Risk Measures in Depth: VaR, CTE, and Coherent Measures
Study advanced risk measure theory including coherent and spectral risk measures for Exam MAS-II.
Value at Risk (VaR)
Value at Risk at confidence level p is the p-th quantile of the loss distribution: VaR_p(X) = inf{x : P(X <= x) >= p}. It answers the question "what loss level will not be exceeded with probability p?" For example, VaR at the 99% level is the loss amount such that there is only a 1% chance of exceeding it. VaR is widely used in banking regulation (Basel framework) and insurance regulation (Solvency II SCR at 99.5%).
Despite its popularity, VaR has significant limitations. It provides no information about the severity of losses beyond the VaR threshold. Two portfolios can have the same VaR but vastly different tail risk profiles: one might have a maximum possible loss just slightly above VaR, while the other could face catastrophic losses far exceeding VaR. This limitation motivates the use of tail-sensitive risk measures like CTE.
Conditional Tail Expectation (CTE)
The Conditional Tail Expectation at level p, also called Tail Value at Risk (TVaR) or Expected Shortfall (ES), is CTE_p(X) = E[X | X > VaR_p(X)] for continuous distributions. It measures the average loss given that the loss exceeds VaR. CTE always exceeds VaR at the same confidence level, providing a more conservative risk measure that accounts for the severity of tail losses.
For continuous distributions, CTE can be computed as CTE_p(X) = (1/(1-p)) * integral from p to 1 of VaR_u(X) du. This formula shows that CTE is a weighted average of all VaR values above the threshold p. For specific distributions, closed-form expressions exist. For the normal distribution with mean mu and standard deviation sigma, CTE_p = mu + sigma * phi(z_p) / (1 minus p), where phi is the standard normal PDF and z_p is the p-th quantile.
Properties of Risk Measures
A coherent risk measure satisfies four axioms. Subadditivity requires rho(X+Y) <= rho(X) + rho(Y), meaning diversification is never penalized. Positive homogeneity requires rho(lambda*X) = lambda*rho(X) for lambda > 0, so scaling a position scales the risk proportionally. Translation invariance requires rho(X + c) = rho(X) + c for any constant c, meaning adding a certain loss increases the risk measure by exactly that amount. Monotonicity requires that if X <= Y almost surely, then rho(X) <= rho(Y), so a position that always produces smaller losses has smaller risk.
VaR satisfies positive homogeneity, translation invariance, and monotonicity, but it violates subadditivity for non-elliptical distributions. This means the VaR of a combined portfolio can exceed the sum of individual VaRs, which contradicts the fundamental principle that diversification should reduce risk. CTE satisfies all four axioms for continuous distributions, making it a coherent risk measure. For discrete distributions, CTE as defined above may not be coherent, and a modified definition using CVaR (the integral formula) is needed.
Why VaR Fails Subadditivity
The failure of VaR subadditivity is best illustrated with an example. Consider two independent risks, each with a 2% probability of a 100 loss and a 98% probability of zero loss. The 97.5% VaR of each risk is 0 (since there is a 98% chance of zero loss). However, the combined portfolio has a probability of about 3.96% of at least one loss occurring. The 97.5% VaR of the combined portfolio is 100, because P(total > 0) = 3.96% > 2.5%. So VaR(X+Y) = 100 > 0 + 0 = VaR(X) + VaR(Y), violating subadditivity.
This failure occurs because VaR is a quantile and does not account for the probability mass beyond the threshold. Combining two risks that each have a small probability of a large loss can push the combined probability of loss above the VaR threshold, even though each individual risk has its VaR at zero. For elliptical distributions (such as multivariate normal), VaR is subadditive, which is why this issue does not arise in the simplest portfolio models.
Spectral and Distortion Risk Measures
Spectral risk measures generalize CTE by allowing a non-uniform weighting of the tail. A spectral risk measure has the form rho_phi(X) = integral from 0 to 1 of phi(p) * VaR_p(X) dp, where the weight function phi is non-negative, non-decreasing, and integrates to 1. The non-decreasing requirement ensures that larger losses receive at least as much weight as smaller losses, which is necessary for coherence. CTE is the special case where phi(p) = 1/(1 minus alpha) for p > alpha and 0 otherwise.
Distortion risk measures use a concave distortion function g applied to the survival function: rho_g(X) = integral from 0 to infinity of g(S(x)) dx, where S(x) = P(X > x). The concavity of g ensures subadditivity. Wang's transform uses g(u) = Phi(Phi^{-1}(u) + lambda), where Phi is the standard normal CDF and lambda is a risk-loading parameter. This transform is used in insurance pricing to adjust the probability of loss upward, producing a risk-loaded premium. The Proportional Hazards transform g(u) = u^{1/rho} for rho >= 1 is another common example.
Capital Allocation
When a firm holds total capital based on a risk measure applied to its aggregate portfolio, it needs to allocate that capital to individual business units for performance measurement. The Euler allocation principle assigns each unit its marginal contribution to total risk: C_i = partial derivative of rho(X) with respect to the weight of unit i, evaluated at the current portfolio composition.
For CTE, the Euler allocation gives C_i = E[X_i | X_total > VaR_p(X_total)], meaning each unit's allocated capital equals its expected loss given that total losses exceed the portfolio VaR. This allocation is additive (the sum of unit allocations equals total capital) and respects diversification. Units whose losses are more correlated with overall extreme losses receive higher capital allocations.
Common Pitfalls
Students often confuse CTE with the expected loss above VaR. CTE is the conditional expectation of the total loss given it exceeds VaR, not the expected excess above VaR. The excess measure E[X minus VaR | X > VaR] = CTE minus VaR, which is sometimes called the mean excess loss at VaR. Another common error is applying the coherence result for CTE (which requires continuity) to discrete distributions without checking whether the distribution is actually continuous.
When computing risk measures for specific distributions, make sure you use the correct formula. The Pareto, lognormal, and exponential distributions each have different CTE formulas that depend on their specific parameters. Do not attempt to derive these from scratch during the exam; memorize or have quick access to the standard results for the distributions on the syllabus.