Question

Jiminy’s Cricket Farm issued a 20-year, 7 percent semiannual bond four years ago. The bond currently sells for 98 percent of its face value. The company’s tax rate is 25 percent.

For the same firm, suppose the book value of the debt issue is $75 million. In addition, the company has a second debt issue on the market, a zero coupon bond with eight years left to maturity; the book value of this issue is $30 million, and the bonds sell for 80 percent of par. What is your best estimate of the aftertax cost of debt now?

Select one:

A. 1.404%

B. 2.81%

C. 2.11%

D. 4.60%

E. 5.41%

Answer 1

Solution

The answer is D. 4.60%

Calculation and explanations are give below:

The Book value or total Face value of the bond = $75 million

Coupon rate 7% works out to 3.5% semiannually

Coupon payment = Face value x Coupon rate = 75 x 3.5% = $ 2.625 million

Remaining time to maturity = 16 years (since the bond was issued 4 years ago)

Thus, there are 16 x 2 = 32 semi-annual payments outstanding on the bond.

Current total capitalisaton or sale value of the bond is 98% of its Face value i.e. 75 x 98% = $73.50 million

We have to find at what rate coupon payments, C = 2.625 and terminal payment, T = 75 over n = 32 payment periods yields a valuation of 73.50.

For the coupon payments the annuity formula is PVA = C x [(1+r)ⁿ - 1] / [r x (1+r)ⁿ]

We need a rate higher than 3.5% since we need to reduce the PV from 75 to 73.50

Lets assume r = 3.60%

We have PVA = 2.625 x [(1+0.036)³² - 1] / [0.036 x (1+0.036)³²]

= 2.625 x [(1.036)³² - 1] / [0.036 x (1.036)³²]

= 2.625 x [(3.101074 - 1] / [0.036 x 3.101074]

= 2.625 x 2.101074 / 0.111639

= 49.40

For the terminal payment the PV = T / (1+r)ⁿ

PV = 75 / (1+0.036)³²

PV = 75 / 3.101074

= 24.19

Hence total PV of cash flows of the bond is = 49.40 + 24.19 = 73.59 which is near to 73.50

To make it more accurate lets work out using 3.61%

We have PVA = 2.625 x [(1+0.0361)³² - 1] / [0.0361 x (1+0.0361)³²]

= 2.625 x [(1.0361)³² - 1] / [0.0361 x (1.0361)³²]

= 2.625 x [(3.110667 - 1] / [0.0361 x 3.110667]

= 2.625 x 2.110667 / 0.112295

= 49.34

For the terminal payment the PV = T / (1+r)ⁿ

PV = 75 / (1+0.0361)³²

PV = 75 / 3.110667

= 24.11

Hence total PV of cash flows of the bond is = 49.34 + 24.11 = 73.45 which is nearer to 73.50

So semi-annual yield to maturity is 3.61% and annual yield is 3.61 x 2 = 7.22%

After tax-cost of debt is = Debt cost x (1 - Tax rate) = 7.22% x (1 - 0.25) = 5.415%

Now for the zero coupon bond, book value or total face value is $ 30 million and its price is 80% of 30 million = $24 million

Time to maturity or n is 8 years.

We need to find what rate or yield reduces 30 million to 24 million over 8 years

Formula for future value is

FV = PV x (1+r)ⁿ

30 = 24 x (1+r)⁸

1.25 = (1+r)⁸

If we take rate of 3% we have (1+0.03)⁸ = 1.03⁸ = 1.26677 which is higher than 1.25

If we take rate of 2% we have (1+0.02)⁸ = 1.02⁸ = 1.171659 which is lower than 1.25

Interpolating we have the yield rate

= 2% + [ (1.25 - 1.171659) / (1.26677 - 1.171659)]

= 2% + 0.078341 / 0.095111

= 2% + 0.82368

= 2.82368%

Since zero coupon bonds do not have coupon payments, there is no tax implication as there is no coupon payment and no after-tax cost.

Book value of first debt issue is 75 million and second debt issue is 30 million. Total issue = 75 +30 = 105 million

Hence the weights of each of these issues is 75/105 and 30/105

Multiplying the above weights by the after tax cost of the first debt issue and the cost of the second issue respectively and adding them up since the total cost of debt would be the issue value weighted average of their individual costs, we have:

(75/105)(5.415%) + (30/105)(2.82368%)

= 3.867857% + 0.806766%

= 4.674623%