Chapter 10, Problem 40P

(a):

To determine

Calculate the rate of return by considering the MARR equal to cost of equity capital.

Explanation of Solution

Type 1: Debt (D) is 20%. Equity (E) is 80%. Cost of equity capital (CI) is 7.5%. Cost of debt capital (DI) is 10%.

Type 2: Debt (D) is 50%. Equity (E) is 50%. Cost of equity capital (CI) is 7.5%. Cost of debt capital (DI) is 10%.

Type 3: Debt (D) is 60%. Equity (E) is 40%. Cost of equity capital (CI) is 7.5%. Cost of debt capital (DI) is 10%.

Investment (IN) is $600,000. Net cash flow (NC) is $90,000. Time (n) is 7.

Rate of return (i) can be calculated using the following formula:

$\text{NI}\times \text{D}=\text{NC}\left(\frac{{\left(1+\text{i}\right)}^{\text{n}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}\right)$        (1)

Substitute the respective values in Equation (1) to calculate the rate of return for plan 1.

$\begin{array}{c}\text{6,000,000}\times \text{0}\text{.8}=\text{90,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{7}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{7}}}\right)\\ \text{480,000}=\text{90,000}\left(\frac{{\left(1+\text{i}\right)}^{\text{7}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{7}}}\right)\\ \left(\frac{{\left(1+\text{i}\right)}^{\text{7}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{7}}}\right)=\frac{\text{480,000}}{\text{90,000}}\\ \left(\frac{{\left(1+\text{i}\right)}^{\text{7}}-1}{\text{i}{\left(1+\text{i}\right)}^{\text{7}}}\right)=\text{5}\text{.3333}\end{array}$

Substitute i as 7% by trial and error method in the above calculation.

$\begin{array}{c}\left(\frac{{\left(1+\text{0}\text{.07}\right)}^{\text{7}}-1}{\text{0}\text{.07}{\left(1+\text{0}\text{.07}\right)}^{\text{7}}}\right)=\text{5}\text{.3333}\\ \left(\frac{1.6058-1}{\text{0}\text{.07}\left(1.6058\right)}\right)=\text{5}\text{.3333}\\ \left(\frac{0.6058}{0.1124}\right)=\text{5}\text{.3333}\\ 5.3897>\text{5}\text{.3333}\end{array}$

The calculated present value factor to the annual value is greater than the actual value of 5.3333. Thus, increase the i to 7.3%.

$\begin{array}{c}\left(\frac{{\left(1+\text{0}\text{.073}\right)}^{\text{7}}-1}{\text{0}\text{.073}{\left(1+\text{0}\text{.073}\right)}^{\text{7}}}\right)=\text{5}\text{.3333}\\ \left(\frac{1.6376-1}{\text{0}\text{.073}\left(1.6376\right)}\right)=\text{5}\text{.3333}\\ \left(\frac{0.6376}{0.1195}\right)=\text{5}\text{.3333}\\ 5.3356\approx \text{5}\text{.3333}\end{array}$

The calculated present value factor to the annual value is nearly equal to the actual value of 5.3333. Thus, cost of capital is verified as 7.3%.

The MARR is equal to the cost of debt capital. Thus, it is 7.5%. The project is accepted if the rate of return is greater than the MARR. Since the rate of return is less than the MARR, type 1 should not be selected.

Table 1 shows the rate of return that is obtained using Equation (1) and the selection decision.

Table -1

Type	Rate of return	MARR	Decision
1	7.3%	7.5%	Reject
2	22.93%	7.5%	Select
3	32.18%	7.5%	Select

(b):

To determine

Calculate the weighted average cost of capital.

Explanation of Solution

Weighted average cost of capital (WAC) can be calculated using the following formula:

$\text{WAC}=\text{D}\left(\text{DI}\right)+\left(\text{E}\right)\left(\text{CI}\right)$        (2)

Substitute the respective values in Equation (2) to calculate the weighted average cost of capital for type 1.

$\begin{array}{c}\text{WAC}=\text{D}\left(\text{DI}\right)+\left(\text{E}\right)\left(\text{CI}\right)\\ =0.2\left(0.1\right)+0.8\left(0.075\right)\\ =0.02+0.06\\ =0.08\end{array}$

Weighted average cost of capital is 8%.

Table 2 shows the weighted average cost of capital, which is equal to MARR that is obtained using Equation (2) and the selection decision.

Table -2

Type	Rate of return	MARR	Decision
1	7.3%	8%	Reject
2	22.93%	8.75%	Select
3	32.18%	9%	Select

(c):

To determine

Calculate half of the weighted average cost of capital and cost of equity capital.

Explanation of Solution

Half of the weighted average cost of capital and cost of equity capital (MARR) can be calculated using the following formula:

${\text{MARR}}_{\text{A}}=\frac{{\text{WAC}}_{\text{A}}+{\text{i}}_{\text{A}}}{2}$        (3)

Substitute the respective values in Equation (3) to calculate the MARR for type 1.

$\begin{array}{c}\text{MARR}=\frac{0.08+0.075}{2}\\ =\frac{0.155}{2}\\ =0.0775\end{array}$

MARR is 7.75%.

Table 3 shows the MARR that is obtained using Equation (3) and the selection decision.

Table -2

Type	Rate of return	MARR	Decision
1	7.3%	7.75%	Reject
2	22.93%	8.125%	Select
3	32.18%	8.25%	Select

From Tables 1 to 3, it is known that type 1 gets rejected and type 2 and type 3 are accepted.

(d):

To determine

Calculate half of the weighted average cost of capital and cost of equity capital.

Explanation of Solution

These three analyses revealed that highly leveraged investment option types 2 and 3 are accepted. Thus, the answer for this question is yes.