Financial Toolbox

fwd2zero

Zero curve given a forward curve

Syntax

• [ZeroRates, CurveDates] = fwd2zero(ForwardRates, CurveDates, 
  Settle, OutputCompounding, OutputBasis, InputCompounding, 
  InputBasis)
  

Arguments

ForwardRates	A number of bonds (NUMBONDS) by 1 vector of annualized implied forward rates, as decimal fractions. In aggregate, the rates in ForwardRates constitute an implied forward curve for the investment horizon represented by CurveDates.
CurveDates	A NUMBONDS-by-1 vector of maturity dates (as serial date numbers) that correspond to the forward rates.
Settle	A serial date number that is the common settlement date for the forward rates.
OutputCompounding	(Optional) Output compounding. A scalar that sets the compounding frequency per year for annualizing the output zero rates. Allowed values are:
	1	annual compounding
	2	semi-annual compounding (default)
	3	compounding three times per year
	4	quarterly compounding
	6	bimonthly compounding
	12	monthly compounding
	365	daily compounding
	-1	continuous compounding
OutputBasis	(Optional) Output day-count basis for annualizing the output zero rates. Allowed values are:
	0	actual/actual (default)
	1	30/360
	2	actual/360
	3	actual/365
InputCompounding	(Optional) A scalar that indicates the compounding frequency per year used for annualizing the input forward rates. Allowed values are the same as for OutputCompounding. Default = OutputCompounding.
InputBasis	(Optional) Input day-count basis used for annualizing the forward rates. Allowed values are the same as for OutputBasis. Default = OutputBasis.

Description

[ZeroRates, CurveDates] = fwd2zero(ForwardRates, CurveDates, Settle, OutputCompounding, OutputBasis, InputCompounding, InputBasis) returns a zero curve given an implied forward curve and its maturity dates.

ZeroRates	A NUMBONDS-by-1 vector of decimal fractions. In aggregate, the rates in ZeroRates constitute a zero curve for the investment horizon represented by CurveDates.
CurveDates	A NUMBONDS-by-1 vector of maturity dates (as serial date numbers) that correspond to the zero rates in ZeroRates. This vector is the same as the input vector CurveDates.

Examples

Given an implied forward curve ForwardRates over a set of maturity dates CurveDates, and a settlement date Settle

• ForwardRates = [0.0469
                  0.0519
                  0.0549
                  0.0535
                  0.0558
                  0.0508
                  0.0560
                  0.0545
                  0.0615
                  0.0486];
  
  CurveDates = [datenum('06-Nov-2000')
                datenum('11-Dec-2000')
                datenum('15-Jan-2001')
                datenum('05-Feb-2001')
                datenum('04-Mar-2001')
                datenum('02-Apr-2001')
                datenum('30-Apr-2001')
                datenum('25-Jun-2001')
                datenum('04-Sep-2001')
                datenum('12-Nov-2001')];
  
  Settle = datenum('03-Nov-2000');
  

Set daily compounding for the zero curve, on an actual/365 basis. The forward curve was compounded annually on an actual/actual basis.

• OutputCompounding = 365;
  OutputBasis = 3;
  InputCompounding = 1;
  InputBasis = 0;
  

Execute the function

• [ZeroRates, CurveDates] = fwd2zero(ForwardRates, CurveDates,... 
  Settle, OutputCompounding, OutputBasis, InputCompounding,... 
  InputBasis)
  

which returns the zero curve ZeroRates at the maturity dates CurveDates.

• ZeroRates =
  
      0.0457
      0.0501
      0.0516
      0.0517
      0.0523
      0.0517
      0.0521
      0.0523
      0.0540
      0.0528
  
  CurveDates =
  
        730796
        730831
        730866
        730887
        730914
        730943
        730971
        731027
        731098
        731167
  

For readability, ForwardRates and ZeroRates are shown here only to the basis point. However, MATLAB computed them at full precision. If you enter ForwardRates as shown, ZeroRates may differ due to rounding.

See Also

zero2fwd and other functions for Term Structure of Interest Rates

fvvar

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