Actuarial Notes

Risk Measure

Study guides: Exam FAM, Exam ASTAM, Exam 9

A Risk Measure \rho assigns a single number \rho(X) to a random loss X — the capital needed to support it, or the premium to charge for it. Exam 9 uses risk measures in both roles: a capital risk measure sets the assets a an insurer holds against X, and a pricing risk measure sets the premium for the loss those assets can actually pay, X \wedge a.

\text{VaR}_p(X) = \inf\{\, x : F(x) \geq p \,\}

\text{TVaR}_p(X) = \frac{1}{1-p}\int_p^1 \text{VaR}_s(X)\, ds

\rho_g(X) = \int_0^\infty g\big(S(x)\big)\, dx

  • Value at Risk is the lower p-quantile of the loss (Percentile). A PML is a VaR: the 1-in-250 PML is \text{VaR}_{0.996}. VaR is easy to explain, always exists and is easy to back-test, but it is blind to how bad losses beyond the quantile are.

  • Tail Value at Risk averages the worst 1-p of outcomes; for any distribution \text{TVaR}_p = \text{VaR}_p + E[(X - \text{VaR}_p)^+]/(1-p). The conditional tail expectation E[X \mid X \geq \text{VaR}_p] equals TVaR for a continuous loss but not when there is a probability mass at \text{VaR}_p. Expected shortfall and CVaR usually mean TVaR, though some actuarial texts use expected shortfall for the unconditional E[(X - \text{VaR}_p)^+] — check the definition in use. The expected policyholder deficit E[(X - a)^+] is the loss that assets a leave unpaid.

  • Moment-based measures — the standard deviation principle E[X] + k\,\sigma(X) and the variance principle E[X] + k\,\text{Var}(X) — measure volatility around the mean rather than the tail, and are the classical basis of Risk Loads (Standard Deviation, Variance).

  • Coherence (Artzner, Delbaen, Eber and Heath, 1999) asks for four properties:

    • monotone: X \leq Y \Rightarrow \rho(X) \leq \rho(Y)
    • subadditive: \rho(X + Y) \leq \rho(X) + \rho(Y)
    • positive homogeneous: \rho(\lambda X) = \lambda\,\rho(X) for \lambda \geq 0
    • translation invariant: \rho(X + c) = \rho(X) + c

    TVaR has all four. VaR fails subadditivity. The standard deviation principle fails monotonicity (with k = 2, a 50/50 chance of 0 or 10 scores 15 — more than a certain loss of 10); the variance principle fails positive homogeneity and subadditivity; (1+\theta)E[X] fails translation invariance.

  • Distortion (spectral) risk measures weight each layer of loss by g(S(x)), where S = 1 - F and g:[0,1] \to [0,1] is increasing with g(0) = 0, g(1) = 1. Those conditions make \rho_g monotone and translation invariant; a concave g also makes it subadditive, hence coherent (and comonotonic additive). Examples: proportional hazard s^{\alpha} (0 < \alpha \leq 1), Wang \Phi(\Phi^{-1}(s) + \lambda), dual 1 - (1-s)^m (m \geq 1), TVaR \min(1, s/(1-p)) and constant cost of capital \nu s + \delta for s > 0. VaR is the distortion equal to 1 for s > 1-p and 0 otherwise — a step, not concave.

  • Basic measures for any organization. Before tail measures come the descriptive ones. Frequency is how often losses occur and Severity how large they are; the expected value is the central tendency. The range, highest outcome minus lowest, is a rough measure that ignores how likely the extremes are. Variance and standard deviation average the deviations from the mean, treating favourable and unfavourable surprises alike. The coefficient of variation, \sigma / \mu, compares risk between distributions with different means. Semivariance and downside standard deviation count only unfavourable deviations, and shortfall risk is the probability of falling below a target, with the probability of ruin the case where the target is the point capital runs out.

  • Two families. The CAS's 2003 ERM overview splits ERM measures into solvency-related measures, which look at the adverse tail (probability of ruin, shortfall risk, VaR, EPD, TVaR) and concern customers and their proxies, regulators and rating agencies; and performance-related measures, which look at volatility near the mean (variance, standard deviation, semi-variance, below-target risk) and concern owners and their proxies, such as stock analysts.

  • Strategic effect. The measure adopted for capital, limits and reinsurance shapes behaviour: a target at one return period can be met by piling risk just beyond it, while TVaR charges for the whole tail but is harder to estimate and does not exist when the mean is infinite. Solvency II uses a 99.5% one-year VaR (Solvency II); the Swiss Solvency Test uses 99% TVaR. See Risk Taxonomies, Risk Modeling and Risk Capital.

VaR, TVaR and CTE for a Catastrophe Portfolio An insurer's annual catastrophe loss X ($ millions) is 0 with probability 0.80, 10 with probability 0.12, 50 with probability 0.06 and 100 with probability 0.02. Compute E[X], \sigma(X), \text{VaR}_{0.95}, \text{TVaR}_{0.95} and \text{CTE}_{0.95}.

Answer

\begin{align*}
E[X] &= 0.12(10) + 0.06(50) + 0.02(100) \\
&= 6.2 \\
E[X^2] &= 0.12(100) + 0.06(2{,}500) + 0.02(10{,}000) \\
&= 362 \\
\sigma(X) &= \sqrt{362 - 6.2^2} \\
&= 17.99
\end{align*}

The CDF steps through F(0) = 0.80, F(10) = 0.92, F(50) = 0.98, F(100) = 1. The first value at which F reaches 0.95 is 50, so \text{VaR}_{0.95} = 50. TVaR averages the worst 5\% of probability — all 2\% at 100 and 3\% of the 6\% mass at 50:

\begin{align*}
\text{TVaR}_{0.95} &= \frac{0.02(100) + 0.03(50)}{0.05} \\
&= 70
\end{align*}

The shortcut agrees: 50 + 0.02(100 - 50)/0.05 = 50 + 20 = 70. CTE conditions on X \geq 50, which takes in the whole mass at 50:

\begin{align*}
\text{CTE}_{0.95} &= \frac{0.06(50) + 0.02(100)}{0.08} \\
&= 62.5
\end{align*}

VaR says a 1-in-20 year costs $50 million; TVaR says the worst 1-in-20 years average $70 million. CTE is lower because the atom at 50 drags in outcomes better than the worst 5\% — the discrete-distribution trap.

Two Catastrophe Treaties: VaR Is Not Subadditive A reinsurer writes two treaties on unrelated perils. Each year exactly one of three states occurs: no loss (probability 0.92), treaty 1 loses 10 (probability 0.04), or treaty 2 loses 10 (probability 0.04). Compare VaR and TVaR at p = 0.95 for each treaty and for the pair.

Answer Each treaty alone: P(X_i = 0) = 0.96 \geq 0.95, so \text{VaR}_{0.95}(X_i) = 0. Its worst 5\% is 4\% at 10 and 1\% at 0:

\begin{align*}
\text{TVaR}_{0.95}(X_i) &= \frac{0.04(10) + 0.01(0)}{0.05} \\
&= 8
\end{align*}

The pair: X_1 + X_2 = 10 with probability 0.08, so F(0) = 0.92 < 0.95 and \text{VaR}_{0.95} = 10. The worst 5\% is all at 10, so \text{TVaR}_{0.95} = 10.

\begin{align*}
\text{VaR: } 10 &> 0 + 0 \\
\text{TVaR: } 10 &\leq 8 + 8
\end{align*}

Under VaR each treaty needs no capital but the pair needs 10 — combining unrelated risks raised measured risk, so limits set treaty by treaty do not control the total. TVaR rewards the diversification. The older CTE definition fails too: E[X_i \mid X_i \geq 0] = 0.4 for each treaty against 10 for the pair.

TVaR and a Proportional Hazard Premium as Distortions For the portfolio in the first example, evaluate \rho_g(X) with (a) g(s) = \min(1,\, s/0.05) and (b) the proportional hazard distortion g(s) = \sqrt{s}.

Answer S(x) is 0.20 on [0, 10), 0.08 on [10, 50), 0.02 on [50, 100) and 0 above, so the integral is a sum of layer width times g(S):

\begin{align*}
\rho_{(a)} &= 10(1) + 40(1) + 50(0.4) \\
&= 70 \\
\rho_{(b)} &= 10\sqrt{0.20} + 40\sqrt{0.08} + 50\sqrt{0.02} \\
&= 4.47 + 11.31 + 7.07 \\
&= 22.86
\end{align*}

(a) reproduces \text{TVaR}_{0.95} = 70, and g(s) = s would return E[X] = 10(0.20) + 40(0.08) + 50(0.02) = 6.2. The proportional hazard premium of 22.86 carries a margin of 16.66 over expected loss, most of it on the remote layers, where \sqrt{s} is many times s.

Comparing Two Locations With Basic Measures A retailer's annual property losses ($000s) over five years were 40, 55, 60, 45 and 50 at Location A and 70, 130, 100, 90 and 110 at Location B. Treating each year equally, find the mean, range, standard deviation and coefficient of variation for each. Which location's losses are relatively more variable?

Answer

\begin{align*}
\mu_A &= 50, \quad \text{range}_A = 60 - 40 = 20 \\
\sigma_A &= \sqrt{\tfrac{(-10)^2 + 5^2 + 10^2 + (-5)^2 + 0^2}{5}} \\
&= \sqrt{50} \\
&= 7.07 \\
\text{CV}_A &= 7.07 / 50 \\
&= 0.141
\end{align*}
\begin{align*}
\mu_B &= 100, \quad \text{range}_B = 130 - 70 = 60 \\
\sigma_B &= \sqrt{\tfrac{(-30)^2 + 30^2 + 0^2 + (-10)^2 + 10^2}{5}} \\
&= \sqrt{400} \\
&= 20 \\
\text{CV}_B &= 20 / 100 \\
&= 0.200
\end{align*}

Location B's standard deviation is almost three times A's, but its average loss is twice as large. The coefficient of variation adjusts for that: 0.20 against 0.14, so B's losses are still relatively more variable, and harder to budget for as a share of their expected cost.

How it appears on the exam

  • DISC-RM: The introduction to risk management covers basic risk measures. Questions ask what a measure shows, such as why the coefficient of variation is used to compare exposures of different sizes, which measure looks only at unfavourable outcomes, or what a VaR figure means, and may ask for a simple calculation of a mean, a standard deviation or a coefficient of variation.
  • Exam 9: Risk measures set capital and premiums, so questions compute VaR, TVaR and distortion measures for a loss distribution, test coherence, and ask how the choice of measure shapes decisions, as in the examples above.

Exam traps

  • Standard deviation needs a common scale: Comparing standard deviations of exposures with different means overstates the risk of the larger one. Use the coefficient of variation.
  • The range ignores probability: Two exposures can have the same range while one reaches its extreme far more often. The range is a rough first look, not a measure of likelihood.
  • VaR is not the worst case: A 99% VaR is the loss exceeded 1% of the time, and it says nothing about how large the losses beyond it are. TVaR averages those tail losses.