Chapter 6, Problem 33QP

Replacement Decisions Suppose we are thinking about replacing an old computer with a new one. The old one cost us $450,000; the new one will cost $580,000. The new machine will be depreciated straight-line to zero over its five-year life. It will probably be worth about $130,000 after five years. The old computer is being depreciated at a rate of $90,000 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 $230,000; in two years it will probably be worth $60,000. The new machine will save us $85,000 per year in operating costs. The tax rate is 38 percent and the discount rate is 14 percent.

1. 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.)
2. 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 determine: Net present value of the old and new computer.

Equivalent Annual Cost:

Equivalent annual cost is that cost which shows the operating and maintaining cost of the assets of whole life.

Net Present Value(NPV):

The net present value is differential amount between the net cash inflow from future investments and net cash outflow in the form of cost that the company has to pay at present as initial cost of the investment.

Explanation of Solution

Given,

Salvage value of the old computer is $230,000.

Salvage value of the computer after two years is $60,000.

Cost of new computer is $580,000.

Estimated life of the computer is 5 years.

Salvage value of the new computer is $130,000.

Operating cost is $85,000.

Discount rate is 14%.

Calculated values,

Book value of the old computer is $270,000.

Annual depreciation on the old computer is $90,000.

Formula to calculate the equivalent annual cost of the old computer,

$\text{Equivalent}\text{annual}\text{cost}=\frac{\text{Net}\text{present}\text{value}}{PVIF{A}_{\text{14\%,}\text{2}\text{year}}}$

Substitute-$133,966 for the net present value and 1.647 for the PVIFA.

$\begin{array}{c}\text{Equivalent}\text{annual}\text{cost}=\frac{-\$133,966}{1.647}\\ =-\$81,339\end{array}$

Formula to calculate the equivalent annual cost of the new computer,

$\text{Equivalent}\text{annual}\text{cost}=\frac{\text{Net}\text{present}\text{value}}{{\text{PVIFA}}_{\text{14\%,}5\text{year}}}$

Substitute, -$205,923 for the net present value and 3.433 for the PVFA,

$\begin{array}{c}\text{Equivalent}\text{annual}\text{cost}=\frac{-\$205,923}{3.433}\\ =-\$59,983\end{array}$

Working notes:

Calculate the operating cash flow of old computer,

$\begin{array}{c}\text{Operating}\text{cash}\text{flow}=\text{Depreciation}\times \text{Tax}\text{rate }\\ =\text{\$90,000}\times \text{0}\text{.38}\\ =\$34,200\end{array}$

Calculate the operating cash flow of new computer,

$\begin{array}{c}\text{Operating}\text{cash}\text{flow}=\left(\text{Annual}\text{savings}\times \left(\text{1}-\text{Tax}\text{rate}\right)\right)+\left(\text{Depreciation}\times \text{Tax}\text{rate}\right)\\ =\left(\$85,000\times \left(1-0.38\right)\right)+\left(\frac{\$580,000}{5\text{years}}\times 0.38\right)\\ =\left(\$85,000\times 0.62\right)+\left(\$116,000\times 0.38\right)\\ =\$96,780\end{array}$

Calculate the present salvage value of new computer,

$\begin{array}{c}\text{Present}\text{salvage}\text{value}=\text{Salvage}\text{value}\times \left(1-\text{Tax}\text{rate}\right)\times \left(\frac{1}{{\left(1+\text{Annual}\text{interest}\text{rate}\right)}^5}\right)\\ =\$130,000\times \left(1-0.38\right)\times \frac{1}{{\left(1+0.14\right)}^5}\\ =\$80,600\times 0.519\\ =\$41,831\end{array}$

Calculate the present salvage value of the old computer,

$\begin{array}{c}\text{Present}\text{salvage}\text{value}=\left[\begin{array}{c}\left(\text{Salvage}\text{value}-\left(\text{Profit}\text{on}\text{tax}\times \text{Tax}\text{rate}\right)\right)\\ \times \frac{1}{{\left(1+\text{Annual}\text{intrest}\text{rate}\right)}^2}\end{array}\right]\\ =\left(\$60,000-\left(\$60,000-\$90,000\times 0.38\right)\right)\times \frac{1}{{\left(1+0.14\right)}^2}\\ =\left(\$60,000-\left(-\$30,000\times 0.38\right)\right)\times \frac{1}{{\left(1+0.14\right)}^2}\\ =\$54,907\end{array}$

Calculate the initial cost of the old computer,

$\begin{array}{c}\text{Initial}\text{cost}=\left(\text{Oportunity}\text{cost}-\left(\text{Profit}\text{on}\text{sale}\times \text{Tax}\text{rate}\right)\right)\\ =\left(\$230,000-\left(\left(\$230,000-\$270,000\right)\times 0.38\right)\right)\\ =\left(\$230,000-\left(-\$15,200\right)\right)\\ =\$245,200\end{array}$

Calculate the net present value of the old computer,

$\begin{array}{c}\text{Net}\text{present}\text{value}=\left[\begin{array}{c}\left(\text{Operating}\text{cash}\text{flow}\times {\text{PVIFA}}_{\text{14\%,2Years}}\right)-\text{Initial}\text{investment}\\ +\text{Salvage}\text{value}\end{array}\right]\\ =\left(\$34,200\times 1.647\right)-\$245,200+\$54,907\\ =\$56,327-\$245,200+\$54,907\\ =-\$133,966\end{array}$

Calculate the net present value of the new computer,

$\begin{array}{c}\text{Net}\text{present}\text{value}=\left[\begin{array}{c}\left(\text{Operating}\text{cash}\text{flow}\times {\text{PVIFA}}_{\text{14\%,2Years}}\right)-\text{Initial}\text{investment}\\ \text{+Salvage}\text{value}\end{array}\right]\\ =\left(\text{\$96,780}\times 3.433\right)-\text{\$580,000}+\text{\$41,831}\\ =\text{\$332,246}-\text{\$580,000}+\text{\$41,831}\\ =-\text{\$205,923}\end{array}$

Conclusion

Hence, equivalent annual cost of old and new computer is -$81,339 and -$59,983.

Summary Introduction

To determine: Incremental net present value.

Net Present Value(NPV):

The net present value is differential amount between the net cash inflow from future investments and net cash outflow in the form of cost that the company has to pay at present as initial cost of the investment.

Explanation of Solution

Formula to calculate the incremental net present value,

Working notes:

Calculate the incremental cash flows,

Conclusion

Hence, net present value is -$72,019.