# zero2fwd

Forward curve given zero curve

In R2017b, the specification of optional input arguments has changed. While the previous ordered inputs syntax is still supported, it may no longer be supported in a future release. Use the new optional name-value pair inputs: `InputCompounding`, `InputBasis`, `OutputCompounding`, and `OutputBasis`.

## Syntax

``[ForwardRates,CurveDates] = zero2fwd(ZeroRates,CurveDates,Settle)``
``[ForwardRates,CurveDates] = zero2fwd(___,Name,Value)``

## Description

example

````[ForwardRates,CurveDates] = zero2fwd(ZeroRates,CurveDates,Settle)` returns an implied forward rate curve given a zero curve and its maturity dates. If either input for `CurveDates` or `Settle` is a datetime array, `CurveDates` is returned as a datetime array. Otherwise, `CurveDates` is returned as a serial date number. `ForwardRates` is the same for any of these input data types. ```

example

````[ForwardRates,CurveDates] = zero2fwd(___,Name,Value)` adds optional name-value pair arguments```

## Examples

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Given a zero curve over a set of maturity dates, a settlement date, and a compounding rate, compute the forward rate curve.

```ZeroRates = [0.0458 0.0502 0.0518 0.0519 0.0524 0.0519 0.0523 0.0525 0.0541 0.0529]; 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'); InputCompounding = 1; InputBasis = 2; OutputCompounding = 1; OutputBasis = 2;```

Execute the function `zero2fwd` to return the forward rate curve `ForwardRates` at the maturity dates `CurveDates`.

```[ForwardRates, CurveDates] = zero2fwd(ZeroRates, CurveDates,... Settle, 'InputCompounding',1,'InputBasis',2,'OutputCompounding',1,'OutputBasis',2)```
```ForwardRates = 10×1 0.0458 0.0506 0.0535 0.0522 0.0541 0.0498 0.0544 0.0531 0.0594 0.0476 ```
```CurveDates = 10×1 730796 730831 730866 730887 730914 730943 730971 731027 731098 731167 ```

Given a zero curve over a set of maturity dates, a settlement date, and a compounding rate, use `datetime` compute the forward rate curve.

```ZeroRates = [0.0458 0.0502 0.0518 0.0519 0.0524 0.0519 0.0523 0.0525 0.0541 0.0529]; 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'); InputCompounding = 1; InputBasis = 2; OutputCompounding = 1; OutputBasis = 2; CurveDates = datetime(CurveDates, 'ConvertFrom', 'datenum','Locale','en_US'); Settle = datetime(Settle,'ConvertFrom','datenum','Locale','en_US'); [ForwardRates, CurveDates] = zero2fwd(ZeroRates, CurveDates,... Settle,'InputCompounding',1,'InputBasis',2,'OutputCompounding',1,'OutputBasis',2)```
```ForwardRates = 10×1 0.0458 0.0506 0.0535 0.0522 0.0541 0.0498 0.0544 0.0531 0.0594 0.0476 ```
```CurveDates = 10x1 datetime 06-Nov-2000 11-Dec-2000 15-Jan-2001 05-Feb-2001 04-Mar-2001 02-Apr-2001 30-Apr-2001 25-Jun-2001 04-Sep-2001 12-Nov-2001 ```

## Input Arguments

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Annualized zero rates, specified as a `NUMBONDS`-by-`1` vector using decimal fractions. In aggregate, the rates constitute an implied zero curve for the investment horizon represented by `CurveDates`. The first element pertains to forward rates from the settlement date to the first curve date.

Data Types: `double`

Maturity dates, specified as a `NUMBONDS`-by-`1` vector using serial date numbers, date character vectors, or datetime arrays, that correspond to the `ZeroRates`.

Data Types: `double` | `datetime` | `char`

Common settlement date for input `ZeroRates`, specified as serial date numbers, date character vectors, or datetime arrays.

Data Types: `double` | `datetime` | `char`

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: ```[ForwardRates,CurveDates] = zero2fwd(ZeroRates,CurveDates,Settle,'InputCompounding',3,'InputBasis',5,'OutputCompounding',4,'OutputBasis',5)```

Compounding frequency of input zero rates, specified as the comma-separated pair consisting of `'InputCompounding'` and allowed values:

• `0` — Simple interest (no compounding)

• `1` — Annual compounding

• `2` — Semiannual compounding (default)

• `3` — Compounding three times per year

• `4` — Quarterly compounding

• `6` — Bimonthly compounding

• `12` — Monthly compounding

• `365` — Daily compounding

• `-1` — Continuous compounding

Note

If `InputCompounding` is not specified, then `InputCompounding` is assigned the value specified for `OutputCompounding`. If either `InputCompounding` or `OutputCompounding` are not specified, the default is `2` (semiannual) for both.

Data Types: `double`

Day count basis of input zero rates, specified as the comma-separated pair consisting of `'InputBasis'` and allowed values:

• 0 = actual/actual

• 1 = 30/360 (SIA)

• 2 = actual/360

• 3 = actual/365

• 4 = 30/360 (PSA)

• 5 = 30/360 (ISDA)

• 6 = 30/360 (European)

• 7 = actual/365 (Japanese)

• 8 = actual/actual (ICMA)

• 9 = actual/360 (ICMA)

• 10 = actual/365 (ICMA)

• 11 = 30/360E (ICMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

Note

If `InputBasis` is not specified, then `InputBasis` is assigned the value specified for `OutputBasis`. If either `InputBasis` or `Outputbasis` are not specified, the default is `0` (actual/actual) for both.

Data Types: `double`

Compounding frequency of output forward rates, specified as the comma-separated pair consisting of `'OutputCompounding'` and allowed values:

• `0` — Simple interest (no compounding)

• `1` — Annual compounding

• `2` — Semiannual compounding (default)

• `3` — Compounding three times per year

• `4` — Quarterly compounding

• `6` — Bimonthly compounding

• `12` — Monthly compounding

• `365` — Daily compounding

• `-1` — Continuous compounding

Note

If `OutputCompounding` is not specified, then `OutputCompounding` is assigned the value specified for `InputCompounding`. If either `InputCompounding` or `OutputCompounding` are not specified, the default is `2` (semiannual) for both.

Data Types: `double`

Day count basis of output forward rates, specified as the comma-separated pair consisting of `'OutputBasis'` and allowed values:

• 0 = actual/actual

• 1 = 30/360 (SIA)

• 2 = actual/360

• 3 = actual/365

• 4 = 30/360 (PSA)

• 5 = 30/360 (ISDA)

• 6 = 30/360 (European)

• 7 = actual/365 (Japanese)

• 8 = actual/actual (ICMA)

• 9 = actual/360 (ICMA)

• 10 = actual/365 (ICMA)

• 11 = 30/360E (ICMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

Note

If `OutputBasis` is not specified, then `OutputBasis` is assigned the value specified for `InputBasis`. If either `InputBasis` or `OutputBasis` are not specified, the default is `0` (actual/actual) for both.

Data Types: `double`

## Output Arguments

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Forward curve for the investment horizon represented by `CurveDates`, returned as a `NUMBONDS`-by-`1` vector of decimal fractions. In aggregate, the rates in `ForwardRates` constitute a forward curve over the dates in `CurveDates`. `ForwardRates` are ordered by ascending maturity.

Maturity dates that correspond to the `ForwardRates`, returned as a `NUMBONDS`-by-`1` vector of maturity dates that correspond to the `ForwardRates`.

`ForwardRates` are expressed as serial date numbers (default) or datetimes (if `CurveDates` or `Settle` are datetime arrays), representing the maturity dates for each rate in `ForwardRates`. These dates are the same dates as those associated with the input `ZeroRates`, but are ordered by ascending maturity.

## Version History

Introduced before R2006a