Class VarBacktest

java.lang.Object

com.quantfinlib.risk.VarBacktest

public final class VarBacktest
extends Object

VaR model validation: do the exceptions (losses beyond VaR) occur at the promised rate, and independently?

• Kupiec POF — likelihood-ratio test that the exception frequency matches 1 - confidence (two-sided: both too many and too few exceptions reject). χ²(1).
• Christoffersen independence — LR test that exceptions do not cluster (a model that is right on average but wrong in crises fails here). χ²(1).
• Conditional coverage — the joint test (POF + independence), χ²(2), with the exact p-value e^{-LR/2}.

A p-value below the chosen significance (e.g. 0.05) rejects the model.

trafficLight(int, int) is the Basel supervisory zone test on the exception count (MAR32, MAR99).

acerbiSzekelyZ2(double[], double[], double[], double) backtests expected shortfall, which FRTB makes the capital measure and which an exception count alone cannot validate.

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final record	VarBacktest.TrafficLight	Result of the Basel traffic-light test.
static final record	VarBacktest.VarBacktestResult
static enum	VarBacktest.Zone	Backtesting zones of the Basel internal models approach (MAR32.8).

Field Summary

Fields

Modifier and Type	Field	Description
static final double	Z2_THRESHOLD_001PCT	Acerbi-Szekely (2014) Z2 at the 0.01% level, T = 250, 97.5% ES.
static final double	Z2_THRESHOLD_5PCT	Acerbi-Szekely (2014) Z2 at the 5% level, T = 250, 97.5% ES.

Method Summary

Modifier and Type	Method	Description
static double	acerbiSzekelyZ2(double[] returns, double[] varForecasts, double[] esForecasts, double confidence)	The Acerbi-Szekely (2014, "Backtesting expected shortfall", Risk, December) unconditional ES test statistic
static VarBacktest.VarBacktestResult	test(double[] returns, double[] varForecasts, double confidence)	Backtests a VaR forecast series.
static VarBacktest.VarBacktestResult	test(double[] returns, double constantVar, double confidence)	Backtests a constant VaR (positive loss fraction) against realized returns.
static VarBacktest.TrafficLight	trafficLight(int exceptions, int observations)	Basel traffic light for 99 % VaR, the supervisory backtest of the bank-wide model (Basel Framework MAR32.8-32.9 and MAR99.17, Table 2).
static VarBacktest.TrafficLight	trafficLight(int exceptions, int observations, double coverage)	Traffic light at another VaR coverage, with the same 95 % / 99.99 % zone boundaries (the Basel text fixes 99 %; other levels are an extension for internal desk monitoring).

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

Z2_THRESHOLD_5PCT

public static final double Z2_THRESHOLD_5PCT

Acerbi-Szekely (2014) Z2 at the 5% level, T = 250, 97.5% ES.

See Also:

• Constant Field Values

Z2_THRESHOLD_001PCT

public static final double Z2_THRESHOLD_001PCT

Acerbi-Szekely (2014) Z2 at the 0.01% level, T = 250, 97.5% ES.

See Also:

• Constant Field Values

Method Details

test

public static VarBacktest.VarBacktestResult test(double[] returns,
 double constantVar,
 double confidence)

Backtests a constant VaR (positive loss fraction) against realized returns.

test

public static VarBacktest.VarBacktestResult test(double[] returns,
 double[] varForecasts,
 double confidence)

Backtests a VaR forecast series.

Parameters:

returns - realized periodic returns

varForecasts - VaR forecast for each period as a positive loss fraction, aligned with returns

confidence - the VaR confidence level (e.g. 0.95, 0.99)

acerbiSzekelyZ2

public static double acerbiSzekelyZ2(double[] returns,
 double[] varForecasts,
 double[] esForecasts,
 double confidence)

The Acerbi-Szekely (2014, "Backtesting expected shortfall", Risk, December) unconditional ES test statistic

  Z2 = 1 − Σ_t L_t·1{L_t > VaR_t} / (T·(1−c)·ES_t)

with losses L_t = −r_t. Under a correct ES forecast E[Z2] = 0; a negative value means realised tail losses exceed the forecast ES. The paper's simulations give thresholds that are nearly independent of the return distribution for T = 250 and c = 97.5%: Z2_THRESHOLD_5PCT = −0.70 and Z2_THRESHOLD_001PCT = −1.80 (the analogue of the Basel amber and red zones). For other settings the p-value must be simulated under the model.

Parameters:

returns - realised returns

varForecasts - VaR forecasts, positive loss fractions

esForecasts - ES forecasts at the same confidence, positive

confidence - e.g. 0.975

trafficLight

public static VarBacktest.TrafficLight trafficLight(int exceptions,
 int observations)

Basel traffic light for 99 % VaR, the supervisory backtest of the bank-wide model (Basel Framework MAR32.8-32.9 and MAR99.17, Table 2).

Plain idea. With a correct 99 % VaR the exception count in a year of 250 days is Binomial(250, 1 %), about 2.5 on average. A count you would reach or exceed with a correct model at most about 5 % of the time puts the model in the amber zone; about 0.01 % of the time, the red zone.

Formal statement. With F the Binomial(n, 1 - coverage) CDF, the zone is GREEN if F(x) < 0.95, AMBER if 0.95 <= F(x) < 0.9999 and RED otherwise (MAR99.17: the amber zone begins where the cumulative probability equals or exceeds 95 %, the red zone where it equals or exceeds 99.99 %).

Worked example. n = 250: F(4) = 89.22 % and F(5) = 95.88 %, so 0-4 exceptions are green and 5 amber; F(9) = 99.97 % and F(10) = 99.99 %, so 10 or more are red. Those are the boundaries and cumulative probabilities of MAR99 Table 2 (tested). Seven exceptions give AMBER with multiplier 1.83.

Parameters:

exceptions - exceptions observed, 0 ≤ exceptions ≤ observations

observations - sample size, > 0 (250 for the regulatory test)

Returns:

zone, cumulative probability and, for 250 observations, the multiplier

trafficLight

public static VarBacktest.TrafficLight trafficLight(int exceptions,
 int observations,
 double coverage)

Traffic light at another VaR coverage, with the same 95 % / 99.99 % zone boundaries (the Basel text fixes 99 %; other levels are an extension for internal desk monitoring). The multiplier is set only for 250 observations at 99 % coverage.

Parameters:

exceptions - exceptions observed, 0 ≤ exceptions ≤ observations

observations - sample size, > 0

coverage - VaR confidence level in (0, 1), e.g. 0.99

Returns:

zone and cumulative probability