Linear interpolation and Probability: Difference between pages

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A straight-line estimation method for determining an intermediate value.
The study of chance providing an objective measure of uncertainty.




__TOC__
Probabilities range between 1 (=100%) and 0 (=0%). 


A probability of 100% means that an event is considered certain to occur.


<span style="color:#4B0082">'''Example 1: Interpolation'''</span>
A probability of 0% means that an event is considered certain not to occur. 


Consider a set of cashflows which has:


Net present value (NPV) of +$4m at a yield of 5%.  
For example, flipping an unbiased coin, the probability of getting a head is often modelled as 50%.


NPV of -$4m at a yield of 6%.


This simple model of a coin flip assumes that the only two possibilities are a head or a tail.  Applying such simple models to financial situations, and treating financial outcomes as simple coin flips, may lead to errors resulting from:


Using linear interpolation, the estimated yield at which the cashflows have an NPV of $0 is given by:
#The coin landing on its edge 'more often than it's supposed to'.
 
#The underlying assumption of an unbiased coin not being a valid one. This kind of assumption is usually much too simple.
5% + ( +4 / ( +4  -  -4) ) x (6 - 5)%
 
5% + ( +4 / +8 ) x 1%
 
5% + 0.5%
 
= '''5.5%'''.
 
5.5% is the estimated internal rate of return (IRR) of the cashflows.
 
 
==Interpolation and Iteration==
Interpolation is often used in conjunction with Iteration.
 
Using iteration, the straight-line estimated IRR of 5.5% would then be used, in turn, to recalculate the NPV at the estimated IRR of 5.5%, producing a recalculated NPV even closer to $0.
 
5.5% and the recalculated NPV would then be used with interpolation once again to further refine the estimate of the IRR.
 
This iteration process can be repeated as often as required until the result converges on a sufficiently stable final figure.
 
 
==Extrapolation==
 
Another closely related linear estimation technique is extrapolation. 
 
This involves the straight-line estimation of values outside the range of the sample data used to do the estimation with.
 
<span style="color:#4B0082">'''Example 2: Extrapolation'''</span>
 
Using the following data to estimate net present value (NPV) at a yield of 7%, using extrapolation:
 
NPV of +$4m at a yield of 5%.
 
NPV of -$4m at a yield of 6%.
 
 
 
'''''Solution'''''
 
Based on the sample data, for every 1% increase in the yield, the NPV moved by:
 
-$4m - $4m = -$8m
 
 
Extrapolating this trend to a yield of 7%, this is a further increase in the yield of 7 - 6 = 1%.
 
The NPV would be modelled to fall from -$4m to:
 
= -$4m - $8m
 
= -$'''12m'''.




== See also ==
== See also ==
* [[Internal rate of return]]
* [[Black swan]]
* [[Interpolation]]
* [[Conditional probability]]
* [[Iteration]]
* [[Confidence interval]]
* [[Linear]]
* [[Frequency distribution]]
* [[Straight line]]
* [[Mutually exclusive]]
 
* [[Poisson distribution]]
[[Category:Corporate_financial_management]]
[[Category:Cash_management]]

Revision as of 15:19, 8 June 2016

The study of chance providing an objective measure of uncertainty.


Probabilities range between 1 (=100%) and 0 (=0%).

A probability of 100% means that an event is considered certain to occur.

A probability of 0% means that an event is considered certain not to occur.


For example, flipping an unbiased coin, the probability of getting a head is often modelled as 50%.


This simple model of a coin flip assumes that the only two possibilities are a head or a tail. Applying such simple models to financial situations, and treating financial outcomes as simple coin flips, may lead to errors resulting from:

  1. The coin landing on its edge 'more often than it's supposed to'.
  2. The underlying assumption of an unbiased coin not being a valid one. This kind of assumption is usually much too simple.


See also

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