A machine cost $ 4,000. It lasts 2 years and has no scrap value (that is, it has no value at the end of those two years of use). In each year, it produces $ 2400 in income. Should the firm invest in the machine if the interest rate is 10%? Should the firm invest in the machine if the interest rate is 20%? Why? What if the machine’s scrap value was $350?
To determine whether the firm should invest in the machine, we need to calculate the net present value (NPV) of the investment. NPV is the difference between the present value of the expected cash inflows and the present value of the initial investment.
Let's calculate the NPV at an interest rate of 10% first:
1. Calculate the present value of the income for each year using the formula:
PV = CF / (1 + r)^n
where PV is the present value, CF is the cash flow, r is the interest rate, and n is the number of years.
In this case, the income is $2400, the interest rate is 10%, and the number of years is 2.
PV(year 1) = $2400 / (1 + 0.1)^1 = $2400 / 1.1 = $2181.82
PV(year 2) = $2400 / (1 + 0.1)^2 = $2400 / 1.21 ≈ $1983.47
2. Now, let's calculate the initial investment's present value. Since the machine costs $4000 and has no scrap value, the present value would be $4000.
3. Calculate the NPV by subtracting the present value of the initial investment from the sum of the present values of the income:
NPV = (PV(income year 1) + PV(income year 2)) - PV(initial investment)
NPV = ($2181.82 + $1983.47) - $4000
NPV ≈ $4165.29 - $4000 ≈ $165.29
Based on the NPV calculation at a 10% interest rate, the NPV is positive, indicating that the investment should be undertaken.
Now, let's repeat the steps for an interest rate of 20%:
1. Calculate the present value of the income for each year:
PV(year 1) = $2400 / (1 + 0.2)^1 = $2400 / 1.2 = $2000
PV(year 2) = $2400 / (1 + 0.2)^2 = $2400 / 1.44 ≈ $1666.67
2. The present value of the initial investment remains $4000.
3. Calculate the NPV:
NPV = ($2000 + $1666.67) - $4000
NPV ≈ $3666.67 - $4000 ≈ -$333.33
At a 20% interest rate, the NPV is negative, indicating that the investment should not be undertaken.
Lastly, let's consider the scenario where the machine has a scrap value of $350. We will assume the same interest rate of 10%:
1. Calculate the present value of the income for each year:
PV(year 1) = $2400 / (1 + 0.1)^1 = $2400 / 1.1 = $2181.82
PV(year 2) = ($2400 + $350 scrap value) / (1 + 0.1)^2 = $2750 / 1.21 ≈ $2272.73
2. The present value of the initial investment remains $4000.
3. Calculate the NPV:
NPV = ($2181.82 + $2272.73) - $4000
NPV ≈ $4454.55 - $4000 ≈ $454.55
With a scrap value of $350, the NPV is still positive at a 10% interest rate, indicating that the investment should be undertaken.
In summary:
- At a 10% interest rate, the firm should invest in the machine because the NPV is positive.
- At a 20% interest rate, the firm should not invest in the machine because the NPV is negative.
- If the machine's scrap value was $350, the firm should still invest in the machine at a 10% interest rate, as the NPV remains positive.