Chapter 5, Problem 11AP

(a)

To determine

Interpretation of the discount factor value of 0.95.

Explanation of Solution

The discount factor implies the rate that the individual expects in the future period of time. This happens due to the fact that the present value of money will be mostly higher than the future value of money. As a result, the discount factor measures the present value of the future cash flows. With the value of 0.95, it means that the individual values the future sum of money only 95 percent of today’s value.

Economics Concept Introduction

Discount factor:  The discount factor is the weighting term which is used to multiply the future income and losses for determining the net present value of the good or service.

(b)

To determine

Interest rate corresponding to the discount factor.

Explanation of Solution

The discount factor implies the rate that the individual expects in the future period of time. The discount factor is given to be 0.95. The interest rate corresponding to the discount rate can be calculated as follows:

$\begin{array}{c}\frac{1}{1+\text{r}}=\delta \\ 1+\text{r}=\frac{1}{0.95}\\ \text{r}=1.0526-1\\ =0.0526\end{array}$

Thus, the interest rate ‘r’ corresponding to the discount rate of 0.95 will be approximately 5.3 percent.

(c)

To determine

NPV of becoming an ophthalmologist and an accountant.

Explanation of Solution

The discount factor implies the rate that the individual expects in the future period of time. The discount factor is given to be 0.95. The NPV of the future flow of income can be calculated by using the formula as follows:

$\text{NPV}=\text{A}{\left(\delta \right)}^{\text{n}}$ Where n is the number of years and $\delta$ is the discount rate. Thus, the NPV of becoming an ophthalmologist and becoming an accountant can be calculated as follows:

$\begin{array}{c}{\text{NPV}}_{\text{Ophthalmologist}}=-5+1\left(0.95\right)+10{\left(0.95\right)}^2+12{\left(0.95\right)}^3\\ =-5+0.95+9.025+10.2885\\ =-5+20.2635\\ =15.26\end{array}$

The net present value of becoming an accountant can also be calculated similarly as follows:

$\begin{array}{c}{\text{NPV}}_{\text{Accountant}}=2+3\left(0.95\right)+4{\left(0.95\right)}^2+5{\left(0.95\right)}^3\\ =2+2.85+3.61+4.28\\ =12.74\end{array}$

Thus, the NPV of becoming an ophthalmologist is 15.26 whereas of becoming an accountant is only 12.74. This means that becoming an Ophthalmologist is more beneficial than becoming an accountant.

(d)

To determine

IRR of becoming an Ophthalmologist and accountant.

Explanation of Solution

The individual will always prefer to choose the option which has the higher back loaded returns when the internal rate of return is higher than the interest rate in the economy. The discount rate in the economy is calculated to be 0.95 and the interest rate corresponding to the discount rate is calculated to be 5.3 percent.

Thus, the individual would choose to prefer an Ophthalmologist only when the IRR of becoming an Ophthalmologist is higher than the current interest rate of 5.3 percent. when the IRR is below the value, then the decision would reverse.

(e)

To determine

NPV and corresponding interest rate of becoming an ophthalmologist and an accountant.

Explanation of Solution

The discount factor implies the rate that the individual expects in the future period of time. The discount factor is given to be 0.6. The interest rate corresponding to the discount rate can be calculated as follows:

$\begin{array}{c}\frac{1}{1+\text{r}}=\delta \\ 1+\text{r}=\frac{1}{0.6}\\ \text{r}=1.67-1\\ =0.67\end{array}$

The NPV of becoming an ophthalmologist and becoming an accountant can be calculated as follows:

$\begin{array}{c}{\text{NPV}}_{\text{Ophthalmologist}}=-5+1\left(0.6\right)+10{\left(0.6\right)}^2+12{\left(0.6\right)}^3\\ =-5+0.6+3.6+2.59\\ =-5+6.79\\ =1.79\end{array}$

The net present value of becoming an accountant can also be calculated similarly as follows:

$\begin{array}{c}{\text{NPV}}_{\text{Accountant}}=2+3\left(0.6\right)+4{\left(0.6\right)}^2+5{\left(0.6\right)}^3\\ =2+1.8+1.44+1.08\\ =6.28\end{array}$

Thus, the NPV of becoming an ophthalmologist is 1.79 whereas of becoming an accountant is only 6.28. This means that becoming an Ophthalmologist is less beneficial than becoming an accountant. So, the individual would prefer to become an accountant.

(e)

To determine

IRR that equates NPV of ophthalmologist equal to that of Accountant.

Explanation of Solution

The IRR that equates the NPV of becoming an accountant and becoming an ophthalmologist can be calculated as follows:

$\begin{array}{c}{\text{NPV}}_{\text{O}}={\text{NPV}}_{\text{A}}\\ -5+{\left(\delta \right)}^1+10{\left(\delta \right)}^2+12{\left(\delta \right)}^3=2+3{\left(\delta \right)}^1+4{\left(\delta \right)}^2+5{\left(\delta \right)}^3\\ -5-2+{\left(\delta \right)}^1-3{\left(\delta \right)}^1+10{\left(\delta \right)}^2-4{\left(\delta \right)}^2+12{\left(\delta \right)}^3-5{\left(\delta \right)}^3=0\\ -7-2{\left(\delta \right)}^1+6{\left(\delta \right)}^2+7{\left(\delta \right)}^3=0\\ \delta =0.853\end{array}$

When the value of discount factor is 0.853, the value of interest rate can be calculated as follows;

$\begin{array}{c}\frac{1}{1+\text{r}}=\delta \\ 1+\text{r}=\frac{1}{0.853}\\ \text{r}=1.17-1\\ =0.17\end{array}$

The interest rate that equates the NPV of becoming an accountant to that of becoming an ophthalmologist is 17.2 percent.