Interest-Rate Term Conversions

This example illustrates spot curve to forward curve conversion and using ratetimes.

Frequently, given a set of rates including their start and end dates, you can find 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.

Consier the folowing rate table.

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 = 5×1

    0.0500
    0.0620
    0.0680
    0.0801
    0.1155

Place these resulting values in a table. 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

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.

Use this alternate syntax 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 = 5×1

     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 the beginning of the first period (time factor = 0).

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 = 5×1

    0.0500
    0.0560
    0.0600
    0.0650
    0.0750

EndTimes = 5×1

     1
     2
     3
     4
     5

StartTimes = 5×1

     0
     0
     0
     0
     0

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

Times = [StartTimes, EndTimes]

Times = 5×2

     0     1
     0     2
     0     3
     0     4
     0     5