# inflationbuild

Build inflation curve from market zero-coupon inflation swap rates

## Syntax

``InflationCurve = inflationbuild(BaseDate,BaseIndexValue,ZCISDates,ZCISRates)``
``myInflationCurve = inflationbuild(___,Name,Value)``

## Description

example

````InflationCurve = inflationbuild(BaseDate,BaseIndexValue,ZCISDates,ZCISRates)` builds an inflation curve from market zero-coupon inflation swap (ZCIS) rates. The `InflationCurve` output is an `inflationcurve` object.```

example

````myInflationCurve = inflationbuild(___,Name,Value)` specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in the previous syntax. For example, ```myInflationCurve = inflationbuild(BaseDate,BaseIndexValue,ZCISDates,ZCISRates,'Seasonality',SeasonalRates)``` builds an `inflationcurve` object from market zero ZCIS dates and rates. ```

## Examples

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This example shows the workflow to build an `inflationcurve` object from zero-coupon inflation swap (ZCIS) rates using `inflationbuild`.

Define the inflation curve parameters.

```BaseDate = datetime(2020,9,20); BaseIndexValue = 100; ZCISTimes = [calyears([1 2 3 4 5 7 10 20 30])]'; ZCISRates = [0.51 0.65 0.87 0.92 0.95 1.42 1.75 2.03 2.54]'./100; ZCISDates = BaseDate + ZCISTimes; SeasonalRates = [-0.19 -0.09 -0.04 0.1 0.16 0.11 0.26 0.17 -0.07 -0.08 -0.14 -0.19]'./100;```

Use `inflationbuild` to create an `inflationcurve` object.

`myInflationCurve = inflationbuild(BaseDate,BaseIndexValue,ZCISDates,ZCISRates,'Seasonality',SeasonalRates)`
```myInflationCurve = inflationcurve with properties: Basis: 0 Dates: [10x1 datetime] InflationIndexValues: [10x1 double] ForwardInflationRates: [9x1 double] Seasonality: [12x1 double] ```

## Input Arguments

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Base date of inflation curve, specified as a scalar datetime, serial date number, date character vector, or date string.

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

Base index value of inflation curve, specified as a scalar numeric.

Data Types: `double`

Market ZCIS maturity dates minus lag, specified as an `NINST`-by-`1` vector of datetimes, serial date numbers, cell array of date character vectors, or date string array.

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

Market ZCIS rates, specified as an `NINST`-by-`1` vector of decimals.

Data Types: `double`

### Name-Value Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: ```myInflationCurve = inflationbuild(BaseDate,BaseIndexValue,ZCISDates,ZCISRates,'Seasonality',SeasonalRates)```

Day count basis, specified as the comma-separated pair consisting of `'Basis'` and a scalar integer.

• 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

Data Types: `double`

Seasonal adjustment rates, specified as the comma-separated pair consisting of `'Seasonality'` and a `12`-by-`1` vector in decimals for each month ordered from January to December. The rates are annualized and continuously compounded seasonal rates that are internally corrected to add to `0`.

Data Types: `double`

First month inflation index, specified as the comma-separated pair consisting of `'FirstMonthIndex'` and a positive numeric.

Data Types: `double`

## Output Arguments

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Inflation curve, returned as an `inflationcurve` object. The object has the following properties:

• `Basis`

• `Dates`

• `InflationIndexValues`

• `ForwardInflationRates`

• `Seasonality`

## Algorithms

Build an inflation curve from a series of breakeven zero-coupon inflation swap (ZCIS) rates:

`$\begin{array}{l}I\left(0,{T}_{1Y}\right)=I\left({T}_{0}\right){\left(}^{1}\\ I\left(0,{T}_{2Y}\right)=I\left({T}_{0}\right){\left(}^{1}\\ I\left(0,{T}_{3Y}\right)=I\left({T}_{0}\right){\left(}^{1}\\ ...\\ I\left(0,{T}_{i}\right)=I\left({T}_{0}\right){\left(}^{1}\end{array}$`

where

• $I\left(0,{T}_{i}\right)$ is the breakeven inflation index reference number for maturity date Ti.

• $I\left({T}_{0}\right)$ is the base inflation index value for the starting date T0.

• $b\left(0;{T}_{0},{T}_{i}\right)$ is the breakeven inflation rate for the ZCIS maturing on Ti.

The ZCIS rates typically have maturities that increase in whole number of years, so the inflation curve is built on an annual basis. From the annual basis inflation curve, the annual unadjusted (that is, not seasonally adjusted) forward inflation rates are computed as follows:

`${f}_{i}=\frac{1}{\left({T}_{i}-{T}_{i-1}\right)}\mathrm{log}\left(\frac{I\left(0,{T}_{i}\right)}{I\left(0,{T}_{i-1}\right)}\right)$`

The unadjusted forward inflation rates are used for interpolating and also for incorporating seasonality to the inflation curve.

For monthly periods that are not a whole number of years, seasonal adjustments can be made to reflect seasonal patterns of inflation within the year. These 12 monthly seasonal adjustments are annualized and they add up to zero to ensure that the cumulative seasonal adjustments are reset to zero every year.

`$\begin{array}{l}I\left(0,{T}_{i}\right)=I\left({T}_{0}\right)\mathrm{exp}\left(\underset{{T}_{0}}{\overset{{T}_{i}}{\int }}f\left(u\right)du\right)\right)\mathrm{exp}\left(\underset{{T}_{0}}{\overset{{T}_{i}}{\int }}s\left(u\right)du\right)\right)\\ I\left(0,{T}_{i}\right)=I\left(0,{T}_{i-1}\right)\mathrm{exp}\left(\left({T}_{i}-{T}_{i-1}\right)\left({f}_{i}+{s}_{i}\right)\right)\end{array}$`

where

• $I\left(0,{T}_{i}\right)$ is the breakeven inflation index reference number.

• $I\left(0,{T}_{i-1}\right)$ is the previous inflation reference number.

• fi is the annual unadjusted forward inflation rate.

• si is the annualized seasonal component for the period $\left[{T}_{i-1},{T}_{i}\right]$.

The first year seasonal adjustment may need special treatment because, typically, the breakeven inflation reference number of the first month is already known. If that is the case, the unadjusted forward inflation rate for the first year needs to be recomputed for the remaining 11 months.

Introduced in R2021a