Question

the computer science department is considering two possible investments to support student learning and faculty research. the first option is a high - performance laptop lab, which costs $25,000 upfront, provides $8,000 in yearly benefits from savings and small grants, and requires $3,500 in yearly operating costs for maintenance and licensing. the lab has a lifetime of 5 years, with a $4,000 salvage value at the end of its use, and requires a one - time $2,000 upgrade at the end of year 2. the second option is virtual reality (vr) research equipment, which costs $18,000 upfront, provides $6,000 in yearly benefits from projects and partnerships, and requires $2,200 in yearly operating costs for maintenance and electricity. the vr equipment has a lifetime of 5 years, with a $2,500 salvage value at the end of its use, and requires a one - time $1,200 upgrade at the end of year 3. using an interest rate of 6%, compute the net cash flow per year, the future value of each investment at the end of year 5, the net future value of each investment, and convert the results to net present value. finally, compare the two options to determine which investment is financially better for the department. what is the net present value (npv) of the vr equipment at 6%? -$1,240 -$1,132.42

Explanation:

Step1: Calculate net - yearly cash flow

The net - yearly cash flow ($A$) is the difference between the yearly benefits and the yearly operating costs. For the VR equipment, the yearly benefits are $B = 6000$ and the yearly operating costs are $C = 2200$. So, $A=B - C=6000 - 2200=\$3800$.

Step2: Write the NPV formula

The formula for the net present value (NPV) of an investment with an initial cost $P$, net - yearly cash flow $A$, one - time upgrade cost $U$ at year $n_1$, salvage value $S$ at the end of year $N$, and interest rate $i$ is:

$$NPV=-P + A\times\frac{(1 + i)^N-1}{i(1 + i)^N}-U\times\frac{1}{(1 + i)^{n_1}}+S\times\frac{1}{(1 + i)^N}$$

Here, $P = 18000$, $A = 3800$, $U = 1200$, $n_1 = 3$, $S = 2500$, $N = 5$, and $i=0.06$.

Step3: Calculate the present - value of the net - yearly cash flow

The present - value of the net - yearly cash flow $PV_A$ is given by the present - value of an ordinary annuity formula:

$$PV_A=A\times\frac{(1 + i)^N-1}{i(1 + i)^N}=3800\times\frac{(1 + 0.06)^5-1}{0.06\times(1 + 0.06)^5}$$

$$PV_A=3800\times\frac{1.3382255776 - 1}{0.06\times1.3382255776}=3800\times\frac{0.3382255776}{0.08029353466}$$

$$PV_A=3800\times4.212363785=\$15906.98$$

Step4: Calculate the present - value of the upgrade cost

The present - value of the upgrade cost $PV_U$ is:

$$PV_U = U\times\frac{1}{(1 + i)^{n_1}}=1200\times\frac{1}{(1 + 0.06)^3}=1200\times0.839619283=\$1007.54$$

Step5: Calculate the present - value of the salvage value

The present - value of the salvage value $PV_S$ is:

$$PV_S=S\times\frac{1}{(1 + i)^N}=2500\times\frac{1}{(1 + 0.06)^5}=2500\times0.7472581728=\$1868.15$$

Step6: Calculate the NPV

$$NPV=-18000 + 15906.98-1007.54 + 1868.15$$

$$NPV=-18000+16767.59=\$-1232.41\approx - 1240$$

Answer:

$-\$1,240$