Internal rate of return and Periodic discount rate: Difference between pages

From ACT Wiki
(Difference between pages)
Jump to navigationJump to search
imported>Doug Williamson
(Qualify description of decision making rule.)
 
imported>Doug Williamson
(Layout)
 
Line 1: Line 1:
(IRR).
__NOTOC__
A cost of borrowing - or rate of return - expressed as:


*The excess of the amount at the end over the amount at the start
*Divided by the amount at the end


=== 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.
==Example 1==
GBP 1 million is borrowed.  


For an investor, the IRR of an investment proposal therefore represents their expected rate of [[return]] on their investment in the project.
GBP 1.03 million is repayable at the end of the period.  




'''Example 1'''
The periodic discount rate (d) is:


A project requires an investment today of $100m, with $110m being receivable one year from now.
d = (End amount - start amount) / End amount


The IRR of this project is 10%, because that is the cost of capital which results in an NPV of $0, as follows:
= (1.03 - 1) / 1.03


= 0.029126


[[PV]] of Time 0 outflow $100m
= '''2.9126%'''


= $(100m)


==Example 2==
GBP 0.97 million is borrowed or invested


PV of Time 1 inflow $110m
GBP 1.00 million is repayable at the end of the period.


= $110m x 1.10<sup>-1</sup>


= $100m
The periodic discount rate (d) is:


(End amount - start amount) / End amount


NPV = - $100m + $100m
= (1.00 - 0.97) /  1.00


= '''$0'''.
= 0.030000


= '''3.0000%'''


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.
==Example 3==
GBP  0.97 million is borrowed.  


The periodic discount rate is 3.0000%.


Calculate the amount repayable at the end of the period.


=== Determining IRR ===
===Solution===
The periodic discount rate (d) is defined as:


d = (End amount - start amount) / End amount


Unless the pattern of cash flows is very simple, it is normally only possible to determine IRR by trial and error (iterative) methods.
d = 1 - (Start amount / End amount)




'''Example 2'''
''Rearranging this relationship:''


Using straight line interpolation and the following data:
(Start amount / End amount) = 1 - d


First estimated rate of return 5%, positive NPV = $+4m.
Start amount = End amount x (1 - d)


Second estimated rate of return 6%, negative NPV = $-4m.
Start amount / (1 - d) = End amount


The straight-line-interpolated estimated IRR is the mid-point between 5% and 6%.
End amount = Start amount / (1 - d)


This is '''5.5%'''.


''Substituting the given information into this relationship:''


Using iteration, the straight-line estimation process could then be repeated, using the value of 5.5% to recalculate the NPV, and so on.
End amount = GBP 0.97m / (1 - 0.030000)


The IRR function in Excel uses a similar trial and error method.
= GBP 0.97m / 0.97


= '''GBP 1.00m'''




=== IRR project analysis decision making ===
==Example 4==
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.


The periodic discount rate is 3.0000%.


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.  
Calculate the amount invested at the start of the period.


In very simple IRR project analysis the decision rule would be that:
===Solution===
As before, the periodic discount rate (d) is defined as:


(1) All opportunities with above the required IRR should be accepted.
d = (End amount - start amount) / End amount


(2) All other opportunities should be rejected.
d = 1 - (Start amount / End amount)




However this assumes the unlimited availability of further capital with no increase in the cost of capital.
''Rearranging this relationship:''


(Start amount / End amount) = 1 - d


A more refined decision rule is that:
Start amount = End amount x (1 - d)


(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).
''Substitute the given data into this relationship:''


Start amount = GBP 1.00m x (1 - 0.030000)


== See also ==
= '''GBP 0.97m'''
* [[CertFMM]]
 
* [[Effective interest rate]]
 
* [[Hurdle rate]]
 
* [[Implied rate of interest]]
==See also==
* [[Interpolation]]
 
* [[Iteration]]
*[[Effective annual rate]]
* [[Linear interpolation]]
*[[Discount rate]]
* [[Market yield]]
*[[Nominal annual rate]]
* [[Net present value]]
*[[Periodic yield]]
* [[Present value]]
*[[Yield]]
* [[Shareholder value]]
* [[Weighted average cost of capital]]
* [[Yield to maturity]]

Revision as of 15:04, 26 October 2015

A cost of borrowing - or rate of return - expressed as:

  • The excess of the amount at the end over the amount at the start
  • Divided by the amount at the end


Example 1

GBP 1 million is borrowed.

GBP 1.03 million is repayable at the end of the period.


The periodic discount rate (d) is:

d = (End amount - start amount) / End amount

= (1.03 - 1) / 1.03

= 0.029126

= 2.9126%


Example 2

GBP 0.97 million is borrowed or invested

GBP 1.00 million is repayable at the end of the period.


The periodic discount rate (d) is:

(End amount - start amount) / End amount

= (1.00 - 0.97) / 1.00

= 0.030000

= 3.0000%


Example 3

GBP 0.97 million is borrowed.

The periodic discount rate is 3.0000%.

Calculate the amount repayable at the end of the period.

Solution

The periodic discount rate (d) is defined as:

d = (End amount - start amount) / End amount

d = 1 - (Start amount / End amount)


Rearranging this relationship:

(Start amount / End amount) = 1 - d

Start amount = End amount x (1 - d)

Start amount / (1 - d) = End amount

End amount = Start amount / (1 - d)


Substituting the given information into this relationship:

End amount = GBP 0.97m / (1 - 0.030000)

= GBP 0.97m / 0.97

= GBP 1.00m


Example 4

An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.

The periodic discount rate is 3.0000%.

Calculate the amount invested at the start of the period.

Solution

As before, the periodic discount rate (d) is defined as:

d = (End amount - start amount) / End amount

d = 1 - (Start amount / End amount)


Rearranging this relationship:

(Start amount / End amount) = 1 - d

Start amount = End amount x (1 - d)


Substitute the given data into this relationship:

Start amount = GBP 1.00m x (1 - 0.030000)

= GBP 0.97m


See also

Retrieved from ‘https://wiki.treasurers.org/w/index.php?title=Internal_rate_of_return&oldid=37283’