Interest-rate evolution is typically represented by a set of interest rates, including the beginning and end of the periods the rates apply to. For zero rates, the start dates are typically at the valuation date, with the rates extending from that valuation date until their respective maturity dates.
Frequently, given a set of rates including their start and end
dates, you may be interested in finding the rates applicable to different
terms (periods). This problem is addressed by the function ratetimes
. This function interpolates
the interest rates given a change in the original terms. The syntax
for calling ratetimes
is
[Rates, EndTimes, StartTimes] = ratetimes(Compounding, RefRates, RefEndDates, RefStartDates, EndDates, StartDates, ValuationDate);
where:
Compounding
represents the frequency
at which the zero rates are compounded when annualized.
RefRates
is a vector of initial
interest rates representing the interest rates applicable to the initial
time intervals.
RefEndDates
is a vector of dates
representing the end of the interest rate terms (period) applicable
to RefRates
.
RefStartDates
is a vector of dates
representing the beginning of the interest rate terms applicable to RefRates
.
EndDates
represent the maturity
dates for which the interest rates are interpolated.
StartDates
represent the starting
dates for which the interest rates are interpolated.
ValuationDate
is the date of observation,
from which the StartTimes
and EndTimes
are
calculated. This date represents time = 0.
The input arguments to this function can be separated into two groups:
The initial or reference interest rates, including the terms for which they are valid
Terms for which the new interest rates are calculated
As an example, consider the rate table specified in Calculating Discount Factors from Rates.
From | To | Rate |
---|---|---|
15 Feb 2000 | 15 Aug 2000 | 0.05 |
15 Feb 2000 | 15 Feb 2001 | 0.056 |
15 Feb 2000 | 15 Aug 2001 | 0.06 |
15 Feb 2000 | 15 Feb 2002 | 0.065 |
15 Feb 2000 | 15 Aug 2002 | 0.075 |
Assuming that the valuation date is February 15, 2000, these
rates represent zero-coupon bond rates with maturities specified in
the second column. Use the function ratetimes
to
calculate the forward rates at the beginning of all periods implied
in the table. Assume a compounding value of 2.
% Reference Rates. RefStartDates = ['15-Feb-2000']; RefEndDates = ['15-Aug-2000'; '15-Feb-2001'; '15-Aug-2001';... '15-Feb-2002'; '15-Aug-2002']; Compounding = 2; ValuationDate = ['15-Feb-2000']; RefRates = [0.05; 0.056; 0.06; 0.065; 0.075]; % New Terms. StartDates = ['15-Feb-2000'; '15-Aug-2000'; '15-Feb-2001';... '15-Aug-2001'; '15-Feb-2002']; EndDates = ['15-Aug-2000'; '15-Feb-2001'; '15-Aug-2001';... '15-Feb-2002'; '15-Aug-2002']; % Find the new rates. Rates = ratetimes(Compounding, RefRates, RefEndDates,... RefStartDates, EndDates, StartDates, ValuationDate)
Rates = 0.0500 0.0620 0.0680 0.0801 0.1155
Place these values in a table like the previous one. Observe the evolution of the forward rates based on the initial zero-coupon rates.
From | To | Rate |
---|---|---|
15 Feb 2000 | 15 Aug 2000 | 0.0500 |
15 Aug 2000 | 15 Feb 2001 | 0.0620 |
15 Feb 2001 | 15 Aug 2001 | 0.0680 |
15 Aug 2001 | 15 Feb 2002 | 0.0801 |
15 Feb 2002 | 15 Aug 2002 | 0.1155 |
ratetimes
)The ratetimes
function
can provide the additional output arguments StartTimes
and EndTimes
,
which represent the time factor equivalents to the StartDates
and EndDates
vectors.
The ratetimes
function uses
time factors for interpolating the rates. These time factors are calculated
from the start and end dates, and the valuation date, which are passed
as input arguments. ratetimes
can
also use time factors directly, assuming time = 0 as the valuation
date. This alternate syntax is:
[Rates, EndTimes, StartTimes] = ratetimes(Compounding,
RefRates, RefEndTimes, RefStartTimes, EndTimes, StartTimes);
Use this alternate version of ratetimes
to
find the forward rates again. In this case, you must first find the
time factors of the reference curve. Use date2time
for
this.
RefEndTimes = date2time(ValuationDate, RefEndDates, Compounding)
RefEndTimes = 1 2 3 4 5
RefStartTimes = date2time(ValuationDate, RefStartDates,... Compounding)
RefStartTimes = 0
These are the expected values, given semiannual discounts (as denoted by a value of
2
in the variable Compounding
), end dates
separated by six-month periods, and the valuation date equal to the date marking
beginning of the first period (time factor = 0
).
Now call ratetimes
with
the alternate syntax.
[Rates, EndTimes, StartTimes] = ratetimes(Compounding,... RefRates, RefEndTimes, RefStartTimes, EndTimes, StartTimes);
Rates = 0.0500 0.0620 0.0680 0.0801 0.1155
EndTimes
and StartTimes
have,
as expected, the same values they had as input arguments.
Times = [StartTimes, EndTimes]
Times = 0 1 1 2 2 3 3 4 4 5
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