Chapter 10, Problem 36QP

Replacement Decisions [LO2] Suppose we are thinking about replacing an old computer with a new one. The old one cost us $1,300,000; the new one will cost $1,560,000. The new machine will be depreciated straight-line to zero over its five-year life. It will probably be worth about $300,000 after five years.

The old computer is being depreciated at a rate of $260,000 per year. It will be completely written off in three years. If we don’t replace it now, we will have to replace it in two years. We can sell it now for $420,000; in two years, it will probably be worth $120,000. The new machine will save us $290,000 per year in operating costs. The tax rate is 38 percent, and the discount rate is 12 percent.

a. Suppose we recognize that if we don’t replace the computer now, we will be replacing it in two years. Should we replace now or should we wait? Hint: What we effectively have here is a decision either to “invest” in the old computer (by not selling it) or to invest in the new one. Notice that the two investments have unequal lives.

b. Suppose we consider only whether we should replace the old computer now without worrying about what’s going to happen in two years. What are the relevant cash flows? Should we replace it or not? Hint: Consider the net change in the firm’s aftertax cash flows if we do the replacement.

Summary Introduction

To find: Whether the replacement of the computer must take place now or later and whether the investment has to be made in the old computer or in the new one.

Introduction:

A decision that concerns the replacement of an asset with a newer version of the same asset is the replacement decision. One of the most significant classifications of the capital budgeting is the replacement decision.

Explanation of Solution

Given information:

Person X is planning to replace the old computer with the new one. The cost of the old computer is $1,300,000 and the cost of the new computer is $1,560,000. The depreciation of the new computer is based on the straight line to zero over the life of 5 years of the computer. The worth of the computer after 5 years will be $300,000.

The old computer is depreciated at a rate of $260,000 for a year and it will be written off in 3 years. If the replacement does not happen today then it will be after 2 years. The old computer can be sold now for $420,000 and after 2 years it will be sold for $120,000. The new computer will save an operating cost of $290,000. The tax rate is 38% and discount rate is 12%.

Explanation:

As the two computers do not have an equal life, the appropriate method to analyze the decision is the equivalent annual cost.

Formula to calculate the operating cash flow using the depreciation tax shield approach:

$\text{OCF = }\{\begin{array}{c}\left[\text{Operating cost of the new computer}\right]\left(\text{1 –Tax rate}\right)\\ \text{+Tax rate}\times \text{Depreciation}\end{array}\}$

Computation of the operating cash flow:

$\begin{array}{c}\text{OCF = }\{\begin{array}{c}\left[\text{Operating cost of the new computer}\right]\left(\text{1 –Tax rate}\right)\\ \text{+Tax rate}\times \text{Depreciation}\end{array}\}\\ =\left\{\left[\$290,000\right]\left(1-0.38\right)+0.38\times \left(\frac{1,560,000}{5}\right)\right\}\\ =\left\{\left[\$290,000\right]\left(0.62\right)+0.38\times \left(312,000\right)\right\}\\ =\$298,360\end{array}$

Hence, the operating cash flow is $298,360.

Note: The cost are in a positive flow thus there is a cash inflow. The cost is in positive because the new computer generates the cost savings. The only initial cash flow for the new computer is its cost of $1,560,000.

Formula to calculate the after tax salvage value:

$\text{After tax salvage value = Market price}-\left(\text{1 –Tax rate}\right)$

Computation of the after tax salvage value:

$\begin{array}{c}\text{After tax salvage value = Market price}\left(\text{1 –Tax rate}\right)\\ =\$300,000\left(\text{1}–\text{0}\text{.38}\right)\\ =\$186,000\end{array}$

Hence, the after tax salvage value is $186,000.

Formula to calculate the net present value of the project:

$\text{NPV}=-\text{Outflow of cash}+\text{Inflow of cash}$

Computation of the net present value:

$\begin{array}{c}\text{NPV}=-\text{Outflow of cash}+\text{Inflow of cash}\\ =-1,560,000+\$298,360\left({\text{PVIFA}}_{r\text{ year}}\right)+\frac{\$186,000}{{1.12}^5}\\ =-1,560,000+\$298,360\left({\text{PVIFA}}_{\text{12\% 5}}\right)+\frac{\$186,000}{{1.12}^5}\\ =-1,560,000+\$298,360\left(3.6048\right)+\frac{\$186,000}{{1.12}^5}\end{array}$

$=\$-378,937.58$

Formula to calculate the equivalent annual cost:

$\text{EAC}=\frac{\text{NPV}}{\left({\text{PVIFA}}_{r\text{ year}}\right)}$

Computation of the equivalent annual cost:

$\begin{array}{c}\text{EAC}=\frac{\text{NPV}}{\left({\text{PVIFA}}_{r\text{ year}}\right)}\\ =\frac{-378,937.58}{\left({\text{PVIFA}}_{\text{12\% 5year}}\right)}\\ =\frac{-378,937.58}{3.6048}\\ =-\$105,120.97\end{array}$

Hence, the equivalent annual cost is - $105,120.97.

In analyzing the old computer the only operating cash flow is the depreciation tax shield and it is calculated as follows:

$\begin{array}{c}\text{Operating cash flow}=\$260,000\left(0.38\right)\\ =\$98,800\end{array}$

The initial cost of the old computer is not easy to find. It can be assumed that since Person X already owns the old computer there is no initial payment, but it can be sold and Person X can obtain the opportunity cost. It is essential to account this opportunity cost. In order to do this the after tax salvage value of the old computer is calculated at present.

The book value of the old computer is essential to do so. The book value is not stated in the information, but it is stated that the depreciation of the old computer is $260,000 for a years and for the next three years. Thus, the book value is assumed to be the overall depreciation amount over the remaining life of the system or $780,000. The after tax salvage value of the computer is calculated as follows:

Formula to calculate the after-tax salvage value:

$\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]$

Computation of the after-tax salvage value:

$\begin{array}{c}\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]\\ =\$420,000+\left(\$780,000-\$420,000\right)\times 0.38\\ =\$420,000+\$136,800\\ =\$556,800\end{array}$

Hence, the after tax salvage value is $556,800.

This is the old computer’s initial cost at present because Person X forgoes the opportunity to sell it at present. Next, the after tax salvage value of the computer in 2 years since it has been purchased has to be calculated. The calculations are as follows:

$\begin{array}{c}\text{After-taxsalvage value = }\left[\begin{array}{l}\text{Total fixed cost +}\left(\begin{array}{l}\text{Ending book value}-\\ \text{Total fixed cost}\end{array}\right)\\ \times \text{Marginal tax rate}\end{array}\right]\\ =\$120,000+\left(\$260,000-\$120,000\right)\times 0.38\\ =\$173,200\end{array}$

Hence, the after tax salvage value is $173,200.

Computation of the net present value of the old computer:

$\begin{array}{c}\text{NPV}=-\text{Outflow of cash}+\text{Inflow of cash}\\ =-556,800+\frac{\$98,800}{1.12}+\frac{\left(\$98,800+\$173,200\right)}{{\left(1.12\right)}^2}\\ =-556,800+\frac{\$98,800}{1.12}+\frac{\left(\$272,000\right)}{1.2544}\\ =-\$251,748.98\end{array}$

Hence, the net present value is - $251,748.98.

Computation of the equivalent annual cost:

$\begin{array}{c}\text{EAC}=\frac{\text{NPV}}{\left({\text{PVIFA}}_{r\text{ year}}\right)}\\ =\frac{-251,748.98}{\left({\text{PVIFA}}_{\text{12\% 2year}}\right)}\\ =-\$148,959.40\end{array}$

Hence, the equivalent annual cost is - $148,959.40.

Even if Person X plans to replace the two systems in 2 years, no matter what is his decision today, Person X should replace it at present because the equivalent annual cost is more positive.

Summary Introduction

To find: The relevant cash flows and whether the system must be replaced or not.

Introduction:

A decision that concerns the replacement of an asset with a newer version of the same asset is the replacement decision. One of the most significant classifications of the capital budgeting is the replacement decision.

Explanation of Solution

Given information:

Person X considers only to replace the old computer. The net change in the company’s after tax cash flow is considered if replacement is considered.

Explanation:

If Person X is concerned with whether to replace or not to replace the machine at present, and not considering the two years, then the net present value is only the appropriate analysis. To compute the net present value of the decision on the system at present, then it is essential to find the difference in the cash flow of the old and the new system. From the previous calculation the cash flow of the computer are as follows:

Time	New computer	Old computer	Difference
0	–$1,560,000	–$556,800	–$1,003,200
1	298,360	98,800	199,560
2	298,360	272,000	26,360
3	298,360	0	298,360
4	298,360	0	298,360
5	484,360	0	484,360

As Person X is concerned about the marginal cash flows, the cash flow decision to replace the old system with the new system are the differential cash flows.

Computation of the net present value for the replacement decision:

$\begin{array}{c}\text{NPV = Cash outflow +Cash inflow}\\ \text{=}-\text{Total cash outflow +}\left(\begin{array}{l}\frac{\text{Cash inflow for year 1}}{{\left(1+\text{r}\right)}^1}\text{+}\\ \frac{\text{Cash inflow for year 2}}{{\left(1+\text{r}\right)}^2}\text{+}\\ \frac{\text{Cash inflow for year 3}}{{\left(1+\text{r}\right)}^3}\text{+}\\ \frac{\text{Cash inflow for year 4}}{{\left(1+\text{r}\right)}^4}\text{+}\\ \frac{\text{Cash inflow for year 5}}{{\left(1+\text{r}\right)}^5}\end{array}\right)\\ =-\$1,003,200+\left(\begin{array}{l}\frac{\$199,560}{{\left(1.12\right)}^1}+\frac{\$26,360}{{\left(1.12\right)}^2}+\\ \frac{\$298,360}{{\left(1.12\right)}^3}+\frac{\$298,360}{{\left(1.12\right)}^4}+\frac{\$484,360}{{\left(1.12\right)}^5}\end{array}\right)\\ =\left[-\$1,003,200+\left(\begin{array}{l}\$178,178.57+\$21,014.30+\\ \$212,366.75+\$189,613.17+\$274,838.87\end{array}\right)\right]\\ =-\$127,188.60\end{array}$

Hence, the net present value is - $127,188.60.

If Person X is not concerned with what will happen in 2 years then the replacement of the old computer should not take place.