# fixedbycir

Price fixed rate note from Cox-Ingersoll-Ross interest-rate tree

## Syntax

``````[Price,PriceTree] = fixedbycir(CIRTree,CouponRate,Settle,Maturity)``````
``````[Price,PriceTree] = fixedbycir(___,Name,Value)``````

## Description

example

``````[Price,PriceTree] = fixedbycir(CIRTree,CouponRate,Settle,Maturity)``` prices a fixed-rate note from a Cox-Ingersoll-Ross (CIR) interest-rate tree using a CIR++ model with the Nawalka-Beliaeva (NB) approach.```

example

``````[Price,PriceTree] = fixedbycir(___,Name,Value)``` adds additional name-value pair arguments.```

## Examples

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Define the `CouponRate` for a fixed-rate note.

`CouponRate = 0.03;`

Create a `RateSpec` using the `intenvset` function.

```Rates = [0.035; 0.042147; 0.047345; 0.052707]; Dates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)]; ValuationDate = datetime(2017,1,1); EndDates = Dates(2:end)'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding); ```

Create a `CIR` tree.

```NumPeriods = length(EndDates); Alpha = 0.03; Theta = 0.02; Sigma = 0.1; Settle = datetime(2017,1,1); Maturity = datetime(2021,1,1); CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)```
```CIRT = struct with fields: FinObj: 'CIRFwdTree' VolSpec: [1x1 struct] TimeSpec: [1x1 struct] RateSpec: [1x1 struct] tObs: [0 1 2 3] dObs: [736696 737061 737426 737791] FwdTree: {1x4 cell} Connect: {[3x1 double] [3x3 double] [3x5 double]} Probs: {[3x1 double] [3x3 double] [3x5 double]} ```

Price the 3% fixed-rate note.

`[Price,PriceTree] = fixedbycir(CIRT,CouponRate,Settle,Maturity) `
```Price = 92.1422 ```
```PriceTree = struct with fields: FinObj: 'CIRPriceTree' tObs: [0 1 2 3 4] dObs: [736696 737061 737426 737791 738157] PTree: {1x5 cell} AITree: {[0] [0 0 0] [0 0 0 0 0] [0 0 0 0 0 0 0] [0 0 0 0 0 0 0]} Connect: {[3x1 double] [3x3 double] [3x5 double]} ```

## Input Arguments

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Interest-rate tree structure, created by `cirtree`

Data Types: `struct`

Coupon annual rate, specified as a `NINST`-by-`1` vector.

Data Types: `double`

Settlement date, specified either as a scalar or a `NINST`-by-`1` vector using a datetime array, string array, or date character vectors.

To support existing code, `fixedbycir` also accepts serial date numbers as inputs, but they are not recommended.

The `Settle` date for every fixed-rate note is set to the `ValuationDate` of the CIR tree. The fixed-rate note argument `Settle` is ignored.

Maturity date, specified as a `NINST`-by-`1` vector using a datetime array, string array, or date character vectors representing the maturity date for each fixed-rate note.

To support existing code, `fixedbycir` also accepts serial date numbers as inputs, but they are not recommended.

### 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: ```[Price,PriceTree] = fixedbycir(CIRTree,CouponRate,Settle,Maturity,'FixedReset',4)```

Frequency of payments per year, specified as the comma-separated pair consisting of `'FixedReset'` and a `NINST`-by-`1` vector.

Data Types: `double`

Day count basis representing the basis used when annualizing the input forward rate tree, specified as the comma-separated pair consisting of `'Basis'` and a `NINST`-by-`1` vector.

• 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`

Notional principal amounts, specified as the comma-separated pair consisting of `'Principal'` and a vector or cell array.

`Principal` accepts a `NINST`-by-`1` vector or `NINST`-by-`1` cell array, where each element of the cell array is a `NumDates`-by-`2` cell array and the first column is dates and the second column is its associated notional principal value. The date indicates the last day that the principal value is valid.

Data Types: `cell` | `double`

End-of-month rule flag for generating dates when `Maturity` is an end-of-month date for a month having 30 or fewer days, specified as the comma-separated pair consisting of `'EndMonthRule'` and a nonnegative integer [`0`, `1`] using a `NINST`-by-`1` vector.

• `0` = Ignore rule, meaning that a payment date is always the same numerical day of the month.

• `1` = Set rule on, meaning that a payment date is always the last actual day of the month.

Data Types: `logical`

Flag to adjust cash flows based on actual period day count, specified as the comma-separated pair consisting of `'AdjustCashFlowsBasis'` and a `NINST`-by-`1` vector of logicals with values of `0` (false) or `1` (true).

Data Types: `logical`

Holidays used in computing business days, specified as the comma-separated pair consisting of `'Holidays'` and MATLAB dates using a `NHolidays`-by-`1` vector.

Data Types: `datetime`

Business day conventions, specified as the comma-separated pair consisting of `'BusinessDayConvention'` and a character vector or a `N`-by-`1` cell array of character vectors of business day conventions. 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 (e.g. 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 non-business day are assumed to be distributed on the following business day.

• `modifiedfollow` — Cash flows that fall on a non-business 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 non-business day are assumed to be distributed on the previous business day.

• `modifiedprevious` — Cash flows that fall on a non-business 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`

## Output Arguments

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Expected fixed-rate note prices at time 0, returned as a `NINST`-by-`1` vector.

Tree structure of instrument prices, returned as a MATLAB structure of trees containing vectors of instrument prices and accrued interest, and a vector of observation times for each node. Within `PriceTree`:

• `PriceTree.tObs` contains the observation times.

• `PriceTree.dObs` contains the observation dates.

• `PriceTree.PTree` contains the clean prices.

• `PriceTree.AITree` contains the accrued interest.

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### Fixed-Rate Note

A fixed-rate note is a long-term debt security with a preset interest rate and maturity, by which the interest must be paid.

The principal may or may not be paid at maturity. In Financial Instruments Toolbox™, the principal is always paid at maturity. For more information, see Fixed-Rate Note.

## References

[1] Cox, J., Ingersoll, J.,and S. Ross. "A Theory of the Term Structure of Interest Rates." Econometrica. Vol. 53, 1985.

[2] Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance, 2006.

[3] Hirsa, A. Computational Methods in Finance. CRC Press, 2012.

[4] Nawalka, S., Soto, G., and N. Beliaeva. Dynamic Term Structure Modeling. Wiley, 2007.

[5] Nelson, D. and K. Ramaswamy. "Simple Binomial Processes as Diffusion Approximations in Financial Models." The Review of Financial Studies. Vol 3. 1990, pp. 393–430.

## Version History

Introduced in R2018a

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