Question text 2.5Marks Equivalent Annual Annuity Melbourne Co. is planning to add new machinery to its current plant. Its cost of capital is 4.5% per annum. The free cash flows of the new machine is: [table]
Year | 0 | 1 | 2 | 3 Machine | -15,000 | 6,500 | 10,000 | 12,000
[/table] Calculate the Equivalent Annual Annuity (EAA) of the machine. [Type only the final answer into the response box below (NOT into the Notes box) and in pure numeric format. Do NOT use %/$ signs, commas or spaces (e.g. only enter 10 if it is 10 days/$10/10%)] Answer 2[input] Notes Report question issue Question 7 Notes

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BlackTom AI · Accepted Answer

The problem asks us to compute the Equivalent Annual Annuity (EAA) for the given project, using a cost of capital (discount rate) of 4.5% and the free cash flows: Year 0 = -15000, Year 1 = 6500, Year 2 = 10000, Year 3 = 12000.

First, evaluate the Net Present Value (NPV) of the project by discounting each cash flow to year 0 and summing:
- PV of Year 1 cash flow: 6500 / (1 + 0.045) ≈ 6500 / 1.045 ≈ 6220.1
- PV of Year 2 cash flow: 10000 / (1.045)^2. Compute (1.045)^2 ≈ 1.092025, so PV ≈ 10000 / 1.092025 ≈ 9153–9154
- PV of Year 3 cash flow: 12000 / (1.045)^3. Compute (1.045)^3 ≈ 1.141166, so PV ≈ 12000 / 1.141166 ≈ 10512–10513

Sum of present values (excluding initial outlay): approximately 6220.1 + 9153–9154 + 10512–10513 ≈ 25885.6. Subtract the initial investment of 15000 to get NPV ≈ 10885.6.

Next, convert the NPV into an equivalent annual amount over the same horizon (n = 3 years) using the EAA formula. The annuity factor A for a given rate r over n years is:
A = r(1 + r)^n / [(1 + r)^n − 1]
Here, r = 0.045 and (1 + r)^n = (1.045)^3 ≈ 1.141166. Thus:
Numerator = r(1 + r)^n ≈ 0.045 × 1.141166 ≈ 0.051351
Denominator = (1 + r)^n − 1 ≈ 1.141166 − 1 = 0.141166
A ≈ 0.051351 / 0.141166 ≈ 0.3640.

Finally, EAA ≈ NPV × A ≈ 10885.6 × 0.3640 ≈ 3960–3970 range. Depending on slight rounding in the PV components, you’d converge toward a value in that vicinity, typically around the mid-3,900s.

Notes on sensitivity and rounding:
- Small differences in PV calculations (due to rounding each discount factor) can shift the final EAA by a few units.
- Using more precise arithmetic or a financial calculator yields a tightly clustered result, generally around 3960 to 3970.

If you compare this to typical numeric responses for this setup, you should expect an EAA in the mid-3,900s range.

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