# InflationBond

## Description

Create and price an `InflationBond`

instrument object for one
or more Inflation Bond instruments using this workflow:

Use

`fininstrument`

to create an`InflationBond`

instrument object for one or more Inflation Bond instruments.Use

`ratecurve`

to specify an interest-rate model for the`InflationBond`

instrument object.Use

`inflationcurve`

to specify an inflation curve model for the`InflationBond`

instrument object.Use

`finpricer`

to specify an`Inflation`

pricing method for one or more`InflationBond`

instruments.Use

`inflationCashflows`

to compute cash flows for each one of the`InflationBond`

instruments.

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

For more information on the available models and pricing methods for an
`InflationBond`

instrument, see Choose Instruments, Models, and Pricers.

## Creation

### Syntax

### Description

creates an `InflationBond`

= fininstrument(`InstrumentType`

,'`CouponRate`

',couponrate_value,'`Maturity`

',maturity_date)`InflationBond`

object for one or more Inflation
Bond instruments by specifying `InstrumentType`

and sets
the properties for
the required name-value pair arguments `CouponRate`

and
`Maturity`

.

sets optional properties using
additional name-value pairs in addition to the required arguments in the
previous syntax. For example, `InflationBond`

= fininstrument(___,`Name,Value`

)```
InflationBond =
fininstrument("InflationBond",'Maturity',Maturity,'CouponRate',CouponRate,'IssueDate',IssueDate)
```

creates a `InflationBond`

option.

### Input Arguments

`InstrumentType`

— Instrument type

string with value `"InflationBond"`

| string array with values of
`"InflationBond"`

| character vector with value
`'InflationBond'`

| cell array of character vectors with values of
`'InflationBond'`

Instrument type, specified as a string with the value of
`"InflationBond"`

, a character vector with the
value of `'InflationBond'`

, an
`NINST`

-by-`1`

string array with
values of `"InflationBond"`

, or an
`NINST`

-by-`1`

cell array of
character vectors with values of `'InflationBond'`

.

**Data Types: **`char`

| `cell`

| `string`

**Name-Value Arguments**

Specify required
and 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: **```
InflationBond =
fininstrument("InflationBond",'Maturity',Maturity,'CouponRate',CouponRate,'IssueDate',IssueDate)
```

**Required**

`InflationBond`

Name-Value Pair
Arguments`CouponRate`

— `InflationBond`

coupon rate

scalar decimal | vector of decimals

`InflationBond`

coupon rate, specified as the
comma-separated pair consisting of `'CouponRate'`

and a scalar decimal or an
`NINST`

-by-`1`

vector of
decimals for an annual rate.

**Data Types: **`double`

`Maturity`

— `InflationBond`

maturity date

datetime array | string array | date character vector

`InflationBond`

maturity date, specified as the
comma-separated pair consisting of `'Maturity'`

and
a scalar or an `NINST`

-by-`1`

vector using a datetime array, string array, or date character
vectors.

To support existing code, `InflationBond`

also
accepts serial date numbers as inputs, but they are not recommended.

If you use date character vectors or strings, the format must be
recognizable by `datetime`

because
the `Maturity`

property is stored as a
datetime.

**Optional**

`InflationBond`

Name-Value Pair
Arguments`Period`

— Frequency of payments per year

`2`

(default) | scalar numeric with value of `0`

,
`1`

, `2`

,
`3`

, `4`

, `6`

, or
`12`

| numeric vector with values of `0`

,
`1`

, `2`

,
`3`

, `4`

, `6`

, or
`12`

Frequency of payments, specified as the comma-separated pair
consisting of `'Period'`

and a scalar integer or an
`NINST`

-by-`1`

vector of
integers. Values for `Period`

are
`1`

, `2`

,
`3`

, `4`

, `6`

, or
`12`

.

**Data Types: **`double`

`Basis`

— Day count basis

`0`

(actual/actual) (default) | scalar integer from `0`

to
`13`

| vector of integers from `0`

to
`13`

Day count basis, specified as the comma-separated pair consisting
of `'Basis'`

and scalar integer or an
`NINST`

-by-`1`

vector of
integers using the following 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

For more information, see Basis.

**Data Types: **`double`

`Principal`

— Initial principal amount

`100`

(default) | scalar numeric | numeric vector

Initial principal amount, specified as the comma-separated pair
consisting of `'Principal'`

and a scalar numeric or
an `NINST`

-by-`1`

numeric vector.

**Data Types: **`double`

`DaycountAdjustedCashFlow`

— Flag indicating whether cash flow adjusts for day count convention

`false`

(default) | scalar logical value of `true`

or
`false`

| vector of logical values of `true`

or
`false`

Flag indicating whether cash flow is adjusted by day count
convention, specified as the comma-separated pair consisting of
`'DaycountAdjustedCashFlow'`

and a scalar
logical or an `NINST`

-by-`1`

vector of logicals with values of `true`

or
`false`

.

**Data Types: **`logical`

`IssueDate`

— Bond issue date

`NaT`

(default) | datetime array | string array | date character vector

Bond issue date, specified as the comma-separated pair consisting
of `'IssueDate'`

and a scalar or an
`NINST`

-by-`1`

vector using a
datetime array, string array, or date character vectors.

To support existing code, `InflationBond`

also
accepts serial date numbers as inputs, but they are not recommended.

If you use date character vectors or strings, the format must be
recognizable by `datetime`

because
the `IssueDate`

property is stored as a
datetime.

`Lag`

— Indexation lag in months

`3`

(default) | scalar numeric | numeric vector

Indexation lag in months, specified as the comma-separated pair
consisting of `'Lag'`

and a scalar numeric or an
`NINST`

-by-`1`

numeric
vector.

**Data Types: **`double`

`BusinessDayConvention`

— Business day conventions for cash flow dates

`"actual"`

(default) | string | string array | character vector | cell array of character vectors

Business day conventions for cash flow dates, specified as the
comma-separated pair consisting of
`'BusinessDayConvention'`

and a scalar string
or character vector or an
`NINST`

-by-`1`

cell array of
character vectors or string array. The selection for business day
convention determines how nonbusiness days are treated. Nonbusiness
days are defined as weekends plus any other date that businesses are
not open (for example, statutory holidays). Values are:

`"actual"`

— Nonbusiness days are effectively ignored. Cash flows that fall on nonbusiness days are assumed to be distributed on the actual date.`"follow"`

— Cash flows that fall on a nonbusiness day are assumed to be distributed on the following business day.`"modifiedfollow"`

— Cash flows that fall on a nonbusiness day are assumed to be distributed on the following business day. However, if the following business day is in a different month, the previous business day is adopted instead.`"previous"`

— Cash flows that fall on a nonbusiness day are assumed to be distributed on the previous business day.`"modifiedprevious"`

— Cash flows that fall on a nonbusiness day are assumed to be distributed on the previous business day. However, if the previous business day is in a different month, the following business day is adopted instead.

**Data Types: **`char`

| `cell`

| `string`

`Holidays`

— Holidays used in computing business days

`NaT`

(default) | datetime array | string array | date character vector

Holidays used in computing business days, specified as the
comma-separated pair consisting of `'Holidays'`

and
dates using an `NINST`

-by-`1`

vector of a datetime array, string array, or date character vectors.
For
example:

H = holidays(datetime('today'),datetime(2025,12,15)); InflationBondObj = fininstrument("InflationBond",'CouponRate',0.34,'Maturity',datetime(2025,12,15),'Holidays',H)

To support existing code, `InflationBond`

also
accepts serial date numbers as inputs, but they are not recommended.

`EndMonthRule`

— End-of-month rule flag for generating dates when `Maturity`

is end-of-month date for month with 30
or fewer days

`true`

(in effect) (default) | scalar logical values of `true`

or
`false`

| vector of logical values with `true`

or
`false`

End-of-month rule flag for generating dates when
`Maturity`

is an end-of-month date for a month
with 30 or fewer days, specified as the comma-separated pair
consisting of `'EndMonthRule'`

and a scalar logical
value or an `NINST`

-by-`1`

vector
of logical values of `true`

or
`false`

.

If you set

`EndMonthRule`

to`false`

, the software ignores the rule, meaning that a payment date is always the same numerical day of the month.If you set

`EndMonthRule`

to`true`

, the software sets the rule on, meaning that a payment date is always the last actual day of the month.

**Data Types: **`logical`

`FirstCouponDate`

— Irregular first coupon date

`NaT`

(default) | datetime array | string array | date character vector

Irregular first coupon date, specified as the comma-separated pair
consisting of `'FirstCouponDate'`

and a scalar or
an `NINST`

-by-`1`

vector using a
datetime array, string array, or date character vectors.

`InflationBond`

also
accepts serial date numbers as inputs, but they are not recommended.

When `FirstCouponDate`

and
`LastCouponDate`

are both specified,
`FirstCouponDate`

takes precedence in
determining the coupon payment structure. If you do not specify
`FirstCouponDate`

, the cash flow payment dates
are determined from other inputs.

If you use date character vectors or strings, the format must be
recognizable by `datetime`

because
the `FirstCouponDate`

property is stored as a
datetime.

`LastCouponDate`

— Irregular last coupon date

`NaT`

(default) | datetime array | string array | date character vector

Irregular last coupon date, specified as the comma-separated pair
consisting of `'LastCouponDate'`

and a scalar or an
`NINST`

-by-`1`

vector using a
datetime array, string array, or date character vectors.

`InflationBond`

also
accepts serial date numbers as inputs, but they are not recommended.

If you specify `LastCouponDate`

but not
`FirstCouponDate`

,
`LastCouponDate`

determines the coupon
structure of the bond. The coupon structure of a bond is truncated
at `LastCouponDate`

, regardless of where it falls,
and is followed only by the bond's maturity cash flow date. If you
do not specify `LastCouponDate`

, the cash flow
payment dates are determined from other inputs.

If you use date character vectors or strings, the format must be
recognizable by `datetime`

because
the `LastCouponDate`

property is stored as a
datetime.

`Name`

— User-defined name for instrument

`" "`

(default) | string | string array | character vector | cell array of character vectors

User-defined name for one of more instruments, specified as the
comma-separated pair consisting of `'Name'`

and a
scalar string or character vector or an
`NINST`

-by-`1`

cell array of
character vectors or string array.

**Data Types: **`char`

| `cell`

| `string`

## Properties

`CouponRate`

— `InflationBond`

coupon annual rate

scalar decimal | vector of decimals

`InflationBond`

coupon annual rate, returned as a scalar
decimal or an `NINST`

-by-`1`

vector of
decimals.

**Data Types: **`double`

`Maturity`

— `InflationBond`

maturity date

datetime | vector of datetimes

`InflationBond`

maturity date, returned as a scalar
datetime or an `NINST`

-by-`1`

vector of
datetimes.

**Data Types: **`datetime`

`Period`

— Frequency of payments per year

`2`

(default) | scalar integer | vector of integers

Frequency of payments per year, returned as a scalar integer or an
`NINST`

-by-`1`

vector of
integers.

**Data Types: **`double`

`Basis`

— Day count basis

`0`

(actual/actual) (default) | scalar integer from `0`

to `13`

| vector of integers from `0`

to
`13`

Day count basis, returned as a scalar integer or an
`NINST`

-by-`1`

vector of integers.

**Data Types: **`double`

`Principal`

— Initial principal amount

`100`

(default) | scalar numeric | numeric vector

Initial principal amount, returned as a scalar numeric or an
`NINST`

-by-`1`

numeric vector.

**Data Types: **`double`

`DaycountAdjustedCashFlow`

— Flag indicating whether cash flow adjusts for day count convention

`false`

(default) | scalar logical value of `true`

or
`false`

| vector of logical values with `true`

or
`false`

Flag indicating whether cash flow adjusts for day count convention,
returned as scalar logical or an
`NINST`

-by-`1`

vector of logicals with
values of `true`

or `false`

.

**Data Types: **`logical`

`IssueDate`

— Bond issue date

`NaT`

(default) | datetime | vector of datetimes

Bond issue date, returned as a datetime or an
`NINST`

-by-`1`

datetime vector.

**Data Types: **`datetime`

`Lag`

— Indexation lag in months

`3`

(default) | scalar numeric | numeric vector

Indexation lag in months, returned as a scalar numeric or an
`NINST`

-by-`1`

numeric vector.

**Data Types: **`double`

`BusinessDayConvention`

— Business day conventions

`"actual"`

(default) | string | string array

Business day conventions, returned as a scalar string or an
`NINST`

-by-`1`

string array.

**Data Types: **`string`

`Holidays`

— Holidays used in computing business days

`NaT`

(default) | datetimes

Holidays used in computing business days, returned as an
`NINST`

-by-`1`

vector of
datetimes.

**Data Types: **`datetime`

`EndMonthRule`

— End-of-month rule flag for generating dates when `Maturity`

is end-of-month date for month with 30 or fewer
days

`true`

(in effect) (default) | scalar logical value of `true`

or
`false`

| vector of logicals with value of `true`

or
`false`

End-of-month rule flag for generating dates when
`Maturity`

is an end-of-month date for a month having
30 or fewer days, returned as a scalar logical or an
`NINST`

-by-`1`

vector of
logicals.

**Data Types: **`logical`

`FirstCouponDate`

— Irregular first coupon date

`NaT`

(default) | datetime | vector of datetimes

Irregular first coupon date, returned as a scalar datetime or an
`NINST`

-by-`1`

vector of datetimes.

**Data Types: **`datetime`

`LastCouponDate`

— Irregular last coupon date

`NaT`

(default) | datetime | vector of datetimes

Irregular last coupon date, returned as a scalar datetime or an
`NINST`

-by-`1`

vector of
datetimes.

**Data Types: **`datetime`

`Name`

— User-defined name for instrument

`" "`

(default) | string | string array

User-defined name for the instrument, returned as a scalar string or an
`NINST`

-by-`1`

string array.

**Data Types: **`string`

## Object Functions

`inflationCashflows` | Compute cash flows for `InflationBond` instrument |

## Examples

### Price Inflation Bond Instrument Using `inflationcurve`

and Inflation Pricer

This example shows the workflow to price an `InflationBond`

instrument when you use an `inflationcurve`

and an `Inflation`

pricing method.

**Create ratecurve Object**

Create a `ratecurve`

object using `ratecurve`

.

Settle = datetime(2021,1,15); Type = "zero"; ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]'; ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]'; ZeroDates = Settle + ZeroTimes; ZeroCurve = ratecurve('zero',Settle,ZeroDates,ZeroRates)

ZeroCurve = ratecurve with properties: Type: "zero" Compounding: -1 Basis: 0 Dates: [10x1 datetime] Rates: [10x1 double] Settle: 15-Jan-2021 InterpMethod: "linear" ShortExtrapMethod: "next" LongExtrapMethod: "previous"

**Create inflationcurve Object**

Create an `inflationcurve`

object using `inflationcurve`

.

BaseDate = datetime(2020,10,1); 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]

**Create InflationBond Instrument Object**

Use `fininstrument`

to create an `InflationBond`

instrument object.

IssueDate = datetime(2021,1,1); Maturity = datetime(2026,1,1); CouponRate = 0.02; InflationBond = fininstrument("InflationBond", 'IssueDate', IssueDate, 'Maturity', Maturity, 'CouponRate', CouponRate,'Name',"inflation_bond_instrument")

InflationBond = InflationBond with properties: CouponRate: 0.0200 Period: 2 Basis: 0 Principal: 100 DaycountAdjustedCashFlow: 0 Lag: 3 BusinessDayConvention: "actual" Holidays: NaT EndMonthRule: 1 IssueDate: 01-Jan-2021 FirstCouponDate: NaT LastCouponDate: NaT Maturity: 01-Jan-2026 Name: "inflation_bond_instrument"

**Create Inflation Pricer Object**

Use `finpricer`

to create an `Inflation`

pricer object and use the `ratecurve`

object with the `'DiscountCurve'`

name-value pair argument and the `inflationcurve`

object with the `'InflationCurve'`

name-value pair argument.

outPricer = finpricer("Inflation",'DiscountCurve',ZeroCurve,'InflationCurve',myInflationCurve)

outPricer = Inflation with properties: DiscountCurve: [1x1 ratecurve] InflationCurve: [1x1 inflationcurve]

**Price InflationBond Instrument**

Use `price`

to compute the price and sensitivities for the `InflationBond`

instrument.

[Price, outPR] = price(outPricer, InflationBond)

Price = 112.1856

outPR = priceresult with properties: Results: [1x1 table] PricerData: []

outPR.Results

`ans=`*table*
Price
______
112.19

### Price Multiple Inflation Bond Instruments Using `inflationcurve`

and Inflation Pricer

This example shows the workflow to price multiple `InflationBond`

instruments when you use an `inflationcurve`

and an `Inflation`

pricing method.

**Create ratecurve Object**

Create a `ratecurve`

object using `ratecurve`

.

Settle = datetime(2020,1,15); Type = "zero"; ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]'; ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]'; ZeroDates = Settle + ZeroTimes; ZeroCurve = ratecurve('zero',Settle,ZeroDates,ZeroRates)

ZeroCurve = ratecurve with properties: Type: "zero" Compounding: -1 Basis: 0 Dates: [10x1 datetime] Rates: [10x1 double] Settle: 15-Jan-2020 InterpMethod: "linear" ShortExtrapMethod: "next" LongExtrapMethod: "previous"

**Create inflationcurve Object**

Create an `inflationcurve`

object using `inflationcurve`

.

BaseDate = datetime(2019,8,1); 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]

**Create InflationBond Instrument Object**

Use `fininstrument`

to create an `InflationBond`

instrument object for three Inflation Bond instruments.

IssueDate = datetime([2020,1,1 ; 2019,12,1 ; 2019,11,1]); Maturity = datetime([2026,1,1 ; 2026,2,1 ; 2026,3,1]); CouponRate = 0.02; InflationBond = fininstrument("InflationBond", 'IssueDate', IssueDate, 'Maturity', Maturity, 'CouponRate', CouponRate,'Name',"inflation_bond_instrument")

`InflationBond=`*3×1 object*
3x1 InflationBond array with properties:
CouponRate
Period
Basis
Principal
DaycountAdjustedCashFlow
Lag
BusinessDayConvention
Holidays
EndMonthRule
IssueDate
FirstCouponDate
LastCouponDate
Maturity
Name

**Create Inflation Pricer Object**

Use `finpricer`

to create an `Inflation`

pricer object and use the `ratecurve`

object with the `'DiscountCurve'`

name-value pair argument and the `inflationcurve`

object with the `'InflationCurve'`

name-value pair argument.

outPricer = finpricer("Inflation",'DiscountCurve',ZeroCurve,'InflationCurve',myInflationCurve)

outPricer = Inflation with properties: DiscountCurve: [1x1 ratecurve] InflationCurve: [1x1 inflationcurve]

**Price InflationBond Instruments**

Use `price`

to compute the prices and sensitivities for the `InflationBond`

instruments.

[Price, outPR] = price(outPricer, InflationBond)

`Price = `*3×1*
112.8769
113.1022
113.3434

`outPR=`*1×3 object*
1x3 priceresult array with properties:
Results
PricerData

outPR.Results

`ans=`*table*
Price
______
112.88

`ans=`*table*
Price
_____
113.1

`ans=`*table*
Price
______
113.34

## More About

### Inflation-Indexed Bond

An *inflation-indexed bond* is a security
that guarantees a return higher than the rate of inflation if it is held to
maturity. Inflation-indexed securities link their capital appreciation, or coupon
payments, to inflation rates

## Algorithms

To price an inflation-indexed bond, use an inflation curve and a nominal discount curve (model-free approach), where the cash flows are discounted using the nominal discount curve.

$$\begin{array}{l}I(0,T){P}_{n}(0,T)=I(0){P}_{r}(0,T)\\ {B}_{TIPS}(0,{T}_{M})=\frac{1}{I({T}_{0})}{\displaystyle \sum _{i=1}^{M}cI(0){P}_{r}(0,{T}_{i})+FI(0){P}_{r}(0,{T}_{M})}\\ \text{}=\frac{1}{I({T}_{0})}{\displaystyle \sum _{i=1}^{M}cI(0,{T}_{i}){P}_{n}(0,{T}_{i})+FI(0,{T}_{M})}{P}_{n}(0,{T}_{M})\end{array}$$

where

*P*_{n}is the nominal zero-coupon bond price.*P*_{r}is the real zero-coupon bond price.*k*is the fixed inflation rate.*I*(0,*T*) is the breakeven inflation index for period (0,*T*).*I*(0) is the inflation index at (*t*= 0).*I*(*T*_{0}) is the base inflation index at the issue date (t =*T*_{0}).*B*_{TIPS}(0,*T*_{M}) is the inflation-indexed bond price.*I*(*T*_{i-1}) is the inflation index at the start date with some lag (for example, three months).*C*is the coupon.*F*is the face value.

## 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] Treasury Inflation-Protected Securities (TIPS) at: https://www.treasurydirect.gov/.

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

## Version History

**Introduced in R2021a**

### R2022b: Serial date numbers not recommended

Although `InflationBond`

supports serial date numbers,
`datetime`

values are recommended instead. The
`datetime`

data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.

To convert serial date numbers or text to `datetime`

values, use the `datetime`

function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)

y = 2021

There are no plans to remove support for serial date number inputs.

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