Chapter 5, Problem 4QP

Calculating Discounted Payback An investment project costs $17,500 and has annual cash f1ows of $4,300 for six years. What is the discounted payback period if the discount rate is 0 percent? What if the discount rate is 10 percent? If it is 17 percent?

Summary Introduction

To determine: Discounted payback period.

Discounted Payback Period:

Discounted payback period is the time period in which the company earns back their investment by discounting the future cash flows.. It is use to determine whether to take this project or not. In this, cash inflow is adjusted according to time value of money.

Explanation of Solution

Compute the discounted payback period.

For discount rate of 0%:

Given,

Initial cash outflow is $17,500.

First four year of cash inflow is $17,200.

Fifth year of cash inflow is $4,300.

Formula to compute discounted payback period, where r is 0.

$\text{Discounted payback period}=4+\frac{\text{Initial cash outflow}-\text{First four year of cash inflow}}{\text{Fifth year of cash inflow}}$

Substitute $17,500 for initial cash outflow, $17,200 for first four year of cash inflow and $4,300 for fifth year of cash inflow.

$\begin{array}{c}\text{Discounted payback period}=4+\frac{\$17,500-\$17,200}{\$4,300}\\ =4.07\end{array}$

The discounted payback period is 4.07.

For discount rate of 10%:

Given,

Initial cash outflow is $17,500.

First five year of cash inflow is $16,300.37.

Sixth year of cash inflow is $2,427.23.

Formula to compute discounted payback period,

$\text{Discounted payback period}=5+\frac{\text{Initial cash outflow}-\text{First five year of cash inflow}}{\text{Fifth year of cash inflow}}$

Substitute $17,500 for initial cash outflow, $16,300.37 for first five year of cash inflow and $2,427.23 for sixth year of cash inflow.

$\begin{array}{c}\text{Discounted payback period}=5+\frac{\$17,500-\$16,300.37}{\$2,427.23}\\ =5.494\end{array}$

The discounted payback period is 5.494 years.

Working notes:

Calculation for present value for first cash inflow,

$\begin{array}{c}\text{Present value for first cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.10\right)}^1}\\ =\$3,909.09\end{array}$

Calculation for present value for second cash inflow,

$\begin{array}{c}\text{Present value for second cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.10\right)}^2}\\ =\$3,553.71\end{array}$

Calculation for present value for third cash inflow,

$\begin{array}{c}\text{Present value for third cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.10\right)}^3}\\ =\$3,230.65\end{array}$

Calculation for present value for fourth cash inflow,

$\begin{array}{c}\text{Present value for fourth cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.10\right)}^4}\\ =\$2,936.96\end{array}$

Calculation for present value for fifth cash inflow,

$\begin{array}{c}\text{Present value for fifth cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.10\right)}^5}\\ =\$2,669.96\end{array}$

Calculation for present value for sixth cash inflow,

$\begin{array}{c}\text{Present value for sixth cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.10\right)}^6}\\ =\$2,427.23\end{array}$

For discount rate of 17%:

The total present value of all cash inflow is $15,433.46 which is lower than initial investment. Hence, one cannot find out payback period because this project is not able to payback all the amount that it has used.

Working notes:

Calculation for present value for first cash inflow,

$\begin{array}{c}\text{Present value for first cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.17\right)}^1}\\ =\$3,675.21\end{array}$

Calculation for present value for second cash inflow,

$\begin{array}{c}\text{Present value for second cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.17\right)}^2}\\ =\$3,141.20\end{array}$

Calculation for present value for third cash inflow,

$\begin{array}{c}\text{Present value for third cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.17\right)}^3}\\ =\$2,684.79\end{array}$

Calculation for present value for fourth cash inflow,

$\begin{array}{c}\text{Present value for fourth cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.17\right)}^4}\\ =\$2,294.69\end{array}$

Calculation for present value for fifth cash inflow,

$\begin{array}{c}\text{Present value for fifth cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.17\right)}^5}\\ =\$1,961.27\end{array}$

Calculation for present value for sixth cash inflow,

$\begin{array}{c}\text{Present value for sixth cash inflow}=\frac{\text{Cash inflow}}{{(1+r)}^i}\\ =\frac{\$4,300}{{\left(1+0.17\right)}^6}\\ =\$1,676.30\end{array}$

Calculation for total of present value,

$\begin{array}{c}\text{Total}\text{of}\text{present}\text{value}=\left(\begin{array}{l}\text{Present value for first cash inflow}\\ +\text{Present value for second cash inflow}\\ +\text{Present value for third cash inflow}\\ +\text{Present value for fourth cash inflow}\\ +\text{Present value for fifth cash inflow}\\ +\text{Present value for sixth cash inflow}\end{array}\right)\\ =\left(\begin{array}{l}\$3,675.21+\$3,141.20+\$2,684.79\\ +\$2,294.69+\$1,961.27+\$1,676.30\end{array}\right)\\ =\$15,433.46\end{array}$

Conclusion

Hence, discounted payback period for discount rate of 0% and 10% is 4.07, 5.494 respectively and there are no payback period of 17% discount rate.