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method, and ( b ) the shortcut method.

SOLUTION

Using the regular method, the yield to maturity is:

(a)

M

â€œ I

V=C-

+-

(1 + r)â€˜ (1 + r I n

r-1

I

= Z(PVIFAr,) + M(PVIF,,n)

$800 = $12O(PVIFA,7)+ $l,OOO(PVIFr,7)

289

COST OF CAPITAL

CHAP. 101

At 17%,

V = $120(3.9215) + $1,000(0.3338) = $470.58 + $333.80 = $804.38

which is close enough to $800; therefore, the yield to maturity or before-tax cost of debt is 17

percent.

The after-tax cost of debt is computed as:

= kj(1 - t )

kd

17% (1 - 0.4) 10.2%

= =

(6) Using the shortcut method:

+ [(M- V ) h ] - $120 + [($l,O00 - $800)/7] - $120 + $28.57 = 16.51%

I -

ki =

(M+ V)/2 ($l,OOO + $800)/2 $900

Therefore, the after-tax cost of debt is computed as:

= 16.51%(1- 0.4) = 9.91%

kd

Cost of Bonds. Assume the same data as in Problem 10.2, but now assume the interest is paid

10.3

semiannually. Calculate the after-tax cost of debt, using ( a ) the regular method, and ( 6 ) the

shortcut method.

SOLUTl0N

Since the interest is paid semiannually, the interest payment is $120 -+ 2 = $60 and the number of

(a)

periods is 14. Using the regular method gives:

V = I(PVIFA,,) + M (PVIF,,)

$800 = $6O(PVIFAj,1,) + $1,000 (PVIFj.1,)

To arrive at a value of $800, first try 8 percent:

V = $60(8.2442) + $1,000(0.3405) = $494.65 + $340.50 = $835.15

Since this is too high, try 9 percent:

V = $60(7.7862) + $1,000(0.2992) = $467.17 + $299.20 = $766.37

Since this value is too low, the cost of debt is somewhere between 8 percent and 9 percent. Using the

interpolation:

8% 9yo

PV $835.15 $835.15

True rate 800.00

PV --

766.37

Difference $ 35.15 $ 68.78

$35.15

+ -(l%) = 8% + 0.51 = 8.51%

True rate = 8%

$68.78

Annual rate = 8.51% x 2 = 17.02%

Therefore, the after-tax cost of debt is computed as:

290 COST OF CAPITAL [CHAP. 10

( 6 ) Using the shortcut method:

$60 + [($1,000 - $800)/14] - $60 + $14.29

k, =

($l,OOo + $800)/2 $900 = 8.25y0

Annual rate = 8.25% X 2 = 16.5%

Therefore, the after-tax cost of debt is computed as:

16.5%(1- 0.4) 9.9%

= =

kd

Cost of Preferred Stock. In its capital structure, ABC Corporation has preferred stock paying a

10.4

dividend of $5 per share and selling for $23, The companyâ€™s tax rate is 40 percent. Calculate ( a )

the before-tax cost of preferred stock, and ( b )the after-tax cost of preferred stock.

SOLUTION

The before-tax cost of preferred stock is:

(a)

(b) The same as the above, since preferred stock dividends are not a tax-deductible expense and are

therefore paid out after taxes

Cost of Retained Earnings. Plato Companyâ€™s common stock is selling for $50. Last yearâ€™s dividend

10.5

was $4.8 per share. Compute the cost of retained earnings (or internal equity) if both earnings

and dividends are expected to grow at ( a ) zero percent and ( b )a constant rate of 9 percent.

SOLUTION

D,= Do(l + g ) = $4.8(1 + 0) = $4.8

(4

DI Do(1 + g ) = $4.8(1 + 0.09) = $5.232

=

k , = -D,g = - $5*232+ 9% = 10.5 + 9% = 19.5%

+

PO $50

Cost of Retained Earnings (or Internal Equity). Epsilon Companyâ€™s last annual dividend was $4

10.6

per share, and both earnings and dividends are expected to grow at a constant rate of 8 percent.

The stock now sells for $50per share. The companyâ€™s beta coefficient is 1.5. The return of a market

portfolio is 12 percent, and the risk-free rate is 8 percent. The companyâ€™s A-rated bonds are

yielding 12 percent. Calculate the cost of retained earnings (or internal equity) using: ( a ) the

Gordonâ€™s growth model; (6) the bond plus method; and ( c ) the capital asset pricing model.

SOLUTION

For the Gordonâ€™s growth model:

(a)

D = Do(1 + g ) = $4.00(1 + 0.08) = $4.32

1

k s = - 1 g = - $4.32

D+ + 8% = 8.64% + 8% = 16.64%

$50

P

O

k, = bond yield + risk premium = 12% + -4% = 16%

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