Tutorial: Basic Internal Rate of Return,IRR, Calculation and Explanation 2

This is the second part of Tutorial: Basic Internal Rate of Return,IRR, Calculation and Explanation. Now, let's see the third example.

Example 3:

Question: Return at the end of period is $455,996.00, Annual saving is $6000.00, Period is 30 years, initial saving is $6000.00 as well.

Solution:
1. First of all, let's see the cash flow below:

Year 0: Initial Saving = PW = $6000.00
Year 1: Annual Saving = AW (Annual Worth) = $6000.00
Year 2: Annual Saving = AW = $6000.00
:
:
:
Year 29: Annual Saving = AW  = $6000.00
Year 30: Maturity = FW = $455,996.00

2. After understand this cash flow, the equation represents this cash flow is:

Saving = Return

PW + AW (P|A, IRR, 29) = FW ( P|F, IRR , 30)

3. In this complicated case, we can only use second method which is mentioned at Tutorial: Basic Internal Rate of Return,IRR, Calculation and Explanation, that is, the trial and error plus linear interpolation approach.

$6000.00 + $6000.00*(P|A, IRR, 29) = $455,996.00*(P|F, IRR, 30)

4. Since the equation of (P|A, IRR, N) = ((1 + IRR)^N - 1) / ( IRR*(1 + IRR)^N), then

(P|A, IRR, 29) = ((1 + IRR)^29 - 1) / ( IRR*(1 + IRR)^29)

5. Divide the equation $6000.00 + $6000.00*(P|A, IRR, 29) = $455,996.00*(P|F, IRR, 30) by  $455,996.00, so

0.013158 + 0.013158*(P|A, IRR, 29) = (P|F, IRR, 30)

6. Rearrange, again.

   0.013158 = (P|F, IRR, 30) - 0.013158*(P|A, IRR, 29)

7. Using previous equations to get the follow value:

(P|F, 4%, 30) = 0.3083
(P|F, 5%, 30) = 0.2314
(P|F, 6%, 30) = 0.1741


(P|A, 4%, 29) = 16.9837
(P|A, 5%, 29) = 15.1411
(P|A, 6%, 29) = 13.5907


(P|F, 4%, 30) - 0.013158*(P|A,  4%, 29) = 0.0848

8. Obviously, the IRR is between 5% and 6%. Now, we can using linear interpolation as we did last time.
(P|F, 5%, 30) - 0.013158*(P|A,  5%, 29) = 0.03217
(P|F, IRR, 30) - 0.013158*(P|A, IRR, 29) = 0.013158
(P|F, 6%, 30) - 0.013158*(P|A,  6%, 29) = - 0.00473

(6% - 5%) / (-0.00473 - 0.03217) = (IRR - 5%) / (0.013158 - 0.03217 )

Therefore, IRR = 5.5%

9. Now, let's check the answer by using IRR = 5.5%.

(P|A, 5.5%, 29) = ((1 + 5.5%)^29 - 1) / ( 5.5%*(1 + 5.5%)^29) = 14.3331
(P|F,  5.5%, 30) =  1/(1 + 5.5%)^30 = 0.2006

PW = $455,996.00*(P|F, IRR, 30) - $6000.00*(P|A, IRR, 29) = 5494.26

Note: A small different reasonable due to the use of linear approximation approximation.
 
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