Tutorial: Intermediate Internal Rate of Return (IRR) - Calculation and Explanation 3

Previous example of 1 and 2 can be find at IRR Tutorial 1 and example 3 can be found at Tutorial 2.

Example 4:
Question: An investment plan with annual amount of 3000 for 15 years, one cash  return every two years as stated below (Year 1 of 1089, Year 3 of 1089 etc) until Year 13, and then annual return of 1742 until Year 29. Lastly, in Year 30, you will be give a lump sum of return of 42,952. What is the IRR of the below investment?

Year 0: -3000
Year 1: -3000 + 1089
Year 2: -3000
Year 3: -3000 + 1089
Year 4: -3000
Year 5: -3000 + 1089
Year 6: -3000
Year 7: -3000 + 1089
Year 8: -3000
Year 9: -3000 + 1089
Year 10: -3000
Year 11: -3000 + 1089
Year 12: -3000
Year 13: -3000 + 1089
Year 14: -3000 + 1742
Year 15: + 1742
Year 16: + 1742
Year 17: + 1742
Year 18: + 1742
.
.
.
Year 29: +1742
Year 30: +42952



Solution:
Step 1: Calculate the net amount of money.
Year 0: -3000
Year 1: -3000 + 1089 = -1911
Year 2: -3000
Year 3: -3000 + 1089 = -1911
Year 4: -3000
Year 5: -3000 + 1089 = -1911
Year 6: -3000
Year 7: -3000 + 1089 = -1911
Year 8: -3000
Year 9: -3000 + 1089 = -1911
Year 10: -3000
Year 11: -3000 + 1089 = -1911
Year 12: -3000
Year 13: -3000 + 1089 = -1911
Year 14: -3000 + 1742 = -1258
Year 15: + 1742
Year 16: + 1742
Year 17: + 1742
Year 18: + 1742
.
.
.
Year 29: +1742
Year 30: +42952
Note: Positive sign means return (Money flow-in to your pocket); Negative sign means investing (Money flow-out from your pocket)

Step 2: Set a time for calculation and write down the equation. I will use Present Worth (at Year 0) in this case.

-3000 -1911*(P|F,IRR,1) - 3000*(P|F.IRR,2) -1911*(P|F,IRR,3) - 3000*(P|F.IRR,4) -1911*(P|F,IRR,5) - 3000*(P|F.IRR,6) -1911*(P|F,IRR,7) - 3000*(P|F.IRR,8) -1911*(P|F,IRR,9) - 3000*(P|F.IRR,10) -1911*(P|F,IRR,11) - 3000*(P|F.IRR,12) -1911*(P|F,IRR,13) - 1258*(P|F.IRR,14) + 1742*(P|F,IRR,14)*(P|A,IRR,15) + 42952*(P|F.IRR,30) = 0

Note: (P|A,IRR,15) calculates the Present Worth at Year 14 of total Annual Worth for 15 years (from Year 15 to year 29). Then, we use the Present Worth at Year 14 as Future Worth to calculate the Present Worth at Year 0.

Step 3: Tried-and-Error – We need to try several times in order to estimate its IRR using interpolation.

When IRR = 5%
(P|F,IRR,1) = 0.952
(P|F,IRR,2) = 0.907
(P|F,IRR,3) = 0.864
(P|F,IRR,4) = 0.823
(P|F,IRR,5) = 0.784
(P|F,IRR,6) = 0.746
(P|F,IRR,7) = 0.711
(P|F,IRR,8) = 0.677
(P|F,IRR,9) = 0.645
(P|F,IRR,10) = 0.614
(P|F,IRR,11) = 0.585
(P|F,IRR,12) = 0.557
(P|F,IRR,13) = 0.530
(P|F,IRR,14) = 0.506
(P|A,IRR,15) = 10.380
(P|A,IRR,30) = 0.231

Then, the answer is
-3000 -1819.272 - 2721 – 1651.104 – 2469 -1498.224 - 2238 -1358.721 - 2031-1232.595 - 1842 -1117.935 - 1671 -1012.83- 636.548 + 9149.4717 + 9921.921= - 7227.845

Note: The answer is negative. This means that the IRR is large. So, we need to try smaller IRR as follows.

When IRR = 2%
(P|F,IRR,1) = 0.980
(P|F,IRR,2) = 0.961
(P|F,IRR,3) = 0.942
(P|F,IRR,4) = 0.924
(P|F,IRR,5) = 0.906
(P|F,IRR,6) = 0.888
(P|F,IRR,7) = 0.871
(P|F,IRR,8) = 0.853
(P|F,IRR,9) = 0.837
(P|F,IRR,10) = 0.820
(P|F,IRR,11) = 0.804
(P|F,IRR,12) = 0.788
(P|F,IRR,13) = 0.773
(P|F,IRR,14) = 0.758
(P|A,IRR,15) = 12.849
(P|F,IRR,30) = 0.552

-3000 – 1872.78 – 2883 – 1800.162 - 2772 - 1731.366 - 2664 - 1664.481 - 2559 - 1599.507 – 2460 – 1536.444 – 2364 – 1477.203 – 953.564 + 16966.2821 + 23709.504 = 9338.2791

Note: The answer is positive. This means that the IRR is small. So, we need to try larger IRR.

After two times tried-and-errors process, the results show that IRR is in the range of 2% to 5%. In other to get more accurate result, it is recommended to obtain the result of IRR of 3% and 4% before doing interpolation to estimate the IRR. Since it is quite straighforward and just repeat the procedure above, I will leave it to you. You are welcome to share your answer via comment box below. That's all for today. More fascinating articles and sharing will be updated from time to time in Xaivier Blog. So, you are welcome to subscribe our feed, look at our sitemap or simply visit our Homepage.

Written by: Xaivier Chia

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