Chapter 10, Problem 25P

(a):

To determine

Calculate cost of debt capital before tax.

Explanation of Solution

Interest expense (IP) is $1,200,000 per year. Initial received payment (RP) is $19,000,000. Face value of the bond (B) is $20,000,000. Time period (n) is 15. Effective tax rate (ET) is 29%. Cost of capital (i) before tax can be calculated as follows:

$\begin{array}{c}\text{RP}=\text{IP}\left(\frac{{\left(\text{1}+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{n}}}\right)+\frac{\text{B}}{{\left(\text{1}+\text{i}\right)}^{\text{n}}}\\ 19,000,000=1,200,000\left(\frac{{\left(\text{1}+\text{i}\right)}^{\text{15}}-1}{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+\text{i}\right)}^{\text{15}}}\end{array}$

Substitute i as 6% by trial and error method in the above calculation.

$\begin{array}{l}19,000,000=1,200,000\left(\frac{{\left(\text{1}+\text{0}\text{.06}\right)}^{\text{15}}-1}{\text{0}\text{.06}{\left(\text{1}+\text{0}\text{.06}\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+\text{0}\text{.06}\right)}^{\text{15}}}\\ 19,000,000=1,200,000\left(\frac{2.396558-1}{\text{0}\text{.06}\left(2.396558\right)}\right)+\frac{\text{20,000,000}}{2.396558}\\ 19,000,000=1,200,000\left(\frac{1.396558}{0.143793}\right)+8,345,301.89\\ 19,000,000=1,200,000\left(9.712281\right)+8,345,301.89\\ 19,000,000=11,654,737.2+8,345,301.89\\ 19,000,000=20,000,039.09\end{array}$

Since the calculated value is greater than the initial received payment, increase the cost of capital from 6% to 6.53%.

$\begin{array}{l}19,000,000=1,200,000\left(\frac{{\left(\text{1}+\text{0}\text{.0653}\right)}^{\text{15}}-1}{\text{0}\text{.0653}{\left(\text{1}+\text{0}\text{.0653}\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+\text{0}\text{.0653}\right)}^{\text{15}}}\\ 19,000,000=1,200,000\left(\frac{2.582729-1}{\text{0}\text{.0653}\left(2.582729\right)}\right)+\frac{\text{20,000,000}}{2.582729}\\ 19,000,000=1,200,000\left(9.384585\right)+7,743,747.02\\ 19,000,000=11,261,502+7,743,747.02\\ 19,000,000\approx 19,005,249.02\end{array}$

The calculated value is equal to the initial received payment. Thus, it is confirmed that the cost of debt capital is 6.53%.

(b):

To determine

Calculate cost of debt capital after tax.

Explanation of Solution

Effective tax rate (ET) is 29%. Cost of capital (i) before tax can be calculated as follows:

$\begin{array}{c}\text{RP}=\left(\text{IP}-\left(\text{IP}\times \text{ET}\right)\right)\left(\frac{{\left(\text{1}+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{n}}}\right)+\frac{\text{B}}{{\left(\text{1}+\text{i}\right)}^{\text{n}}}\\ 19,000,000=\left(1,200,000-\left(1,200,000\times 0.29\right)\right)\left(\frac{{\left(\text{1}+\text{i}\right)}^{\text{15}}-1}{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+\text{i}\right)}^{\text{15}}}\\ 19,000,000=\left(1,200,000-348,000\right)\left(\frac{{\left(\text{1}+\text{i}\right)}^{\text{15}}-1}{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+\text{i}\right)}^{\text{15}}}\\ 19,000,000=852,000\left(\frac{{\left(\text{1}+\text{i}\right)}^{\text{15}}-1}{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+\text{i}\right)}^{\text{15}}}\end{array}$

Substitute i as 5% by trial and error method in the above calculation.

$\begin{array}{l}19,000,000=852,000\left(\frac{{\left(\text{1}+0.05\right)}^{\text{15}}-1}{0.05{\left(\text{1}+0.05\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+0.05\right)}^{\text{15}}}\\ 19,000,000=852,000\left(\frac{2.078928-1}{0.05\left(2.078928\right)}\right)+\frac{\text{20,000,000}}{2.078928}\\ 19,000,000=852,000\left(\frac{1.078928}{0.103946}\right)+\text{9,620,342}\text{.79}\\ 19,000,000=852,000\left(10.379697\right)+\text{9,620,342}\text{.79}\\ 19,000,000=8,843,502.696+\text{9,620,342}\text{.79}\\ 19,000,000>18,463,845.49\end{array}$

Since the calculated value is less than the initial received payment, decrease the cost of capital from 5% to 4.73%.

$\begin{array}{l}19,000,000=852,000\left(\frac{{\left(\text{1}+0.0473\right)}^{\text{15}}-1}{0.0473{\left(\text{1}+0.0473\right)}^{\text{15}}}\right)+\frac{\text{20,000,000}}{{\left(\text{1}+0.0473\right)}^{\text{15}}}\\ 19,000,000=852,000\left(\frac{2.000168-1}{0.0473\left(2.000168\right)}\right)+\frac{\text{20,000,000}}{2.000168}\\ 19,000,000=852,000\left(\frac{1.000168}{0.094608}\right)+9,999,160.07\\ 19,000,000=852,000\left(10.571706\right)+9,999,160.07\\ 19,000,000=9,007,093.51+9,999,160.07\\ 19,000,000\approx 19,006,253.58\end{array}$

The calculated value is equal to the initial received payment. Thus, it is confirmed that the cost of debt capital is 4.73%.

(c):

To determine

Spreadsheet function.

Explanation of Solution

Spreadsheet function for cost of debt capital before tax is given below.

= RATE(15,-1200000,19000000,-20000000)

Spreadsheet function for cost of debt capital after tax is given below.

= RATE(15,-1200000*(1-0.29),19000000,-20000000)