10.12: Problem

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Problem $\PageIndex{1}$

Assume our company has a bond outstanding with 20-years remaining until maturity. This bond has a 7.5% coupon rate. Our marginal tax rate is 35%. Find our after-tax cost of debt if the bond price is:

$1135
$875

Answer

First we must find the YTM and then plug it into the formula:

k_i = YTM(1 – T)

Remember that bonds pay interest semi-annually so that we must set our calculators to 2 Periods per Year and adjust the N and PMT to reflect the semi-annual framework.

Part a

40 N
-1135 PV
37.50 PMT
1000 FV
I/Y ⇒ 6.30%

k_i = 6.30%(1 – 0.35) = 4.10%

Part b

40 N
-875 PV
37.50 PMT
1000 FV
I/Y ⇒ 8.84%

k_i = 8.86%(1 – 0.35) = 5.75%

Problem $\PageIndex{2}$

If the par value of our preferred stock is $30 and the dividend rate is 5% of par while the current price is $16.50, what is the cost of preferred stock?

Answer: k_p = D/P = (0.05)(30)/16.50 = 1.50/16.50 = 9.09%

Problem $\PageIndex{3}$

The price of our common stock is $25. The constant growth rate in dividends is 8% and our current dividend (D0) is $0.75. Also, the risk-free rate of interest is 5% and the expected return on the market is 12%. Beta for this stock is 0.8. Finally, we estimate a risk premium of 5% for stocks relative to bonds and the current YTM on our long-term debt is 9%. Find the estimated cost of capital for common stock under each of the 3 methods.

Answer

Dividend Valuation Approach
k_s = (D₁/P) + g = [(0.75)×(1.08)/25] + 0.08 = 11.24%

Security Market Line Approach
k_s = k_rf + β(k_m – k_rf) = 5% + 0.8(12% – 5%) = 5% + 5.6% = 10.60%

Bond Yield plus Risk Premium Approach
k_s = YTM + RP = 9% + 5% = 14.00%

Average k_s = (11.24% + 10.60% + 14.00%)/3 = 11.95%

Problem $\PageIndex{4}$

You have the following information about XYZ Corp:

Asset	Book Value	Market Value
Bonds	$20,000,000	$24,000,000
Preferred Stock	$4,000,000	$5,000,000
Common Stock	$10,000,000	$35,000,000

Constant growth on common	6.5%
YTM on bonds	11%
Beta	1.35
Treasury bond yield	5%
Price of common stock	$34
Tax rate	40%
Coupon rate on bonds	10%
Risk prem. stocks over bonds	5%
Expected market return (k_m)	12%
Expected Common Dividend (D₁)	2.75
Number of pref. shares	100,000
Per share dividend on preferred	$6.50

What is the marginal cost of capital for this firm?
If you have a capital budgeting project that will generate after tax cash flows of $25,000 per year for the next four years and costs $75,000, should you take it?

Answer

Part a

Step 1 ⇒ Solve for Market Value Weights

MV_debt = 24,000,000
MV_pref = 5,000,000
MV_com = 35,000,000
MV_total = 64,000,000

W_debt = 24,000,000/64,000,000 = 0.38
W_pref = 5,000,000/64,000,000 = 0.08
W_com = 35,000,000/64,000,000 = 0.55

Step 2 ⇒ Solve for After-tax Cost of Debt

k_i = YTM(1 – T) = 11%(1 – 0.40) = 6.60%

Step 3 ⇒ Solve for Cost of Preferred Stock

k_p = D/P = 6.50/50 = 13%

Note that the price per share for preferred stock is found by taking the total market value of preferred stock divided by the number of shares ⇒ $5,000,000/100,000 = $50

Step 4 ⇒ Solve for Cost of Common Stock

Dividend Valuation Approach
k_s = (D₁/P) + g = (2.75/34) + 0.065 = 14.59%

Security Market Line Approach
k_s = k_rf + β(k_m – k_rf) = 5% + 1.35(12% – 5%) = 14.45%

Bond Yield Plus Risk Premium Approach
k_s = YTM + RP = 11% + 5% = 16.00%

Cost of Common Stock Financing
(14.59% + 14.45% + 16.00%)/3 = 15.01%

Step 5 ⇒ Solve for Marginal Cost of Capital (MCC)

MCC = W_debt(k_i) + W_pref(k_p) + W_com(k_s)
= (0.38)(6.60%) + (0.08)(13.00%) + (0.55)(15.01%)
= 11.80%

Part b

Solve for IRR ⇒ IRR = 12.59% > 11.80% ⇒ Accept Project
Solve for NPV@11.80% ⇒ $1254.70 > $0 ⇒ Accept Project

Note – Since there is no crossover problem and it is a single project instead of mutually exclusive, we can use either IRR or NPV to make our decision.

Problem $\PageIndex{5}$

The following information is available about ACME Inc.
Balance Sheet:

LT 10% Coupon Bonds (10,000 bonds)	$10,000,000
Preferred Stock (40,000 shares) ($50 par with a 10% dividend)	2,000,000
Common Stock (1,000,000 shares)	20,000,000

The market values are $1060 for each $1000 par value bond, $53 for each share of preferred, and $41.25 for each share of common. The bonds are recorded on the balance sheet at their par value and mature in 10 years.

Beta	1.3
Current Treasury bond rate	6%
Risk Premium for stocks over bonds	5%
Tax Rate	40%
Growth rate in dividends	10%
Expected market return	13%
Dividend (D₀)	2.25

What are the appropriate weights for the opportunity cost of capital?
What are the appropriate costs of debt, preferred, and common (use an average of the 3 methods for common)?
What is the marginal cost of capital?

Answer: Part a

MV_debt = 10,000*1060 = $10,600,000
MV_pref = 40,000*53 = $ 2,120,000
MV_com = 1,000,000*41.25 = $41,250,000
MV_total = $53,970,000

W_debt = 0.20
W_pref = 0.04
W_com = 0.76

Part b

k_i = YTM(1 – T) = 9.07%(1 – 0.40) = 5.44%

Find YTM
20 N
-1060 PV
50 PMT
1000 FV
I/Y ⇒ 9.07%

k_p = D/P = $5/$53 = 9.43%

Div Val Approach ⇒ k_s = (D₁/P) + g = [(2.25)(1.10)/41.25] + 0.10 = 16.00%
SML Approach ⇒ k_s = k_rf + β(k_m – k_rf) = 6% + 1.3(13% – 6%) = 15.10%
BY + RP Approach ⇒ k_s = YTM + RP = 9.06% + 5% = 14.06%
Average of Three Approaches ⇒ (16.00% + 15.10% + 14.06%)/3 = 15.05%

Part c

MCC = W_debt(k_i) + W_pref(k_p) + W_com(k_s)
= (0.20)(5.44%) + (0.04)(9.43%) + 0.76(15.05%)
= 12.90%

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