# Internal rate of return

*Investment and funding appraisal.*

(IRR).

## Contents

## Overview of internal rate of return (IRR)

IRR is a percentage summary of the cash flows of a project, for example, an IRR of 10%.

The IRR summarises the timing, as well as the amounts, of the cashflows.

For an investor, the IRR of an investment proposal represents their expected rate of return on their investment in the project.

A greater IRR is normally more attractive for an investor.

The IRR is driven by the expected future cash flows from the project.

The IRR 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 0.

**Example 1: IRR - single period 10%**

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**.

**Example 2: IRR - single period 5%**

A project requires an investment today of $100m, with $105m being receivable one year from now.

The IRR of this project is 5%, 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 $105m

= $105m x 1.05^{-1}

= $100m

NPV = - $100m + $100m

= **$0**.

**Example 3: IRR - two periods 5%**

A project requires an investment today of $100m, with $5m being receivable one year from now, and $105m two years from now.

The IRR of this project is 5%, 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 $5m

= $5m x 1.05^{-1}

= $4.76m

PV of Time 2 inflow $105m

= $105m x 1.05^{-2}

= $95.24m

NPV = - $100m + $4.76m + $95.24m

= **$0**.

**Example 4: IRR - three periods 5%**

A project requires an investment today of $100m, with $5m being receivable one year from now, a further $5m two years from now, and $105m three years from now.

The IRR of this project is 5%, 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 $5m

= $5m x 1.05^{-1}

= $4.76m

PV of Time 2 inflow $5m

= $5m x 1.05^{-2}

= $4.54m

PV of Time 3 inflow $105m

= $105m x 1.05^{-3}

= $90.70m

NPV = - $100m + $4.76m + $4.54m + $90.70m

= **$0**.

## 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).

## Excel's =IRR() function

Excel's =IRR() function returns the IRR for a block of cells within a single row or column, specified as a range.

**Example 5: =IRR() function**

Cell A1 contains -100.

Cell A2 contains 110.

=IRR(A1:A2)

will return **10%**.

(This is the result we saw in Example 1 above.)

## Determining IRR manually

Unless the pattern of cash flows is very simple, it is normally only possible to determine IRR manually by trial and error (iterative) methods.

**Example 6: 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.

## See also

- Discounted cash flow
- Effective interest rate
- Funding
- Hurdle rate
- IBR
- Implied rate of interest
- Interpolation
- Investment appraisal
- IRI
- Iteration
- Linear interpolation
- Market yield
- Net present value
- Present value
- Return on investment
- Shareholder value
- Time value of money
- Weighted average cost of capital
- Yield to maturity