Conditional value-at-risk

The section Measuring Risk explains and visualizes the relationships between the formulation of expected value, value-at-risk and conditional value-at-risk for discrete probability distribution.

In this section, CVaR models are defined for both path-based and RJT models.

Path-based model

Given decision strategy $Z,$ we define the cumulative distribution of compatible paths' probabilities as

\[F_Z(t) = ∑_{𝐬∈𝐒∣\mathcal{U}(𝐬)≤t} x(𝐬) p(𝐬).\]

Given a probability level $α∈(0, 1],$ we define the value-at-risk as

\[\operatorname{VaR}_α(Z)=u_α=\sup \{\mathcal{U}(𝐬)∣𝐬∈𝐒, F_Z(\mathcal{U}(𝐬))<α\}.\]

Then, we have the paths that have path utility less than and equal to the value-at-risk as

\[𝐒_{α}^{<}=\{𝐬∈𝐒∣\mathcal{U}(𝐬)<u_α\},\]

\[𝐒_{α}^{=}=\{𝐬∈𝐒∣\mathcal{U}(𝐬)=u_α\}.\]

We define conditional value-at-risk as

\[\operatorname{CVaR}_α(Z)=\frac{1}{α}\left(∑_{𝐬∈𝐒_α^{<}} x(𝐬) \ p(𝐬) \ \mathcal{U}(𝐬) + \left(α - ∑_{𝐬'∈𝐒_α^{<}} x(𝐬') \ p(𝐬') \right) u_α \right).\]

We can form the conditional value-at-risk as an optimization problem. We have the following pre-computed parameters.

Lower and upper bound of the value-at-risk

\[\operatorname{VaR}_0(Z)=u^-=\min\{\mathcal{U}(𝐬)∣𝐬∈𝐒\}, \tag{11}\]

\[\operatorname{VaR}_1(Z)=u^+=\max\{\mathcal{U}(𝐬)∣𝐬∈𝐒\}. \tag{12}\]

A "large number", specifically the largest difference between path utilities

\[M=u^+-u^-. \tag{13}\]

A "small number", specifically half of the smallest positive difference between path utilities

\[ϵ=\frac{1}{2} \min\{|\mathcal{U}(𝐬)-\mathcal{U}(𝐬^′)| \mid |\mathcal{U}(𝐬)-\mathcal{U}(𝐬^′)| > 0, 𝐬, 𝐬^′∈𝐒\}. \tag{14}\]

The objective is to minimize the variable $η$ whose optimal value is equal to the value-at-risk, that is, $\operatorname{VaR}_α(Z)=\min η.$

We define the constraints as follows:

\[η-\mathcal{U}(𝐬)≤M λ(𝐬),\quad ∀𝐬∈𝐒 \tag{14}\]

\[η-\mathcal{U}(𝐬)≥(M+ϵ) λ(𝐬) - M,\quad ∀𝐬∈𝐒 \tag{15}\]

\[η-\mathcal{U}(𝐬)≤(M+ϵ) \bar{λ}(𝐬) - ϵ,\quad ∀𝐬∈𝐒 \tag{16}\]

\[η-\mathcal{U}(𝐬)≥M (\bar{λ}(𝐬) - 1),\quad ∀𝐬∈𝐒 \tag{17}\]

\[\bar{ρ}(𝐬) ≤ \bar{λ}(𝐬),\quad ∀𝐬∈𝐒 \tag{18}\]

\[x(𝐬) \ p(𝐬) - (1 - λ(𝐬)) ≤ ρ(𝐬) ≤ λ(𝐬),\quad ∀𝐬∈𝐒 \tag{19}\]

\[ρ(𝐬) ≤ \bar{ρ}(𝐬) ≤ x(𝐬) \ p(𝐬),\quad ∀𝐬∈𝐒 \tag{20}\]

\[∑_{𝐬∈𝐒}\bar{ρ}(𝐬) = α \tag{21}\]

\[\bar{λ}(𝐬), λ(𝐬)∈\{0, 1\},\quad ∀𝐬∈𝐒 \tag{22}\]

\[\bar{ρ}(𝐬),ρ(𝐬)∈[0, 1],\quad ∀𝐬∈𝐒 \tag{23}\]

\[η∈[u^-, u^+] \tag{24}\]

We can express the conditional value-at-risk objective as

\[\operatorname{CVaR}_α(Z)=\frac{1}{α}∑_{𝐬∈𝐒}\bar{ρ}(𝐬) \mathcal{U}(𝐬)\tag{25}.\]

RJT model

CVaR formulation for the RJT model is close to that of path-based model. We denote the possible utility values with $u ∈ U$ and suppose we can define the probability $p(u)$ of attaining a given utility value. In the presence of a single value node, we define $p(u) = ∑_{s_{C_v}∈ \text{\{} S_{C_v} \vert U(s_{C_v})=u \text{\}} }µ(s_{C_v})$. We can then pose the constraints

\[η-u≤M λ(u),\quad ∀u∈U \tag{26}\]

\[η-u≥(M+ϵ) λ(u) - M,\quad ∀u∈U \tag{27}\]

\[η-u≤(M+ϵ) \bar{λ}(u) - ϵ,\quad ∀u∈U \tag{28}\]

\[η-u≥M (\bar{λ}(u) - 1),\quad ∀u∈U \tag{29}\]

\[\bar{ρ}(u) ≤ \bar{λ}(u),\quad ∀u∈U \tag{30}\]

\[p(u) - (1 - λ(u)) ≤ ρ(u) ≤ λ(u),\quad ∀u∈U \tag{31}\]

\[ρ(u) ≤ \bar{ρ}(u) ≤ p(u),\quad ∀u∈U \tag{32}\]

\[∑_{u∈U}\bar{ρ}(u) = α \tag{33}\]

\[\bar{λ}(u), λ(u)∈\{0, 1\},\quad ∀u∈U \tag{34}\]

\[\bar{ρ}(u),ρ(u)∈[0, 1],\quad ∀u∈U \tag{35}\]

\[η∈\mathbb{R} \tag{36}\]

where $α$ is the probability level in $CVaR_α$.

Finally, $CVaR_α$ can be obtained as $1/α ∑_{u∈U} \bar{ρ}(u)u$.

More details, including explanations of variables and constraints, can be found from Herrala et al. (2024)^[1].

Convex Combination

We can combine expected value and conditional value-at-risk using a convex combination at a fixed probability level $α∈(0, 1]$ as follows

\[w \operatorname{E}(Z) + (1-w) \operatorname{CVaR}_α(Z), \tag{37}\]

where the parameter $w∈[0, 1]$ expresses the decision maker's risk tolerance.

References