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 determine the annual discount rate at which both machines are equally economical, we compare their Net Present Cost (NPC) over the 8-year lifespan.
Let:
- Machine A:
Initial cost = N3,000
Resale value = N600 - Machine B:
Initial cost = N4,000
Resale value = N1,900
Let the annual discount rate be r.
Step 1: Net Present Cost Formula
The Net Present Cost (NPC) of a machine is: NPC=Initial Cost−Resale Value(1+r)n\text{NPC} = \text{Initial Cost} – \frac{\text{Resale Value}}{(1 + r)^n}
where:
- rr is the annual discount rate,
- n=8n = 8 is the lifespan in years.
Set the NPCs equal to each other for the machines: 3000−600(1+r)8=4000−1900(1+r)83000 – \frac{600}{(1 + r)^8} = 4000 – \frac{1900}{(1 + r)^8}
Step 2: Solve the Equation
Move all terms to one side: 3000−4000=600−1900(1+r)83000 – 4000 = \frac{600 – 1900}{(1 + r)^8} −1000=−1300(1+r)8-1000 = \frac{-1300}{(1 + r)^8}
Multiply both sides by (1+r)8(1 + r)^8: −1000(1+r)8=−1300-1000(1 + r)^8 = -1300 (1+r)8=13001000=1.3(1 + r)^8 = \frac{1300}{1000} = 1.3
Now take the 8th root of both sides: 1+r=(1.3)1/81 + r = (1.3)^{1/8} 1+r≈1.033431 + r \approx 1.03343 r≈0.03343=3.343%r \approx 0.03343 = 3.343\%
✅ Final Answer:
Annual discount rate ≈ 3.34%
🔍 Explanation (300+ words):
When businesses or individuals choose between two capital investments with different initial costs and resale values, they often use Net Present Cost (NPC) to compare options. The NPC helps us account for the time value of money — the concept that a naira today is worth more than a naira in the future due to potential earning capacity.
In this scenario, both machines serve the same purpose for the same lifespan (8 years). The goal is to find a discount rate (interest rate) at which the present value of the total cost of owning either machine is the same — this is when it’s equally economical to choose either option.
Machine A has a lower initial cost (N3,000), but also a lower resale value (N600). Machine B is more expensive up front (N4,000), but has a higher resale value (N1,900). To properly compare these, we calculate each machine’s net present cost by subtracting the discounted resale value from the initial cost.
Setting both NPCs equal and solving gives us a discount rate where the buyer is indifferent between machines. Mathematically, we solve: 3000−600(1+r)8=4000−1900(1+r)83000 – \frac{600}{(1 + r)^8} = 4000 – \frac{1900}{(1 + r)^8}
This simplifies to: (1+r)8=1.3(1 + r)^8 = 1.3
Solving gives r≈3.34%r \approx 3.34\%. This is the break-even discount rate: below this rate, the cheaper Machine A is better; above it, the higher-resale Machine B becomes more cost-effective.