RiskMeasures

Julia library for computing risk measures for random variables. The random variable represents profits or rewards that are to be maximized. The computed risk value is also better when greater.

The following risk measures are currently supported

VaR: Value at risk
CVaR: Conditional value at risk
ERM: Entropic risk measure
EVaR: Entropic value at risk
expectile: A coherent elicitable risk measure
UBSR: Utility-based shortfall risk for a given utility
Choquet: Choquet and distortion risk measures for a choquet capacity and distortion functions

When supported, the risk measure returns also the optimal distribution

General assumptions

Random variables represent rewards (greater value is preferred)
Risk measures become less risk-averse with the increasing value of the risk parameter α or β

Supported distributions

General discrete distributions (DiscreteNonParametric)
Transformed discrete distributions ('DiscreteAffineDistribution')
Mixtures of random variables ('MixtureModel')

Examples

Using arrays

using RiskMeasures

x = [1, 5, 6, 7, 20]
p = [0.1, 0.1, 0.2, 0.5, 0.1]

VaR(x, p, 0.1)   # value at risk
CVaR(x, p, 0.1)  # conditional value at risk
EVaR(x, p, 0.1)  # entropic value at risk
ERM(x, p, 0.1)   # entropic risk measure
expectile(x, p, 0.1)   # expectile risk measure
UBSR(x, p, z -> (z ≥ 0 ? 0 : -1), 0.1)  # utility-based shortfall risk that equals to VaR
β = 0.1; UBSR(x, p, z -> (-exp(-β * z)), 0.1)  # utility-based shortfall risk that equals to ERM
choquet_risk(x, p, cvar_capacity, 0.5) # choquet risk measure
choquet_distortion_risk(x, p, cvar_distortion, 0.5) # law-invariant choquet (distortion) risk measure

Using random variables

using RiskMeasures
using Distributions

X = [1, 5, 6, 7, 20]
P = [0.1, 0.1, 0.2, 0.5, 0.1]
x̃ = DiscreteNonParametric(X, P)

VaR(x̃, 0.1)   # value at risk
CVaR(x̃, 0.1)  # conditional value at risk
EVaR(x̃, 0.1)  # entropic value at risk
ERM(x̃, 0.1)   # entropic risk measure
expectile(x̃, 0.1)   # expectile risk measure
UBSR(x̃, z -> (z ≥ 0 ? 0 : -1), 0.1)  # utility-based shortfall risk that equals to VaR
β = 0.1; UBSR(x̃, z -> (-exp(-β * z)), 0.1)  # utility-based shortfall risk that equals to ERM
choquet_risk(x̃, cvar_capacity, 0.5) # choquet risk measure
choquet_distortion_risk(x̃, cvar_distortion, 0.5) # law-invariant choquet (distortion) risk measure

Using transformed random variables

We can also compute risk measures of transformed random variables

using RiskMeasures
using Distributions

x̃ = DiscreteNonParametric([1, 5, 6, 7, 20], [0.1, 0.1, 0.2, 0.5, 0.1])
VaR(5*x̃ + 10, 0.1)   # value at risk
CVaR(x̃ - 10, 0.1)    # conditional value at risk

Using mixture models

using RiskMeasures
using Distributions

x̃ = DiscreteNonParametric([-5, -3, 2, 7, 33], [0., 0.2, 0.1, 0.5, 0.2])
ỹ = DiscreteNonParametric([1, 5, 6, 7, 20], [0.1, 0.1, 0.2, 0.5, 0.1])
m̃ = MixtureModel([x̃, ỹ], [0.3, 0.7])

VaR(m̃, 0.4)
CVaR(m̃, 0.4)

Please see the documentation and unit tests for more examples of how this package can be used to compute the risk.

Most risk measure functions return tuples that include additional statistics and values. For example, this includes the distribution that attains the risk value and the optimal ERM-β in EVaR.

Functions

Value at Risk

RiskMeasures.VaR — Function

VaR(x̃, α)

Compute the value at risk at risk level α for the random variable x̃.

Risk must satisfy $α ∈ [0,1]$ and α=0.5 computes the median and α=0 computes the essential infimum (smallest value with positive probability) and α=1 returns infinity.

Solves for

\[\max \{t ∈ \mathbb{R} : \mathbb{P}[x̃ < t] \le α \}\]

In general, this function is neither convex nor concave in the random variable x̃.