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 (r) at which both machines are equally economical, we need to compare the present value (PV) of their net costs over 8 years, considering both purchase price and resale value.
Step 1: Define the present value of net cost
For both machines, we calculate the present value of net cost as:
$$
\text{Net Cost} = \text{Purchase Price} – \text{Present Value of Resale Value}
$$
So, for each machine:
- Machine A:
- Cost = N3,000
- Resale Value = N600 after 8 years
- Machine B:
- Cost = N4,000
- Resale Value = N1,900 after 8 years
We equate the net present costs of both machines:
$$
3,000 – \frac{600}{(1 + r)^8} = 4,000 – \frac{1,900}{(1 + r)^8}
$$
Step 2: Solve for the discount rate $r$
Subtract both sides:
$$
3,000 – \frac{600}{(1 + r)^8} = 4,000 – \frac{1,900}{(1 + r)^8}
$$
Bring like terms together:
$$
4,000 – 3,000 = \frac{1,900 – 600}{(1 + r)^8}
$$
$$
1,000 = \frac{1,300}{(1 + r)^8}
$$
Multiply both sides:
$$
(1 + r)^8 = \frac{1,300}{1,000} = 1.3
$$
Now solve for $r$:
$$
1 + r = (1.3)^{1/8}
$$
$$
1 + r = 1.0335
$$
$$
r = 0.0335 \text{ or } 3.35\%
$$
✅ Final Answer: 3.35%
Explanation (300+ words):
This problem involves comparing the net present cost of two capital investments (machines), each with different initial costs and resale values at the end of 8 years. The goal is to determine the discount rate at which these two options become financially equivalent.
In economics and finance, the present value (PV) concept is used to determine how much a future amount of money is worth today, taking into account the time value of money. The further in the future a cash flow is, the less valuable it is today. This principle is crucial when evaluating long-term investments like machinery.
For both machines, we assume:
- No recurring operational costs.
- No tax implications.
- Only the initial purchase and final resale value matter.
Machine A is cheaper to buy (N3,000) but has a lower resale value (N600). Machine B is more expensive (N4,000) but returns more at the end of its life (N1,900). Over 8 years, the real value of these resale amounts depends on the discount rate applied.
We used the present value formula:
$$
\text{PV} = \frac{\text{Future Value}}{(1 + r)^n}
$$
Where:
- PV is the present value of the resale amount.
- r is the annual discount rate.
- n is the number of years (8 years in this case).
By equating the net present cost of both machines and solving for $r$, we found the breakeven discount rate where the total economic cost is the same.
At a 3.35% annual discount rate, it doesn’t matter which machine you choose—they both result in the same economic cost when time value of money is considered.