Annualisation Conventions: Tools
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Overview
The Nematrian website
provides a web service, MnRestateYieldOrDiscount
that converts interest or discount rates quoted using one annualisation
convention to equivalent rates using a different annualisation convention. With
this function, the relevant annualisation convention is defined by a parameter
(taking only integral values) that if zero means continuous compounding, if
positive () means expressed
as an interest rate compounded times per annum
and if negative () means expressed
as a discount rate compounded times per annum.
Usually, derivative
pricing algorithms concentrate on continuously compounded rates (i.e. ‘forces’
of interest), since this simplifies the mathematical presentation. For example,
if interest rates vary through time then the accumulated value after time of
an investment of now (i.e. at time
), if the
continuously compounded interest rate between now and then is at time is:
where:
This corresponds to the
natural way in which we would expect to average a quantity through time. If we
used other annualisation conventions then the averaging process involves more
complicated mathematical formulae.
The impact of different
annualisation conventions is shown in Illustratative
table of annualisation conventions.
Exact computation of
present values of cash flows generally requires not just information about the
annualisation convention applicable to the interest rate or discount rate in
question, but also precise information about when the cash flows will occur. In
real life, the ‘expected’ date on which these may occur may not be their actual
date of payment, e.g. the specified date may not be a business day. Moreover,
if, say, coupon payments are specified as annual percentages of nominal amounts
but are paid more frequently than annually, say quarterly, then it become
necessary to specify how these annual amounts are apportioned between the
different quarter dates. These types of conventions are usually called day
count conventions.
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