Linear interpolation and Nominal annual rate: Difference between pages

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A straight-line estimation method for determining an intermediate value.
The rate of return - or cost of borrowing - ''per [[conventional year]]'' named or quoted in a market, under the quoting convention for the given market.


Market quotations are usually given per annum, and for this reason the term ''nominal rate'' is sometimes used interchangeably with nominal annual rate.


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


Consider a set of cashflows which has:
Different conventional market bases for expressing nominal annual rates have arisen in different markets over time.


Net present value (NPV) of +$4m at a yield of 5%.  
Often this was originally for local or domestic convenience, but it can lead to difficulties of comparability between different markets.  


Net present value (NPV) of -$4m at a yield of 6%.
For this reason it is essential to know and understand the basis on which any given nominal annual rate is being quoted.  


It is always best to be explicit about [[day count conventions| conventions]] when dealing, rather than assuming and hoping for the best.


Using linear interpolation, the estimated yield at which the cashflows have an NPV of $0 is given by:


5% + ( +4 / ( +4  -  -4 = +8 ) ) x ( 6 - 5 )%
==Illustration==


= '''5.5%'''.
Most rates in wholesale financial markets are quoted as nominal annual rates.


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


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


==Interpolation and Iteration==
GBP overnight interest payable at the periodic rate of 0.014% per day, would conventionally be quoted on a nominal annual basis as:
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.
= number of times the period fits into a conventional year x periodic rate


5.5% and the recalculated NPV would then be used with interpolation once again to further refine the estimate of the IRR.
= 365 x 0.014%  


This iteration process can be repeated as often as required until the result converges on a sufficiently stable final figure.
= 5.11%.




==Extrapolation==
==Other names for Nominal annual rates==
The nominal annual rate is sometimes also known as a 'quoted rate', 'market rate, or 'nominal rate'.


Another closely related linear estimation technique is extrapolation.
The nominal annual rate should not be confused with the [[effective annual rate]], which is usually different.


This involves the straight-line estimation of values outside the range of the data used to do the estimation with.


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


Using the following data to estimate net present value (NPV) at a yield of 7%, using extrapolation:
Nominal annual interest rates and yields are often denoted by 'R'.


NPV of +$4m at a yield of 5%.  
This distinguishes them from periodic interest rates and yields (r), and from effective annual rates (EAR).


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


==Conversion==


Using the notation above:


'''''Solution'''''
R = nominal annual interest rate or yield


Based on the sample data, for every 1% increase in the yield, the NPV moved by:
r = periodic interest rate or yield


-$4m - $4m = -$8m
n = number of times the period fits into a conventional year




Extrapolating this trend to a yield of 7%, this is a further increase in the yield of 7 - 6 = 1%.
====To convert from a periodic rate (r) to a nominal annual rate (R)====


The NPV would be modelled to fall from -$4m to:
R = r x n


= -$4m - $8m


= -$'''12m'''.
<span style="color:#4B0082">'''Example 2: Calculating the nominal annual rate'''</span>
 
The periodic rate for GBP interest is 0.01% per day.
 
Calculate the nominal annual rate.
 
 
R = r x 365
 
= 0.01% x 365
 
= 3.65%
 
 
====To convert from a nominal annual rate (R) to a periodic rate (r)====
 
r = R / n
 
 
<span style="color:#4B0082">'''Example 3: Calculating periodic rate'''</span>
 
GBP interest is quoted at a nominal annual rate of 5.11% for daily interest calculation.
 
Calculate the periodic rate per day.
 
 
r = R / n
 
= 5.11% / 365
 
= 0.014%




== See also ==
== See also ==
* [[CertFMM]]
* [[ACT/365 fixed]]
* [[Internal rate of return]]
* [[Compound]]
* [[Interpolation]]
* [[Continuously compounded rate of return]]
* [[Iteration]]
* [[Coupon rate]]
* [[Linear]]
* [[Daily rate]]
* [[Straight line]]
* [[Day count conventions]]
* [[Effective annual rate]]
* [[Forward yield]]
* [[Money market]]
* [[Nominal]]
* [[Nominal annual yield]]
* [[Nominal rate]]
* [[Periodic discount rate]]
* [[Periodic rate of interest]]
* [[Periodic yield]]
* [[Quarterly rate]]
* [[Rate of return]]
* [[Semi-annual rate]]
 
[[Category:Manage_risks]]

Latest revision as of 21:16, 4 March 2022

The rate of return - or cost of borrowing - per conventional year named or quoted in a market, under the quoting convention for the given market.

Market quotations are usually given per annum, and for this reason the term nominal rate is sometimes used interchangeably with nominal annual rate.


Different conventional market bases for expressing nominal annual rates have arisen in different markets over time.

Often this was originally for local or domestic convenience, but it can lead to difficulties of comparability between different markets.

For this reason it is essential to know and understand the basis on which any given nominal annual rate is being quoted.

It is always best to be explicit about conventions when dealing, rather than assuming and hoping for the best.


Illustration

Most rates in wholesale financial markets are quoted as nominal annual rates.


Example 1: Overnight interest

GBP overnight interest payable at the periodic rate of 0.014% per day, would conventionally be quoted on a nominal annual basis as:

= number of times the period fits into a conventional year x periodic rate

= 365 x 0.014%

= 5.11%.


Other names for Nominal annual rates

The nominal annual rate is sometimes also known as a 'quoted rate', 'market rate, or 'nominal rate'.

The nominal annual rate should not be confused with the effective annual rate, which is usually different.


Notation

Nominal annual interest rates and yields are often denoted by 'R'.

This distinguishes them from periodic interest rates and yields (r), and from effective annual rates (EAR).


Conversion

Using the notation above:

R = nominal annual interest rate or yield

r = periodic interest rate or yield

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


To convert from a periodic rate (r) to a nominal annual rate (R)

R = r x n


Example 2: Calculating the nominal annual rate

The periodic rate for GBP interest is 0.01% per day.

Calculate the nominal annual rate.


R = r x 365

= 0.01% x 365

= 3.65%


To convert from a nominal annual rate (R) to a periodic rate (r)

r = R / n


Example 3: Calculating periodic rate

GBP interest is quoted at a nominal annual rate of 5.11% for daily interest calculation.

Calculate the periodic rate per day.


r = R / n

= 5.11% / 365

= 0.014%


See also

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