Risk Measures: VaR, TVaR, and Expected Shortfall for STAM

Why Risk Measures Matter

Risk measures quantify the potential for adverse outcomes in a portfolio of insurance liabilities or financial assets. They serve several critical purposes: determining the amount of capital an insurer must hold to remain solvent, pricing reinsurance and catastrophe bonds, setting regulatory requirements, and communicating risk to stakeholders. A good risk measure should be intuitive, mathematically tractable, and capture the essential features of the loss distribution that matter for decision-making.

Two risk measures dominate the actuarial syllabus and regulatory practice: Value at Risk (VaR) and Tail Value at Risk (TVaR). Both are defined relative to a confidence level p (typically 95% or 99%), but they capture different aspects of the tail. Understanding their properties, computation, and limitations is essential for Exam STAM.

Value at Risk (VaR)

VaR at level p is the p-th percentile (quantile) of the loss distribution: VaR_p(X) = F^{-1}(p) = inf{x : F(x) >= p}. It answers the question "what loss level will not be exceeded with probability p?" Equivalently, there is a (1-p) probability that losses will exceed VaR_p. For example, VaR at the 99% level is the loss that is exceeded only 1% of the time.

For an exponential distribution with mean theta, VaR_p = -theta * ln(1-p). With theta = 2000 and p = 0.95, VaR_{0.95} = -2000 * ln(0.05) = 2000 * 2.996 = 5991.5. For a Pareto distribution with parameters alpha and theta, VaR_p = theta * ((1-p)^{-1/alpha} - 1). For a lognormal with parameters mu and sigma, VaR_p = exp(mu + sigma * z_p), where z_p is the p-th percentile of the standard normal.

VaR is simple to interpret and widely used in banking regulation (Basel framework) and insurance solvency requirements. However, it has significant limitations. It provides no information about the magnitude of losses beyond VaR. Two portfolios can have the same VaR but very different tail risks: one might have losses just slightly above VaR in the worst cases, while the other could face catastrophic losses far exceeding VaR.

VaR and the Subadditivity Problem

A critical limitation of VaR is that it is not subadditive. Subadditivity requires that the risk measure of a combined portfolio not exceed the sum of individual risk measures: rho(X+Y) <= rho(X) + rho(Y). This property reflects the benefit of diversification. VaR can violate subadditivity, meaning that combining two portfolios can produce a VaR larger than the sum of their individual VaRs, which incorrectly penalizes diversification.

A classic example involves two independent risks, each with a 4% probability of a loss of 100 and 96% probability of zero loss. VaR at 95% for each risk is 0 (since P(X > 0) = 4% < 5%). The combined portfolio has P(loss = 200) = 0.04^2 = 0.0016, P(loss = 100) = 2*0.04*0.96 = 0.0768, and P(loss = 0) = 0.96^2 = 0.9216. Since P(loss > 0) = 0.0784 > 0.05, VaR at 95% for the combined portfolio is 100. So VaR(X+Y) = 100 > 0 + 0 = VaR(X) + VaR(Y), violating subadditivity.

Tail Value at Risk (TVaR)

TVaR at level p (also called Conditional Tail Expectation, CTE, or Expected Shortfall) addresses VaR's limitations by measuring the average loss in the tail beyond VaR. For continuous distributions, TVaR_p(X) = E[X | X > VaR_p(X)] = (1/(1-p)) * integral from VaR_p to infinity of x * f(x) dx. TVaR is always at least as large as VaR at the same level, and it captures information about the severity of extreme losses.

For an exponential distribution with mean theta, TVaR_p = VaR_p + theta = -theta*ln(1-p) + theta. This follows from the memoryless property: given that an exponential loss exceeds VaR_p, the expected excess is still theta, so the conditional expected total is VaR_p + theta.

For a Pareto distribution with parameters alpha > 1 and theta, TVaR_p = theta*((1-p)^{-1/alpha} - 1) * alpha/(alpha-1). This can be derived by computing the conditional mean above the p-th percentile. Notice that TVaR is a multiple alpha/(alpha-1) of VaR for the Pareto, so heavier tails (smaller alpha) produce a larger ratio of TVaR to VaR.

TVaR as a Coherent Risk Measure

TVaR satisfies all four axioms of a coherent risk measure for continuous distributions. Subadditivity holds: TVaR(X+Y) <= TVaR(X) + TVaR(Y), so diversification is always recognized. Positive homogeneity means TVaR(c*X) = c*TVaR(X) for c > 0, so scaling losses scales the risk measure proportionally. Translation invariance means TVaR(X + c) = TVaR(X) + c, so adding a certain cost increases the risk measure by that amount. Monotonicity means that if X <= Y always, then TVaR(X) <= TVaR(Y).

These properties make TVaR theoretically superior to VaR for risk management and capital allocation. In insurance regulation, Solvency II and the Swiss Solvency Test use VaR and TVaR respectively, reflecting different regulatory philosophies. The actuarial literature generally favors TVaR because of its coherence and tail sensitivity.

Computing TVaR from Empirical Data

When working with empirical data (a finite sample of losses), TVaR at level p is computed by averaging all observations that exceed the empirical VaR. If you have n observations sorted in ascending order as x_{(1)} <= x_{(2)} <= ... <= x_{(n)}, the empirical VaR at level p is x_{(ceil(n*p))}. TVaR is the average of all observations above this threshold.

For example, with 100 observations and p = 0.95, VaR is x_{(95)} and TVaR is the average of x_{(96)}, x_{(97)}, x_{(98)}, x_{(99)}, and x_{(100)}. With only 5 observations in the tail, the TVaR estimate can be quite variable. This highlights a practical challenge: tail risk measures require substantial data in the tail region for reliable estimation, and insurance data often has limited observations of extreme events.

Common Pitfalls

Students often confuse TVaR with the expected excess above VaR. TVaR is the expected total loss given that losses exceed VaR, not the expected excess: TVaR_p = E[X | X > VaR_p], not E[X - VaR_p | X > VaR_p]. The expected excess (sometimes called mean excess loss at VaR) is TVaR_p - VaR_p. Be sure to read the problem carefully to determine which quantity is asked for.

Another common mistake is applying the TVaR formula for continuous distributions to discrete data without adjustment. For discrete distributions, TVaR may need to account for the probability mass exactly at VaR. The general formula is TVaR_p = VaR_p + e(VaR_p) * [P(X > VaR_p)/(1-p)], where e is the mean excess function. For continuous distributions, P(X > VaR_p) = 1-p and this simplifies to the standard formula.

Study Tips

Exam STAM tests computation of VaR and TVaR from parametric distributions (exponential, Pareto, lognormal) and from empirical data. Memorize the VaR formulas for each major distribution family, and practice deriving TVaR by integrating the tail. Know the subadditivity counterexample for VaR and be able to explain why TVaR is preferred as a coherent risk measure. Practice computing both measures from a small set of ordered data to prepare for empirical questions. Understanding the relationship TVaR_p = VaR_p + mean excess at VaR provides a useful shortcut for distributions with known mean excess functions.