The risk of the project determines the discount rate, and in this case, Geothermal’s WACC is more reflective of the risk of the project in question.. If it is the book value, then 12.5%
Trang 1Solutions to Chapter 12 The Cost of Capital
1 The yield to maturity on the bonds (since maturity is now 19 years) is the interest rate that solves the following equation:
90 × annuity factor(r, 19 years) + 1000/(1 + r)19 = 1050
The solution can be obtained most easily from a financial calculator: Set n = 19,
FV = 1000, PV = (-)1050, PMT = 90 Compute the interest rate as 8.46% The after-tax cost of debt is therefore 8.46% ×(1 – 30) = 5.92%
2 r = DIV/P0 = $4/$40 = 10 = 10%
3 WACC = × rdebt × (1 – Tc) + × rpreferred + × requity
= 3 × 8.46% × (1 – 30) + 2 × 10% + 5 × 12.5% = 10.03%
4 r = DIV1/P0 + g = + g = + 05 = 1375 = 13.75%
5 The total value of the firm is $80 million The weights for each security class are:
Preferred: P/V = 10/80 = 125
Common: E/V = 50/80 = 625
WACC = × rdebt × (1 – Tc) + × rpreferred + × requity
= 25 × 8% × (1 – 35) + 125 × 10% + 625 × 15% = 11.925%
6 Executive Fruit should use the WACC of Geothermal, not its own WACC, when evaluating an investment in geothermal power production The risk of the project determines the discount rate, and in this case, Geothermal’s WACC is more
reflective of the risk of the project in question The proper discount rate, therefore,
is not 12.3% It is more likely to be 11.4%
7 a First calculate the company’s WACC
WACC = 30% × (1- 40%)×6% + 70% ×11% = 8.78%
Free cash flow = operating cash flow -investment
Trang 2= 68 – 30 million
= 38 million Value of business = 38 /(8.78% - 4%) = 794.98 million
b Equity = Value of business - Debt
= 794.98 × (1-30%)
= 556.49 million
8 The rate on Buildwell’s debt is 5 percent The cost of equity capital is the required rate of return on equity, which can be calculated from the CAPM as 4% + 80 ×
8% = 10.4%
The weighted average cost of capital, with a tax rate of zero, is
WACC = × rdebt + × requity
= 30 × (1 – 0) × 5% + 70 × 10.4% = 8.78%
9 IRR, which is 12%, exceeds the cost of capital Therefore, BCCI should accept the project The present value of the project cash flows is
$100,000 × annuity factor(8.78%, 7 years) = $507,032
Thus the most BCCI should pay for the project is $507,032 If it does, the
project’s NPV will be zero but the project will earn enough to meet the cost of capital If BCCI pays $507,032 for the project, then the project’s IRR will be 8.78%, just equal to the cost of capital
10 The company doesn’t pay tax, so EBITDA is operating cash flow, therefore the forecasted free cash flows are:
Year
Since free cash flow from year 5 onward will remain unchanged at year-4 levels, the horizon value at year 4 is:
Horizon value = (free cash flow at year 5)/ (r –g)
= 100 / (8.78% - 0)
= 1138.85 million
Trang 3Free cash flow at year 5 $100 million
$1,138.85
The company’s total value is:
1.0878 (1.0878) (1.0878) (1.0878) (1.0878)
Since the capital structure is 30% debt, the value of the firm’s debt is:
0.30× $1,094.53 million = $328.36 million
The value of the equity is:
0.70 × $1,094.53 million = $766.17 million
Debt $ 5.5 million 1.10 × par value of $5 million
Equity $15.0 million $30 per share × 500,000 shares*
*Number of shares = = 500,000
WACC = × rdebt + × requity
= × (1 – 3) × 9% + × 15% = 12.67%
12 Because the firm is all-equity financed, asset beta = equity beta = 8 The WACC is the same as the cost of equity which may be calculated using the CAPM:
requity = rf + β(rm – rf) = 5% + 8 × 10% = 13%
13 The 12.5% value calculated by the analyst is the current yield of the firm’s
outstanding debt: interest payments/bond value This calculation neglects the fact that bonds selling at discounts from or premiums over par value provide expected returns determined in part by expected price appreciation or depreciation The analyst should be using yield to maturity instead of current yield to calculate cost
of debt [This answer assumes the value of the debt provided is the market value
If it is the book value, then 12.5% would be the average coupon rate of
outstanding debt, which also would be a poor estimate of the required rate of return on the firm’s debt The coupon rate was set when the debt was issued We have no idea how long ago the debt was issued Don’t use the coupon rate as an estimate of the bond’s required rate of return!]
14 a Using the recent growth rate of 30% and the dividend yield of 2%, one
estimate would be:
Trang 4DIV1/P0 + g = 02 + 30 = 32 = 32%
In this calculation, we’ve assumed that the current dividend yield is the next expected dividend divided by the current price, DIV1/P0 However, if the
dividend yield was the most recent past dividend, DIV0/P0, then with 30%
dividend growth, DIV1/P0 would be 02 × 1.3 = 026 and the estimated required rate of return would be 026 + 3 = 326, or 32.6%
Another estimate, based on the CAPM, would be
r = rf + β(rm – rf) = 4% + 1.2 × 8% = 13.6%
b The estimate of 32% seems far less reasonable It is based on an historic growth rate that is impossible to sustain No company can grow at 30% forever The [DIV1/P0 + g] rule requires that the growth rate of dividends per share must be viewed as highly stable over the foreseeable future In other words, it requires us to use the sustainable growth rate
15 a The 9% coupon bond has a yield to maturity of 10% and sells for 93.86% of
face value:
n = 10, i = 10%, PMT = 90, FV = 1000, compute PV = $938.55
The market value of the issue is therefore
.9386 × $20 million = $18.77 million
The 10% coupon bond sells for 92.8% of par value, and has a yield to maturity of 11.0%:
n = 15, PV = (−)928, PMT = 100, FV = 1000, compute i = 11.00%
The market value of the issue is
.928 × $25 million = $23.20 million
The weighted average before-tax cost of debt is therefore
× 10% + × 11% = 10.55%
b The after-tax cost of debt is (1 – 30) × 10.55% = 7.39%
16 The bonds must be selling below par value, because the YTM is greater than the coupon rate
Trang 5The price per $1000 par value is
80 × annuity factor(9%, 10 years) + 1000/1.0910 = $935.82
The total market value of the bonds is
$10 million par value × = $9.36 million
Book value of the preferred shares is $2 million and the par value per share is $20 Thus there are 100,000 shares of preferred stock (=$2 million/$20 per share) Preferred shares are selling at $15 per share, for total market value of $1.5 million
The market value of 1 million common shares selling at $20/share is $20 million The book value of the common shares is sum of the common stock plus retained earnings, which also happens to equal $20 million
Therefore, the market value capital structure is:
17 The yield to maturity on debt is rdebt = 9%
The rate on preferred stock is rpreferred = $2/$15 = 133 = 13.3%
Trang 6The rate on common stock is
requity = rf + β(rm – rf) = 4% + 1.5 × 7% = 14.5%
Using the capital structure derived in the previous problem, we can calculate WACC as:
WACC = × rdebt + × requity + × rpreferred
= 303 × (1 – 4) × 9% + 648 × 14.5% + 049 × 13.3% = 11.68%
18 The IRR on the computer project is less than the WACC of firms in the computer industry Therefore, the project should be rejected However, the WACC of the firm (based on its existing mix of projects) is only 11.68% If the firm uses this figure as the hurdle rate, it will incorrectly go ahead with the venture in home computers The discount rate for a project is determined by the risk of the project University Product's WACC is irrelevant to the analysis of the investment in the computer project
19 a r = rf + β(rm – rf) = 4% + 1.5 × 7% = 14.5%
b Total market value of Muskoka Real Estate is $6 million and the market value
of the debt is $2 million Thus the market value of its equity is $6 - $2, or $4 million The current capital structure is 1/3 debt, 2/3 equity
Weighted average beta = × 0 + × 1.5 = 1.0
c WACC = ×rdebt + × requity
= × (1 – 4) × 4% + × 14.5% = 10.47%
d If the company wishes to expand its present business then the WACC is a reasonable estimate of the discount rate since the risk of the proposed project
is similar to the risk of the existing projects Use a discount rate of 10.47%
e The WACC of optical projects should be based on the risk of those projects Using a beta of 1.2, the discount rate for the new venture is
r = 4 + 1.2 × 7 = 12.4%
20 a Equity
Market value = 10 million shares × $15/share = $150 million
rE = rf + rf + β(rm – rf) = 2.5% + 1.2×6.5% = 10.3%
Debt
Market value per bond =
semi-annual coupon payment × PVIFA(6-month rB, no of payments)
Trang 7+ face value × PVIF(6-month rB, nọ of periods to maturity)
rB = required rate of return on 10-year Gov't debt +150 basis points
Required rate of return on 10-year Gov't debt = (1 + 04/2)2 - 1 =.0404
rB = 0404 + 0150 = 0.0554 = 5.54%
6-month required rate of return = (1.0554)1/2 - 1 = 0273
Face value = 1000
Coupon payment= 06/2 × 1000 = 30
Nọ of payments = 2 payments/year × 10 years = 20
Market value per bond
= 30 PVIFẶ0273,20) + 1000 × PVIF(.0273, 20) = $1,041.2
Market value of all bonds = 20,000 bonds × $1,041.2 = $20,824,000
WACC Calculation
Market Value Market
Weight Before-taxRequired
Rate of Return
After-tax Required Rate
of Return
Weight × After-tax Return
b βU =
To find βdebt, use the CAPM:
rdebt = rf + βdebt(rm – rf)
βdebt = = = 468
βU =
= = 1.14
As expected, the unlevered beta is lower than the levered equity betạ With
no debt in the capital structure, the equity is less riskỵ
c Relever the equity beta to reflect the new capital structure of 50% debt:
βlevered = βU + [βU - βdebt] × D/E × (1 - TC)
= 1.14 + [1.14 - 468] × 5/.5 × (1 - 35) = 1.58
As expected, moving from a debt/equity ratio of 122/.878, or about 139 to 5/.5, or 1, increases the riskiness of the equitỵ The levered equity beta increases from 1.2 to 1.58 The new required rate of return to equity is:
rE = rf + β(rm – rf) = 2.5% + 1.58×6.5% = 12.77%
Trang 8The new WACC is:
WACC = × rdebt ×(1 - TC)+ × requity = 5 × 5.54% (1 - 35) + 5 × 12.77% = 8.19%
21 a NOTE: this question reports two tax rates for Premier Pizza There should be
only one tax rate For the purposes of answering this question, the tax rate is assumed to be 35%
The annual free cash flows expected from Premier Pizza are:
Calculation of Annual Cash Flow Annual Cash Flow
2 Operating costs .7 × 10 million = $ 7.0 million
9.Capital expenditures 05×$10 million= $ 0.5 million
The annual cash flows from Premier Pizza are a constant growing perpetuity Using the perpetuity formula, the present value of the cash flows is:
PV(annual cash flows) = annual cash flow
required rate of return - growth rate
Required rate of return on cash flows:
WACC = × rdebt ×(1 - TC)+ × requity
= 25 × 06 × (1-.35) + 75 × 15 = 12225
PV(annual cash flows) = $1.625 million = $19.76 million
12225 - 04
If Boris and Isabelle offer $19.76 million for Premier Pizza, the NPV of their investment will be zero The offer is the investment in the firm:
NPV = - investment + PV(future firm’s free cash flows)
= -19.76 million + 19.76 million = 0
The project will earn the required rate of return, just enough return to
compensate all investors This is the maximum amount they should offer If they can purchase Premier Pizza for less than $19.76 million, the investment will earn more than the required rate of return, with the bulk of the extra return going to the equity investors in the project
Trang 9b Fresh Foods current WACC
rEQUITY = rf + β(rm – rf) = 03 + 8 × 07 = 086
rDEBT = 05
D/V = 4, E/V = 6
WACC = × rdebt ×(1 - TC)+ × requity
= 4 × 05 × (1 - 35) + 6 × 086 = 0646
Using Fresh Food's current WACC as the required rate of return on Premier Pizza gives a valuation of:
PV(annual cash flows) = $1.625 million = $66.06 million
0646 - 04
This says that Premier Pizza is worth over 3 times more owned by Fresh Foods than by Boris and Isabelle, even though both groups expect the same cash flows! The difference comes solely from the different discount rates
Question: What is the appropriate discount rate for Fresh Foods to use in valuing Premier Pizza? If you use Fresh Foods' current WACC, you are assuming that riskiness of the pizza manufacturing business is the same as the riskiness of the grocery retail business Why should it necessarily be the case?
Without further investigation, the better assumption would be that the
appropriate discount rate for Fresh Foods to use for Premier Pizza's cash flows is 12.225%, the rate used by Boris and Isabella We know that Boris and Isabelle investigated the risks of the business before determining the appropriate discount rate Under this assumption, the maximum Fresh Foods should be willing to pay is also $19.76 million
22 a The current promised YTM if there is no default is
YTM = (1000+60-569)/569 = 86.3%
b If default occurs,
YTM = [.3 × (1000+60) - 569]/569 = -44.1%
c The current expected YTM is
YTM = probability of default × YTM if default occurs + probability of no default × YTM if no default
= 6 × -44.1% + 4× 86.3% = 8.06%
Obviously when the probability of default is high, the promised YTM is too high, 86.3% in this case compared to 8.06% of expected return, so it is not a good measure of expected return
Trang 1023 If Big Oil does not pay taxes, the after-tax and before-tax costs of debt are
identical WACC would then become:
WACC = × rdebt + × requity
= 243 × 9% + 757 × 13.5% = 12.41%
If Big Oil issues new equity and uses the proceeds to pay off all of its debt, the cost of equity will fall There is no longer any leverage, so the equity becomes safer and commands a lower risk premium In fact, with all-equity financing, the cost of equity would be the same as the firm’s WACC, which is 12.41%, lower than the previous value of 13.5% (We use the WACC derived in the absence of interest tax shields since, for the all-equity firm, there is no interest tax shield.)
24 The net effect of Big Oil’s transaction is to leave the firm with $200 million more debt (because of the borrowing) and $200 million less equity (because of the dividend payout) Total assets and business risk are unaffected The WACC will remain unaffected, since business risk is unchanged However, the cost of equity will rise With the now higher leverage, the business risk is spread over a smaller equity base, so each share is now riskier
The new financing mix for the firm would be E = 1,000 and D = 585.7 Therefore,
= = 369 and = = 631
If the cost of debt is still 9%, then we can solve for the new cost of equity as follows We use the fact that, even at the new financing mix, WACC must still be 12.41%
WACC = × rdebt + × requity
= 369 × 9% + 631 × requity = 12.41%
We solve to find that requity = 14.40%
25 Even if the WACC were lower when the firm’s tax rate is higher, this does not imply that the firm would be worth more The after-tax cash flows that the firm would generate for its owners also would be lower and this would reduce the value of the firm, even if those cash flows were discounted at a lower rate If the tax authority is collecting more income from the firm, the value of the firm will fall
26 This reasoning is faulty in that it implicitly treats the discount rate for the project
as the cost of debt if the project is debt financed, and as the cost of equity if the project is equity financed In fact, if the project poses risk comparable to the risk
of the firm’s other projects, the proper discount rate is the firm’s cost of capital,