Internal Rate of Return

Calculating the IRR

Given a collection of pairs (time, cash flow), a rate of return for which the net present value is zero is an internal rate of return.

LEARNING OBJECTIVE

Calculate a project's internal rate of return

KEY POINTS

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 the function in which $NPV = 0$ .
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.
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.

TERMS

cost of capital
the rate of return that capital could be expected to earn in an alternative investment of equivalent risk
net present value profile
a graph of the sum of all cash inflows and outflows adjusted for the time value of money at different discount rates

Given a collection of pairs (time, cash flow) involved in a project, the internal rate of return follows from the net present value as a function of the rate of return. A rate of return for which this function is zero is an 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:

$\mathrm{NPV}=\sum_{n=0}^{N} \frac{C_{n}}{(1+r)^{n}}$

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 the majority 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:

Year ( $n$ )	Cash flow ( $C_n$ )
0	-4000
1	1200
2	1410
3	1875
4	1050

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:

$\mathrm{NPV}=-4000+\frac{1200}{(1+r)^{1}}+\frac{1410}{(1+r)^{2}}+\frac{1875}{(1+r)^{3}}+\frac{1050}{(1+r)^{4}}=0$

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.