Interest-Rate Term Conversions

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.

Spot Curve to Forward Curve Conversion

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)

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

Alternative Syntax (`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.

StartDates = ['15-Feb-2000'];
EndDates  = ['15-Aug-2000'; '15-Feb-2001'; '15-Aug-2001';... 
'15-Feb-2002'; '15-Aug-2002'];
Compounding = 2;
ValuationDate = ['15-Feb-2000'];
Rates = [0.05; 0.056; 0.06; 0.065; 0.075];
[Disc, EndTimes, StartTimes] = rate2disc(Compounding, Rates,... 
EndDates, StartDates, ValuationDate);

[Rates, EndTimes, StartTimes] = ratetimes(Compounding,... 
RefRates, RefEndTimes, RefStartTimes, EndTimes, StartTimes)

Rates =

    0.0500
    0.0560
    0.0600
    0.0650
    0.0750


EndTimes =

     1
     2
     3
     4
     5


StartTimes =

     0
     0
     0
     0
     0

EndTimes and StartTimes have, as expected, the same values they had as input arguments.

Times = [StartTimes, EndTimes]