# unconditionalNormal

Unconditional expected shortfall (ES) backtest by Acerbi-Szekely with critical values for normal distributions

## Description

example

TestResults = unconditionalNormal(ebt) runs the unconditional expected shortfall (ES) backtest by Acerbi-Szekely (2014) using precomputed critical values and assuming that the returns distribution is standard normal.

example

TestResults = unconditionalNormal(ebt,Name,Value) adds an optional name-value pair argument for TestLevel.

## Examples

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Create an esbacktest object.

ebt = esbacktest(Returns,VaRModel1,ESModel1,'VaRLevel',VaRLevel)
ebt =
esbacktest with properties:

PortfolioData: [1966x1 double]
VaRData: [1966x1 double]
ESData: [1966x1 double]
PortfolioID: "Portfolio"
VaRID: "VaR"
VaRLevel: 0.9750

Generate the TestResults report for the unconditional ES backtest that assumes the returns distribution is standard normal.

TestResults = unconditionalNormal(ebt,'TestLevel',0.99)
TestResults=1×9 table
PortfolioID    VaRID    VaRLevel    UnconditionalNormal     PValue      TestStatistic    CriticalValue    Observations    TestLevel
___________    _____    ________    ___________________    _________    _____________    _____________    ____________    _________

"Portfolio"    "VaR"     0.975            reject           0.0054099      -0.38265         -0.34639           1966          0.99

## Input Arguments

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esbacktest (ebt) object, which contains a copy of the given data (the PortfolioData, VarData, and ESData properties) and all combinations of portfolio ID, VaR ID, and VaR levels to be tested. For more information on creating an esbacktest object, see esbacktest.

### 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: TestResults = unconditionalNormal(ebt,'TestLevel',0.99)

Test confidence level, specified as the comma-separated pair consisting of 'TestLevel' and a numeric value between 0.5 and 0.9999.

Data Types: double

## Output Arguments

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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 the given data.

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

• 'VaRLevel' — VaR level for the corresponding VaR data column.

• 'UnconditionalNormal'— Categorical array with categories 'accept' and 'reject' that indicate the result of the unconditional normal test.

• 'PValue'— P-value of the unconditional normal test, interpolated from the precomputed critical values under the assumption that the returns follow a standard normal distribution.

Note

p-values < 0.0001 are truncated to the minimum (0.0001) and p-values > 0.5 are displayed as a maximum (0.5).

• 'TestStatistic'— Unconditional normal test statistic.

• 'CriticalValue'— Precomputed critical value for the corresponding test level and number of observations. Critical values are obtained under the assumption that the returns follow a standard normal distribution.

• 'Observations'— Number of observations.

• 'TestLevel'— Test confidence level.

Note

For the test results, the terms 'accept' and 'reject' are used for convenience. Technically, a test does not accept a model; rather, a test fails to reject it.

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### Unconditional Test by Acerbi and Szekely

The unconditional test (also known as the second Acerbi-Szekely test) scales the losses by the corresponding ES value.

The unconditional test statistic is based on the unconditional relationship

$E{S}_{t}=-{E}_{t}\left[\frac{{X}_{t}{I}_{t}}{{p}_{VaR}}\right]$

where

Xt is the portfolio outcome, that is, the portfolio return or portfolio profit and loss for period t.

PVaR is the probability of VaR failure defined as 1-VaR level.

ESt is the estimated expected shortfall for period t.

It is the VaR failure indicator on period t with a value of 1 if Xt < -VaR, and 0 otherwise.

The unconditional test statistic is defined as

${Z}_{uncond}=\frac{1}{N{p}_{VaR}}\sum _{t=1}^{N}\frac{{X}_{t}{I}_{t}}{E{S}_{t}}+1$

The critical values for the unconditional test statistic, which form the basis for table-based tests, are stable across a range of distributions. The esbacktest class runs the unconditional test against precomputed critical values under two distributional assumptions: normal distribution (thin tails) using unconditionalNormal and t distribution with 3 degrees of freedom (heavy tails) using unconditionalT).

## References

[1] Acerbi, C., and B. Szekely. Backtesting Expected Shortfall. MSCI Inc. December, 2014.

## Version History

Introduced in R2017b