# minBiasRelative

Minimally biased relative test for Expected Shortfall (ES) backtest by Acerbi-Szekely

## Syntax

``TestResults = minBiasRelative(ebts)``
``[TestResults,SimTestStatistic] = minBiasRelative(ebts,Name,Value)``

## Description

example

````TestResults = minBiasRelative(ebts)` runs the relative version of the minimally biased Expected Shortfall (ES) back test by Acerbi-Szekely (2017) using the `esbacktestbysim` object. ```

example

````[TestResults,SimTestStatistic] = minBiasRelative(ebts,Name,Value)` specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax. ```

## Examples

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Create an `esbacktestbysim` object.

```load ESBacktestBySimData rng('default'); % for reproducibility ebts = esbacktestbysim(Returns,VaR,ES,"t",... 'DegreesOfFreedom',10,... 'Location',Mu,... 'Scale',Sigma,... 'PortfolioID',"S&P",... 'VaRID',["t(10) 95%","t(10) 97.5%","t(10) 99%"],... 'VaRLevel',VaRLevel);```

Generate the `TestResults` and the `SimTestStatistic` reports for the `minBiasRelative` ES backtest.

`[TestResults,SimTestStatistic] = minBiasRelative(ebts)`
```TestResults=3×10 table PortfolioID VaRID VaRLevel MinBiasRelative PValue TestStatistic CriticalValue Observations Scenarios TestLevel ___________ _____________ ________ _______________ ______ _____________ _____________ ____________ _________ _________ "S&P" "t(10) 95%" 0.95 reject 0.003 -0.10509 -0.056072 1966 1000 0.95 "S&P" "t(10) 97.5%" 0.975 reject 0 -0.15603 -0.073324 1966 1000 0.95 "S&P" "t(10) 99%" 0.99 reject 0 -0.26716 -0.104 1966 1000 0.95 ```
```SimTestStatistic = 3×1000 0.0860 0.0284 -0.0480 0.0176 0.0262 0.0309 -0.0107 0.0361 -0.0171 -0.0154 -0.0247 0.0047 0.0055 0.0217 0.0073 0.0519 0.0388 0.1023 0.0516 -0.0326 -0.0203 0.0192 -0.0022 -0.0198 -0.0205 0.0036 0.0285 0.0462 -0.0134 -0.0335 -0.0301 0.0223 -0.0291 -0.0494 -0.0246 -0.0075 0.0060 0.0516 0.0498 -0.0020 -0.0008 -0.0060 -0.1238 -0.0222 0.0447 0.0352 -0.0422 -0.0667 0.0429 0.0079 0.1145 0.0177 -0.0741 0.0357 0.0505 0.0275 -0.0136 0.0421 -0.0190 -0.0230 -0.0074 0.0098 0.0209 0.0229 -0.0012 0.0561 0.0421 0.1078 0.0530 -0.0306 -0.0167 0.0193 0.0014 -0.0214 -0.0214 -0.0224 0.0185 0.0730 -0.0089 -0.0278 -0.0458 0.0348 -0.0066 -0.0522 -0.0304 -0.0095 -0.0073 0.0490 0.0575 -0.0118 -0.0051 0.0058 -0.1318 -0.0280 0.0349 0.0473 -0.0522 -0.0894 0.0420 0.0120 0.1435 -0.0195 -0.0915 0.0478 0.0796 0.0397 -0.0022 0.0282 -0.0055 -0.0587 0.0631 0.0314 0.0446 0.0340 0.0034 0.0706 0.0652 0.1414 0.0783 0.0148 -0.0196 0.0057 0.0395 -0.0479 -0.0352 -0.0644 0.0034 0.0960 -0.0064 -0.0081 -0.0651 0.0436 0.0241 -0.0357 -0.0170 0.0242 -0.0282 0.0730 0.0449 -0.0388 0.0169 0.0506 -0.1160 -0.0663 0.0338 0.0610 -0.0815 -0.1285 0.0363 0.0209 ```

## Input Arguments

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

### Name-Value Pair 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: `minBiasRelative(ebts,'TestLevel',0.99)`

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

Data Types: `double`

## Output Arguments

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Results, returned as a table where the rows correspond to all combinations of portfolio IDs, VaR IDs, 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

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

• `'PValue'`p-value for the `minBiasRelative` test

• `'TestStatistic'``minBiasRelative` test statistic

• `'CriticalValue'`— Critical value for `minBiasRelative` test

• `'Observations'`— Number of observations

• `'Scenarios'` — Number of scenarios simulated to obtain p-values

• `'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.

Simulated values of the test statistic, returned as a `NumVaRs`-by-`NumScenarios` numeric array.

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### Minimally Biased Test, Relative Version by Acerbi and Szekely

The relative version of the Acerbi-Szekely test ([1]) computes the `TestStatistic` in the units of data.

The absolute version of the minimally biased test statistic is given by

`${Z}_{minbias}^{rel}=\frac{1}{N}\sum _{t=1}^{N}\frac{1}{E{S}_{t}}\left(E{S}_{t}-Va{R}_{t}-\frac{1}{{p}_{VaR}}\left({X}_{t}+Va{R}_{t}\right)_\right)$`

where

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

VaRt is the essential VaR for period t.

ESt is the expected shortfall for period t.

pVaR is the probability of VaR failure, defined as 1 — VaR level.

N is the number of periods in the test window (t = 1,...N).

(x)_ is the negative part function defined as (x)_ = max(0,-x).

### Significance of the Test

Negative values of the test statistic indicate risk underestimation.

The minimally biased test is a one-sided test that rejects the model when there is evidence that the model underestimates risk (for technical details, see Acerbi-Szekely [1] and [2]). The test rejects the model when the p-value is less than `1` minus the test confidence level. For more information on the steps to simulate the test statistics and details on the computation of the p-values and critical values, see `simulate`.

ES backtests are necessarily approximated in that they are sensitive to errors in the predicted VaR. However, the minimally biased test has only a small sensitivity to VaR errors and the sensitivity is prudential, in the sense that VaR errors lead to a more punitive ES test. For details, see Acerbi-Szekely ([1] and [2]). When distribution information is available using the minimally biased test is recommended.

## References

[1] Acerbi, Carlo, and Balazs Szekely. "General Properties of Backtestable Statistics." SSRN Electronic Journal. (January, 2017).

[2] Acerbi, Carlo, and Balazs Szekely. "The Minimally Biased Backtest for ES." Risk. (September, 2019).

[3] Acerbi, C. and D. Tasche. “On the Coherence of Expected Shortfall.” Journal of Banking and Finance. Vol. 26, 2002, pp. 1487-1503.

[4] Rockafellar, R. T. and S. Uryasev. "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking and Finance. Vol. 26, 2002, pp. 1443-1471.

Introduced in R2020b