR you obtain 7.18%.
Viewed another way, if earnings had grown at the rate of 10% per year for 10 years, then EPS would have increased from $1.00 to $2.59, found as follows: Using a financial calculator, input N = 10, I/YR = 10, PV = 1, PMT = 0, and FV = ?. Solving for FV you obtain $2.59. This formulation recognizes the "interest on interest" phenomenon.
24 For the same stated rate, daily compounding is best. You would earn more "interest on interest."
25 False. One can find the present value of an embedded annuity and add this PV to the PVs of the other individual cash flows to determine the present value of the cash flow stream.
26 The concept of a perpetuity implies that payments will be received forever. FV (Perpetuity) = PV (Perpetuity)(1 + I)¥ = ¥ .
27 The annual percentage rate (APR) is the periodic rate times the number of periods per year. It is also called the nominal, or stated, rate. With the "Truth in Lending" law, Congress required that financial institutions disclose the APR so the rate charged would be more "transparent" to consumers. The APR is only equal to the effective annual rate when compounding occurs annually. If more frequent compounding occurs, the effective rate is always greater than the annual percentage rate. Nominal rates can be compared with one another, but only if the instruments being compared use the same number of compounding periods per year. If this is not the case, then the instruments being compared should be put on an effective annual rate basis for comparisons.
28 A loan amortization schedule is a table showing precisely how a loan will be repaid. It gives the required payment on each payment date and a breakdown of the payment, showing how much is interest and how much is repayment of principal. These schedules can be used for any loans that are paid off in installments over time such as automobile loans, home mortgage loans, student loans, and many business loans.
Solutions to EndofChapter Problems
21 0 1 2 3 4 5
     
PV = 10,000 FV_{5} = ?
FV_{5} = $10,000(1.10)^{5}
= $10,000(1.61051) = $16,105.10.
Alternatively, with a financial calculator enter the following: N = 5, I/YR = 10, PV = 10000, and PMT = 0. Solve for FV = $16,105.10.
22 0 5 10 15 20
    
PV = ? FV_{20} = 5,000
With a financial calculator enter the following: N = 20, I/YR = 7, PMT = 0, and FV = 5000. Solve for PV = $1,292.10.
23 0 18
 
PV = 250,000 FV_{18} = 1,000,000
With a financial calculator enter the following: N = 18, PV = 250000, PMT = 0, and FV = 1000000. Solve for I/YR = 8.01% ≈ 8%.
24 0 N = ?
 
PV = 1 FV_{N} = 2
$2 = $1(1.065)^{N}.
With a financial calculator enter the following: I/YR = 6.5, PV = 1, PMT = 0, and FV = 2. Solve for N = 11.01 ≈ 11 years.
25 0 1 2 N – 2 N – 1 N
   · · ·   
PV = 42,180.53 5,000 5,000 5,000 5,000 FV = 250,000
Using your financial calculator, enter the following data: I/YR = 12; PV = 42180.53; PMT = 5000; FV = 250000; N = ? Solve for N = 11. It will take 11 years to accumulate $250,000.
26 Ordinary annuity:
0 1 2 3 4 5
     
300 300 300 300 300
FVA_{5} = ?
With a financial calculator enter the following: N = 5, I/YR = 7, PV = 0, and PMT = 300. Solve for FV = $1,725.22.
Annuity due:
0 1 2 3 4 5
     
300 300 300 300 300
With a financial calculator, switch to "BEG" and enter the following: N = 5, I/YR = 7, PV = 0, and PMT = 300. Solve for FV = $1,845.99. Don’t forget to switch back to "END" mode.
27 0 1 2 3 4 5 6
      
100 100 100 200 300 500
PV = ? FV = ?
Using a financial calculator, enter the following: CF_{0} = 0; CF_{1} = 100; N_{j} = 3; CF_{4} = 200 (Note calculator will show CF_{2} on screen.); CF_{5} = 300 (Note calculator will show CF_{3} on screen.); CF_{6} = 500 (Note calculator will show CF_{4} on screen.); and I/YR = 8. Solve for NPV = $923.98.
To solve for the FV of the cash flow stream with a calculator that doesn’t have the NFV key, do the following: Enter N = 6, I/YR = 8, PV = 923.98, and PMT = 0. Solve for FV = $1,466.24. You can check this as follows:
0 1 2 3 4 5 6
      
100 100 100 200 300 500
324.00
233.28
125.97
136.05
146.93
$1,466.23
28 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = 20000, and FV = 0. Solve for PMT = $444.89.
EAR = _{ }– 1.0
= (1.01)^{12} – 1.0
= 12.68%.
Alternatively, using a financial calculator, enter the following: NOM% = 12 and P/YR = 12. Solve for EFF% = 12.6825%. Remember to change back to P/YR = 1 on your calculator.
29 a. 0 1
  $500(1.06) = $530.00.
500 FV = ?
Using a financial calculator, enter N = 1, I/YR = 6, PV = 500, PMT = 0, and FV = ? Solve for FV = $530.00.
b. 0 1 2
   $500(1.06)^{2} = $561.80.
500 FV = ?
Using a financial calculator, enter N = 2, I/YR = 6, PV = 500, PMT = 0, and FV = ? Solve for FV = $561.80.
c. 0 1
  $500(1/1.06) = $471.70.
PV = ? 500
Using a financial calculator, enter N = 1, I/YR = 6, PMT = 0, and FV = 500, and PV = ? Solve for PV = $471.70.
d. 0 1 2
   $500(1/1.06)^{2} = $445.00.
PV = ? 500
Using a financial calculator, enter N = 2, I/YR = 6, PMT = 0, FV = 500, and PV = ? Solve for PV = $445.00.
210 a. 0 1 2 3 4 5 6 7 8 9 10
           $500(1.06)^{10} = $895.42.
500 FV = ?
Using a financial calculator, enter N = 10, I/YR = 6, PV = 500, PMT = 0, and FV = ? Solve for FV = $895.42.
b. 0 1 2 3 4 5 6 7 8 9 10
           $500(1.12)^{10} = $1,552.92.
500 FV = ?
Using a financial calculator, enter N = 10, I/YR = 12, PV = 500, PMT = 0, and FV = ? Solve for FV = $1,552.92.
c. 0 1 2 3 4 5 6 7 8 9 10
           $500/(1.06)^{10} = $279.20.
PV = ? 500
Using a financial calculator, enter N = 10, I/YR = 6, PMT = 0, FV = 500, and PV = ? Solve for PV = $279.20.
d. 0 1 2 3 4 5 6 7 8 9 10
          
PV = ? 1,552.90
$1,552.90/(1.12)^{10} = $499.99.
Using a financial calculator, enter N = 10, I/YR = 12, PMT = 0, FV = 1552.90, and PV = ? Solve for PV = $499.99.
$1,552.90/(1.06)^{10} = $867.13.
Using a financial calculator, enter N = 10, I/YR = 6, PMT = 0, FV = 1552.90, and PV = ? Solve for PV = $867.13.
e. The present value is the value today of a sum of money to be received in the future. For example, the value today of $1,552.90 to be received 10 years in the future is about $500 at an interest rate of 12%, but it is approximately $867 if the interest rate is 6%. Therefore, if you had $500 today and invested it at 12%, you would end up with $1,552.90 in 10 years. The present value depends on the interest rate because the interest rate determines the amount of interest you forgo by not having the money today.
211 a. 2000 2001 2002 2003 2004 2005
     
6 12 (in millions)
With a calculator, enter N = 5, PV = 6, PMT = 0, FV = 12, and then solve for I/YR = 14.87%.
b. The calculation described in the quotation fails to consider the compounding effect. It can be demonstrated to be incorrect as follows:
$6,000,000(1.20)^{5} = $6,000,000(2.48832) = $14,929,920,
which is greater than $12 million. Thus, the annual growth rate is less than 20%; in fact, it is about 15%, as shown in part a.
212 These problems can all be solved using a financial calculator by entering the known values shown on the time lines and then pressing the I/YR button.
a. 0 1
 
+700 749
With a financial calculator, enter: N = 1, PV = 700, PMT = 0, and FV = 749. I/YR = 7%.
b. 0 1
 
700 +749
With a financial calculator, enter: N = 1, PV = 700, PMT = 0, and FV = 749. I/YR = 7%.
c. 0 10
 
+85,000 201,229
With a financial calculator, enter: N = 10, PV = 85000, PMT = 0, and FV = 201229. I/YR = 9%.
d. 0 1 2 3 4 5
     
+9,000 2,684.80 2,684.80 2,684.80 2,684.80 2,684.80
With a financial calculator, enter: N = 5, PV = 9000, PMT = 2684.80, and FV = 0. I/YR = 15%.
213 a. ?
 
200 400
With a financial calculator, enter I/YR = 7, PV = 200, PMT = 0, and FV = 400. Then press the N key to find N = 10.24. Override I/YR with the other values to find N = 7.27, 4.19, and 1.00.
b. ?
  Enter: I/YR = 10, PV = 200, PMT = 0, and FV = 400.
200 400 N = 7.27.
c. ?
  Enter: I/YR = 18, PV = 200, PMT = 0, and FV = 400.
200 400 N = 4.19.
d. ?
  Enter: I/YR = 100, PV = 200, PMT = 0, and FV = 400.
200 400 N = 1.00.
214 a. 0 1 2 3 4 5 6 7 8 9 10
          
400 400 400 400 400 400 400 400 400 400
FV = ?
With a financial calculator, enter N = 10, I/YR = 10, PV = 0, and PMT = 400. Then press the FV key to find FV = $6,374.97.
b. 0 1 2 3 4 5
     
200 200 200 200 200
FV = ?
With a financial calculator, enter N = 5, I/YR = 5, PV = 0, and PMT = 200. Then press the FV key to find FV = $1,105.13.
c. 0 1 2 3 4 5
     
400 400 400 400 400
FV = ?
With a financial calculator, enter N = 5, I/YR = 0, PV = 0, and PMT = 400. Then press the FV key to find FV = $2,000.
d. To solve part d using a financial calculator, repeat the procedures discussed in parts a, b, and c, but first switch the calculator to "BEG" mode. Make sure you switch the calculator back to "END" mode after working the problem.
1. 0 1 2 3 4 5 6 7 8 9 10
          
400 400 400 400 400 400 400 400 400 400 FV = ?
With a financial calculator on BEG, enter: N = 10, I/YR = 10, PV = 0, and PMT = 400. FV = $7,012.47.
2. 0 1 2 3 4 5
     
200 200 200 200 200 FV = ?
With a financial calculator on BEG, enter: N = 5, I/YR = 5, PV = 0, and PMT = 200. FV = $1,160.38.
3. 0 1 2 3 4 5
     
400 400 400 400 400 FV = ?
With a financial calculator on BEG, enter: N = 5, I/YR = 0, PV = 0, and PMT = 400. FV = $2,000.
215 a. 0 1 2 3 4 5 6 7 8 9 10
          
PV = ? 400 400 400 400 400 400 400 400 400 400
With a financial calculator, simply enter the known values and then press the key for the unknown. Enter: N = 10, I/YR = 10, PMT = 400, and FV = 0. PV = $2,457.83.
b. 0 1 2 3 4 5
     
PV = ? 200 200 200 200 200
With a financial calculator, enter: N = 5, I/YR = 5, PMT = 200, and FV = 0. PV = $865.90.
c. 0 1 2 3 4 5
     
PV = ? 400 400 400 400 400
With a financial calculator, enter: N = 5, I/YR = 0, PMT = 400, and FV = 0. PV = $2,000.00.
d. 1. 0 1 2 3 4 5 6 7 8 9 10
          
400 400 400 400 400 400 400 400 400 400
PV = ?
With a financial calculator on BEG, enter: N = 10, I/YR = 10, PMT = 400, and FV = 0. PV = $2,703.61.
2. 0 1 2 3 4 5
     
200 200 200 200 200
PV = ?
With a financial calculator on BEG, enter: N = 5, I/YR = 5, PMT = 200, and FV = 0. PV = $909.19.
3. 0 1 2 3 4 5
     
400 400 400 400 400
PV = ?
With a financial calculator on BEG, enter: N = 5, I/YR = 0, PMT = 400, and FV = 0. PV = $2,000.00.
216 PV = $100/0.07 = $1,428.57. PV = $100/0.14 = $714.29.
When the interest rate is doubled, the PV of the perpetuity is halved.
217 0 1 2 3 4 30
     · · · 
85,000 8,273.59 8,273.59 8,273.59 8,273.59 8,273.59
With a calculator, enter N = 30, PV = 85000, PMT = 8273.59, FV = 0, and then solve for I/YR = 9%.
218 a. Cash Stream A Cash Stream B
0 1 2 3 4 5 0 1 2 3 4 5
           
PV = ? 100 400 400 400 300 PV = ? 300 400 400 400 100
With a financial calculator, simply enter the cash flows (be sure to enter CF_{0} = 0), enter I/YR = 8, and press the NPV key to find NPV = PV = $1,251.25 for the first problem. Override I/YR = 8 with I/YR = 0 to find the next PV for Cash Stream A. Repeat for Cash Stream B to get NPV = PV = $1,300.32.
b. PV_{A} = $100 + $400 + $400 + $400 + $300 = $1,600.
PV_{B} = $300 + $400 + $400 + $400 + $100 = $1,600.
219 a. Begin with a time line:
40 41 64 65
  · · ·  
5,000 5,000 5,000
Using a financial calculator input the following: N = 25, I/YR = 9, PV = 0, PMT = 5000, and solve for FV = $423,504.48.
b. 40 41 69 70
  · · ·  
5,000 5,000 5,000
FV = ?
Using a financial calculator input the following: N = 30, I/YR = 9, PV = 0, PMT = 5000, and solve for FV = $681,537.69.
c. 1. 65 66 67 84 85
   · · ·  
423,504.48 PMT PMT PMT PMT
Using a financial calculator, input the following: N = 20, I/YR = 9, PV = 423504.48, FV = 0, and solve for PMT = $46,393.42.
2. 70 71 72 84 85
   · · ·  
681,537.69 PMT PMT PMT PMT
Using a financial calculator, input the following: N = 15, I/YR = 9, PV = 681537.69, FV = 0, and solve for PMT = $84,550.80.
220 Contract 1: PV =
= $2,727,272.73 + $2,479,338.84 + $2,253,944.40 + $2,049,040.37
= $9,509,596.34.
Using your financial calculator, enter the following data: CF_{0} = 0; CF_{14} = 3000000; I/YR = 10; NPV = ? Solve for NPV = $9,509,596.34.
Contract 2: PV =
= $1,818,181.82 + $2,479,338.84 + $3,005,259.20 + $3,415,067.28
= $10,717,847.14.
Alternatively, using your financial calculator, enter the following data: CF_{0} = 0; CF_{1} = 2000000; CF_{2} = 3000000; CF_{3} = 4000000; CF_{4} = 5000000; I/YR = 10; NPV = ? Solve for NPV = $10,717,847.14.
Contract 3: PV =
= $6,363,636.36 + $826,446.28 + $751,314.80 + $683,013.46
= $8,624,410.90.
Alternatively, using your financial calculator, enter the following data: CF_{0} = 0; CF_{1} = 7000000; CF_{2} = 1000000; CF_{3} = 1000000; CF_{4} = 1000000; I/YR = 10; NPV = ? Solve for NPV = $8,624,410.90.
Contract 2 gives the quarterback the highest present value; therefore, he should accept Contract 2.
221 a. If Crissie expects a 7% annual return on her investments:
1 payment 10 payments 30 payments
N = 10 N = 30
I/YR = 7 I/YR = 7
PMT = 9500000 PMT = 5500000
FV = 0 FV = 0
PV = $61,000,000 PV = $66,724,025 PV = $68,249,727
Crissie should accept the 30year payment option as it carries the highest present value ($68,249,727).
b. If Crissie expects an 8% annual return on her investments:
1 payment 10 payments 30 payments
N = 10 N = 30
I/YR = 8 I/YR = 8
PMT = 9500000 PMT = 5500000
FV = 0 FV = 0
PV = $61,000,000 PV = $63,745,773 PV = $61,917,808
Crissie should accept the 10year payment option as it carries the highest present value ($63,745,773).
c. If Crissie expects a 9% annual return on her investments:
1 payment 10 payments 30 payments
N = 10 N = 30
I/YR = 9 I/YR = 9
PMT = 9500000 PMT = 5500000
FV = 0 FV = 0
PV = $61,000,000 PV = $60,967,748 PV = $56,505,097
Crissie should accept the lumpsum payment option as it carries the highest present value ($61,000,000).
d. The higher the interest rate, the more useful it is to get money rapidly, because it can be invested at those high rates and earn lots more money. So, cash comes fastest with #1, slowest with #3, so the higher the rate, the more the choice is tilted toward #1. You can also think about this another way. The higher the discount rate, the more distant cash flows are penalized, so again, #3 looks worst at high rates, #1 best at high rates.
222 a. This can be done with a calculator by specifying an interest rate of 5% per period for 20 periods with 1 payment per period.
N = 10 ´ 2 = 20, I/YR = 10/2 = 5, PV = 10000, FV = 0. Solve for PMT = $802.43.
b. Set up an amortization table:
Beginning Payment of Ending
Period Balance Payment Interest Principal Balance
1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57
2 9,697.57 802.43 484.88 317.55 9,380.02
$984.88
Because the mortgage balance declines with each payment, the portion of the payment that is applied to interest declines, while the portion of the payment that is applied to principal increases. The total payment remains constant over the life of the mortgage.
c. Jan must report interest of $984.88 on Schedule B for the first year. Her interest income will decline in each successive year for the reason explained in part b.
d. Interest is calculated on the beginning balance for each period, as this is the amount the lender has loaned and the borrower has borrowed. As the loan is amortized (paid off), the beginning balance, hence the interest charge, declines and the repayment of principal increases.
223 a. 0 1 2 3 4 5
     
500 FV = ?
With a financial calculator, enter N = 5, I/YR = 12, PV = 500, and PMT = 0, and then press FV to obtain FV = $881.17.
b. 0 1 2 3 4 5 6 7 8 9 10
          
500 FV = ?
With a financial calculator, enter N = 10, I/YR = 6, PV = 500, and PMT = 0, and then press FV to obtain FV = $895.42.
Alternatively, FV_{N} = PV = $500
= $500(1.06)^{10} = $895.42.
c. 0 4 8 12 16 20
     
500 FV = ?
With a financial calculator, enter N = 20, I/YR = 3, PV = 500, and PMT = 0, and then press FV to obtain FV = $903.06.
Alternatively, FV_{N} = $500 = $500(1.03)^{20} = $903.06.
d. 0 12 24 36 48 60
     
500 FV = ?
With a financial calculator, enter N = 60, I/YR = 1, PV = 500, and PMT = 0, and then press FV to obtain FV = $908.35.
Alternatively, FV_{N} = $500 = $500(1.01)^{60} = $908.35.
e. 0 365 1,825
  · · · 
500 FV = ?
With a financial calculator, enter N = 1825, I/YR = 12/365 = 0.032877, PV = 500, and PMT = 0, and then press FV to obtain FV = $910.97.
f. The FVs increase because as the compounding periods increase, interest is earned on interest more frequently.
224 a. 0 2 4 6 8 10
     
PV = ? 500
With a financial calculator, enter N = 10, I/YR = 6, PMT = 0, and FV = 500, and then press PV to obtain PV = $279.20.
Alternatively, PV = FV_{N} = $500
= $500 = $279.20.
b. 0 4 8 12 16 20
     
PV = ? 500
With a financial calculator, enter N = 20, I/YR = 3, PMT = 0, and FV = 500, and then press PV to obtain PV = $276.84.
Alternatively, PV = $500 = $500 = $276.84.
c. 0 1 2 12
   · · · 
PV = ? 500
With a financial calculator, enter N = 12, I/YR = 1, PMT = 0, and FV = 500, and then press PV to obtain PV = $443.72.
Alternatively, PV = $500 = $500 = $443.72.
d. The PVs for parts a and b decline as periods/year increases. This occurs because, with more frequent compounding, a smaller initial amount (PV) is required to get to $500 after 5 years. For part c, even though there are 12 periods/year, compounding occurs over only 1 year, so the PV is larger.
225 a. 0 1 2 3 9 10
    · · ·  
400 400 400 400 400
FV = ?
Enter N = 5 ´ 2 = 10, I/YR = 12/2 = 6, PV = 0, PMT = 400, and then press FV to get FV = $5,272.32.
b. Now the number of periods is calculated as N = 5 ´ 4 = 20, I/YR = 12/4 = 3, PV = 0, and PMT = 200. The calculator solution is $5,374.07. The solution assumes that the nominal interest rate is compounded at the annuity period.
c. The annuity in part b earns more because the money is on deposit for a longer period of time and thus earns more interest. Also, because compounding is more frequent, more interest is earned on interest.
226 Using the information given in the problem, you can solve for the maximum car price attainable.
Financed for 48 months Financed for 60 months
N = 48 N = 60
I/YR = 1 (12%/12 = 1%) I/YR = 1
PMT = 350 PMT = 350
FV = 0 FV = 0
PV = 13,290.89 PV = 15,734.26
You must add the value of the down payment to the present value of the car payments. If financed for 48 months, you can afford a car valued up to $17,290.89 ($13,290.89 + $4,000). If financing for 60 months, you can afford a car valued up to $19,734.26 ($15,734.26 + $4,000).
227 a. Bank A: I_{NOM} = Effective annual rate = 4%.
Bank B:
Effective annual rate = – 1.0 = (1.000096)^{365} – 1.0
= 1.035618 – 1.0 = 0.035618 = 3.5618%.
With a financial calculator, you can use the interest rate conversion feature to obtain the same answer. You would choose Bank A because its EAR is higher.
b. If funds must be left on deposit until the end of the compounding period (1 year for Bank A and 1 day for Bank B), and you think there is a high probability that you will make a withdrawal during the year, then Bank B might be preferable. For example, if the withdrawal is made after 6 months, you would earn nothing on the Bank A account but (1.000096)^{365/2} – 1.0 = 1.765% on the Bank B account.
Ten or more years ago, most banks were set up as described above, but now virtually all are computerized and pay interest from the day of deposit to the day of withdrawal, provided at least $1 is in the account at the end of the period.
228 Here you want to have an effective annual rate on the credit extended that is 2% more than the bank is charging you, so you can cover overhead.
First, we must find the EAR = EFF% on the bank loan. Enter NOM% = 6, P/YR = 12, and press EFF% to get EAR = 6.1678%.
So, to cover overhead you need to charge customers a nominal rate so that the corresponding EAR = 8.1678%. To find this nominal rate, enter EFF% = 8.1678, P/YR = 12, and press NOM% to get I_{NOM} = 7.8771%. (Customers will be required to pay monthly, so P/YR = 12.)
Alternative solution: We need to find the effective annual rate (EAR) the bank is charging first. Then, we can add 2% to this EAR to calculate the nominal rate that you should quote your customers.
Bank EAR: EAR = (1 + I_{NOM}/M)^{M} – 1 = (1 + 0.06/12)^{12} – 1 = 6.1678%.
So, the EAR you want to earn on your receivables is 8.1678%.
Nominal rate you should quote customers:
8.1678% = (1 + I_{NOM}/12)^{12} – 1
1.081678 = (1 + I_{NOM}/12)^{12}
1.006564 = 1 + I_{NOM}/12
I_{NOM} = 0.006564(12) = 7.8771%.
229 I_{NOM} = 12%, daily compounding 360day year.
Cost per day = 0.12/360 = 0.0003333 = 0.03333%.
Customers’ credit period = 90 days.
If you loaned $1, after 90 days a customer would owe you (1 + 0.12/360)^{90} ´ $1 = $1.030449. So, the required markup would be 3.0449% or approximately 3%.
230 a. Using the information given in the problem, you can solve for the length of time required to reach $1 million.
Erika: I/YR = 6; PV = 30000; PMT = 5000; FV = 1000000; and then solve for N = 38.742182. Therefore, Erika will be 25 + 38.74 = 63.74 years old when she becomes a millionaire.
Kitty: I/YR = 20; PV = 30000; PMT = 5000; FV = 1000000; and then solve for N = 16.043713. Therefore, Kitty will be 25 + 16.04 = 41.04 years old when she becomes a millionaire.
b. Using the 16.0437 year target, you can solve for the required payment:
N = 16.0437; I/YR = 6; PV = 30000; FV = 1000000; then solve for PMT = $35,825.33.
If Erika wishes to reach the investment goal at the same time as Kitty, she will need to contribute $35,825.33 per year.
c. Erika is investing in a relatively safe fund, so there is a good chance that she will achieve her goal, albeit slowly. Kitty is investing in a very risky fund, so while she might do quite well and become a millionaire shortly, there is also a good chance that she will lose her entire investment. High expected returns in the market are almost always accompanied by a lot of risk. We couldn’t say whether Erika is rational or irrational, just that she seems to have less tolerance for risk than Kitty does.
231 a. 0 1 2 3 4
    
PV = ? 10,000 10,000 10,000 10,000
With a calculator, enter N = 4, I/YR = 5, PMT = 10000, and FV = 0. Then press PV to get PV = $35,459.51.
b. At this point, we have a 3year, 5% annuity whose value is $27,232.48. You can also think of the problem as follows:
$35,459.51(1.05) – $10,000 = $27,232.49.
232 0 1 2 3 4 5 6
      
1,500 1,500 1,500 1,500 1,500 ?
FV = 10,000
With a financial calculator, get a "ballpark" estimate of the years by entering I/YR = 8, PV = 0, PMT = 1500, and FV = 10000, and then pressing the N key to find N = 5.55 years. This answer assumes that a payment of $1,500 will be made 55/100th of the way through Year 5.
Now find the FV of $1,500 for 5 years at 8% as follows: N = 5, I/YR = 8, PV = 0, PMT = 1500, and solve for FV = $8,799.90. Compound this value for 1 year at 8% to obtain the value in the account after 6 years and before the last payment is made; it is $8,799.90(1.08) = $9,503.89. Thus, you will have to make a payment of $10,000 – $9,503.89 = $496.11 at Year 6.
233 Begin with a time line:
0 1 2 3
   
5,000 5,500 6,050
FV = ?
Use a financial calculator to calculate the present value of the cash flows and then determine the future value of this present value amount:
Step 1: CF_{0} = 0, CF_{1} = 5000, CF_{2} = 5500, CF_{3} = 6050, I/YR = 7. Solve for NPV = $14,415.41.
Step 2: Input the following data: N = 3, I/YR = 7, PV = 14415.41, PMT = 0, and solve for FV = $17,659.50.
234 a. With a financial calculator, enter N = 3, I/YR = 10, PV = 25000, and FV = 0, and then press the PMT key to get PMT = $10,052.87. Then go through the amortization procedure as described in your calculator manual to get the entries for the amortization table.
Beginning Repayment Remaining
Year Balance Payment Interest of Principal Balance
1 $25,000.00 $10,052.87 $2,500.00 $7,552.87 $17,447.13
2 17,447.13 10,052.87 1,744.71 8,308.16 9,138.97
3 9,138.97 10,052.87 913.90 9,138.97 0
$30,158.61 $5,158.61 $25,000.00
b. % Interest % Principal
Year 1: $2,500/$10,052.87 = 24.87% $7,552.87/$10,052.87 = 75.13%
Year 2: $1,744.71/$10,052.87 = 17.36% $8,308.16/$10,052.87 = 82.64%
Year 3: $913.90/$10,052.87 = 9.09% $9,138.97/$10,052.87 = 90.91%
These percentages change over time because even though the total payment is constant the amount of interest paid each year is declining as the balance declines.
235 a. Using a financial calculator, enter N = 3, I/YR = 7, PV = 90000, and FV = 0, then solve for PMT = $34,294.65.
3year amortization schedule:
Beginning Principal Ending
Period Balance Payment Interest Repayment Balance
1 $90,000.00 $34,294.65 $6,300.00 $27,994.65 $62,005.35
2 62,005.35 34,294.65 4,340.37 29,954.28 32,051.07
3 32,051.07 34,294.65 2,243.58 32,051.07 0
No. Each payment would be $34,294.65, which is significantly larger than the $7,500 payments that could be paid (affordable).
b. Using a financial calculator, enter N = 30, I/YR = 7, PV = 90000, and FV = 0, then solve for PMT = $7,252.78.
Yes. Each payment would now be $7,252.78, which is less than the $7,500 payment given in the problem that could be made (affordable).
c. 30year amortization with balloon payment at end of Year 3:
Beginning Principal Ending
Period Balance Payment Interest Repayment Balance
1 $90,000.00 $7,252.78 $6,300.00 $ 952.78 $89,047.22
2 89,047.22 7,252.78 6,233.31 1,019.47 88,027.75
3 88,027.75 7,252.78 6,161.94 1,090.84 86,936.91
The loan balance at the end of Year 3 is $86,936.91 and the balloon payment is $86,936.91 + $7,252.78 = $94,189.69.
236 a. Begin with a time line:
0 1 2 3 4 5 6 6mos.
0 1 2 3 Years
      
1,000 1,000 1,000 1,000 1,000 FVA = ?
Since the first payment is made 6 months from today, we have a 5period ordinary annuity. The applicable interest rate is 4%/2 = 2%. First, we find the FVA of the ordinary annuity in period 5 by entering the following data in the financial calculator: N = 5, I/YR = 4/2 = 2, PV = 0, and PMT = 1000. We find FVA_{5} = $5,204.04. Now, we must compound this amount for 1 semiannual period at 2%.
$5,204.04(1.02) = $5,308.12.
b. Here’s the time line:
0 1 2 3 4 Qtrs
    
PMT =? PMT = ? FV = 10,000
= $9,802.96
Step 1: Discount the $10,000 back 2 quarters to find the required value of the 2period annuity at the end of Quarter 2, so that its FV at the end of the 4^{th} quarter is $10,000.
Using a financial calculator enter N = 2, I/YR = 1, PMT = 0, FV = 10000, and solve for PV = $9,802.96.
Step 2: Now you can determine the required payment of the 2period annuity with a FV of $9,802.96.
Using a financial calculator, enter N = 2, I/YR = 1, PV = 0, FV = 9802.96, and solve for PMT = $4,877.09.
237 a. Using the information given in the problem, you can solve for the length of time required to pay off the card.
I/YR = 1.5 (18%/12); PV = 350; PMT = 10; FV = 0; and then solve for N = 50 months.
b. If Simon makes monthly payments of $30, we can solve for the length of time required before the account is paid off.
I/YR = 1.5; PV = 350; PMT = 30; FV = 0; and then solve for N = 12.92 ≈ 13 months.
With $30 monthly payments, Simon will only need 13 months to pay off the account.
c. Total payments @ $10.month: 50 ´ $10 = $500.00
Total payments @ $30/month: 12.921 ´ $30 = 387.62
Extra interest = $112.38
238 0 1 2 3
12/31/04 12/31/05 12/31/06 12/31/07 12/31/08
    
34,000.00 35,020.00 36,070.60 37,152.72 38,267.30
100,000.00
20,000.00
Payment will be made
Step 1: Calculate salary amounts (20042008):
2004: $34,000
2005: $34,000(1.03) = $35,020.00
2006: $35,020(1.03) = $36,070.60
2007: $36,070.60(1.03) = $37,152.72
2008: $37,152.72(1.03) = $38,267.30
Step 2: Compound back pay, pain and suffering, and legal costs to 12/31/06 payment date:
$34,000(1.07)^{2} + $155,020(1.07)^{1}
$38,960.60 + $165,871.40 = $204,798.00.
Step 3: Discount future salary back to 12/31/06 payment date:
$36,070.60 + $37,152.72/(1.07)^{1} + $38,267.30/(1.07)^{2}
$36,070.60 + $34,722.17 + $33,424.14 = $104,217.91.
Step 4: City must write check for $204,798.00 + $104,217.91 = $309,014.91.
239 1. Will save for 10 years, then receive payments for 25 years. How much must he deposit at the end of each of the next 10 years?
2. Wants payments of $40,000 per year in today’s dollars for first payment only. Real income will decline. Inflation will be 5%. Therefore, to find the inflated fixed payments, we have this time line:
0 5 10
  
40,000 FV = ?
Enter N = 10, I/YR = 5, PV = 40000, PMT = 0, and press FV to get FV = $65,155.79.
3. He now has $100,000 in an account that pays 8%, annual compounding. We need to find the FV of the $100,000 after 10 years. Enter N = 10, I/YR = 8, PV = 100000, PMT = 0, and press FV to get FV = $215,892.50.
4. He wants to withdraw, or have payments of, $65,155.79 per year for 25 years, with the first payment made at the beginning of the first retirement year. So, we have a 25year annuity due with PMT = 65,155.79, at an interest rate of 8%. Set the calculator to "BEG" mode, then enter N = 25, I/YR = 8, PMT = 65155.79, FV = 0, and press PV to get PV = $751,165.35. This amount must be on hand to make the 25 payments.
5. Since the original $100,000, which grows to $215,892.50, will be available, we must save enough to accumulate $751,165.35  $215,892.50 = $535,272.85.
So, the time line looks like this:
Retires
50 51 52 59 60 61 83 84 85
   · · ·    · · ·   
$100,000 PMT PMT PMT PMT
65,155.79 65,155.79 65,155.79 65,155.79
+ 215,892.50
 751,165.35 = PVA(due)
Need to accumulate $535,272.85 = FVA_{10}
6. The $535,272.85 is the FV of a 10year ordinary annuity. The payments will be deposited in the bank and earn 8% interest. Therefore, set the calculator to "END" mode and enter N = 10, I/YR = 8, PV = 0, FV = 535272.85, and press PMT to find PMT = $36,949.61.
240 Step 1: Determine the annual cost of college. The current cost is $15,000 per year, but that is escalating at a 5% inflation rate:
College Current Years Inflation Cash
Year Cost from Now Adjustment Required
1 $15,000 5 (1.05)^{5} $19,144.22
2 15,000 6 (1.05)^{6} 20,101.43
3 15,000 7 (1.05)^{7} 21,106.51
4 15,000 8 (1.05)^{8} 22,161.83
Now put these costs on a time line:
13 14 15 16 17 18 19 20 21
        
19,144 –20,101 –21,107 –22,162
How much must be accumulated by age 18 to provide these payments at ages 18 through 21 if the funds are invested in an account paying 6%, compounded annually?
With a financial calculator enter: CF_{0} = 19144, CF_{1} = 20101, CF_{2} = 21107, CF_{3} = 22162, and I/YR = 6. Solve for NPV = $75,500.00.
Thus, the father must accumulate $75,500 by the time his daughter reaches age 18.
Step 2: The daughter has $7,500 now (age 13) to help achieve that goal. Five years hence, that $7,500, when invested at 6%, will be worth $10,037: $7,500(1.06)^{5} = $10,036.69 ≈ $10,037.
Step 3: The father needs to accumulate only $75,500 – $10,037 = $65,463. The key to completing the problem at this point is to realize the series of deposits represent an ordinary annuity rather than an annuity due, despite the fact the first payment is made at the beginning of the first year. The reason it is not an annuity due is there is no interest paid on the last payment that occurs when the daughter is 18.
Using a financial calculator, N = 6, I/YR = 6, PV = 0, and FV = 65463. PMT = $9,384.95 ≈ $9,385.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
241 a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
Chapter 3
Financial Statements, Cash Flow, and Taxes
After reading this chapter, students should be able to:
Briefly explain the history of accounting and financial statements, and how financial statements are used.
List the types of information found in a corporation’s annual report.
Explain what a balance sheet is, the information it provides, and how assets and claims on assets are arranged on a balance sheet.
Explain what an income statement is and the information it provides.
Differentiate between net cash flow and accounting profit.
Identify the purpose of the statement of cash flows, list the factors affecting a firm’s cash position that are reflected in this statement, and identify the three categories of activities that are separated out in this statement.
Specify the changes reported in a firm’s statement of retained earnings.
Discuss what questions can be answered by looking through the financial statements, and explain why investors need to be cautious when they review financial statements.
Discuss how certain modifications to the accounting data are needed and used for corporate decision making and stock valuation purposes. In the process, explain the terms: net operating working capital, total operating capital, NOPAT, operating cash flow, and free cash flow; and explain how each is calculated.
Define the terms Market Value Added (MVA) and Economic Value Added (EVA), explain how each is calculated, and differentiate between them.
Explain why financial managers must be concerned with taxation, and list some of the most important elements of the current tax law, such as the differences between the treatment of dividends and interest paid and interest and dividend income received.
Lecture Suggestions
The goal of financial management is to take actions that will maximize the value of a firm’s stock. These actions will show up, eventually, in the financial statements, so a general understanding of financial statements is critically important.
Note that Chapter 3 provides a bridge between accounting, which students have just covered, and financial management. Unfortunately, many nonaccounting students did not learn as much as they should have in their accounting courses, so we find it necessary to spend more time on financial statements than we would like. Also, at Florida and many other schools, students vary greatly in their knowledge of accounting, with accounting majors being wellgrounded because they have had more intense introductory courses and, more importantly, because they are taking advanced financial accounting concurrently with finance. This gives the accountants a major, and somewhat unfair, advantage over the others in dealing with Chapters 3 and 4 on exams. We know of no good solution to this problem, but what we do is pitch the coverage of this material to the nonaccountants. If we pitch the lectures (and exams) to the accountants, they simply blow away and demoralize our nonaccountants, and we do not want that. Perhaps Florida has more of a difference between accounting and nonaccounting students, but at least for us there really is a major difference.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 3, which appears at the end of this chapter solution. For other suggestions about the lecture, please see the "Lecture Suggestions" in Chapter 2, where we describe how we conduct our classes.
DAYS ON CHAPTER: 2 OF 58 DAYS (50minute periods)
Answers to EndofChapter Questions
31 The four financial statements contained in most annual reports are the balance sheet, income statement, statement of retained earnings, and statement of cash flows.
32 Accountants translate physical quantities into numbers when they construct the financial statements. The numbers shown on balance sheets generally represent historical costs. When examining a set of financial statements, one should keep in mind the physical reality that lies behind the numbers, and the fact that the translation from physical assets to numbers is far from precise.
33 Bankers and investors use financial statements to make intelligent decisions about what firms to extend credit or invest in, managers need financial statements to operate their businesses efficiently, and taxing authorities need them to assess taxes in a reasonable way.
34 No, because the $20 million of retained earnings would probably not be held as cash. The retained earnings figure represents the reinvestment of earnings by the firm. Consequently, the $20 million would be an investment in all of the firm’s assets.
35 The balance sheet shows the firm’s financial position on a specific date, for example, December 31, 2005. It shows each account balance at that particular point in time. For example, the cash account shown on the balance sheet would represent the cash the firm has on hand and in the bank on December 31, 2005. The income statement, on the other hand, reports on the firm’s operations over a period of time, for example, over the last 12 months. It reports revenues and expenses that the firm has incurred over that particular time period. For example, the sales figures reported on the income statement for the period ending December 31, 2005, would represent the firm’s sales over the period from January 1, 2005, through December 31, 2005, not just sales for December 31, 2005.
36 Investors need to be cautious when they review financial statements. While companies are required to follow GAAP, managers still have quite a lot of discretion in deciding how and when to report certain transactions. Consequently, two firms in exactly the same operating situation may report financial statements that convey different impressions about their financial strength. Some variations may stem from legitimate differences of opinion about the correct way to record transactions. In other cases, managers may choose to report numbers in a way that helps them present either higher earnings or more stable earnings over time. As long as they follow GAAP, such actions are not illegal, but these differences make it harder for investors to compare companies and gauge their true performances.
Unfortunately, there have also been cases where managers overstepped the bounds and reported fraudulent statements. Indeed, a number of highprofile executives have faced criminal charges because of their misleading accounting practices.
37 The emphasis in accounting is on the determination of accounting income, or net income, while the emphasis in finance is on net cash flow. Net cash flow is the actual net cash that a firm generates during some specified period. The value of an asset (or firm) is determined by the cash flows generated. Cash is necessary to purchase assets to continue operations and to pay dividends. Thus, financial managers should strive to maximize cash flows available to investors over the long run.
Although companies with relatively high accounting profits generally have a relatively high cash flow, the relationship is not precise. A business’s net cash flow generally differs from net income because some of the expenses and revenues listed on the income statement are not paid out or received in cash during the year.
Most other companies have little if any noncash revenues, but this item can be important for construction companies that work on multiyear projects, report income on a percentage of completion basis, and then are paid only after the project is completed. Also, if a company has a substantial amount of deferred taxes, which means that taxes actually paid are less than that reported in the income statement, then this amount could also be added to net income when estimating the net cash flow. The relationship between net cash flow and net income can be expressed as:
Net cash flow = Net income + Noncash charges – Noncash revenues.
The primary examples of noncash charges are depreciation and amortization. These items reduce net income but are not paid out in cash, so we add them back to net income when calculating net cash flow. Likewise, some revenues may not be collected in cash during the year, and these items must be subtracted from net income when calculating net cash flow. Typically, depreciation and amortization represent the largest noncash items, and in many cases the other noncash items roughly net to zero. For this reason, many analysts assume that net cash flow equals net income plus depreciation and amortization.
38 Operating cash flow arises from normal, ongoing operations, whereas net cash flow reflects both operating and financing decisions. Thus, operating cash flow is defined as the difference between sales revenues and operating expenses paid, after taxes on operating income. Operating cash flow can be calculated as follows:
Operating cash flow = EBIT (1 – T) + Depreciation and amortization
= NOPAT + Depreciation and amortization.
Note that net cash flow can be calculated as follows:
Net cash flow = Net income + Depreciation and amortization.
Thus, the difference between the two equations is that net cash flow includes aftertax interest expense.
39 NOPAT is the profit a company would generate if it had no debt and held only operating assets. Net income is the profit available to common stockholders; thus, both interest and taxes have been deducted. NOPAT is a better measure of the performance of a company’s operations than net income because debt lowers income. In order to get a true reflection of a company’s operating performance, one would want to take out debt to get a clearer picture of the situation.
310 Free cash flow is the cash flow actually available for distribution to investors after the company has made all the investments in fixed assets, new products, and operating working capital necessary to sustain ongoing operations. It is defined as net operating profit after taxes (NOPAT) minus the amount of net investment in operating working capital and fixed assets necessary to sustain the business. It is the most important measure of cash flows because it shows the exact amount available to all investors (stockholders and debtholders). The value of a company’s operations depends on expected future free cash flows. Therefore, managers make their companies more valuable by increasing their free cash flow. Net income, on the other hand, reflects accounting profit but not cash flow. Therefore, investors ought to focus on cash flow rather than accounting profit.
311 Yes. Negative free cash flow is not necessarily bad. It depends on why the free cash flow was negative. If free cash flow was negative because NOPAT was negative, this is definitely bad, and it suggests that the company is experiencing operating problems. However, many highgrowth companies have positive NOPAT but negative free cash flow because they must invest heavily in operating assets to support rapid growth. There is nothing wrong with a negative cash flow if it results from profitable growth.
312 Double taxation refers to the fact that corporate income is subject to an income tax, and then stockholders are subject to a further personal tax on dividends received. Income could even be subject to triple taxation; therefore, corporations that receive dividend income can exclude some of the dividends from its taxable income. This provision in the Tax Code minimizes the amount of triple taxation that would otherwise occur.
313 Because interest paid is tax deductible but dividend payments are not, the aftertax cost of debt is lower than the aftertax cost of equity. This encourages the use of debt rather than equity. This point is discussed in detail in later chapters: "The Cost of Capital" and "Capital Structure and Leverage."
Solutions to EndofChapter Problems
31 NI = $3,000,000; EBIT = $6,000,000; T = 40%; Interest = ?
Need to set up an income statement and work from the bottom up.
EBIT $6,000,000
Interest 1,000,000
EBT $5,000,000 EBT =
Taxes (40%) 2,000,000
NI $3,000,000
Interest = EBIT – EBT = $6,000,000 – $5,000,000 = $1,000,000.
32 EBITDA = $7,500,000; NI = $1,800,000; Int = $2,000,000; T = 40%; DA = ?
EBITDA $7,500,000
DA 2,500,000 EBITDA – DA = EBIT; DA = EBITDA – EBIT
EBIT $5,000,000 EBIT = EBT + Int = $3,000,000 + $2,000,000
Int 2,000,000 (Given)
EBT $3,000,000
Taxes (40%) 1,200,000
NI $1,800,000 (Given)
33 NI = $3,100,000; DEP = $500,000; AMORT = 0; NCF = ?
NCF = NI + DEP and AMORT = $3,100,000 + $500,000 = $3,600,000.
34 NI = $50,000,000; R/E_{Y/E} = $810,000,000; R/E_{B/Y} = $780,000,000; Dividends = ?
R/E_{B/Y} + NI – Div = R/E_{Y/E}
$780,000,000 + $50,000,000 – Div = $810,000,000
$830,000,000 – Div = $810,000,000
$20,000,000 = Div.
35 Statements b and d will decrease the amount of cash on a company’s balance sheet. Statement a will increase cash through the sale of common stock. This is a source of cash through financing activities. On one hand, Statement c would decrease cash; however, it is also possible that Statement c would increase cash, if the firm receives a tax refund.
36 Ending R/E = Beg. R/E + Net income  Dividends
$278,900,000 = $212,300,000 + Net income  $22,500,000
$278,900,000 = $189,800,000 + Net income
Net income = $89,100,000.
37 a. From the statement of cash flows the change in cash must equal cash flow from operating activities plus longterm investing activities plus financing activities. First, we must identify the change in cash as follows:
Cash at the end of the year $25,000
– Cash at the beginning of the year – 55,000
Change in cash $30,000
The sum of cash flows generated from operations, investment, and financing must equal a negative $30,000. Therefore, we can calculate the cash flow from operations as follows:
CF from operations + CF from investing + CF from financing = D in cash
CF from operations  $250,000 + $170,000 = $30,000
CF from operations = $50,000.
b. To determine the firm’s net income for the current year, you must realize that cash flow from operations is determined by adding sources of cash (such as depreciation and amortization and increases in accrued liabilities) and subtracting uses of cash (such as increases in accounts receivable and inventories) from net income. Since we determined that the firm’s cash flow from operations totaled $50,000 in part a of this problem, we can now calculate the firm’s net income as follows:
NI + +  =
NI + $10,000 + $25,000 – $100,000 = $50,000
NI – $65,000 = $50,000
NI = $115,000.
38 EBIT = $750,000; DEP = $200,000; AMORT = 0; 100% Equity; T = 40%; NI = ?; NCF = ?; OCF = ?
First, determine net income by setting up an income statement:
EBIT $750,000
Interest 0
EBT $750,000
Taxes (40%) 300,000
NI $450,000
NCF = NI + DEP and AMORT = $450,000 + $200,000 = $650,000.
OCF = EBIT(1 – T) + DEP and AMORT = $750,000(0.6) + $200,000 = $650,000.
Note that NCF = OCF because the firm is 100% equity financed.
39 MVA = (P_{0} ´ Number of common shares)  BV of equity
$130,000,000 = $60X  $500,000,000
$630,000,000 = $60X
X = 10,500,000 common shares.
310 a. NOPAT = EBIT(1 – T)
= $4,000,000,000(0.6)
= $2,400,000,000.
b. NCF = NI + DEP and AMORT
= $1,500,000,000 + $3,000,000,000
= $4,500,000,000.
c. OCF = NOPAT + DEP and AMORT
= $2,400,000,000 + $3,000,000,000
= $5,400,000,000.
d. FCF = NOPAT – Net Investment in Operating Capital
= $2,400,000,000 – $1,300,000,000
= $1,100,000,000.
311 Working up the income statement you can calculate the new sales level would be $12,681,482.
Sales $12,681,482 $5,706,667/(1  0.55)
Operating costs (excl. D&A) 6,974,815 $12,681,482 ´ 0.55
EBITDA $ 5,706,667 $4,826,667 + $880,000
Depr. & amort. 880,000 $800,000 ´ 1.10
EBIT $ 4,826,667 $4,166,667 + $660,000
Interest 660,000 $600,000 ´ 1.10
EBT $ 4,166,667 $2,500,000/(1  0.4)
Taxes (40%) 1,666,667 $4,166,667 ´ 0.40
Net income $ 2,500,000
312 a. Because we’re interested in net cash flow available to common stockholders, we exclude common dividends paid.
CF_{05} = NI available to common stockholders + Depreciation and amortization
= $372 + $220 = $592 million.
The net cash flow number is larger than net income by the current year’s depreciation and amortization expense, which is a noncash charge.
b. Balance of RE, December 31, 2004 $1,374
Add: NI, 2005 372
Less: Div. paid to common stockholders (146)
Balance of RE, December 31, 2005 $1,600
The RE balance on December 31, 2005 is $1,600 million.
c. $1,600 million.
d. Cash + Equivalents = $15 million.
e. Total current liabilities = $620 million.
313 a. NOPAT_{05} = EBIT(1 – T)
= $150,000,000(0.6)
= $90,000,000.
b. = Current assets –
= $360,000,000 – ($90,000,000 + $60,000,000)
= $210,000,000.
= $372,000,000 – $180,000,000 = $192,000,000.
c. Operating capital_{04} =
= $250,000,000 + $210,000,000
= $460,000,000.
Operating capital_{05} = $300,000,000 + $192,000,000
= $492,000,000.
d. FCF_{05} = NOPAT – Net investment in operating capital
= $90,000,000 – ($492,000,000 – $460,000,000)
= $58,000,000.
e. The large increase in dividends for 2005 can most likely be attributed to a large increase in free cash flow from 2004 to 2005, since FCF represents the amount of cash available to be paid out to stockholders after the company has made all investments in fixed assets, new products, and operating working capital necessary to sustain the business.
314 a. Sales revenues $12,000,000
Costs except deprec. and amort. (75%) 9,000,000
EBITDA $ 3,000,000
Depreciation and amortization 1,500,000
EBT $ 1,500,000
Taxes (40%) 600,000
Net income $ 900,000
Add back deprec. and amort. 1,500,000
Net cash flow $ 2,400,000
b. If depreciation and amortization doubled, taxable income would fall to zero and taxes would be zero. Thus, net income would decrease to zero, but net cash flow would rise to $3,000,000. Menendez would save $600,000 in taxes, thus increasing its cash flow:
D CF = T(D Depreciation and amortization) = 0.4($1,500,000) = $600,000.
c. If depreciation and amortization were halved, taxable income would rise to $2,250,000 and taxes to $900,000. Therefore, net income would rise to $1,350,000, but net cash flow would fall to $2,100,000.
d. You should prefer to have higher depreciation and amortization charges and higher cash flows. Net cash flows are the funds that are available to the owners to withdraw from the firm and, therefore, cash flows should be more important to them than net income.
e. In the situation where depreciation and amortization doubled, net income fell by 100%. Since many of the measures banks and investors use to appraise a firm’s performance depend on net income, a decline in net income could certainly hurt both the firm’s stock price and its ability to borrow. For example, earnings per share is a common number looked at by banks and investors, and it would have declined by 100%, even though the firm’s ability to pay dividends and to repay loans would have improved.
Comprehensive/Spreadsheet Problem
Note to Instructors:
The solution to this problem is not provided to students at the back of their text. Instructors can access the Excel file on the textbook’s Web site or the Instructor’s Resource CD.
315 a.
b.
c.
The difference in these two measures is notes payable. Notes payable is deducted from current assets to arrive at net working capital, while it is not deducted from current assets to calculate net operating working capital because notes payable is an interestbearing current liability.
d.
e.
The firm's market value exceeds its book value by $14.895 million. This means that management has added this much to shareholder value over the company's history. It would have to be compared to the MVA of other companies before declaring the performance good, bad, or indifferent.
f. An increase in the firm's dividend payout ratio would have no effect on its corporate taxes paid because dividends are paid with aftertax dollars. However, the company's shareholders would pay additional taxes on the additional dividends they would receive. As of 12/05, dividends are generally taxed at a maximum rate of 15 percent.