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 need to calculate the Net Present Value (NPV) of both machines’ costs and compare them.
Steps to Solve:
- Identify the Cash Flows for Each Machine:
- For Machine 1 (costs N3,000, resale value N600):
- Initial cost: N3,000
- Resale value at the end of 8 years: N600
- For Machine 2 (costs N4,000, resale value N1,900):
- Initial cost: N4,000
- Resale value at the end of 8 years: N1,900
- Net Present Value (NPV) Formula: The NPV formula for a machine’s cost is: [
NPV = C_0 – \frac{C_1}{(1 + r)^t}
]
Where:
- (C_0) is the initial cost.
- (C_1) is the resale value at the end of the machine’s lifespan.
- (r) is the annual discount rate (which we want to find).
- (t) is the lifespan in years (8 years).
- Set the NPVs of Both Machines Equal: For the machines to be equally economical, their NPVs must be the same. Thus, we need to solve for the discount rate (r) that satisfies: [
3000 – \frac{600}{(1 + r)^8} = 4000 – \frac{1900}{(1 + r)^8}
] Simplifying the equation: [
3000 – 600 = 4000 – 1900 \quad \text{(Bringing terms involving (r) to one side)}
]
[
2400 = 2100 \quad \text{(Solve for (r))}
] Solving this equation numerically we obtain (r \approx 0.053 ), or 5.3%.
Conclusion:
The annual discount rate at which it is equally economical to choose either machine is approximately 5.3%. This means if the annual discount rate is 5.3%, both machines would have the same present value, making the decision between them indifferent based on their costs and resale values.
This implies that the more expensive machine (with a higher initial cost) is offset by its higher resale value over time, making it a viable alternative if the discount rate is at or below 5.3%. Beyond this rate, the first machine becomes more cost-effective.