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Build inflation curve from market zero-coupon inflation swap rates

builds an inflation curve from market zero-coupon inflation swap (ZCIS) rates. The
`InflationCurve`

= inflationbuild(`BaseDate`

,`BaseIndexValue`

,`ZCISDates`

,`ZCISRates`

)`InflationCurve`

output is an `inflationcurve`

object.

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(___,`Name,Value`

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

builds an `inflationcurve`

object from market zero ZCIS dates and rates.

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

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

where

$$I(0,{T}_{i})$$ is the breakeven inflation index reference number for maturity date

*T*_{i}.$$I({T}_{0})$$ is the base inflation index value for the starting date

*T*_{0}.$$b(0;{T}_{0},{T}_{i})$$ is the breakeven inflation rate for the ZCIS maturing on

*T*_{i}.

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:

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

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.

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

where

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

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

*f*_{i}is the annual unadjusted forward inflation rate.*s*_{i}is the annualized seasonal component for the period $$[{T}_{i-1},{T}_{i}]$$.

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.