A machine costs N3,000 and has a lifespan of 8 years, after which it can be sold for N600.
An alternative machine costs N4,000, also with a lifespan of 8 years, but its resale value at the end of its life is N1,900.
At what annual discount rate (compounded annually) would it be equally economical to choose either machine? (Ignore taxes.)
The Correct Answer and Explanation is:
To find the annual discount rate at which both machines are equally economical, we need to compare the present value (PV) of their costs over their 8-year lifespans.
Let’s define:
- C₁ = N3,000, S₁ = N600 (resale value), lifespan = 8 years
- C₂ = N4,000, S₂ = N1,900 (resale value), lifespan = 8 years
- Let r be the annual discount rate (compounded annually)
Step 1: Net Present Cost Formula
The present cost of each machine is:
$$
\text{Net Present Cost} = \text{Initial Cost} – \frac{\text{Salvage Value}}{(1 + r)^n}
$$
Where:
- $r$ = annual discount rate
- $n$ = lifespan = 8 years
Set the net present cost of both machines equal:
$$
3000 – \frac{600}{(1 + r)^8} = 4000 – \frac{1900}{(1 + r)^8}
$$
Step 2: Solve the Equation
Move all terms to one side:
$$
3000 – 4000 = \frac{600 – 1900}{(1 + r)^8}
$$
$$
-1000 = \frac{-1300}{(1 + r)^8}
$$
Multiply both sides by $(1 + r)^8$:
$$
-1000(1 + r)^8 = -1300
$$
Divide both sides by -1000:
$$
(1 + r)^8 = \frac{1300}{1000} = 1.3
$$
Take the 8th root of both sides:
$$
1 + r = (1.3)^{1/8}
$$
Using a calculator:
$$
(1.3)^{1/8} \approx 1.0332
$$
$$
r = 1.0332 – 1 = 0.0332 \text{ or } 3.32\%
$$
✅ Final Answer: 3.32% annual discount rate
🔍 Explanation (300+ words)
To determine which machine is more economical, we use the present value approach, which helps compare money spent or saved at different points in time. Since both machines last 8 years and provide similar services, the main difference lies in their initial cost and salvage (resale) value.
The concept of discounting helps us find how much future money (like resale value) is worth in today’s terms. A higher discount rate makes future money less valuable today.
We calculate each machine’s net present cost, which is the initial cost minus the present value of the resale value. This is because we “recover” some money at the end by selling the machine.
For both machines to be equally economical, their net present costs must be the same. We set up an equation that expresses this equality and solve it to find the discount rate.
The solution involves solving an equation where the only unknown is the discount rate $r$. Using algebra and exponentiation, we find that the value of $r$ that satisfies the condition is approximately 3.32% per year, compounded annually.
This means that if the time value of money (or cost of capital) is exactly 3.32%, both machines are financially equivalent over their lifespan. If the actual discount rate is higher than 3.32%, the cheaper machine (N3,000) is preferable. If the rate is lower, the more expensive machine with the higher salvage value (N4,000) becomes more economical in the long run.