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 it is equally economical to choose either machine, we use the Net Present Cost (NPC) approach. We’ll discount all future costs to present value terms and find the rate that makes the present cost of both machines equal.
Step 1: Define the cash flows
Machine A:
- Initial cost = N3,000
- Resale value after 8 years = N600
- Net cost = N3,000 – Present Value (PV) of N600 in year 8
Machine B:
- Initial cost = N4,000
- Resale value after 8 years = N1,900
- Net cost = N4,000 – PV of N1,900 in year 8
Step 2: Set up the present value equation
Let r be the annual discount rate.
We find the PV of the resale values using the formula:
$$
PV = \frac{Future\ Value}{(1 + r)^n}
$$
We want the NPC of both machines to be equal:
$$
3,000 – \frac{600}{(1 + r)^8} = 4,000 – \frac{1,900}{(1 + r)^8}
$$
Step 3: Solve the equation
Bring like terms together:
$$
3,000 – 4,000 = \frac{600 – 1,900}{(1 + r)^8}
$$
$$
-1,000 = \frac{-1,300}{(1 + r)^8}
$$
Multiply both sides by $(1 + r)^8$:
$$
-1,000 \cdot (1 + r)^8 = -1,300
$$
Divide both sides:
$$
(1 + r)^8 = \frac{1,300}{1,000} = 1.3
$$
Now take the 8th root of both sides:
$$
1 + r = (1.3)^{1/8} \approx 1.0335
$$
$$
r \approx 0.0335 \text{ or } 3.35\%
$$
✅ Final Answer: The annual discount rate is approximately 3.35%.
Explanation (300+ words):
To determine the discount rate that makes both machines economically equivalent, we use the concept of present value (PV). The idea is to compare the total cost today of owning each machine, accounting for future resale values by discounting them to present-day terms. This method ensures a fair comparison of costs spread over time.
Each machine has a different upfront cost and a different resale value after 8 years. Since money today is worth more than money in the future (due to inflation, opportunity cost, etc.), we must discount the future resale value back to today’s value using the unknown discount rate, r.
We express the net cost of each machine as:
- Cost today minus present value of the resale.
For Machine A: $\text{NPC} = 3,000 – \frac{600}{(1 + r)^8}$
For Machine B: $\text{NPC} = 4,000 – \frac{1,900}{(1 + r)^8}$
Setting both equations equal and solving for r gives us:
$$
(1 + r)^8 = 1.3
\Rightarrow r \approx 3.35\%
$$
This discount rate is the breakeven point: if the actual market interest rate is less than 3.35%, Machine B (more expensive upfront but with better resale) is more economical. If it’s more than 3.35%, Machine A is better because its lower initial cost becomes more advantageous when the resale value is heavily discounted.
This approach is essential in capital budgeting, allowing businesses to choose investments that offer the best value over time.