Entropic value at risk

In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value-at-risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value-at-risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called 'entropic value-at-risk'. The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class. EVaR 1 − α ( X ) := inf z > 0 { z − 1 ln ⁡ ( M X ( z ) α ) } . {displaystyle { ext{EVaR}}_{1-alpha }(X):=inf _{z>0}left{z^{-1}ln left({frac {M_{X}(z)}{alpha }} ight) ight}.}     (1) Pr ( X ≥ a ) ≤ e − z a M X ( z ) , ∀ z > 0. {displaystyle Pr(Xgeq a)leq e^{-za}M_{X}(z),quad forall z>0.}     (2) EVaR 1 − α ( X ) = sup Q ∈ ℑ ( E Q ( X ) ) , {displaystyle { ext{EVaR}}_{1-alpha }(X)=sup _{Qin Im }(E_{Q}(X)),}     (3) M X ( z ) = sup 0 < α ≤ 1 { α exp ⁡ ( z EVaR 1 − α ( X ) ) } . {displaystyle M_{X}(z)=sup _{0<alpha leq 1}{alpha exp(z{ ext{EVaR}}_{1-alpha }(X))}.}     (4) θ − 1 ln ⁡ M X ( θ ) = a X ( 1 , θ ) = sup 0 < α ≤ 1 { EVaR 1 − α ( X ) + θ − 1 ln ⁡ α } . {displaystyle heta ^{-1}ln M_{X}( heta )=a_{X}(1, heta )=sup _{0<alpha leq 1}{{ ext{EVaR}}_{1-alpha }(X)+ heta ^{-1}ln alpha }.}     (5) VaR ( X ) ≤ CVaR ( X ) ≤ EVaR ( X ) . {displaystyle { ext{VaR}}(X)leq { ext{CVaR}}(X)leq { ext{EVaR}}(X).}     (6) E ( X ) ≤ EVaR 1 − α ( X ) ≤ esssup ( X ) {displaystyle { ext{E}}(X)leq { ext{EVaR}}_{1-alpha }(X)leq { ext{esssup}}(X)}     (7) EVaR 1 − α ( X ) = μ + σ − 2 ln ⁡ α . {displaystyle { ext{EVaR}}_{1-alpha }(X)=mu +sigma {sqrt {-2ln alpha }}.}     (8) EVaR 1 − α ( X ) = inf t > 0 { t ln ⁡ ( t e t − 1 b − e t − 1 a b − a ) − t ln ⁡ α } . {displaystyle { ext{EVaR}}_{1-alpha }(X)=inf _{t>0}leftlbrace tln left(t{frac {e^{t^{-1}b}-e^{t^{-1}a}}{b-a}} ight)-tln alpha ight brace .}     (9) min w ∈ W ρ ( G ( w , ψ ) ) , {displaystyle min _{{oldsymbol {w}}in {oldsymbol {W}}} ho (G({oldsymbol {w}},{oldsymbol {psi }})),}     (10) min w ∈ W , t > 0 { t ln ⁡ M G ( w , ψ ) ( t − 1 ) − t ln ⁡ α } . {displaystyle min _{{oldsymbol {w}}in {oldsymbol {W}},t>0}left{tln M_{G({oldsymbol {w}},{oldsymbol {psi }})}(t^{-1})-tln alpha ight}.}     (11) G ( w , ψ ) = g 0 ( w ) + ∑ i = 1 m g i ( w ) ψ i , g i : R n → R , i = 0 , 1 , … , m , {displaystyle G({oldsymbol {w}},{oldsymbol {psi }})=g_{0}({oldsymbol {w}})+sum _{i=1}^{m}g_{i}({oldsymbol {w}})psi _{i},qquad g_{i}:mathbb {R} ^{n} o mathbb {R} ,i=0,1,ldots ,m,}     (12) min w ∈ W , t > 0 { g 0 ( w ) + t ∑ i = 1 m ln ⁡ M g i ( w ) ψ i ( t − 1 ) − t ln ⁡ α } . {displaystyle min _{{oldsymbol {w}}in {oldsymbol {W}},t>0}leftlbrace g_{0}({oldsymbol {w}})+tsum _{i=1}^{m}ln M_{g_{i}({oldsymbol {w}})psi _{i}}(t^{-1})-tln alpha ight brace .}     (13) min w ∈ W , t ∈ R { t + 1 α E [ g 0 ( w ) + ∑ i = 1 m g i ( w ) ψ i − t ] + } . {displaystyle min _{{oldsymbol {w}}in {oldsymbol {W}},tin mathbb {R} }leftlbrace t+{frac {1}{alpha }}{ ext{E}}left_{+} ight brace .}     (14) ER g , β ( X ) := sup Q ∈ ℑ E Q ( X ) {displaystyle { ext{ER}}_{g,eta }(X):=sup _{Qin Im }{ ext{E}}_{Q}(X)}     (15) ER g , β ( X ) = inf t > 0 , μ ∈ R { t [ μ + E P ( g ∗ ( X t − μ + β ) ) ] } {displaystyle { ext{ER}}_{g,eta }(X)=inf _{t>0,mu in mathbb {R} }leftlbrace tleft ight brace }     (16) g ( x ) = { x ln ⁡ x x > 0 0 x = 0 + ∞ x < 0 {displaystyle g(x)={egin{cases}xln x&x>0\0&x=0\+infty &x<0end{cases}}}     (17) g ( x ) = { 0 0 ≤ x ≤ 1 α + ∞ otherwise {displaystyle g(x)={egin{cases}0&0leq xleq {frac {1}{alpha }}\+infty &{ ext{otherwise}}end{cases}}}     (18) In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value-at-risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value-at-risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called 'entropic value-at-risk'. The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class. Let ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},P)} be a probability space with Ω {displaystyle Omega } a set of all simple events, F {displaystyle {mathcal {F}}} a σ {displaystyle sigma } -algebra of subsets of Ω {displaystyle Omega } and P {displaystyle P} a probability measure on F {displaystyle {mathcal {F}}} . Let X {displaystyle X} be a random variable and L M + {displaystyle mathbf {L} _{M^{+}}} be the set of all Borel measurable functions X : Ω → R {displaystyle X:Omega o mathbb {R} } whose moment-generating function M X ( z ) {displaystyle M_{X}(z)} exists for all z ≥ 0 {displaystyle zgeq 0} . The entropic value-at-risk (EVaR) of X ∈ L M + {displaystyle Xin mathbf {L} _{M^{+}}} with confidence level 1 − α {displaystyle 1-alpha } is defined as follows: In finance, the random variable X ∈ L M + , {displaystyle Xin mathbf {L} _{M^{+}},} in the above equation, is used to model the losses of a portfolio.