Chapter 10, Problem 27P

(a):

To determine

Calculate the face value.

Explanation of Solution

Debt capital (D) is $2,500,000. Required return (RR) is $2,500,000. Selling is at 12^th year. Returns from the bond (r) is 4.2%. Discount rate (i) is 3% quarterly on the face value. Effective tax rate (ET) is 35%.

Face value (F) of the bond that would give $2,500,000 can be calculated as follows:

$\begin{array}{c}\text{F}=\frac{\text{RR}}{1-\text{i}}\\ =\frac{2,500,000}{1-0.03}\\ =\frac{2,500,000}{0.97}\\ =2,577,319.59\end{array}$

The face value to obtain $2,500,000 million return is $2,577,319.59.

(b):

To determine

Cost of debt capital after effective tax rate.

Explanation of Solution

Debt capital (D) is $2,500,000. Time period (n) is 20. Returns from the bond (r) is 4.2%. Discount rate (i) is 3% quarterly on the face value. Effective tax rate (ET) is 35%. Number of quarter in a year (Q) is 4. The cost of debt capital (i) can be calculated as follows:

$\begin{array}{l}\text{D}=\frac{\text{F}\times \text{r}}{\text{Q}}\times \left(1-\text{ET}\right)\left(\frac{{\left(1+\text{i}\right)}^{\text{n}\times \text{Q}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}\times \text{Q}}}\right)-\frac{\text{F}}{{\left(1+\text{i}\right)}^{\text{n}\times \text{Q}}}\\ \text{2,500,000}=\frac{\text{2,577,319}\text{.59}\times \text{0}\text{.042}}{\text{4}}\times \left(1-0.35\right)\left(\frac{{\left(1+\text{i}\right)}^{20\times \text{4}}-1}{\text{i}{\left(1+\text{i}\right)}^{20\times \text{4}}}\right)-\frac{\text{2,577,319}\text{.59}}{{\left(1+\text{i}\right)}^{20\times \text{4}}}\\ \text{2,500,000}=27,061.86\times \left(0.65\right)\left(\frac{{\left(1+\text{i}\right)}^{80}-1}{\text{i}{\left(1+\text{i}\right)}^{80}}\right)-\frac{\text{2,577,319}\text{.59}}{{\left(1+\text{i}\right)}^{80}}\end{array}$

Substitute i as 0.7% by trial and error method in the above calculation.

$\begin{array}{l}\text{2,500,000}=27,061.86\times \left(0.65\right)\left(\frac{{\left(1+\text{0}\text{.007}\right)}^{80}-1}{\text{0}\text{.007}{\left(1+\text{0}\text{.007}\right)}^{80}}\right)+\frac{\text{2,577,319}\text{.59}}{{\left(1+\text{0}\text{.007}\right)}^{80}}\\ \text{2,500,000}=17,590.21\left(\frac{1.74726-1}{\text{0}\text{.007}\left(1.74726\right)}\right)+\frac{\text{2,577,319}\text{.59}}{1.74726}\\ \text{2,500,000}=17,590.21\left(\frac{0.74726}{0.01223}\right)+1,475,063.58\\ \text{2,500,000}=17,590.21\left(61.10057\right)+1,475,063.58\\ \text{2,500,000}=1,074,771.86+1,475,063.58\\ \text{2,500,000}<2,549,835.44\end{array}$

Since the calculated value is greater than the present value of quarterly debt value, increase the i to 0.734%.

$\begin{array}{l}\text{2,500,000}=27,061.86\times \left(0.65\right)\left(\frac{{\left(1+\text{0}\text{.00734}\right)}^{80}-1}{\text{0}\text{.00734}{\left(1+\text{0}\text{.00734}\right)}^{80}}\right)+\frac{\text{2,577,319}\text{.59}}{{\left(1+\text{0}\text{.00734}\right)}^{80}}\\ \text{2,500,000}=17,590.21\left(\frac{1.795091-1}{\text{0}\text{.00734}\left(1.795091\right)}\right)+\frac{\text{2,577,319}\text{.59}}{1.795091}\\ \text{2,500,000}=17,590.21\left(\frac{0.795091}{0.013176}\right)+1,435,759.85\\ \text{2,500,000}=17,590.21\left(60.343883\right)+1,435,759.85\\ \text{2,500,000}\approx 2,497,221.42\end{array}$

The calculated value is nearly equal to the present value of quarterly debt value. Thus, the cost of debt capital is 0.743% per quarter. Thus, the annual cost of debt is 2.936%$\left(0.734\times 4\right)$.

The spreadsheet function to calculate the cost of debt capital is given below:

= RATE(80,-17590,2500000,-2577320)*4

The above function gives the value of 2.93%.