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

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)

Warning: This is package is in development and the computed values should be treated with caution.

Examples

using RiskMeasures
using Distributions

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

We can also compute risk measures of transformed random variables

VaR(5*x̃ + 10, 0.1)   # value at risk
CVaR(x̃ - 10, 0.1)  # conditional value at risk

Many methods methods VaR, CVaR, and EVaR also return additional statistics and values, such as the distribution that attains the risk value and the optimal β in EVaR. These are returned as named tuples.

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=1 computes the essential infimum (smallest value with positive probability) and α=0 returns infinity.

Solves for $\inf \{x ∈ \mathbb{R} : \mathbb{P}[x̃ ≤ x] > 1-α \}$

In general, this function is neither convex nor concave.

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Conditional Value at Risk

RiskMeasures.CVaR — Function

CVaR(x̃, α)

Compute the conditional value at risk at level α for the random variable x̃. Also compute the equivalent random variable with the same support but a different distribution.

Returns a named tuple with value and the solution random variable that solves the robust CVaR formulation. That is if ỹ is the solution, then the support of x̃ and ỹ are the same and $\mathbb{E}[ỹ] = \operatorname{CVaR}_{α}[x̃]$.

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Entropic Value at Risk

RiskMeasures.EVaR — Function

EVaR(x̃, α; βmin = 1e-5, βmax = 100, reciprocal = false, distribution = false)

Compute the EVaR risk measure of the random variable x̃ with risk level α in [0,1].

When α = 1, the function computes the expected value, and when α = 0, then the function computes the essential infimum (a minimum value with positive probability).

The function solves

\[\max_{β ∈ [βmin, βmax]} \operatorname{ERM}_β (x̃) - β^{-1} \log (1/(α)).\]

Large values of βmax may cause the computation to overflow.

If reciprocal = false, then the quasi-concave problem above is solved directly. If reciprocal = true, then the optimization is reformulated in terms of λ = 1/β to get a concave function that can be solved (probably) more efficiently

The function implicitly assumes that all elements of the probability space have non-zero probability.

Returns the EVaR and the value β that attains the maximum above.

See: Ahmadi-Javid, A. “Entropic Value-at-Risk: A New Coherent Risk Measure.” Journal of Optimization Theory and Applications 155(3), 2012.

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Entropic Risk Measure

RiskMeasures.ERM — Function

ERM(x̃, β [, x̃min, check_inputs = true])

Compute the entropic risk measure of the random variable x̃ with risk level β.

The optional x̃min parameter is used as an offset in order to avoid overflows when computing the exponential function. If not provided, the minimum value of x̃ is used instead.

Assumes a maximization problem. Using β=0 computes expectation and β=Inf computes the essential infimum (smallest value with positive probability).

More details: https://en.wikipedia.org/wiki/Entropic_risk_measure

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RiskMeasures.softmin — Function

softmin(x̃, β; x̃min check_inputs = true)

Compute a weighted softmin function for random variable x̃ with risk level β. This can be seen as an approximation of the arg min function and not the min function.

The operator computes a distribution p such that

\[p_i = \frac{e^{-β x_i}}{\mathbb{E}[e^{-β x̃}]}\]

The optional x̃min parameter is used as an offset in order to avoid overflows when computing the exponential function. If not provided, the minimum value of x̃ is used instead.

The value β must be positive

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Expectile

RiskMeasures.expectile — Function

expectile(x̃, α)

Compute the expectile risk measure of the random variable x̃ with risk level α ∈ (0,1). When α = 1/2, the function computes the expected value. Expectile is only coherent when α ∈ (0,1/2].

Notice the range for α does not include 0 or 1.

The function solves

\[\argmin_{x ∈ Real} α \mathbb{E}((x̃ - x)^2_+) - (1-α) \mathbb{E}((x̃ - x)^2_-)\]

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Essential Infimum

RiskMeasures.essinf — Function

essinf(x̃; distribution = false)

Compute the essential infimum of the random variable x̃, which is the minimum value with positive probability

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