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# RW7eCh14

CHAPTER14

COST OF CAPITAL

Learning Objectives

LO1 How to determine a firm’s cost of equity capital.

LO2 How to determine a firm’s cost of debt.

LO3 How to determine a firm’s overall cost of capital.

LO4 How to correctly include flotation costs in capital budgeting projects.

LO5 Some of the pitfalls associated with a firm’s overall cost of capital and what to do about them.

Answers to Concepts Review and Critical Thinking Questions

1. (LO3) 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. (LO3) Book values for debt are likely to be much closer to market values than are equity book values.

3. (LO5) No. The cost of capital depends on the risk of the project, not the source of the money.

4. (LO3) Interest expense is tax-deductible. There is no difference between pretax and aftertax equity costs.

5. (LO1) 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. (LO1) 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 dividend discount model 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. (LO2) 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.

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8.

9.

(LO5) 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. (LO5) 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. (LO5) 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.

1. (LO1) With the information given, we can find the cost of equity using the dividend growth model. Using this

model, the cost of equity is:

2.

RE = [\$2.40(1.055)/\$52] + .055 = .1037 or 10.37% (LO1) Here we have information to calculate the cost of equity using the CAPM. The cost of equity is:

RE = .053 + 1.05(.12 – .053) = .1234 or 12.34%

(LO1) 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 = .05 + 0.85(.08) = .1180 or 11.80% 3.

And using the dividend growth model, the cost of equity is

RE = [\$1.60(1.06)/\$37] + .06 = .1058 or 10.58%

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 two percent higher than average, and the estimate from the dividend growth model is about one percent higher 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 = (.1180 + .1058)/2 = .1119 or 11.19%

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4. 5. 6. 7. (LO1) 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 = (\$1.12 – 1.05)/\$1.05 = .0667 or 6.67% g2 = (\$1.19 – 1.12)/\$1.12 = .0625 or 6.25% g3 = (\$1.30 – 1.19)/\$1.19 = .0924 or 9.24% g4 = (\$1.43 – 1.30)/\$1.30 = .1000 or 10.00% So, the average arithmetic growth rate in dividends was: g = (.0667 + .0625 + .0924 + .1000)/4 = .0804 or 8.04% Using this growth rate in the dividend growth model, we find the cost of equity is: RE = [\$1.43(1.0804)/\$45.00] + .0804 = .1147 or 11.47% Calculating the geometric growth rate in dividends, we find: \$1.43 = \$1.05(1 + g)4 g = .0803 or 8.03% The cost of equity using the geometric dividend growth rate is: RE = [\$1.43(1.0803)/\$45.00] + .0803 = .1146 or 11.46% (LO1) The cost of preferred stock is the dividend payment divided by the price, so: RP = \$6/\$96 = .0625 or 6.25% (LO2) The pretax cost of debt is the YTM of the company’s bonds, so: P0 = \$1,070 = \$35(PVIFAR%,30) + \$1,000(PVIFR%,30) R = 3.137% YTM = 2 × 3.137% = 6.27% And the aftertax cost of debt is: RD = .0627(1 – .35) = .0408 or 4.08% (LO2) a. The pretax cost of debt is the YTM of the company’s bonds, so: b. c. P0 = \$950 = \$40(PVIFAR%,46) + \$1,000(PVIFR%,46) R = 4.249% YTM = 2 × 4.249% = 8.50% The aftertax cost of debt is: RD = .0850(1 – .35) = .0552 or 5.52% The after-tax rate is more relevant because that is the actual cost to the company.

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8.

9.

10.

11.

(LO2) The book value of debt is the total par value of all outstanding debt, so: BVD = \$80,000,000 + 35,000,000 = \$115,000,000 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 = .95(\$80,000,000) + .61(\$35,000,000) MVD = \$76,000,000 + 21,350,000 MVD = \$97,350,000 The YTM of the zero coupon bonds is: PZ = \$610 = \$1,000(PVIFR%,14) R = 3.594% YTM = 2 × 3.594% = 7.19% So, the aftertax cost of the zero coupon bonds is: RZ = .0719(1 – .35) = .0467 or 4.67% 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 = .0552(\$76/\$97.35) + .0467(\$21.35/\$97.35) = .0534 or 5.34% (LO3) a. Using the equation to calculate the WACC, we find: b. WACC = .60(.14) + .05(.06) + .35(.08)(1 – .35) = .1052 or 10.52% Since interest is tax deductible and dividends are not, we must look at the after-tax cost of debt, which is: .08(1 – .35) = .0520 or 5.20% Hence, on an after-tax basis, debt is cheaper than the preferred stock. (LO3) Here we need to use the debt-equity ratio to calculate the WACC. Doing so, we find: WACC = .15(1/1.65) + .09(.65/1.65)(1 – .35) = .1140 or 11.40% (LO3) Here we have the WACC and need to find the debt-equity ratio of the company. Setting up the WACC equation, we find: WACC = .0890 = .12(E/V) + .079(D/V)(1 – .35) Rearranging the equation, we find: .0890(V/E) = .12 + .079(.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 .0890(D/E + 1) = .12 + .05135(D/E)

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12.

Now we can solve for D/E as: .06765(D/E) = .031 D/E = .8234 (LO3) 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: b. BVE = 11,000,000(\$6) = \$66,000,000 BVD = \$70,000,000 + 55,000,000 = \$125,000,000 So, the total value of the company is: V = \$66,000,000 + 125,000,000 = \$191,000,000 And the book value weights of equity and debt are: E/V = \$66,000,000/\$191,000,000 = .3455 D/V = 1 – E/V = .6545 The market value of equity is the share price times the number of shares, so: MVE = 11,000,000(\$68) = \$748,000,000 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(\$70,000,000) + 1.04(\$55,000,000) = \$122,300,000

This makes the total market value of the company:

V = \$748,000,000 + 122,300,000 = \$870,300,000

And the market value weights of equity and debt are:

E/V = \$748,000,000/\$870,300,000 = .8595

D/V = 1 – E/V = .1405

c. The market value weights are more relevant.

13. (LO3) 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)/\$68] + .06 = .1239 or 12.39%

Next, we need to find the YTM on both bond issues. Doing so, we find:

P1 = \$930 = \$35(PVIFAR%,42) + \$1,000(PVIFR%,42)

R = 3.838%

YTM = 3.838% × 2 = 7.68%

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14.

b.

15.

P2 = \$1,040 = \$40(PVIFAR%,12) + \$1,000(PVIFR%,12) R = 3.584% YTM = 3.584% × 2 = 7.17% 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(\$70,000,000)/\$122,300,000 = .5323 wD2 = 1.04(\$55,000,000)/\$122,300,000 = .4677 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)[(.5323)(.0768) + (.4677)(.0717)] = .0484 or 4.84% Using these costs we have found and the weight of debt we calculated earlier, the WACC is: WACC = .8595(.1239) + .1405(.0484) = .1133 or 11.33% (LO3) a. Using the equation to calculate WACC, we find: WACC = .094 = (1/2.05)(.14) + (1.05/2.05)(1 – .35)RD RD = .0772 or 7.72% Using the equation to calculate WACC, we find: WACC = .094 = (1/2.05)RE + (1.05/2.05)(.068) RE = .1213 or 12.13% (LO3) We will begin by finding the market value of each type of financing. We find: MVD = 8,000(\$1,000)(0.92) = \$7,360,000

MVE = 250,000(\$57) = \$14,250,000

MVP = 15,000(\$93) = \$1,395,000

And the total market value of the firm is:

V = \$7,360,000 + 14,250,000 + 1,395,000 = \$23,005,000

Now, we can find the cost of equity using the CAPM. The cost of equity is: RE = .045 + 1.05(.08) = .1290 or 12.90%

The cost of debt is the YTM of the bonds, so:

P0 = \$920 = \$32.50(PVIFAR%,40) + \$1,000(PVIFR%,40)

R = 3.632%

YTM = 3.632% × 2 = 7.26%

And the aftertax cost of debt is:

RD = (1 – .35)(.0726) = .0472 or 4.72%

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The cost of preferred stock is: RP = \$5/\$93 = .0538 or 5.38% Now we have all of the components to calculate the WACC. The WACC is: WACC = .0472(7.36/23.005) + .1290(14.25/23.005) + .0538(1.395/23.005) = .0983 or 9.83% Notice that we didn’t include the (1 – tC) term in the WACC equation. We used the aftertax cost of debt in the

equation, so the term is not needed here.

16. (LO3)

a. We will begin by finding the market value of each type of financing. We find:

MVD = 105,000(\$1,000)(0.93) = \$97,650,000

MVE = 9,000,000(\$34) = \$306,000,000

MVP = 250,000(\$91) = \$22,750,000

And the total market value of the firm is:

V = \$97,650,000 + 306,000,000 + 22,750,000 = \$426,400,000

So, the market value weights of the company’s financing is:

D/V = \$97,650,000/\$426,400,000 = .2290

P/V = \$22,750,000/\$426,400,000 = .0534

E/V = \$306,000,000/\$426,400,000 = .7176

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 + 1.25(.085) = .1563 or 15.63%

The cost of debt is the YTM of the bonds, so:

P0 = \$930 = \$37.5(PVIFAR%,30) + \$1,000(PVIFR%,30)

R = 4.163%

YTM = 4.163% × 2 = 8.33%

And the aftertax cost of debt is:

RD = (1 – .35)(.0833) = .0541 or 5.41%

The cost of preferred stock is:

RP = \$6/\$91 = .0659 or 6.59%

Now we can calculate the WACC as:

WACC = .0541(.2290) + .1563(.7176) + .0659(.0534) = .1280 or 12.80%

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17. (LO1)

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 lower 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 + .80(.11 – .05) = .0980 < .10, so accept W

E[X] = .05 + .90(.11 – .05) = .1040 < .12, so accept X

E[Y] = .05 + 1.45(.11 – .05) = .1370 > .13, so reject Y

E[Z] = .05 + 1.60(.11 – .05) = .1460 < .15, so accept Z

c. Project W would be incorrectly rejected; Project Y would be incorrectly accepted.

18. (LO4)

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 = .05(.75/1.75) + .08(1/1.75) = .0671 or 6.71%

c. The total cost of the equipment including floatation costs is:

Amount raised(1 – .0671) = \$20,000,000

Amount raised = \$20,000,000/(1 – .0671) = \$21,439,510

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. (LO4) We first need to find the weighted average floatation cost. Doing so, we find:

fT = .65(.09) + .05(.06) + .30(.03) = .071 or 7.1%

And the total cost of the equipment including floatation costs is:

Amount raised(1 – .071) = \$45,000,000

Amount raised = \$45,000,000/(1 – .071) = \$48,413,125

20. (LO3, 5) Using the debt-equity ratio to calculate the WACC, we find:

WACC = (.90/1.90)(.048) + (1/1.90)(.13) = .0912 or 9.12%

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 = 9.12% + 2.00% = 11.12%

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:

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PV of future CF = \$2,700,000/(.1112 – .04) = \$37,943,787 The project should only be undertaken if its cost is less than \$37,943,787 since costs less than this amount will

result in a positive NPV.

For the 1st printing of the textbook, please note the following amendment to the question printed in the textbook:

21. ‘The project cost \$1.5 million’ should read ‘The project cost \$15 million’.

21. (LO4) The total cost of the equipment including floatation costs was:

Total costs = \$15,000,000 + 850,000 = \$15,850,000

Using the equation to calculate the total cost including floatation costs, we get:

Amount raised(1 – fT) = Amount needed after floatation costs

\$15,850,000(1 – fT) = \$15,000,000

fT = .0536 or 5.36%

Now, we know the weighted average floatation cost. The equation to calculate the percentage floatation costs

is:

fT = .0536 = .07(E/V) + .03(D/V)

We can solve this equation to find the debt-equity ratio as follows:

.0536(V/E) = .07 + .03(D/E)

We must recognize that the V/E term is the equity multiplier, which is (1 + D/E), so:

.0536(D/E + 1) = .08 + .03(D/E)

D/E = 0.6929

22. (LO3) Division A:

Using the CAPM, the cost of equity is:

RE = .06 + 0.90(.05)

RE = .105 or 10.5%

Division B:

Using the CAPM, the cost of equity is:

RE = .06 + 1.50(.05)

RE = .135 or 13.5%

Division A has a lower cost of capital than the firm’s overall cost of 12%.

23. (LO3, 5)

Project A:

Cost of capital = 12% + 2% = 14%

NPV = -\$200,000 + \$71,104 PVIFA(14%, 4) = -200,000 + 71,104({1 – [1/(1 + .14)]4 } / .14 )

= \$7,177

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Project B:

Cost of capital = 12% - 2% = 10%

4 NPV = -\$200,000 + \$146,410[PVIF(10%,3) + PVIF(10%,4)] = -200,000 + 146,410[1/(1.103) + 1/(1.10)] = \$10,000

Both Project A and B have a positive NPV, so both Project A and B should be chosen.

24. (LO3, 4) 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 = (1/1.7)(.14) + (0.7/1.7)[(.20/1.2)WACC + (1/1.2)(.08)(1 – .35)]

Solving for WACC, we find:

WACC = .0824 + .4118[(.20/1.2)WACC + .0433]

WACC = .0824 + (.0686)WACC + .0178

(.9314)WACC = .1002

WACC = .1076 or 10.76%

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 = (1/1.7)(.08) + (0.7/1.7)[(.20/1.2)(0) + (1/1.2)(.04)] = .0608 or 6.08%

The total amount we need to raise to fund the new equipment will be:

Amount raised = \$45,000,000/(1 – .0608)

Amount raised = \$47,912,317

Since the cash flows go to perpetuity, we can calculate the present value using the equation for the PV of a

perpetuity. The NPV is:

NPV = –\$47,912,317 + (\$6,200,000/.1076)

NPV = \$9,719,777

A1. Evaluating as an all equity project:

NPV = -63.6M + [13.5M(1 – (1/1.216)10)/0.216] = -\$9.941873M

Equity flotation costs would only make the NPV more negative.

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Additional effects of flotation costs: Debt flotation costs = \$42M(1/.988 – 1) = \$510,121

Recall from the example of debt flotation costs in Section 14A that the flotation cost tax shield will be amortized over the life of the issue. So amortized over 10 years, the annual tax shield is:

(\$510,121/10) x 0.40 = \$20,405.

Additional equity required = 63.6M – 42M = \$21.6M

Equity flotation costs = \$21.6M(1/.988 – 1) = \$262,348

Annual tax shield = (262,348/10) x 0.40 = \$10,494

The equity flotation costs have been amortized over the life of the project, consistent with the treatment of the debt flotation costs for the project.

Combining the tax shields from the flotation costs of debt and equity, the net flotation cost is:

-510,121 + 20,405(1– (1.12)10)/0.12 –262,348 + 10,494(1– (1/1.216)10)/0.216 =

-510,121 + 115,293 – 262,348 + 41,352 = -\$615,824

Annual interest after tax = \$42M x 0.12 x (1-0.40) = \$3.024M

NPV (loan) = 42M – 3.024M(1-(1/1.12)10)/0.12 – 42M/(1.12)10

= 42M – 17.086274M – 13.522876M

= \$11.390850M = \$11,390,850

Combining all of these effects

APV = -\$9.941873M – 0.615824M + 11.39085M = \$.833153M = \$833,153

A2. In this case, the annual interest after tax = \$42M x 0.084 x (1-0.40) = \$2.1168M

NPV of the loan will be increased by the amount (3.024M – 2.1168M)(1-(1/1.12)10)/0.12 = \$5.125882M Also, the flotation cost of the loan is now zero. In question A1 the net flotation cost of the loan was –510,121 + [20,405(1-(1/1.12)10)/0.12] = -\$394,828

Therefore the incremental gain from the government loan is \$5,125,882 + \$394,828 = \$5,520,710

A3. Evaluating as an all equity project:

NPV = -28M + [9M(1 – (1/1.20)5)/0.20] = -\$1.084491M

There are no equity flotation costs in this problem.

Additional effects of the three year loan:

Annual interest after tax = \$22.5M x 0.10 x (1-0.30) = \$1.575M

NPV (loan) = 22.5M – 1.575M(1-(1/1.10)3)/0.10 – 22.5M/(1.10)3

= 22.5M – 3.91679M – 16.90458M

= \$1.678625M = \$1,678,625

Combining all these effects:

APV = -\$1.084491M + 1.678625M = \$.594134M = \$594,134

The investment is now worthwhile for the firm, as the APV is positive. This APV calculation implicitly assumes that the firm can replace the loan amount with equity for the last two years of the project.

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B1. With ROA = 37% and Total Capital = \$12M → Earnings after tax = \$4.44M.

EVA = Earnings after tax – (WACC)(Total Capital)

EVAInitial = \$4.44M – (0.175)(\$12M) = \$2.34M

Additional after tax earnings = \$1.3M

EVAA = (\$4.44M + \$1.3M) – (0.175)(\$12M + \$5.5M) = \$2.6775M = \$2,677,500

Additional after tax earnings = \$450,000

EVAB = (\$4.44M + \$0.45M) – (0.175)(\$12M + \$3M) = \$2.265M = \$2,265,000

Since Project A raises the EVA from \$2,300,000 to \$2,677,500 it should be accepted. On the other hand, accepting Project B on its own would lower the EVA for the firm from \$2,300,000 to \$2,265,000. Project B should therefore not be accepted based on the EVA criterion.

110185B2. WACC = ?0.068??1-0.36?+?0.113?= 0.0870922 295295

EVAProject = \$1,250,000(1 – 0.36) – (0.0870922)(\$6,000,000) = \$277,447.

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