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''Investment and funding appraisal.''
(IRR).  
(IRR).  




== Definition of IRR ==
== Overview of internal rate of return (IRR) ==
 
IRR is an accounting method for calculating the return forecast to be achieved on a (potential) investment by equating the net present value (NPV) of its cash outflows and inflows over time to zero.
 
 
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,


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.
:including any initial investment outflow at Time 0,


For an investor, the IRR of an investment proposal therefore represents their expected rate of [[return]] on their investment in the project.
:results in a [[net present value]] (NPV) of 0.




<span style="color:#4B0082">'''Example 1: IRR'''</span>
<span style="color:#4B0082">'''Example 1: IRR - single period 10%'''</span>


A project requires an investment today of $100m, with $110m being receivable one year from now.
A project requires an investment today of $100m, with $110m being receivable one year from now.
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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.
<span style="color:#4B0082">'''Example 2: IRR - single period 5%'''</span>
 
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<sup>-1</sup>
 
= $100m
 
 
NPV = - $100m + $100m
 
= '''$0'''.
 
 
<span style="color:#4B0082">'''Example 3: IRR - two periods 5%'''</span>
 
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


This is another way to define the IRR.
= $5m x 1.05<sup>-1</sup>


= $4.76m




== Determining IRR ==
PV of Time 2 inflow $105m


= $105m x 1.05<sup>-2</sup>


Unless the pattern of cash flows is very simple, it is normally only possible to determine IRR by trial and error (iterative) methods.
= $95.24m




<span style="color:#4B0082">'''Example 2: Straight line interpolation'''</span>
NPV = - $100m + $4.76m + $95.24m


Using straight line interpolation and the following data:
= '''$0'''.
 
 
<span style="color:#4B0082">'''Example 4: IRR - three periods 5%'''</span>
 
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<sup>-1</sup>
 
= $4.76m
 
 
PV of Time 2 inflow $5m
 
= $5m x 1.05<sup>-2</sup>


First estimated rate of return 5%, positive NPV = $+4m.
= $4.54m


Second estimated rate of return 6%, negative NPV = $-4m.


The straight-line-interpolated estimated IRR is the mid-point between 5% and 6%.
PV of Time 3 inflow $105m


This is '''5.5%'''.
= $105m x 1.05<sup>-3</sup>


= $90.70m


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


= '''$0'''.




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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.  
Target or required IRRs for ''investment'' 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:
In very simple IRR investment project analysis the decision rule would be that:


(1) All opportunities with above the required IRR should be accepted.
(1) All opportunities with above the required IRR should be accepted.
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(2) All other opportunities remain eligible for further consideration (rather than automatically being accepted).
(2) All other opportunities remain eligible for further consideration (rather than automatically being accepted).
For ''borrowing'' or ''funding'' opportunities, the appropriate comparator rate is the organisation's cost of borrowing, for borrowings of comparable risk.
The IRR decision rule for evaluating borrowing opportunities is the opposite of that for investments, as described above.
For borrowing opportunities, a ''lower'' IRR indicates a potentially more cost-effective borrowing, that warrants further consideration.
== 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.
<span style="color:#4B0082">'''Example 5: =IRR() function'''</span>
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.
<span style="color:#4B0082">'''Example 6: Straight line interpolation'''</span>
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 ==
== See also ==
* [[CertFMM]]
* [[Comparator]]
* [[Compound Annual Growth Rate]]
* [[Cost of capital]]
* [[Cost of debt]]
* [[Discount]]
* [[Discount rate]]
* [[Discounted cash flow]]
* [[Effective interest rate]]
* [[Effective interest rate]]
* [[Funding]]
* [[Hurdle rate]]
* [[Hurdle rate]]
* [[IBR]]
* [[Implied rate of interest]]
* [[Implied rate of interest]]
* [[Incremental borrowing rate]]
* [[Interest rate implicit in a lease]]
* [[Interpolation]]
* [[Interpolation]]
* [[Investment appraisal]]
* [[Iteration]]
* [[Iteration]]
* [[Linear interpolation]]
* [[Linear interpolation]]
* [[Market yield]]
* [[Market yield]]
* [[Net present value]]
* [[Net present value]]  (NPV)
* [[Present value]]
* [[Opportunity cost]]
* [[Payback period]]
* [[Present value]]  (PV)
* [[Profitability index]]  (PI)
* [[Return on investment]]
* [[Shareholder value]]
* [[Shareholder value]]
* [[Time value of money]]  (TVM)
* [[Total shareholder return]]  (TSR)
* [[Weighted average cost of capital]]
* [[Weighted average cost of capital]]
* [[Yield]]
* [[Yield to maturity]]
* [[Yield to maturity]]
[[Category:Corporate_finance]]
[[Category:Financial_products_and_markets]]
[[Category:Investment]]
[[Category:Long_term_funding]]
[[Category:The_business_context]]

Latest revision as of 20:31, 2 February 2024

Investment and funding appraisal.

(IRR).


Overview of internal rate of return (IRR)

IRR is an accounting method for calculating the return forecast to be achieved on a (potential) investment by equating the net present value (NPV) of its cash outflows and inflows over time to zero.


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 for investment 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 investment 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).


For borrowing or funding opportunities, the appropriate comparator rate is the organisation's cost of borrowing, for borrowings of comparable risk.

The IRR decision rule for evaluating borrowing opportunities is the opposite of that for investments, as described above.

For borrowing opportunities, a lower IRR indicates a potentially more cost-effective borrowing, that warrants further consideration.


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