13. A project has the following cash flows. What is the internal rate of return?

Question

13. A project has the following cash flows. What is the internal rate of return?
Year	   0     cash flow -$443,600
            1               $224,800
            2               $224,800
            3               $ 67,200

(Please calculate the solution and show your work)

Bot · Answer

The internal rate of return (IRR) is the discount rate that makes the net present value (NPV) of a project equal to zero. To find the IRR, we'll need to find the discount rate "r" that equals the following NPV equation:

NPV = (Cash flow in year 0) + (Cash flow in year 1) / (1 + r)^1 + (Cash flow in year 2) / (1 + r)^2 + (Cash flow in year 3) / (1 + r)^3 = 0

Plugging in the numbers, we get:

0 = -443,600 + 224,800 / (1 + r)^1 + 224,800 / (1 + r)^2 + 67,200 / (1 + r)^3

IRR can't be directly solved algebraically, so we'll need to use trial and error or an IRR calculator. Here, I will demonstrate the trial and error method:

Starting with an estimated rate (r) of 10% (0.1):
NPV = -443,600 + 224,800 / (1 + 0.1)^1 + 224,800 / (1 + 0.1)^2 + 67,200 / (1 + 0.1)^3
NPV = -443,600 + 224,800 / 1.1 + 224,800 / 1.21 + 67,200 / 1.331
NPV ≈ -443,600 + 204,363.64 + 185,785.12 + 50,494.75
NPV ≈ -3,057

The NPV is not yet close enough to zero, so we try a slightly larger discount rate:

Trying r = 12% (0.12):
NPV = -443,600 + 224,800 / (1 + 0.12)^1 + 224,800 / (1 + 0.12)^2 + 67,200 / (1 + 0.12)^3
NPV = -443,600 + 224,800 / 1.12 + 224,800 / 1.2544 + 67,200 / 1.404928
NPV ≈ -443,600 + 200,714.29 + 179,215.17 + 47,827.49
NPV ≈ -15,843

Still not close enough, so we try r = 14%:

Trying r = 14% (0.14):
NPV = -443,600 + 224,800 / (1 + 0.14)^1 + 224,800 / (1 + 0.14)^2 + 67,200 / (1 + 0.14)^3
NPV = -443,600 + 224,800 / 1.14 + 224,800 / 1.2996 + 67,200 / 1.481544
NPV ≈ -443,600 + 197,193.86 + 173,045.41 + 45,316.33
NPV ≈ 28,955

Now, we overshot a bit, but we're getting closer. Let's try r = 13.5%:

Trying r = 13.5% (0.135):
NPV = -443,600 + 224,800 / (1 + 0.135)^1 + 224,800 / (1 + 0.135)^2 + 67,200 / (1 + 0.135)^3
NPV = -443,600 + 224,800 / 1.135 + 224,800 / 1.287225 + 67,200 / 1.459886375
NPV ≈ -443,600 + 198,122.14 + 174,537.20 + 44,260.31
NPV ≈ 8,319

Getting even closer, we can estimate that the IRR is around 13.5%. To get a more accurate result, we could continue iterating or use an IRR calculator.