inflationcurve

Createinflationcurveobject for interest-rate curve from dates and data

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Description

Build aninflationcurveobject usinginflationcurve.

After creating ainflationcurveobject, you can use the associated object functionindexvalues.

To price anInflationBond,YearYearInflationSwap, orZeroCouponInflationSwapinstrument, you must create aninflationcurveobject and then create anInflationpricer object.

For more information on this workflow, seeGet Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available instruments, models, and pricing methods, seeChoose Instruments, Models, and Pricers.

Creation

Syntax

inflationcurve_obj = inflationcurve(Dates,InflationIndexValues)

inflationcurve_obj = inflationcurve(___,Name,Value)

Description

example

inflationcurve_obj= inflationcurve(Dates,InflationIndexValues)creates aninflationcurveobject.

example

inflationcurve_obj= inflationcurve(___,Name,Value)creates aninflationcurveobject using name-value pairs and any of the arguments in the previous syntax. For example,myInflationCurve = inflationcurve(InflationDates,InflationIndexValues,'Basis',4)creates aninflationcurveobject. You can specify multiple name-value pair arguments.

Input Arguments

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`Dates`—Dates corresponding to`InflationIndexValues`
datetime array|serial date number|cell array of date character vectors|date string array

Dates corresponding toInflationIndexValues, specified as a datetime array, serial date numbers, cell array of date character vectors, or date string array. The first date is the base date.

If you use a date character vector or date string, the format must be recognizable bydatetimebecause theDatesproperty is stored as a datetime.

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

`InflationIndexValues`—Inflation index values for the curve
vector of positive values

Inflation index values for the curve, specified as a vector of positive values. The first value is the base index value.

Data Types:double

Name-Value Arguments

Specify optional pairs of arguments asName1=Value1,...,NameN=ValueN, whereNameis the argument name andValueis 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 encloseNamein quotes.

Example:myInflationCurve = inflationcurve(InflationDates,InflationIndexValues,'Basis',4)

`Basis`—Day count basis
`0`(actual/actual)(default) |integer from`0`to`13`

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

For more information, seeBasis.

Data Types:double

`Seasonality`—Seasonal adjustment rates
`12`-by-`1`vector of`0`s (no seasonality)(default) |decimal

Seasonal adjustment rates, specified as the comma-separated pair consisting of'Seasonality'and a12-by-1每个月从1月订购向量小数uary to December. The rates are annualized and continuously compounded seasonal rates that are internally corrected to add to0.

Data Types:double

Properties

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`Basis`—Day count basis
`0`(actual/actual)(default) |integer from`0`to`13`

Day count basis of the instrument, returned as a scalar integer.

Data Types:double

`Dates`—Dates corresponding to`InflationIndexValues`
datetime

Dates corresponding toInflationIndexValues, returned as a datetime array.

Data Types:datetime

`InflationIndexValues`—Inflation index values for the curve
vector

Inflation index values for the curve, returned as vector.

Data Types:double

`ForwardInflationRates`—Forward inflation rates
vector

Forward inflation rates, returned as vector.

Data Types:double

`Seasonality`—Seasonal adjustment rates
vector

Seasonal adjustment rates, returned as a12-by-1vector.

Data Types:double

Object Functions

indexvalues Calculate index values forinflationcurveobject

Examples

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Create`inflationcurve`Object

Open Live Script

Create aninflationcurveobject usinginflationcurve.

BaseDate = datetime(2020, 9, 20); InflationTimes = [0 calyears([1 2 3 4 5 7 10 20 30])]'; InflationIndexValues = [100 102 103.5 105 106.8 108.2 111.3 120.1 130.4 150.2]'; InflationDates = BaseDate + InflationTimes; myInflationCurve = inflationcurve(InflationDates,InflationIndexValues)

myInflationCurve = inflationcurve with properties: Basis: 0 Dates: [10x1 datetime] InflationIndexValues: [10x1 double] ForwardInflationRates: [9x1 double] Seasonality: [12x1 double]

Algorithms

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

$\begin{array}{l} I (0, T_{1 Y}) = I (T_{0}) (^{1 + b (0; T_{0}, T_{1 Y})) T_{1 Y} - T_{0}} \\ I (0, T_{2 Y}) = I (T_{0}) (^{1 + b (0; T_{0}, T_{2 Y})) T_{2 Y} - T_{0}} \\ I (0, T_{3 Y}) = I (T_{0}) (^{1 + b (0; T_{0}, T_{3 Y})) T_{3 Y} - T_{0}} \\ ... \\ I (0, T_{i}) = I (T_{0}) (^{1 + b (0; T_{0}, T_{i})) T_{i} - T_{0}} \end{array}$

where

$I (0, T_{i})$ is the breakeven inflation index reference number for maturity dateT_i.
$I (T_{0})$ is the base inflation index value for the starting dateT₀.
$b (0; T_{0}, T_{i})$ is the breakeven inflation rate for the ZCIS maturing onT_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:

$f_{i} = \frac{1}{(T_{i} - T_{i - 1})} \log (\frac{I (0, T_{i})}{I (0, T_{i - 1})})$

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.

$\begin{array}{l} I (0, T_{i}) = I (T_{0}) \exp (\int_{T_{0}}^{T_{i}} f (u) d u)) \exp (\int_{T_{0}}^{T_{i}} s (u) d u)) \\ I (0, T_{i}) = I (0, T_{i - 1}) \exp ((T_{i} - T_{i - 1}) (f_{i} + s_{i})) \end{array}$

where

$I (0, T_{i})$ 损益平衡通货膨胀指数参考号码。
$I (0, T_{i - 1})$ is the previous inflation reference number.
f_iis the annual unadjusted forward inflation rate.
s_iis 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.

References

[1] Brody, D. C., Crosby, J., and Li, H. "Convexity Adjustments in Inflation-Linked Derivatives."Risk Magazine. November 2008, pp. 124–129.

[2] Kerkhof, J. "Inflation Derivatives Explained: Markets, Products, and Pricing."Fixed Income Quantitative Research, Lehman Brothers, July 2005.

[3] Zhang, J. X. "Limited Price Indexation (LPI) Swap Valuation Ideas."Wilmott Magazine. no. 57, January 2012, pp. 58–69.

Version History

Introduced in R2021a

inflationcurve

Description