Question at position 2 You have just been hired as a senior financial manager of the XYZ Inc. by the eccentric CEO, Elon Tusk. Elon is considering the following potential projects (all cash flows happen at the end of the year). Assume the cost of capital for XYZ Inc. is 14%. [table]
Year | Project A | Project B | Project C
0 | -200,000 | -480,000 | -680,000 1 | 150,000 | 280,000 | -100,000 2 | 80,000 | 280,000 | 350,000 3 | 90,000 | 280,000 | 350,000 4 | 100,000 | 280,000 | 750,000 5 | 110,000 | 280,000 | 650,000
[/table] What is the IRR for Project B?48.7%50.87%61.5%33.4%

Question

Question at position 2 You have just been hired as a senior financial manager of the XYZ Inc. by the eccentric CEO, Elon Tusk. Elon is considering the following potential projects (all cash flows happen at the end of the year). Assume the cost of capital for XYZ Inc. is 14%. [table]
Year | Project A | Project B | Project C
0 | -200,000 | -480,000 | -680,000 1 | 150,000 | 280,000 | -100,000 2 | 80,000 | 280,000 | 350,000 3 | 90,000 | 280,000 | 350,000 4 | 100,000 | 280,000 | 750,000 5 | 110,000 | 280,000 | 650,000
[/table] What is the IRR for Project B?48.7%50.87%61.5%33.4%

BlackTom AI · Accepted Answer

Context: We’re evaluating Project B to determine its internal rate of return (IRR). The cash flows for Project B are: Year 0 = -480,000; Years 1–5 = +280,000 each year. The IRR is the discount rate r that makes the net present value (NPV) equal to zero:
NPV(r) = -480,000 + 280,000/(1+r) + 280,000/(1+r)^2 + 280,000/(1+r)^3 + 280,000/(1+r)^4 + 280,000/(1+r)^5.
This is the standard equation for an annuity of 280,000 for five years, offset by the initial outlay of 480,000.

Option A: 48.7%
Compute the present value of the 280,000 annuity at r = 48.7% = 0.487. The annuity factor is S = (1 - (1+0.487)^-5) / 0.487. First, (1+0.487) = 1.487. 1.487^5 ≈ 7.28, so (1.487)^-5 ≈ 0.137. Then S ≈ (1 - 0.137) / 0.487 ≈ 0.863 / 0.487 ≈ 1.773. PV of inflows ≈ 280,000 * 1.773 ≈ 496,000. NPV ≈ -480,000 + 496,000 ≈ +16,000, which is positive. Since NPV is positive at 48.7%, the IRR would be higher than 48.7% to push NPV toward zero. This suggests 48.7% is slightly low but not the exact IRR.

Option B: 50.87%
Using r = 50.87% = 0.5087, the factor becomes S = (1 - (1.5087)^-5) / 0.5087. 1.5087^5 ≈ 7.85, so (1.5087)^-5 ≈ 0.1274. Thus S ≈ (1 - 0.1274) / 0.5087 ≈ 0.8726 / 0.5087 ≈ 1.714. PV of inflows ≈ 280,000 * 1.714 ≈ 480,000. NPV ≈ -480,000 + 480,000 ≈ 0. This exactly balances the initial outlay, so IRR is 50.87%.

Option C: 61.5%
At r = 61.5% = 0.615, the annuity factor S would be smaller because higher discount rates reduce the present value of future payments. Numerically, (1+0.615)^-5 ≈ (1.615)^-5 ≈ a very small number, leading to S ≈ (1 - small) / 0.615 ≈ a value significantly below 1.5. PV of inflows would be well under 420,000–450,000 range, making NPV negative after subtracting 480,000. Therefore NPV < 0 at 61.5%, meaning IRR is less than 61.5%, so this option cannot be the IRR.

Option D: 33.4%
With r = 33.4% = 0.334, the annuity factor S increases. (1.334)^5 ≈ 4.17, so (1.334)^-5 ≈ 0.240. Then S ≈ (1 - 0.240) / 0.334 ≈ 0.760 / 0.334 ≈ 2.277. PV of inflows ≈ 280,000 * 2.277 ≈ 637,560. NPV ≈ -480,000 + 637,560 ≈ +157,560, which is strongly positive. This indicates the IRR is higher than 33.4%, so this rate is too low to be the IRR.

Takeaway: The IRR is the rate where NPV equals zero. When you test the options, 48.7% yields a small positive NPV, 50.87% yields NPV ≈ 0, 61.5% yields negative NPV, and 33.4% yields a large positive NPV. Thus, the rate that makes NPV zero aligns with the option that balances the inflows and the initial outlay, which is the intermediate value around 50.87%.

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