Internal rate of return
(IRR).
Definition of IRR
The internal rate of return of a set of cash flows is the cost of capital which, when applied to discount all of the cash flows (including any initial investment outflow at Time 0) results in a net present value (NPV) of NIL.
For an investor, the IRR of an investment proposal therefore represents their expected rate of return on their investment in the project.
Example 1: IRR
A project requires an investment today of $100m, with $110m being receivable one year from now.
The IRR of this project is 10%, because that is the cost of capital which results in an NPV of $0, as follows:
PV of Time 0 outflow $100m
= $(100m)
PV of Time 1 inflow $110m
= $110m x 1.10-1
= $100m
NPV = - $100m + $100m
= $0.
If the project had been funded by borrowing all the required money at the IRR of 10%, there would have been exactly the right amount of surplus from the project to repay the borrowing and interest, with neither a deficit nor a surplus.
This is another way to define the IRR.
Determining IRR
Unless the pattern of cash flows is very simple, it is normally only possible to determine IRR by trial and error (iterative) methods.
Example 2: Straight line interpolation
Using straight line interpolation and the following data:
First estimated rate of return 5%, positive NPV = $+4m.
Second estimated rate of return 6%, negative NPV = $-4m.
The straight-line-interpolated estimated IRR is the mid-point between 5% and 6%.
This is 5.5%.
Using iteration, the straight-line estimation process could then be repeated, using the value of 5.5% to recalculate the NPV, and so on.
The IRR function in Excel uses a similar trial and error method.
Project decision making with IRR
Target or required IRRs are set based on the investor's weighted average cost of capital, appropriately adjusted for the risk of the proposal under review.
In very simple IRR project analysis the decision rule would be that:
(1) All opportunities with above the required IRR should be accepted.
(2) All other opportunities should be rejected.
However this assumes the unlimited availability of further capital with no increase in the cost of capital.
A more refined decision rule is that:
(1) All opportunities with IRRs BELOW the required IRR should still be REJECTED; while
(2) All other opportunities remain eligible for further consideration (rather than automatically being accepted).
