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 (compounded annually) at which it would be equally economical to choose either machine, we need to compare the present value (PV) of the costs and the resale values of both machines. The objective is to identify the rate at which the net present cost of both machines is the same.
Step 1: Define the variables
- Machine 1:
- Initial cost = N3,000
- Lifespan = 8 years
- Resale value after 8 years = N600
- Machine 2:
- Initial cost = N4,000
- Lifespan = 8 years
- Resale value after 8 years = N1,900
Step 2: Formula for Present Value (PV)
The Present Value (PV) of a cash flow can be calculated using the formula:
[
PV = \frac{F}{(1 + r)^n}
]
Where:
- ( F ) is the future value (cash inflow or outflow),
- ( r ) is the annual discount rate (which we are trying to find),
- ( n ) is the number of years.
Step 3: Present Value of Costs for Machine 1 and Machine 2
The net present cost of each machine is the sum of the initial cost and the present value of the resale value, subtracted from the initial cost.
For Machine 1:
- Initial cost: N3,000
- Resale value at year 8: N600
The Present Value of the resale value for Machine 1 is:
[
PV_{\text{resale1}} = \frac{600}{(1 + r)^8}
]
Thus, the total net present cost of Machine 1 is:
[
\text{Net Cost Machine 1} = 3,000 – \frac{600}{(1 + r)^8}
]
For Machine 2:
- Initial cost: N4,000
- Resale value at year 8: N1,900
The Present Value of the resale value for Machine 2 is:
[
PV_{\text{resale2}} = \frac{1,900}{(1 + r)^8}
]
Thus, the total net present cost of Machine 2 is:
[
\text{Net Cost Machine 2} = 4,000 – \frac{1,900}{(1 + r)^8}
]
Step 4: Set the Net Present Costs Equal to Each Other
To find the rate at which both machines are equally economical, set the net present costs equal:
[
3,000 – \frac{600}{(1 + r)^8} = 4,000 – \frac{1,900}{(1 + r)^8}
]
Step 5: Solve for ( r )
Simplifying the equation:
[
3,000 – 4,000 = -\frac{600}{(1 + r)^8} + \frac{1,900}{(1 + r)^8}
]
[
-1,000 = \frac{1,900 – 600}{(1 + r)^8}
]
[
-1,000 = \frac{1,300}{(1 + r)^8}
]
Now, solving for ( (1 + r)^8 ):
[
(1 + r)^8 = \frac{1,300}{1,000} = 1.3
]
Taking the 8th root of both sides:
[
1 + r = (1.3)^{1/8}
]
Using a calculator to evaluate ( (1.3)^{1/8} ):
[
1 + r \approx 1.0336
]
Thus:
[
r \approx 0.0336 \text{ or } 3.36\%
]
Final Answer:
The annual discount rate at which it would be equally economical to choose either machine is approximately 3.36%.
Explanation:
The concept behind this calculation is comparing the present value of both machines’ costs, considering the initial purchase price and the resale value at the end of their 8-year lifespans. The key idea is that the more valuable a machine’s resale price, the lower its net cost becomes when adjusted for time (through discounting). At a specific annual discount rate, the present values of the total costs for both machines become equal. This rate, which we computed to be around 3.36%, represents the point at which both machines are equally economically attractive when considering the time value of money.