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:

Let’s analyze the problem carefully and solve it step-by-step.

Problem Summary:

Machine A:
Initial cost: N3,000
Lifespan: 8 years
Resale value after 8 years: N600
Machine B:
Initial cost: N4,000
Lifespan: 8 years
Resale value after 8 years: N1,900

Goal:

Find the annual discount rate (compounded annually) at which the present values of both machines are equal, meaning it is equally economical to choose either machine.

Step 1: Understand the Concept

The present value (PV) of a machine cost is its initial cost minus the discounted value of its resale (salvage) value after 8 years. The formula for present value of a future amount is:

$$
\text{PV of resale value} = \frac{\text{Resale value}}{(1 + r)^8}
$$

where $r$ is the annual discount rate.

So, the net present cost (NPC) of a machine is:

$$
\text{NPC} = \text{Initial cost} – \text{PV of resale value}
$$

We want to find $r$ such that:

$$
\text{NPC of Machine A} = \text{NPC of Machine B}
$$

That is,

$$
3000 – \frac{600}{(1+r)^8} = 4000 – \frac{1900}{(1+r)^8}
$$

Step 2: Set up the equation

$$
3000 – \frac{600}{(1+r)^8} = 4000 – \frac{1900}{(1+r)^8}
$$

Rearranging terms:

$$
3000 – 4000 = -\frac{1900}{(1+r)^8} + \frac{600}{(1+r)^8}
$$

$$
-1000 = -\frac{1900 – 600}{(1+r)^8}
$$

$$
-1000 = -\frac{1300}{(1+r)^8}
$$

Multiply both sides by -1:

$$
1000 = \frac{1300}{(1+r)^8}
$$

Multiply both sides by $(1+r)^8$:

$$
1000 (1+r)^8 = 1300
$$

Divide both sides by 1000:

$$
(1+r)^8 = \frac{1300}{1000} = 1.3
$$

Step 3: Solve for $r$

Take the 8th root of both sides:

$$
1 + r = (1.3)^{1/8}
$$

Using a calculator,

$$
(1.3)^{1/8} \approx 1.0331
$$

Therefore,

$$
r = 1.0331 – 1 = 0.0331 = 3.31\%
$$

Final Answer:

The annual discount rate at which it is equally economical to choose either machine is approximately 3.31% per year (compounded annually).

Explanation (300+ words):

This problem is a classic example of comparing two investment alternatives by analyzing their net present costs considering time value of money. The key is recognizing that the value of money changes over time due to the discount rate, which reflects the opportunity cost of capital or the minimum acceptable return.

Each machine involves an initial cash outflow (purchase cost) and a cash inflow (resale value) after 8 years. To fairly compare these two machines, we discount the future resale value to its present value equivalent using the annual discount rate. The net present cost (NPC) is the initial purchase cost minus the present value of the resale value.

By setting the NPC of Machine A equal to that of Machine B, we find the discount rate that equalizes the cost-effectiveness of both machines. This discount rate represents the point at which you are indifferent between choosing the cheaper machine with a lower resale value and the more expensive machine with a higher resale value.

The algebraic manipulation leads us to an equation where the eighth power of $(1 + r)$ equals 1.3, representing the growth factor corresponding to the difference in net costs over 8 years. Taking the eighth root brings us to the annual compounded rate.

The result, approximately 3.31%, means:

If the discount rate is less than 3.31%, the cheaper machine with the lower resale value (Machine A) is more economical.
If the discount rate is greater than 3.31%, the more expensive machine with the higher resale value (Machine B) becomes more economical because future money is discounted more heavily, making the higher salvage value less impactful in present terms.

In practical business or engineering decisions, knowing this rate helps decide which asset to purchase depending on expected cost of capital or required return on investment.

If you want, I can also help you with alternative methods or verify with other financial concepts!