# bin

Binomial test for value-at-risk (VaR) backtesting

## Syntax

``TestResults = bin(vbt)``
``TestResults = bin(vbt,Name,Value)``

## Description

example

````TestResults = bin(vbt)` generates the binomial test results for value-at-risk (VaR) backtesting.```

example

````TestResults = bin(vbt,Name,Value)` adds an optional name-value pair argument for `TestLevel`.```

## Examples

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Create a `varbacktest` object.

```load VaRBacktestData vbt = varbacktest(EquityIndex,Normal95)```
```vbt = varbacktest with properties: PortfolioData: [1043x1 double] VaRData: [1043x1 double] PortfolioID: "Portfolio" VaRID: "VaR" VaRLevel: 0.9500 ```

Generate the `bin` test results.

`TestResults = bin(vbt)`
```TestResults=1×9 table PortfolioID VaRID VaRLevel Bin ZScoreBin PValueBin Observations Failures TestLevel ___________ _____ ________ ______ _________ _________ ____________ ________ _________ "Portfolio" "VaR" 0.95 accept 0.68905 0.49079 1043 57 0.95 ```

Use the `varbacktest` constructor with name-value pair arguments to create a `varbacktest` object.

```load VaRBacktestData vbt = varbacktest(EquityIndex,... [Normal95 Normal99 Historical95 Historical99 EWMA95 EWMA99],... 'PortfolioID','Equity',... 'VaRID',{'Normal95' 'Normal99' 'Historical95' 'Historical99' 'EWMA95' 'EWMA99'},... 'VaRLevel',[0.95 0.99 0.95 0.99 0.95 0.99])```
```vbt = varbacktest with properties: PortfolioData: [1043x1 double] VaRData: [1043x6 double] PortfolioID: "Equity" VaRID: ["Normal95" "Normal99" "Historical95" ... ] VaRLevel: [0.9500 0.9900 0.9500 0.9900 0.9500 0.9900] ```

Generate the `bin` test results using the `TestLevel` optional argument.

`TestResults = bin(vbt,'TestLevel',0.90)`
```TestResults=6×9 table PortfolioID VaRID VaRLevel Bin ZScoreBin PValueBin Observations Failures TestLevel ___________ ______________ ________ ______ _________ _________ ____________ ________ _________ "Equity" "Normal95" 0.95 accept 0.68905 0.49079 1043 57 0.9 "Equity" "Normal99" 0.99 reject 2.0446 0.040896 1043 17 0.9 "Equity" "Historical95" 0.95 accept 0.9732 0.33045 1043 59 0.9 "Equity" "Historical99" 0.99 accept 0.48858 0.62514 1043 12 0.9 "Equity" "EWMA95" 0.95 accept 0.9732 0.33045 1043 59 0.9 "Equity" "EWMA99" 0.99 reject 3.6006 0.0003175 1043 22 0.9 ```

## Input Arguments

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`varbacktest` (`vbt`) object, contains a copy of the given data (the `PortfolioData` and `VarData` properties) and all combinations of portfolio ID, VaR ID, and VaR levels to be tested. For more information on creating a `varbacktest` object, see `varbacktest`.

### Name-Value Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: ```TestResults = bin(vbt,'TestLevel',0.99)```

Test confidence level, specified as the comma-separated pair consisting of `'TestLevel'` and a numeric between `0` and `1`.

Data Types: `double`

## Output Arguments

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Bin test results, returned as a table where the rows correspond to all combinations of portfolio ID, VaR ID, and VaR levels to be tested. The columns correspond to the following information:

• `'PortfolioID'` — Portfolio ID for given data

• `'VaRID'` — VaR ID for each of the VaR data columns provided

• `'VaRLevel'` — VaR level for corresponding VaR data column

• `'Bin'` — Categorical array with categories `accept` and `reject` that indicate the result of the `bin` test

• `'ZScoreBin'` — Z-score of the number of failures

• `'PValueBin'` — P-value of the `bin` test

• `'Observations'` — Number of observations

• `'Failures'` — Number of failures.

• `'TestLevel'` — Test confidence level.

Note

For `bin` test results, the terms `accept` and `reject` are used for convenience, technically a `bin` test does not accept a model. Rather, the test fails to reject it.

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### Binomial Test (Bin)

The `bin` function performs a binomial test to assess if the number of failures is consistent with the VaR confidence level.

The binomial test is based on a normal approximation to the binomial distribution.

## Algorithms

The result of the binomial test is based on a normal approximation to a binomial distribution. Suppose:

• N is the number of observations.

• p = `1``VaRLevel` is the probability of observing a failure if the model is correct.

• x is the number of failures.

If the failures are independent, then the number of failures is distributed as a binomial distribution with parameters N and p. The expected number of failures is N*p, and the standard deviation of the number of failures is

`$\sqrt{Np\left(1-p\right)}$`

The test statistic for the `bin` test is the z-score, defined as:

`$ZScoreBin=\frac{\left(x-Np\right)}{\sqrt{Np\left(1-p\right)}}$`

The z-score approximately follows a standard normal distribution. This approximation is not reliable for small values of N or small values of p, but for typical uses in VaR backtesting analyses (N = `250` or much larger,p in the range 1–10%) the approximation gives results in line with other tests.

The tail probability of the `bin` test is the probability that a standard normal distribution exceeds the absolute value of the z-score

`$TailProbability=1-F\left(|ZScoreBin|\right)$`

where F is the standard normal cumulative distribution. When too few failures are observed, relative to the expected failures, PValueBin is (approximately) the probability of observing that many failures or fewer. For too many failures, this is (approximately) the probability of observing that many failures or more.

The p-value of the `bin` test is defined as two times the tail probability. This is because the binomial test is a two-sided test. If alpha is defined as 1 minus the test confidence level, the test rejects if the tail probability is less than one half of alpha, or equivalently if

## References

[1] Jorion, P. Financial Risk Manager Handbook. 6th Edition. Wiley Finance, 2011.