Linear interpolation and Periodic yield: Difference between pages

From ACT Wiki
(Difference between pages)
Jump to navigationJump to search
imported>Doug Williamson
(Improve calculations.)
 
imported>Doug Williamson
(Link with The Treasurer.)
 
Line 1: Line 1:
A straight-line estimation method for determining an intermediate value.
__NOTOC__
Periodic yield is a rate of return - or cost of borrowing - expressed as the proportion by which the amount at the end of the period exceeds the amount at the start.  




__TOC__
<span style="color:#4B0082">'''Example 1'''</span>


GBP 1 million is borrowed or invested.


<span style="color:#4B0082">'''Example 1: Interpolation'''</span>
GBP 1.03 million is repayable at the end of the period.


Consider a set of cashflows which has:


Net present value (NPV) of +$4m at a yield of 5%.
The periodic yield (r) is:


NPV of -$4m at a yield of 6%.
r = (End amount / Start amount) - 1


Which can also be expressed as:


Using linear interpolation, the estimated yield at which the cashflows have an NPV of $0 is given by:
r = (End / Start) - 1


5% + ( +4 / ( +4  -  -4) ) x (6 - 5)%
''or''


5% + ( +4 / +8 ) x 1%
r = <math>\frac{End}{Start}</math> - 1


5% + 0.5%


= '''5.5%'''.
= <math>\frac{1.03}{1}</math> - 1


5.5% is the estimated internal rate of return (IRR) of the cashflows.
= 0.03


= '''3%'''


==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.
<span style="color:#4B0082">'''Example 2'''</span>


5.5% and the recalculated NPV would then be used with interpolation once again to further refine the estimate of the IRR.
GBP  0.97 million is borrowed or invested.  


This iteration process can be repeated as often as required until the result converges on a sufficiently stable final figure.
GBP 1.00 million is repayable at the end of the period.  




==Extrapolation==
The periodic yield (r) is:


Another closely related linear estimation technique is extrapolation. 
r = <math>\frac{End}{Start}</math> - 1


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>
= <math>\frac{1.00}{0.97}</math> - 1


Using the following data to estimate net present value (NPV) at a yield of 7%, using extrapolation:
= 0.030928


NPV of +$4m at a yield of 5%.
= '''3.0928%'''


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


''Check:''


Amount at end = 0.97 x 1.030928 = 1.00, as expected.


'''''Solution'''''


Based on the sample data, for every 1% increase in the yield, the NPV moved by:
<span style="color:#4B0082">'''Example 3'''</span>


-$4m - $4m = -$8m
GBP  0.97 million is invested.


The periodic yield is 3.0928%.


Extrapolating this trend to a yield of 7%, this is a further increase in the yield of 7 - 6 = 1%.
Calculate the amount repayable at the end of the period.


The NPV would be modelled to fall from -$4m to:


= -$4m - $8m
'''''Solution'''''


= -$'''12m'''.
The periodic yield (r) is defined as:


r = <math>\frac{End}{Start}</math> - 1


== See also ==
 
* [[CertFMM]]
''Rearranging this relationship:''
* [[Internal rate of return]]
 
* [[Interpolation]]
1 + r = <math>\frac{End}{Start}</math>
* [[Iteration]]
 
* [[Linear]]
 
* [[Straight line]]
End = Start x (1 + r)
 
 
''Substituting the given information into this relationship:''
 
End = GBP 0.97m x (1 + 0.030928)
 
= '''GBP 1.00m'''
 
 
<span style="color:#4B0082">'''Example 4'''</span>
 
An investment will pay out a single amount of GBP 1.00m at its final maturity after one period.
 
The periodic yield is 3.0928%.
 
Calculate the amount invested at the start of the period.
 
 
'''''Solution'''''
 
As before, the periodic yield (r) is defined as:
 
r = <math>\frac{End}{Start}</math> - 1
 
 
''Rearranging this relationship:''
 
1 + r = <math>\frac{End}{Start}</math>
 
 
Start = <math>\frac{End}{(1 + r)}</math>
 
 
''Substitute the given data into this relationship:''
 
Start = <math>\frac{1.00}{(1  +  0.030928)}</math>
 
 
= '''GBP 0.97m'''
 
 
''Check:''
 
Amount at start = 0.97 x 1.030928 = 1.00, as expected.
 
 
====Effective annual rate====
 
The periodic yield (r) is related to the [[effective annual rate]] (EAR), and each can be calculated from the other.
 
 
'''''Conversion formulae (r to EAR and EAR to r):'''''
 
EAR = (1 + r)<sup>n</sup> - 1
 
r = (1 + EAR)<sup>(1/n)</sup> - 1
 
 
Where:
 
EAR = effective annual rate or yield
 
r = periodic interest rate or yield, as before
 
n = number of times the period fits into a calendar year
 
 
====Periodic discount rate====
 
The periodic yield (r) is also related to the [[periodic discount rate]] (d), and each can be calculated from the other.
 
 
'''''Conversion formulae (r to d and d to r):'''''
 
d = r / (1 + r)
 
r = d / (1 - d)
 
 
Where:
 
d = periodic discount rate
 
r = periodic interest rate or yield
 
 
==See also==
 
*[[Effective annual rate]]
*[[Discount rate]]
*[[Nominal annual rate]]
*[[Nominal annual yield]]
*[[Periodic discount rate]]
*[[Yield]]
*[[Forward yield]]
*[[Zero coupon yield]]
*[[Par yield]]
 
 
===Other resources===
[[Media:2013_09_Sept_-_Simple_solutions.pdf| The Treasurer students, Simple solutions]]

Revision as of 22:14, 22 November 2015

Periodic yield is a rate of return - or cost of borrowing - expressed as the proportion by which the amount at the end of the period exceeds the amount at the start.


Example 1

GBP 1 million is borrowed or invested.

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


The periodic yield (r) is:

r = (End amount / Start amount) - 1

Which can also be expressed as:

r = (End / Start) - 1

or

r = <math>\frac{End}{Start}</math> - 1


= <math>\frac{1.03}{1}</math> - 1

= 0.03

= 3%


Example 2

GBP 0.97 million is borrowed or invested.

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


The periodic yield (r) is:

r = <math>\frac{End}{Start}</math> - 1


= <math>\frac{1.00}{0.97}</math> - 1

= 0.030928

= 3.0928%


Check:

Amount at end = 0.97 x 1.030928 = 1.00, as expected.


Example 3

GBP 0.97 million is invested.

The periodic yield is 3.0928%.

Calculate the amount repayable at the end of the period.


Solution

The periodic yield (r) is defined as:

r = <math>\frac{End}{Start}</math> - 1


Rearranging this relationship:

1 + r = <math>\frac{End}{Start}</math>


End = Start x (1 + r)


Substituting the given information into this relationship:

End = GBP 0.97m x (1 + 0.030928)

= 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 yield is 3.0928%.

Calculate the amount invested at the start of the period.


Solution

As before, the periodic yield (r) is defined as:

r = <math>\frac{End}{Start}</math> - 1


Rearranging this relationship:

1 + r = <math>\frac{End}{Start}</math>


Start = <math>\frac{End}{(1 + r)}</math>


Substitute the given data into this relationship:

Start = <math>\frac{1.00}{(1 + 0.030928)}</math>


= GBP 0.97m


Check:

Amount at start = 0.97 x 1.030928 = 1.00, as expected.


Effective annual rate

The periodic yield (r) is related to the effective annual rate (EAR), and each can be calculated from the other.


Conversion formulae (r to EAR and EAR to r):

EAR = (1 + r)n - 1

r = (1 + EAR)(1/n) - 1


Where:

EAR = effective annual rate or yield

r = periodic interest rate or yield, as before

n = number of times the period fits into a calendar year


Periodic discount rate

The periodic yield (r) is also related to the periodic discount rate (d), and each can be calculated from the other.


Conversion formulae (r to d and d to r):

d = r / (1 + r)

r = d / (1 - d)


Where:

d = periodic discount rate

r = periodic interest rate or yield


See also


Other resources

The Treasurer students, Simple solutions

Retrieved from ‘https://wiki.treasurers.org/w/index.php?title=Linear_interpolation&oldid=37336’