# gaussianDownsideVariance (3) - Linux Manuals

## gaussianDownsideVariance: Statistics tool for gaussian-assumption risk measures.

## NAME

QuantLib::GenericGaussianStatistics - Statistics tool for gaussian-assumption risk measures.

## SYNOPSIS

#include <ql/math/statistics/gaussianstatistics.hpp>

Inherits Stat.

### Public Types

typedef Stat::value_type **value_type**

### Public Member Functions

**GenericGaussianStatistics** (const Stat &s)

**Gaussian risk measures**

Real **gaussianDownsideVariance** () const

Real **gaussianDownsideDeviation** () const

Real **gaussianRegret** (Real target) const

Real **gaussianPercentile** (Real percentile) const

Real **gaussianTopPercentile** (Real percentile) const

Real **gaussianPotentialUpside** (Real percentile) const

*gaussian-assumption Potential-Upside at a given percentile *

Real **gaussianValueAtRisk** (Real percentile) const

*gaussian-assumption Value-At-Risk at a given percentile *

Real **gaussianExpectedShortfall** (Real percentile) const

*gaussian-assumption Expected Shortfall at a given percentile *

Real **gaussianShortfall** (Real target) const

*gaussian-assumption Shortfall (observations below target) *

Real **gaussianAverageShortfall** (Real target) const

*gaussian-assumption ***Average** Shortfall (averaged shortfallness)

## Detailed Description

### template<class Stat> class QuantLib::GenericGaussianStatistics< Stat >

Statistics tool for gaussian-assumption risk measures.This class wraps a somewhat generic statistic tool and adds a number of gaussian risk measures (e.g.: value-at-risk, expected shortfall, etc.) based on the mean and variance provided by the underlying statistic tool.

## Member Function Documentation

### Real gaussianDownsideVariance () const

returns the downside variance, defined as [ ac{N}{N-1} imes ac{ um_{i=1}^{N} heta imes x_i^{2}}{ um_{i=1}^{N} w_i} ], where $ heta $ = 0 if x > 0 and $ heta $ =1 if x <0

### Real gaussianDownsideDeviation () const

returns the downside deviation, defined as the square root of the downside variance.

### Real gaussianRegret (Real target) const

returns the variance of observations below target [ ac{um w_i (min(0, x_i-target))^2 }{um w_i}. ]

See Dembo, Freeman 'The Rules Of Risk', Wiley (2001)

### Real gaussianPercentile (Real percentile) const

gaussian-assumption y-th percentile, defined as the value x such that [ y = ac{1}{qrt{2

i}} int_{-infty}^{x} \xp (-u^2/2) du ]

**Precondition:**

- percentile must be in range (0-100%) extremes excluded

### Real gaussianTopPercentile (Real percentile) const

**Precondition:**

- percentile must be in range (0-100%) extremes excluded

### Real gaussianPotentialUpside (Real percentile) const

gaussian-assumption Potential-Upside at a given percentile

**Precondition:**

- percentile must be in range [90-100%)

### Real gaussianValueAtRisk (Real percentile) const

gaussian-assumption Value-At-Risk at a given percentile

**Precondition:**

- percentile must be in range [90-100%)

### Real gaussianExpectedShortfall (Real percentile) const

gaussian-assumption Expected Shortfall at a given percentile

Assuming a gaussian distribution it returns the expected loss in case that the loss exceeded a VaR threshold,

[ mathrm{E}

average of observations below the given percentile $ p $. Also know as conditional value-at-risk.

See Artzner, Delbaen, Eber and Heath, 'Coherent measures of risk', Mathematical Finance 9 (1999)

**Precondition:**

- percentile must be in range [90-100%)

## Author

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