Internal Rate of Return

Given the (period, cash flow) pairs (n, C_n) where n is a positive integer, the total number of periods N, and the net present value NPV, the internal rate of return is given by r in:

\(NPV = \sigma^N _{n=0} \dfrac{C_n}{(1+r)^n}\)

Calculating IRR NPV formula with r as IRR

The period is usually given in years, but the calculation may be made simpler if r is calculated using the period in which most of the problem is defined (e.g., using months if most of the cash flows occur at monthly intervals) and converted to a yearly period thereafter. Any fixed time can be used in place of the present (e.g., the end of one interval of an annuity); the value obtained is zero if and only if the NPV is zero.

For example, if an investment may be given by the sequence of cash flows:

Calculating IRR Cash flows and time

Because the internal rate of return on an investment or project is the "annualized effective compounded return rate" or "rate of return" that makes the net present value of all cash flows (both positive and negative) from a particular investment equal to zero, then the IRR r is given by the formula:

\( NPW = -4000 + \dfrac{1200}{(1+r)^1} + \dfrac{1410}{(1+r)^2} + \dfrac{1875}{(1+r)^3} + \dfrac{1050}{(1+r)^4} = 0 \).

Calculating IRR IRR is the rate at which NPV = 0.

In this case, the answer is 14.3%. If the IRR is greater than the cost of capital, accept the project. If the IRR is less than the cost of capital, reject the project.