fwd2zero (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.

`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

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