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 compare their Net Present Costs (NPC).
Step 1: Calculate NPC using Present Value Formula
The NPC of a machine is the initial cost minus the present value of the salvage value:
[
\text{NPC} = C – \frac{S}{(1 + r)^n}
]
where:
- ( C ) = Initial cost,
- ( S ) = Salvage value,
- ( n ) = Lifespan (8 years),
- ( r ) = Discount rate.
For the first machine:
[
\text{NPC}_1 = 3,000 – \frac{600}{(1 + r)^8}
]
For the second machine:
[
\text{NPC}_2 = 4,000 – \frac{1,900}{(1 + r)^8}
]
Since both machines should be equally economical, we equate their NPCs:
[
3,000 – \frac{600}{(1 + r)^8} = 4,000 – \frac{1,900}{(1 + r)^8}
]
Step 2: Solve for ( r )
Rearranging,
[
4,000 – 3,000 = \frac{1,900}{(1 + r)^8} – \frac{600}{(1 + r)^8}
]
[
1,000 = \frac{1,300}{(1 + r)^8}
]
[
(1 + r)^8 = \frac{1,300}{1,000} = 1.3
]
Taking the 8th root:
[
1 + r = 1.3^{\frac{1}{8}}
]
[
r = 1.3^{\frac{1}{8}} – 1
]
Using logarithms or a calculator:
[
r \approx 0.033 or 3.3\% \text{ per annum}
]
Conclusion:
At a discount rate of 3.3% per annum, the present costs of both machines are equal, making them equally economical.