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)
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.
To calculate the internal rate of return (IRR) for the given cash flows, you need to find the discount rate that makes the net present value (NPV) of the cash flows equal to zero.
The formula for NPV is:
NPV = CF0 + CF1/(1+r)^1 + CF2/(1+r)^2 + CF3/(1+r)^3 + ...
Where:
- CF0, CF1, CF2, CF3, ... represent the cash flows at different time periods.
- r is the discount rate.
In this case, the cash flows are as follows:
Year 0: -$443,600
Year 1: $224,800
Year 2: $224,800
Year 3: $67,200
We can substitute these values into the NPV formula:
0 = -443,600 + 224,800/(1+r)^1 + 224,800/(1+r)^2 + 67,200/(1+r)^3
To solve for the IRR, we need to find the discount rate (r) that satisfies this equation. There are various numerical methods to calculate the IRR, such as trial and error, or using a financial calculator or spreadsheet software.
One way to find the IRR using trial and error is to start by assuming a discount rate, and then iterating until the NPV is close to zero.
Let's assume that the discount rate is 10%. Substitute the values into the NPV formula:
0 = -443,600 + 224,800/(1+0.10)^1 + 224,800/(1+0.10)^2 + 67,200/(1+0.10)^3
After evaluating the equation, if NPV is not equal to zero, you can try a different discount rate and repeat the process until you find the rate that results in NPV close to zero.
Another option is to use financial software or a spreadsheet program that has built-in functions to calculate the IRR. Most spreadsheet software, such as Microsoft Excel, Google Sheets, or Apple Numbers, have a function specifically for calculating IRR.
By inputting the cash flows into a spreadsheet and using the IRR function, you can quickly obtain the IRR value.
Please note that the calculated IRR might be an approximation due to the iterative nature of the calculation method.