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**Internal Rate of Return **

The internal rate of return (IRR) is one of the most frequently used concepts in capital budgeting and in security analysis. The IRR definition is one that all analysts know by heart. For a project with one investment outlay, made initially, the IRR is the discount rate that makes the present value of the future after tax cash flows equal that investment outlay. Written out in equation form, the IRR solves this equation:

$IRR=\sum _{t=1}^{n}\frac{C{F}_{t}}{(1+IRR{)}^{t}}=Outlay$

Where IRR is the internal rate of return. The left-hand side of this equation is the present value of the project’s future cash flows, which, discounted at the IRR, equals the investment outlay. This equation will also be seen rearranged as

$IRR=\sum _{t=1}^{n}\frac{C{F}_{t}}{(1+IRR{)}^{t}}-Outlay=0$

In this form above Equation looks like the NPV equation, except that the discount rate is the IRR instead of r (the required rate of return). Discounted at the IRR, the NPV is equal to zero.

In below example, we want to find a discount rate that makes the total present value of all cash flows, the NPV, equal zero. In equation form, the IRR is the discount rate that solves this equation:

$IRR=-50+\frac{16}{(1+IRR{)}^{1}}+\frac{16}{(1+IRR{)}^{2}}+\frac{16}{(1+IRR{)}^{3}}+\frac{16}{(1+IRR{)}^{4}}+\frac{20}{(1+IRR{)}^{5}}=0$

For above cash flows With 10% required return the IRR is 19.52 percent.

The decision rule for the IRR is to invest if the IRR exceeds the required rate of return for a project: Invest if IRR > r Do not invest if IRR < r

Many investments have cash flow patterns in which the outlays occur at time zero and at future dates. Thus, it is common to define the IRR as the discount rate that makes the present values of all cash flows sum to zero:

More general version of IRR equation

$IRR=\sum _{t=1}^{n}\frac{C{F}_{t}}{(1+IRR{)}^{t}}=0$

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