inflationcurve

Create inflationcurve object for interest-rate curve from dates and data

Since R2021a

Description

Build an inflationcurve object using inflationcurve.

After creating a inflationcurve object, you can use the associated object function indexvalues.

To price an InflationBond, YearYearInflationSwap, or ZeroCouponInflationSwap instrument, you must create an inflationcurve object and then create an Inflation pricer object.

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

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

Creation

Syntax

inflationcurve_obj = inflationcurve(Dates,InflationIndexValues)

inflationcurve_obj = inflationcurve(___,Name,Value)

Description

inflationcurve_obj = inflationcurve(Dates,InflationIndexValues) creates an inflationcurve object.

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

Input Arguments

Dates — Dates corresponding to InflationIndexValues

datetime array | serial date number | cell array of date character vectors | date string array

Dates corresponding to InflationIndexValues, 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 by datetime because the Dates property 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 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: 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, see Basis.

Data Types: double

Seasonality — Seasonal adjustment rates

12-by-1 vector of 0s (no seasonality) (default) | decimal

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

Properties

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 to InflationIndexValues, 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 a 12-by-1 vector.

Data Types: double

Object Functions

indexvalues

Calculate index values for inflationcurve object

Examples

Create inflationcurve Object

Create an inflationcurve object using inflationcurve.

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:

$$\begin{array}{l}I(0,T_{1Y})=I(T_0){(}^1\\ I(0,T_{2Y})=I(T_0){(}^1\\ I(0,T_{3Y})=I(T_0){(}^1\\ \mathrm{...}\\ I(0,T_i)=I(T_0){(}^1\end{array}$$

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 T₀.

• $$b(0;T_0,T_i)$$ 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:

$$f_i=\frac{1}{(T_i-T_{i-1})}\log \left(\frac{I(0,T_i)}{I(0,T_{i-1})}\right)$$

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.

$$\begin{array}{l}I(0,T_i)=I(T_0)\exp \left({\displaystyle \underset{T_0}{\overset{T_i}{\int }}f(u)du)}\right)\exp \left({\displaystyle \underset{T_0}{\overset{T_i}{\int }}s(u)du)}\right)\\ I(0,T_i)=I(0,T_{i-1})\exp ((T_i-T_{i-1})(f_i+s_i))\end{array}$$

where

• $$I(0,T_i)$$ 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.

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