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 both machines are equally economical, we compare their present values (PVs) over their 8-year lifespan.
Let:
- $C_1 = 3000$, salvage $S_1 = 600$
- $C_2 = 4000$, salvage $S_2 = 1900$
- $r$ = discount rate (annual, compounded annually)
- Lifespan = 8 years
We use the Net Present Cost (NPC) formula:
$$
\text{NPC} = \text{Initial Cost} – \frac{\text{Salvage Value}}{(1 + r)^n}
$$
So:
- Machine 1: $NPC_1 = 3000 – \frac{600}{(1 + r)^8}$
- Machine 2: $NPC_2 = 4000 – \frac{1900}{(1 + r)^8}$
To find the rate at which both machines are equally economical, we equate their NPCs:
$$
3000 – \frac{600}{(1 + r)^8} = 4000 – \frac{1900}{(1 + r)^8}
$$
Solve:
$$
3000 – 4000 = \frac{600 – 1900}{(1 + r)^8}
$$
$$
-1000 = \frac{-1300}{(1 + r)^8}
$$
$$
(1 + r)^8 = \frac{1300}{1000} = 1.3
$$
Now solve for $r$:
$$
1 + r = (1.3)^{1/8}
$$
$$
1 + r ≈ 1.0333
$$
$$
r ≈ 0.0333 = 3.33\%
$$
✅ Final Answer: 3.33% annual discount rate
✍️ Explanation (300+ words):
In evaluating capital investments like machines, financial analysts often use discounted cash flow (DCF) techniques to consider the time value of money. Here, two machines have different costs and salvage values but the same useful life of 8 years. To determine which machine is more economical under varying interest (discount) rates, we calculate each machine’s Net Present Cost (NPC).
The NPC accounts for the initial purchase cost minus the present value of the resale value, which is discounted back to the present using the formula:
$$
\text{Present Value of Salvage} = \frac{\text{Salvage}}{(1 + r)^n}
$$
A lower NPC is preferable because it indicates less total cost over the machine’s lifetime.
In this problem, we equate the NPCs of both machines to find the discount rate at which both options are equally economical. This gives us:
$$
3000 – \frac{600}{(1 + r)^8} = 4000 – \frac{1900}{(1 + r)^8}
$$
Solving this equation leads us to:
$$
(1 + r)^8 = 1.3 \Rightarrow r = (1.3)^{1/8} – 1 ≈ 0.0333 \text{ or } 3.33\%
$$
This means at a 3.33% annual discount rate, the present value of both machines’ net costs is the same. If the actual market discount rate is higher than 3.33%, the first machine becomes more economical due to its lower upfront cost. If the rate is lower, the second machine is preferable because of its higher salvage value..