Chapter 24, Problem 13QP

Summary Introduction

To find: The period when the machine must be purchased.

Introduction:

The variations between the present value of the cash outflows and the present value of the cash inflows is the net present value. In capital budgeting, the net present value is utilized to analyze the profitability of a project or an investment.

Answer to Problem 13QP

The company should purchase a machine at 2 years from the present at which the net present value should be high.

Explanation of Solution

Given information:

The company of Person X is planning to make its investment in a new machine. The new machine increases the cash flow by $273,000 per year. Person X has a belief that the technology that is utilized in the machine has a ten year life, it means no matter when the machine is purchased, it will be obsolete in ten years from now.

The current price of the machine is $1,400,000. There will be a decline in the machine to $95,000 for a year until it reaches $925,000. The required rate of return is 14%.

Formula to calculate the net present value:

$\text{Net present value}=\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)$

Note: The net present value is calculated by the above formula and this formula is used if the machine is bought today. PVIFA is the present value interest factor of an annuity.

Computation of the net present value:

$\begin{array}{c}{\text{Net present value}}_0=\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)\\ =-\$1,400,000+\$273,000\left({\text{PVIFA}}_{14\%,10\text{ years}}\right)\\ =-\$1,400,000+\$273,000\left(5.2161\right)\\ =\$23,999.57\end{array}$

Note: The value of the PVIFA at 14% for 10 years is 5.2161.

Hence, the current net present value is $23,999.57.

Note: It is not essential to buy the machine today, but it is essential to buy the machine when there is a higher net present value. It is necessary to compute the net present value every year. In order to make a right decision, the net present value for every year has to be taken at general date.

Formula to calculate the net present value:

$\text{Net present value}=\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)$

Note: The net present value is calculated using the above formula and this formula is used only if the machine is bought today. PVIFA is the present value interest factor of an annuity.

Computation of the net present value for the Year 1:

$\begin{array}{c}{\text{Net present value}}_1=\frac{\left(\text{Cost of the machine}+\text{Present value of the increased cash savings}\right)}{\left(1+r\right)}\\ =\frac{-\$1,305,000+\$273,000\left({\text{PVIFA}}_{14\%,9\text{ years}}\right)}{\left(1+0.14\right)}\\ =\frac{-\$1,305,000+\$273,000\left(4.9464\right)}{1.14}\\ =\$39,795.78\end{array}$

Note: The value of PVIFA at 14% for 9 years is 4.9464.

Hence, the net present value at year 1 is $39,795.78.

Computation of the net present value for the Year 2:

$\begin{array}{c}{\text{Net present value}}_2=\frac{\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)}{{\left(1+r\right)}^2}\\ =\frac{-\$1,210,000+\$273,000\left({\text{PVIFA}}_{14\%,8\text{ years}}\right)}{{\left(1+0.14\right)}^2}\\ =\frac{-\$1,210,000+\$273,000\left(4.6389\right)}{1.2996}\\ =\$43,413.12\end{array}$

Note: The value of the PVIFA at 14% for 8 years is 4.6389.

Hence, the net present value at year 2 is $43,413.12.

Computation of the net present value for the Year 3:

$\begin{array}{c}{\text{Net present value}}_3=\frac{\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)}{{\left(1+r\right)}^3}\\ =\frac{-\$1,115,000+\$273,000\left({\text{PVIFA}}_{14\%,7\text{ years}}\right)}{{\left(1+0.14\right)}^3}\\ =\frac{-\$1,115,000+\$273,000\left(4.2883\right)}{1.481544}\\ =\$37,600\end{array}$

Note: The value of the PVIFA at 14% for 7 years is 4.2883.

Hence, the net present value at year 3 is $37,600.

Computation of the net present value for the Year 4:

$\begin{array}{c}{\text{Net present value}}_4=\frac{\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)}{{\left(1+r\right)}^4}\\ =\frac{-\$1,020,000+\$273,000\left({\text{PVIFA}}_{14\%,6\text{ years}}\right)}{{\left(1+0.14\right)}^4}\\ =\frac{-\$1,020,000+\$273,000\left(3.8887\right)}{1.68896016}\\ =\$24,639.48\end{array}$

Note: The value of the PVIFA at 14% for 6 years is 3.8887.

Hence, the net present value at year 4 is $24,639.48.

Computation of the net present value for the Year 5:

$\begin{array}{c}{\text{Net present value}}_5=\frac{\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)}{{\left(1+r\right)}^5}\\ =\frac{-\$925,000+\$273,000\left({\text{PVIFA}}_{14\%,5\text{ years}}\right)}{{\left(1+0.14\right)}^5}\\ =\frac{-\$925,000+\$273,000\left(3.4331\right)}{1.925414582}\\ =\$6,355.15\end{array}$

Note: The value of the PVIFA at 14% for 5 years is 3.4331.

Hence, the net present value at year 5 is $6,355.15.

Computation of the net present value for the Year 6:

$\begin{array}{c}{\text{Net present value}}_6=\frac{\left(\text{Cost of the machine}+\text{Present value of the increased cash flow}\right)}{{\left(1+r\right)}^6}\\ =\frac{-\$925,000+\$273,000\left({\text{PVIFA}}_{14\%,4\text{ years}}\right)}{{\left(1+0.14\right)}^6}\\ =\frac{-\$925,000+\$273,000\left(2.9137\right)}{2.194972624}\\ =-\$59,024.22\end{array}$

Note: The value of the PVIFA at 14% for 4 years is 2.9137.

Hence, the net present value at year 6 is -$59,024.22.

The company must buy the machine at 2 years from now as the net present value remains the highest.