Investment Science: Internal rate of return of a cash flow stream

Saturday, August 21, 2010


In this article, we will see the definition of Internal rate of return or IRR, Internal rate of return is often used to analyze investment decisions in that given two cash flow streams, generally the one with the highest internal rate of return is preferred.

Definition: To keep it simple, just keep this in mind, the internal rate of return of a cash flow stream is that return which will make the present value of the cash flow stream equal to zero.

From this definition, it's very obvious that Internal rate of return calculation doesn't depend on interest rate, whereas Net Present Value calculation is.

We know that the Net Present Value of a cash flow stream (x1, x2, x3,....,xn) at an yearly interest rate 'r' for 'n' periods is given by

The internal rate of return is that value of 'r' (IRR) which will make the Net Present value of the cash flow stream equal to zero.


Example:

Find the internal rate of return of a cash flow stream (-2, 1, 1, 1) at an annual interest rate 10% (does this matter to our computation).

The internal rate of the above cash flow stream is given by

-2  + 1 / (1 + IRR) + 1 / (1+ IRR) ^ 2 + 1 / (1 + IRR) ^ 3 = 0

Solving the above equation, we get IRR = 0.2338 or 23.38%

Net Present Value and Internal rate of return are often compared to analyze different investments which raises a question, given a cash flow stream, which analysis would yield the right results, well we will look into that in a future article, it's fun.

The last question/thought of the day: Is there an unique positive solution for Internal rate of return? There is one if the cash flow stream changes from negative to positive or vice versa, for example internal rate of return for the below cash flow streams are unique, positive and equal, please refer Investment Science by Luenberger if you want to explore more on this.

1. Cash flow stream 1: (-1, 0, 3), Internal rate of return IRR = 73.21%

2. Cash flow stream 2: (1, 0, -3) (This equivalent to -(CF1)), Internal rate of return IRR = 73.21% (Obviously, but just remember the sign change in cash flows)

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