Chapter 10, Problem 26P

(a):

To determine

Calculate cost of debt.

Explanation of Solution

Borrowing (B) is $800,000. Effective rate (ER) is 8%. Time period (n) is 8 years. Effective tax rate (ET) is 40%. Cost of debt (i) can be calculated as follows:

$\begin{array}{c}\text{B}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{n}}}{{\left(\text{1}+\text{i}\right)}^{\text{n}}-1}\right)=\left(\text{B}\left(\frac{\text{i}{\left(\text{1}+\text{ER}\right)}^{\text{n}}}{{\left(\text{1}+\text{ER}\right)}^{\text{n}}-1}\right)-\frac{\text{B}}{\text{n}}\right)-\left(\text{B}\left(\frac{\text{i}{\left(\text{1}+\text{ER}\right)}^{\text{n}}}{{\left(\text{1}+\text{ER}\right)}^{\text{n}}-1}\right)-\frac{\text{B}}{\text{n}}\right)\text{ET+}\frac{\text{B}}{\text{n}}\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=\left(\begin{array}{l}\left(\text{800,000}\left(\frac{\text{0}\text{.08}{\left(\text{1}+\text{0}\text{.08}\right)}^{\text{8}}}{{\left(\text{1}+\text{0}\text{.08}\right)}^{\text{8}}-1}\right)-\frac{\text{800,000}}{\text{8}}\right)\\ -\left(\text{800,000}\left(\frac{\text{0}\text{.08}{\left(\text{1}+\text{0}\text{.08}\right)}^{\text{8}}}{{\left(\text{1}+\text{0}\text{.08}\right)}^{\text{8}}-1}\right)-\frac{\text{800,000}}{\text{8}}\right)\text{0}\text{.4+}\frac{\text{800,000}}{\text{8}}\end{array}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=\left(\begin{array}{l}\left(\text{800,000}\left(\frac{\text{0}\text{.08}\left(1.85093\right)}{1.85093-1}\right)-\text{100,000}\right)\\ -\left(\text{800,000}\left(\frac{\text{0}\text{.08}\left(1.85093\right)}{1.85093-1}\right)-\text{100,000}\right)\text{0}\text{.4}+\text{100,000}\end{array}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=\left(\begin{array}{l}\left(\text{800,000}\left(\frac{0.14807}{0.85093}\right)-\text{100,000}\right)\\ -\left(\text{800,000}\left(\frac{0.14807}{0.85093}\right)-\text{100,000}\right)\text{0}\text{.4}+\text{100,000}\end{array}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=\left(\begin{array}{l}\left(\text{800,000}\left(0.17401\right)-\text{100,000}\right)\\ -\left(\text{800,000}\left(0.17401\right)-\text{100,000}\right)\text{0}\text{.4}+\text{100,000}\end{array}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=\left(\begin{array}{l}\left(139,208-\text{100,000}\right)\\ -\left(139,208-\text{100,000}\right)\text{0}\text{.4}+\text{100,000}\end{array}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=\left(39,208-\left(39,208\right)\text{0}\text{.4}+\text{100,000}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{8}}}{{\left(\text{1}+\text{i}\right)}^{\text{8}}-1}\right)=123,524.8\end{array}$

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

$\begin{array}{c}\text{800,000}\left(\frac{\text{0}\text{.05}{\left(\text{1}+\text{.05}\right)}^{\text{8}}}{{\left(\text{1}+\text{0}\text{.05}\right)}^{\text{8}}-1}\right)=123,524.8\\ \text{800,000}\left(\frac{\text{0}\text{.05}\left(1.477455\right)}{1.477455-1}\right)=123,524.8\\ \text{800,000}\left(\frac{0.073873}{0.477455}\right)=123,524.8\\ \text{800,000}\left(0.154722\right)=123,524.8\\ 123,777.6>123,524.8\end{array}$

Since the annual value of the borrowing is greater than the calculated value, decrease the cost of capital from 6% to 4.95%.

$\begin{array}{c}\text{800,000}\left(\frac{\text{0}\text{.0495}{\left(\text{1}+\text{.0495}\right)}^{\text{8}}}{{\left(\text{1}+\text{0}\text{.0495}\right)}^{\text{8}}-1}\right)=123,524.8\\ \text{800,000}\left(\frac{\text{0}\text{.0495}\left(1.471836\right)}{1.471836-1}\right)=123,524.8\\ \text{800,000}\left(\frac{0.072856}{0.471836}\right)=123,524.8\\ \text{800,000}\left(0.15441\right)=123,524.8\\ 123,528\approx 123,524.8\end{array}$

The calculated value is equal to the annual value of the borrowing. Thus, it is confirmed that the cost of debt capital is 4.95%.

Borrowing (B) is $800,000. Time period (n) is 10 years. Effective tax rate (ET) is 40%. Bond rate (BR) is 6%. Cost of debt (i) can be calculated as follows:

$\begin{array}{c}\text{B}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{n}}}{{\left(\text{1}+\text{i}\right)}^{\text{n}}-1}\right)=\left(\text{B}\times \text{BR}\right)-\left(\text{B}\times \text{BR}\times \text{ET}\right)+\text{B}\left(\frac{\text{i}}{{\left(\text{1}+\text{i}\right)}^{\text{n}}-1}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{10}}}{{\left(\text{1}+\text{i}\right)}^{\text{10}}-1}\right)=\left(\text{800,000}\times \text{0}\text{.06}\right)-\left(\text{800,000}\times \text{0}\text{.06}\times \text{0}\text{.4}\right)+\text{800,000}\left(\frac{\text{i}}{{\left(\text{1}+\text{i}\right)}^{\text{10}}-1}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{10}}}{{\left(\text{1}+\text{i}\right)}^{\text{10}}-1}\right)=48,000-19,200+\text{800,000}\left(\frac{\text{i}}{{\left(\text{1}+\text{i}\right)}^{\text{10}}-1}\right)\\ \text{800,000}\left(\frac{\text{i}{\left(\text{1}+\text{i}\right)}^{\text{10}}}{{\left(\text{1}+\text{i}\right)}^{\text{10}}-1}\right)=28,800+\text{800,000}\left(\frac{\text{i}}{{\left(\text{1}+\text{i}\right)}^{\text{10}}-1}\right)\end{array}$

Substitute i as 4% by trial and error method in the above calculation.

$\begin{array}{l}\text{800,000}\left(\frac{\text{0}\text{.04}{\left(\text{1}+0.\text{04}\right)}^{\text{10}}}{{\left(\text{1}+\text{0}\text{.04}\right)}^{\text{10}}-1}\right)=28,800+\text{800,000}\left(\frac{0.04}{{\left(\text{1}+\text{0}\text{.04}\right)}^{\text{10}}-1}\right)\\ \text{800,000}\left(\frac{\text{0}\text{.04}\left(1.480244\right)}{1.480244-1}\right)=28,800+\text{800,000}\left(\frac{0.04}{1.480244-1}\right)\\ \text{800,000}\left(\frac{0.05921}{0.480244}\right)=28,800+\text{800,000}\left(\frac{0.04}{0.480244}\right)\\ \text{800,000}\left(0.123291\right)=28,800+\text{800,000}\left(0.083291\right)\\ 98,632.8=28,800+66,632.8\\ 98,632.8>95,432.8\end{array}$

Since the annual value of the borrowing is greater than the calculated value, decrease the cost of capital from 4% to 3.6%.

$\begin{array}{c}\text{800,000}\left(\frac{\text{0}\text{.036}{\left(\text{1}+0.\text{036}\right)}^{\text{10}}}{{\left(\text{1}+\text{0}\text{.036}\right)}^{\text{10}}-1}\right)=28,800+\text{800,000}\left(\frac{0.036}{{\left(\text{1}+\text{0}\text{.036}\right)}^{\text{10}}-1}\right)\\ \text{800,000}\left(\frac{\text{0}\text{.036}\left(1.424287\right)}{1.424287-1}\right)=28,800+\text{800,000}\left(\frac{0.036}{1.424287-1}\right)\\ \text{800,000}\left(\frac{0.051274}{0.424287}\right)=28,800+\text{800,000}\left(\frac{0.036}{0.424287}\right)\\ \text{800,000}\left(0.120847\right)=28,800+\text{800,000}\left(0.084848\right)\\ 96,677.6=28,800+67,878.4\\ 96,677.6\approx 96,678.4\end{array}$

The calculated value is equal to the annual value of the borrowing. Thus, it is confirmed that the cost of debt capital is 3.6%. Thus, bond financing is cheaper than the borrowing from bank.

(b):

To determine

Comparing the cost of capital.

Explanation of Solution

Before tax the cost of capital from bond financing is 6%, whereas the cost of capital from bank financing is 8%. Thus, bond financing is cheaper before tax analysis. Thus, the result before tax analysis is the same as the after tax analysis.