1. It is the minimum rate of return the firm must earn overall on its existing assets. If it earns more than this, value is created.
2. Book values for debt are likely to be much closer to market values than are equity book values.
3. No. The cost of capital depends on the risk of the project, not the source of the money.
4. Interest expense is tax-deductible. There is no difference between pretax and aftertax equity costs.
5. The primary advantage of the DCF model is its simplicity. The method is disadvantaged in that (1) the model is applicable only to firms that actually pay dividends; many do not; (2) even if a firm does pay dividends, the DCF model requires a constant dividend growth rate forever; (3) the estimated cost of equity from this method is very sensitive to changes in g, which is a very uncertain parameter; and (4) the model does not explicitly consider risk, although risk is implicitly considered to the extent that the market has impounded the relevant risk of the stock into its market price. While the share price and most recent dividend can be observed in the market, the dividend growth rate must be estimated. Two common methods of estimating g are to use analysts’ earnings and payout forecasts or to determine some appropriate average historical g from the firm’s available data.
6. Two primary advantages of the SML approach are that the model explicitly incorporates the relevant risk of the stock and the method is more widely applicable than is the DCF model, since the SML doesn’t make any assumptions about the firm’s dividends. The primary disadvantages of the SML method are (1) three parameters (the risk-free rate, the expected return on the market, and beta) must be estimated, and (2) the method essentially uses historical information to estimate these parameters. The risk-free rate is usually estimated to be the yield on very short maturity T-bills and is, hence, observable; the market risk premium is usually estimated from historical risk premiums and, hence, is not observable. The stock beta, which is unobservable, is usually estimated either by determining some average historical beta from the firm and the market’s return data, or by using beta estimates provided by analysts and investment firms.
7. The appropriate aftertax cost of debt to the company is the interest rate it would have to pay if it were to issue new debt today. Hence, if the YTM on outstanding bonds of the company is observed, the company has an accurate estimate of its cost of debt. If the debt is privately-placed, the firm could still estimate its cost of debt by (1) looking at the cost of debt for similar firms in similar risk classes, (2) looking at the average debt cost for firms with the same credit rating (assuming the firm’s private debt is rated), or (3) consulting analysts and investment bankers. Even if the debt is publicly traded, an additional complication is when the firm has more than one issue outstanding; these issues rarely have the same yield because no two issues are ever completely homogeneous.
8. a. This only considers the dividend yield component of the required return on equity.
b. This is the current yield only, not the promised yield to maturity. In addition, it is based on the book value of the liability, and it ignores taxes.
c. Equity is inherently more risky than debt (except, perhaps, in the unusual case where a firm’s assets have a negative beta). For this reason, the cost of equity exceeds the cost of debt. If taxes are considered in this case, it can be seen that at reasonable tax rates, the cost of equity does exceed the cost of debt.
9. RSup = .12 + .75(.08) = .1800 or 18.00%
Both should proceed. The appropriate discount rate does not depend on which company is investing; it depends on the risk of the project. Since Superior is in the business, it is closer to a pure play. Therefore, its cost of capital should be used. With an 18% cost of capital, the project has an NPV of $1 million regardless of who takes it.
10. If the different operating divisions were in much different risk classes, then separate cost of capital figures should be used for the different divisions; the use of a single, overall cost of capital would be inappropriate. If the single hurdle rate were used, riskier divisions would tend to receive more funds for investment projects, since their return would exceed the hurdle rate despite the fact that they may actually plot below the SML and, hence, be unprofitable projects on a risk-adjusted basis. The typical problem encountered in estimating the cost of capital for a division is that it rarely has its own securities traded on the market, so it is difficult to observe the market’s valuation of the risk of the division. Two typical ways around this are to use a pure play proxy for the division, or to use subjective adjustments of the overall firm hurdle rate based on the perceived risk of the division.
Solutions to Questions and Problems
NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.
Basic
1. With the information given, we can find the cost of equity using the dividend growth model. Using this model, the cost of equity is:
RE = [$2.45(1.06)/$45] + .06 = .1177 or 11.77%
2. Here we have information to calculate the cost of equity using the CAPM. The cost of equity is:
RE = .045 + 1.30 (.12 – .045) = .1425 or 14.25%
3. We have the information available to calculate the cost of equity using the CAPM and the dividend growth model. Using the CAPM, we find:
RE = .04 + 1.15(.08) = .1320 or 13.20%
And using the dividend growth model, the cost of equity is
RE = [$1.80(1.05)/$34] + .05 = .1056 or 10.56%
Both estimates of the cost of equity seem reasonable. If we remember the historical return on large capitalization stocks, the estimate from the CAPM model is about one percent higher than average, and the estimate from the dividend growth model is about one percent lower than the historical average, so we cannot definitively say one of the estimates is incorrect. Given this, we will use the average of the two, so:
RE = (.1320 + .1056)/2 = .1188 or 11.88%
4. To use the dividend growth model, we first need to find the growth rate in dividends. So, the increase in dividends each year was:
g1 = ($.91 – .78)/$.78 = .1667 or 16.67%
g2 = ($.93 – .91)/$.91 = .0220 or 2.20%
g3 = ($1.00 – .93)/$.93 = .0753 or 7.53%
g4 = ($1.22 – 1.00)/$1.00 = .2200 or 22.00%
So, the average arithmetic growth rate in dividends was:
g = (.1667 + .0220 + .0753 + .2200)/4 = .1210 or 12.10%
Using this growth rate in the dividend growth model, we find the cost of equity is:
RE = [$1.22(1.1210)/$45.00] + .1210 = .1514 or 15.14%
Calculating the geometric growth rate in dividends, we find:
$1.22 = $0.78(1 + g)4
g = .1183 or 11.83%
The cost of equity using the geometric dividend growth rate is:
RE = [$1.22(1.1183)/$45.00] + .1183 = 14.86%
5. The cost of preferred stock is the dividend payment divided by the price, so:
RP = $6/$92 = .0652 or 6.52%
6. The pretax cost of debt is the YTM of the company’s bonds, so:
P0 = $1,050 = $40(PVIFAR%,24) + $1,000(PVIFR%,24)
R = 3.683%
YTM = 2 × 3.683% = 7.37%
And the aftertax cost of debt is:
RD = .0737(1 – .35) = .0479 or 4.79%
7. a. The pretax cost of debt is the YTM of the company’s bonds, so:
P0 = $1,080 = $50(PVIFAR%,46) + $1,000(PVIFR%,46)
R = 4.58%
YTM = 2 × 4.58% = 9.16%
b. The aftertax cost of debt is:
RD = .0916(1 – .35) = .0595 or 5.95%
c. The after-tax rate is more relevant because that is the actual cost to the company.
8. The book value of debt is the total par value of all outstanding debt, so:
BVD = $20M + 80M = $100M
To find the market value of debt, we find the price of the bonds and multiply by the number of bonds. Alternatively, we can multiply the price quote of the bond times the par value of the bonds. Doing so, we find:
MVD = 1.08($20M) + .58($80M) = $68M
The YTM of the zero coupon bonds is:
PZ = $580 = $1,000(PVIFR%,7)
R = 8.09%
So, the aftertax cost of the zero coupon bonds is:
RZ = .0809(1 – .35) = .0526 or 5.26%
The aftertax cost of debt for the company is the weighted average of the aftertax cost of debt for all outstanding bond issues. We need to use the market value weights of the bonds. The total aftertax cost of debt for the company is:
RD = .0595($21.6/$68) + .0526($46.4/$68) = .0548 or 5.48%
9. a. Using the equation to calculate the WACC, we find:
WACC = .50(.16) + .05(.075) + .45(.09)(1 – .35) = .1101 or 11.01%
b. Since interest is tax deductible and dividends are not, we must look at the after-tax cost of debt, which is:
.09(1 – .35) = .0585 or 5.85%
Hence, on an after-tax basis, debt is cheaper than the preferred stock.
10. Here we need to use the debt-equity ratio to calculate the WACC. Doing so, we find:
WACC = .18(1/1.60) + .10(.60/1.60)(1 – .35) = .1369 or 13.69%
11. Here we have the WACC and need to find the debt-equity ratio of the company. Setting up the WACC equation, we find:
WACC = .1150 = .16(E/V) + .085(D/V)(1 – .35)
Rearranging the equation, we find:
.115(V/E) = .16 + .085(.65)(D/E)
Now we must realize that the V/E is just the equity multiplier, which is equal to:
V/E = 1 + D/E
.115(D/E + 1) = .16 + .05525(D/E)
Now we can solve for D/E as:
.05975(D/E) = .0450
D/E = .7531
12. a. The book value of equity is the book value per share times the number of shares, and the book value of debt is the face value of the company’s debt, so:
BVE = 9.5M($5) = $47.5M
BVD = $75M + 60M = $135M
So, the total value of the company is:
V = $47.5M + 135M = $182.5M
And the book value weights of equity and debt are:
E/V = $47.5/$182.5 = .2603
D/V = 1 – E/V = .7397
b. The market value of equity is the share price times the number of shares, so:
MVE = 9.5M($53) = $503.5M
Using the relationship that the total market value of debt is the price quote times the par value of the bond, we find the market value of debt is:
MVD = .93($75M) + .965($60M) = $127.65M
This makes the total market value of the company:
V = $503.5 + 127.65M = $631.15
And the market value weights of equity and debt are:
E/V = $503.5/$631.15 = .7978
D/V = 1 – E/V = .2022
c. The market value weights are more relevant.
13. First, we will find the cost of equity for the company. The information provided allows us to solve for the cost of equity using the dividend growth model, so:
RE = [$4.10(1.06)/$53] + .06 = .1420 or 14.20%
Next, we need to find the YTM on both bond issues. Doing so, we find:
P1 = $930 = $40(PVIFAR%,20) + $1,000(PVIFR%,20)
R = 4.54%
YTM = 4.54% × 2 = 9.08%
P2 = $965 = $37.5(PVIFAR%,12) + $1,000(PVIFR%,12)
R = 4.13%
YTM = 4.13% × 2 = 8.25%
To find the weighted average aftertax cost of debt, we need the weight of each bond as a percentage of the total debt. We find:
wD1 = .93($75M)/$127.65M = .546
wD2 = .965($60M)/$127.65M = .454
Now we can multiply the weighted average cost of debt times one minus the tax rate to find the weighted average aftertax cost of debt. This gives us:
RD = (1 – .35)[(.546)(.0908) + (.454)(.0825)] = .0566 or 5.66%
Using these costs we have found and the weight of debt we calculated earlier, the WACC is:
WACC = .7978(.1420) + .2022(.0566) = .1247 or 12.47%
14. a. Using the equation to calculate WACC, we find:
WACC = .105 = (1/1.8)(.15) + (.8/1.8)(1 – .35)RD
RD = .0750 or 7.50%
b. Using the equation to calculate WACC, we find:
WACC = .105 = (1/1.8)RE + (.8/1.8)(.064)
RE = .1378 or 13.78%
15. We will begin by finding the market value of each type of financing. We find:
MVD = 4,000($1,000)(1.03) = $4.12M
MVE = 90,000($57) = $5.13M
MVP = 13,000($104) = $1.352M
And the total market value of the firm is:
V = $4.12M + 5.13M + 1.352M = $10.602M
Now, we can find the cost of equity using the CAPM. The cost of equity is:
RE = .06 + 1.10(.08) = .1480 or 14.80%
The cost of debt is the YTM of the bonds, so:
P0 = $1,030 = $35(PVIFAR%,40) + $1,000(PVIFR%,40)
R = 3.36%
YTM = 3.36% × 2 = 6.72%
And the aftertax cost of debt is:
RD = (1 – .35)(.0672) = .0437 or 4.37%
The cost of preferred stock is:
RP = $6/$104 = .0577 or 5.77%
Now we have all of the components to calculate the WACC. The WACC is:
WACC = .0437(4.12/10.602) + .1480(5.13/10.602) + .0577(1.352/10.602) = 9.60%
Notice that we didn’t include the (1 – tC) term in the WACC equation. We simply used the aftertax cost of debt in the equation, so the term is not needed here.
16. a. We will begin by finding the market value of each type of financing. We find:
MVD = 120,000($1,000)(0.93) = $111.6M
MVE = 9M($34) = $306M
MVP = 500,000($83) = $41.5M
And the total market value of the firm is:
V = $111.6M + 306M + 41.5M = $459.1M
So, the market value weights of the company’s financing is:
D/V = $111.6M/$459.1M = .2431
P/V = $41.5M/$459.1M = .0904
E/V = $306M/$459.1M = .6665
b. For projects equally as risky as the firm itself, the WACC should be used as the discount rate.
First we can find the cost of equity using the CAPM. The cost of equity is:
RE = .05 + 1.20(.10) = .1700 or 17.00%
The cost of debt is the YTM of the bonds, so:
P0 = $930 = $42.5(PVIFAR%,30) + $1,000(PVIFR%,30)
R = 4.69%
YTM = 4.69% × 2 = 9.38%
And the aftertax cost of debt is:
RD = (1 – .35)(.0938) = .0610 or 6.10%
The cost of preferred stock is:
RP = $7/$83 = .0843 or 8.43%
Now we can calculate the WACC as:
WACC = .1700(.6665) + .0843(.0904) + .0610 (.2431) = 13.58%
17. a. Projects X, Y and Z.
b. Using the CAPM to consider the projects, we need to calculate the expected return of the project given its level of risk. This expected return should then be compared to the expected return of the project. If the return calculated using the CAPM is higher than the project expected return, we should accept the project, if not, we reject the project. After considering risk via the CAPM:
E[W] = .05 + .60(.12 – .05) = .0920 < .11, so accept W
E[X] = .05 + .90(.12 – .05) = .1130 < .13, so accept X
E[Y] = .05 + 1.20(.12 – .05) = .1340 < .14, so accept Y
E[Z] = .05 + 1.70(.12 – .05) = .1690 > .16, so reject Z
- Project W would be incorrectly rejected; Project Z would be incorrectly accepted.
18. a. He should look at the weighted average flotation cost, not just the debt cost.
b. The weighted average floatation cost is the weighted average of the floatation costs for debt and equity, so:
fT = .04(.9/1.9) + .10(1/1.9) = .072 or 7.20%
c. The total cost of the equipment including floatation costs is:
Amount raised(1 – .072) = $15M
Amount raised = $15M/(1 – .072) = $16,156,463
Even if the specific funds are actually being raised completely from debt, the flotation costs, and hence true investment cost, should be valued as if the firm’s target capital structure is used.
19. We first need to find the weighted average floatation cost. Doing so, we find:
fT = .60(.11) + .10(.07) + .30(.04) = .085 or 8.5%
And the total cost of the equipment including floatation costs is:
Amount raised(1 – .08500) = $25M
Amount raised = $25M/(1 – .0850) = $27,322,404
Intermediate
20. Using the debt-equity ratio to calculate the WACC, we find:
WACC = (.65/1.65)(.055) + (1/1.65)(.15) = .1126 or 11.26%
Since the project is riskier than the company, we need to adjust the project discount rate for the additional risk. Using the subjective risk factor given, we find:
Project discount rate = 11.26% + 2.00% = 13.26%
We would accept the project if the NPV is positive. The NPV is the PV of the cash outflows plus the PV of the cash inflows. Since we have the costs, we just need to find the PV of inflows. The cash inflows are a growing perpetuity. If you remember, the equation for the PV of a growing perpetuity is the same as the dividend growth equation, so:
PV of future CF = $3,500,000/(.1326 – .05) = $42,385,321
The project should only be undertaken if its cost is less than $42,385,321 since costs less than this amount will result in a positive NPV.
21. The total cost of the equipment including floatation costs was:
Total costs = $2.1M + 128,000 = $2.228M
Using the equation to calculate the total cost including floatation costs, we get:
Amount raised(1 – fT) = Amount needed after floatation costs
$2.228M(1 – fT) = $2.1M
fT = .0575 or 5.75%
Now, we know the weighted average floatation cost. The equation to calculate the percentage floatation costs is:
fT = .0575 = .08(E/V) + .03(D/V)
We can solve this equation to find the debt-equity ratio as follows:
.0575(V/E) = .08 + .03(D/E)
We must recognize that the V/E term is the equity multiplier, which is (1 + D/E), so:
.0575(D/E + 1) = .08 + .03(D/E)
D/E = .8215
Challenge
22. We can use the debt-equity ratio to calculate the weights of equity and debt. The debt of the company has a weight for long-term debt and a weight for accounts payable. We can use the weight given for accounts payable to calculate the weight of accounts payable and the weight of long-term debt. The weight of each will be:
Accounts payable weight = .20/1.20 = .17
Long-term debt weight = 1/1.20 = .83
Since the accounts payable has the same cost as the overall WACC, we can write the equation for the WACC as:
WACC = (1/2.3)(.17) + (1.3/2.3)[(.20/1.2)WACC + (1/1.2)(.09)(1 – .35)]
Solving for WACC, we find:
WACC = .0739 + .5652[(.20/1.2)WACC + .0488]
WACC = .0739 + (.0942)WACC + .0276
(.9058)WACC = .1015
WACC = .1132 or 11.32%
We will use basically the same equation to calculate the weighted average floatation cost, except we will use the floatation cost for each form of financing. Doing so, we get:
Flotation costs = (1/2.3)(.08) + (1.3/2.3)[(.20/1.2)(0) + (1/1.2)(.04)] = .0529 or 5.29%
The total amount we need to raise to fund the new equipment will be:
Amount raised cost = $45,000,000/(1 – .0529)
Amount raised = $47,511,935
Since the cash flows go to perpetuity, we can calculate the future cash inflows using the equation for the PV of a perpetuity. The NPV is:
NPV = –$47,511,935 + ($5,700,000/.1132)
NPV = –$47,511,935 + 50,372,552 = $2,860,617
23. The $7 million cost of the land 3 years ago is a sunk cost and irrelevant; the $9.6M appraised value of the land is an opportunity cost and is relevant. The relevant market value capitalization weights are:
MVD = 15,000($1,000)(0.92) = $13.8M
MVE = 300,000($75) = $22.5M
MVP = 20,000($72) = $1.44M
The total market value of the company is:
V = $13.8M + 22.5M + 1.44M = $37.74M
Next we need to find the cost of funds. We have the information available to calculate the cost of equity using the CAPM, so:
RE = .05 + 1.3(.08) = .1540 or 15.40%
The cost of debt is the YTM of the company’s outstanding bonds, so:
P0 = $920 = $35(PVIFAR%,30) + $1,000(PVIFR%,30)
R = 3.96%
YTM = 3.96% × 2 = 7.92%
And the aftertax cost of debt is:
RD = (1 – .35)(.0792) = .0515 or 5.15%
The cost of preferred stock is:
RP = $5/$72 = .0694 or 6.94%
a. The weighted average floatation cost is the sum of the weight of each source of funds in the capital structure of the company times the floatation costs, so:
fT = ($22.5/$37.74)(.09) + ($1.44/$37.74)(.07) + ($13.8/$37.74)(.04) = .0710 or 7.10%
The initial cash outflow for the project needs to be adjusted for the floatation costs. To account for the floatation costs:
Amount raised(1 – .0710) = $15,000,000
Amount raised = $15,000,000/(1 – .0710) = $16,145,593
So the cash flow at time zero will be:
CF0 = –$9,600,000 – 16,145,593 – 900,000 = – $26,645,593
There is an important caveat to this solution. This solution assumes that the increase in net working capital does not require the company to raise outside funds; therefore the floatation costs are not included. However, this is an assumption and the company could need to raise outside funds for the NWC. If this is true, the initial cash outlay includes these floatation costs, so:
Total cost of NWC including floatation costs:
$900,000/(1 – .0710) = $968,736
This would make the total initial cash flow:
CF0 = –$9,600,000 – 16,145,593 – 968,736 = – $26,714,329
b. To find the required return on this project, we first need to calculate the WACC for the company. The company’s WACC is:
WACC = [($22.5/$37.74)(.1540) + ($1.44/$37.74)(.0694) + ($13.8/$37.74)(.0515)] = .1133
The company wants to use the subjective approach to this project because it is located overseas. The adjustment factor is 2 percent, so the required return on this project is:
Project required return = .1133 + .02 = .1333
c. The annual depreciation for the equipment will be:
$15,000,000/8 = $1,875,000
So, the book value of the equipment at the end of five years will be:
BV5 = $15,000,000 – 5($1,875,000) = $5,625,000
So, the aftertax salvage value will be:
Aftertax salvage value = $5,000,000 + .35($5,625,000 – 5,000,000) = $5,218,750
d. Using the tax shield approach, the OCF for this project is:
OCF = [(P – v)Q – FC](1 – t) + tCD
OCF = [($10,000 – 9,000)(12,000) – 400,000](1 – .35) + .35($15M/8) = $8,196,250
e. The accounting breakeven sales figure for this project is:
QA = (FC + D)/(P – v) = ($400,000 + 1,875,000)/($10,000 – 9,000) = 2,275 units
f. We have calculated all cash flows of the project. We just need to make sure that in Year 5 we add back the aftertax salvage value and the recovery of the initial NWC. The cash flows for the project are:
Year Flow Cash
0 –$26,645,593
1 8,196,250
2 8,196,250
3 8,196,250
4 8,196,250
5 14,315,000
Using the required return of 13.33 percent, the NPV of the project is:
NPV = –$26,645,593 + $8,196,250(PVIFA13.33%,4) + $14,315,000/1.13335
NPV = $5,225,827.13
And the IRR is:
NPV = 0 = –$26,645,593 + $8,196,250(PVIFAIRR%,4) + $14,315,000/(1 + IRR)5
IRR = 20.51%
If the initial NWC is assumed to be financed from outside sources, the cash flows are:
Year Flow Cash
0 –$26,714,329
1 8,196,250
2 8,196,250
3 8,196,250
4 8,196,250
5 14,315,000
With this assumption, and the required return of 13.33 percent, the NPV of the project is:
NPV = –$26,714,329 + $8,196,250(PVIFA13.33%,4) + $14,315,000/1.13335
NPV = $5,175,091.57
And the IRR is:
NPV = 0 = –$26,714,329 + $8,196,250(PVIFAIRR%,4) + $14,315,000/(1 + IRR)5
IRR = 20.40%
