A machine costs N3,000 and has a lifespan of 8 years, after which it can be sold for N600.

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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 at which it would be equally economical to choose either machine, we need to calculate the net present value (NPV) for each machine and set them equal to each other. The NPV is the present value of all costs and benefits over the lifespan of the machine, discounted by the annual rate.

Step 1: Net Present Value (NPV) Formula

The formula for NPV is: NPV=∑Ct(1+r)tNPV = \sum \frac{C_t}{(1 + r)^t}

Where:

  • CtC_t is the net cash flow in year tt,
  • rr is the annual discount rate,
  • tt is the year (1 through 8).

For both machines, we have:

  1. Initial cost: The amount spent upfront at t=0t = 0.
  2. Annual costs: Since there are no recurring costs given in the problem, we will assume no costs other than the initial purchase price.
  3. Resale value: The amount obtained from selling the machine at the end of its life, which is the cash flow in year 8.

Step 2: NPV Calculation for the Two Machines

Machine 1:

  • Initial cost: N3,000
  • Resale value: N600 at the end of 8 years

The NPV for Machine 1 is: NPV1=−3000+600(1+r)8NPV_1 = -3000 + \frac{600}{(1 + r)^8}

Machine 2:

  • Initial cost: N4,000
  • Resale value: N1,900 at the end of 8 years

The NPV for Machine 2 is: NPV2=−4000+1900(1+r)8NPV_2 = -4000 + \frac{1900}{(1 + r)^8}

Step 3: Set the NPVs Equal to Each Other

At the discount rate where both machines are equally economical, the NPVs will be the same. Therefore, we set the NPVs of Machine 1 and Machine 2 equal: −3000+600(1+r)8=−4000+1900(1+r)8-3000 + \frac{600}{(1 + r)^8} = -4000 + \frac{1900}{(1 + r)^8}

Simplifying: 1000=1900−600(1+r)81000 = \frac{1900 – 600}{(1 + r)^8} 1000=1300(1+r)81000 = \frac{1300}{(1 + r)^8}

Now, solve for rr: (1+r)8=13001000=1.3(1 + r)^8 = \frac{1300}{1000} = 1.3 1+r=(1.3)1/81 + r = (1.3)^{1/8} r=(1.3)1/8−1r = (1.3)^{1/8} – 1

Using a calculator to find (1.3)1/8(1.3)^{1/8}, we get: r≈0.0357r \approx 0.0357

So, the annual discount rate is approximately 3.57%.

Conclusion:

At a discount rate of about 3.57%, both machines would be equally economical. This means that if the annual discount rate is 3.57%, the NPV of both machines would be the same, and the decision would depend on factors other than cost, such as preferences or available alternatives.

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