Value-at-Risk
Value-at-Risk at confidence level $p \in (0,1)$ is the $p$-quantile of the loss distribution $F_L$:
\[\text{VaR}_p(L) = F_L^{-1}(p) = \inf\{l : F_L(l) \ge p\}\]For continuous distributions, this is the unique value satisfying $P(L \le \text{VaR}_p) = p$.
Properties:
- Easy to compute and explain
- Widely used: Solvency II SCR at $p = 99.5\%$, Basel III at $p = 99\%$ (10-day)
- Not subadditive in general: $\text{VaR}_p(L_1 + L_2)$ can exceed $\text{VaR}_p(L_1) + \text{VaR}_p(L_2)$
Counterexample to subadditivity: Let $L_1, L_2$ be independent Pareto($\alpha = 1.1$, $\theta = 1$). Each has $\text{VaR}{99\%}(L_i) \approx 99$. But $\text{VaR}{99\%}(L_1 + L_2) > 198$ for heavy enough tails because the joint tail is concentrated.
For a Normal loss, VaR is coherent (Normal is subadditive), which is why VaR is adequate for market risk models dominated by Gaussian assumptions but fails for fat-tailed insurance losses.
Tail Value-at-Risk (TVaR / CVaR / ES)
The Tail Value-at-Risk averages all VaR values above the threshold:
\[\text{TVaR}_p(L) = \frac{1}{1-p}\int_p^1 \text{VaR}_u(L)\,du\]For continuous distributions, this equals the Conditional Value-at-Risk:
\[\text{CVaR}_p(L) = E[L \mid L > \text{VaR}_p(L)]\]but for discrete/mixed distributions the two can differ. The more general Expected Shortfall (ES) uses the TVaR integral definition.
The TVaR can be expressed in terms of the stop-loss premium $\pi(d) = E[(L-d)_+]$:
\[\text{TVaR}_p(L) = \text{VaR}_p(L) + \frac{\pi(\text{VaR}_p(L))}{1 - p}\]For common distributions:
| Distribution | $\text{TVaR}_p$ |
|---|---|
| Normal($\mu, \sigma^2$) | $\mu + \sigma\,\phi(z_p)/(1-p)$ |
| Lognormal($\mu, \sigma^2$) | $e^{\mu + \sigma^2/2}\Phi(\sigma - z_p)/(1-p)$ |
| Exponential($\theta$) | $\text{VaR}_p + \theta$ |
| Pareto($\alpha, \theta$) | $\text{VaR}_p \cdot \alpha/(\alpha - 1)$ |
where $\phi$ and $\Phi$ are the standard Normal pdf and cdf, and $z_p = \Phi^{-1}(p)$.
Coherent Risk Measures: Artzner Axioms
A risk measure $\rho: \mathcal{L} \to \mathbb{R}$ is coherent if it satisfies four axioms for all losses $L, L_1, L_2 \in \mathcal{L}$ and constant $c$:
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Translation invariance: $\rho(L + c) = \rho(L) + c$. Adding a certain loss increases required capital by exactly that amount.
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Monotonicity: If $L_1 \le L_2$ a.s., then $\rho(L_1) \le \rho(L_2)$. Larger losses require more capital.
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Positive homogeneity: $\rho(\lambda L) = \lambda\,\rho(L)$ for $\lambda > 0$. Scaling the portfolio scales the risk.
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Subadditivity: $\rho(L_1 + L_2) \le \rho(L_1) + \rho(L_2)$. Merging risks cannot increase total capital — diversification is rewarded.
Artzner et al.’s representation theorem: $\rho$ is coherent if and only if there exists a set $\mathcal{Q}$ of probability measures (the generalized scenarios) such that:
\[\rho(L) = \sup_{Q \in \mathcal{Q}} E_Q[L]\]TVaR is coherent; VaR is not. The standard deviation premium $\rho(L) = E[L] + \lambda\,\text{SD}(L)$ is coherent only for $\lambda \ge 0$ when losses are bounded below.
Distortion Risk Measures
A distortion risk measure with distortion function $g: [0,1] \to [0,1]$ (increasing, $g(0)=0$, $g(1)=1$) is:
\[\rho_g(L) = -\int_{-\infty}^0 g(1 - F_L(l))\,dl + \int_0^{\infty} [1 - g(F_L(l))]\,dl\]Equivalently, $\rho_g(L) = \int_0^1 \text{VaR}_u(L)\,dg(u)$, a Choquet integral with respect to the distorted probability.
Wang transform: The distortion $g(u) = \Phi(\Phi^{-1}(u) + \lambda)$ for Sharpe ratio parameter $\lambda$ yields:
\[\rho_\lambda(L) = E_Q[L], \quad \text{where } Q \text{ shifts the quantile by } \lambda\]For Normal $L \sim N(\mu, \sigma^2)$: $\rho_\lambda(L) = \mu + \lambda\sigma$, recovering the standard deviation premium.
| Distortion $g(u)$ | Resulting $\rho$ |
|---|---|
| $g(u) = \mathbf{1}[u = 1]$ | $\text{ess}\sup L$ |
| $g(u) = \mathbf{1}[u > 1-p]/(1-p)$ (truncated) | $\text{TVaR}_p(L)$ |
| $g(u) = u^{1/\rho}$ (power) | Proportional hazards transform |
| $g(u) = \Phi(\Phi^{-1}(u)+\lambda)$ | Wang transform |
A distortion risk measure is coherent if and only if $g$ is concave.
Regulatory Capital: Solvency II
Under Solvency II (EU), the Solvency Capital Requirement (SCR) is calibrated so that:
\[\text{SCR} = \text{VaR}_{99.5\%}(L_{\text{net}}) - \text{Best Estimate Liabilities}\]where $L_{\text{net}}$ is the net asset value shock over a one-year horizon.
Risk aggregation: The standard formula uses a correlation matrix $\rho_{ij}$ to aggregate module-level SCRs:
\[\text{SCR} = \sqrt{\sum_{i,j} \rho_{ij}\,\text{SCR}_i\,\text{SCR}_j}\]This is equivalent to assuming multivariate Normal risks — a linear correlation aggregation that ignores tail dependence. Internal models may use copulas or historical simulation.
Swiss Solvency Test (SST): Uses TVaR at 99% rather than VaR at 99.5%, reflecting the coherence advantage and providing more stability across model choices.
The choice between VaR and TVaR for regulation involves a trade-off: TVaR is more sensitive to extreme tail assumptions (model risk), while VaR ignores the shape of the tail above the threshold. For well-diversified portfolios the two are numerically close; for concentrated catastrophe books they diverge substantially.