Severity Distributions
The severity (individual loss size) $X$ is modeled by heavy-tailed distributions that capture the possibility of large claims.
Lognormal: If $\ln X \sim N(\mu, \sigma^2)$ then:
\[f_X(x) = \frac{1}{x\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0\] \[E[X] = e^{\mu + \sigma^2/2}, \quad \text{Var}(X) = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)\]Pareto (single-parameter): $F(x) = 1 - (\theta/(\theta+x))^\alpha$. The mean excess loss $e(d) = E[X-d \mid X>d]$ is linear in $d$:
\[e(d) = \frac{\theta + d}{\alpha - 1}, \quad \alpha > 1\]making the Pareto the canonical heavy-tailed severity model.
Gamma: Shape $\alpha$, rate $\beta$. Closed under convolution for equal-rate gammas: sum of $n$ independent $\text{Gamma}(\alpha_i, \beta)$ is $\text{Gamma}(\sum \alpha_i, \beta)$.
Weibull: $F(x) = 1 - e^{-(x/\theta)^\tau}$. For $\tau < 1$ the hazard decreases (light-tailed); for $\tau > 1$ it increases. At $\tau = 1$, exponential.
| Distribution | Tail behavior | Mean excess | Typical use |
|---|---|---|---|
| Lognormal | Moderate heavy | Increasing | General property |
| Pareto | Heavy (power) | Linear in $d$ | Large liability |
| Gamma | Light-medium | Decreasing | Workers’ comp |
| Weibull | Flexible | Varies | Reliability, WC |
Frequency Distributions and the (a, b, 0) Class
The frequency $N$ (number of claims) follows a counting distribution. The $(a, b, 0)$ class satisfies the recursion:
\[\frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \quad k = 1, 2, 3, \ldots\]This single recursion generates three canonical distributions:
| Distribution | $a$ | $b$ | $p_0$ |
|---|---|---|---|
| Poisson($\lambda$) | $0$ | $\lambda$ | $e^{-\lambda}$ |
| Negative Binomial($r, p$) | $1-p$ | $(r-1)(1-p)$ | $p^r$ |
| Binomial($m, q$) | $-q/(1-q)$ | $(m+1)q/(1-q)$ | $(1-q)^m$ |
The Poisson is the default model ($\text{Var}(N) = E[N]$). The Negative Binomial handles overdispersion ($\text{Var}(N) > E[N]$), while the Binomial handles underdispersion. Zero-modified and zero-truncated variants extend coverage to portfolios with excess zeros.
Compound Distributions and the Panjer Recursion
The aggregate loss for a portfolio is:
\[S = X_1 + X_2 + \cdots + X_N = \sum_{i=1}^{N} X_i\]where $N$ is the claim count (frequency) and $X_i$ are i.i.d. severities independent of $N$. The mean and variance are:
\[E[S] = E[N]\,E[X], \quad \text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,(E[X])^2\]For discrete severities (or after discretization), the Panjer recursion computes the aggregate distribution exactly:
\[f_S(s) = \frac{1}{1 - a\,f_X(0)} \sum_{x=1}^{s} \left(a + \frac{bx}{s}\right) f_X(x)\, f_S(s-x), \quad s = 1, 2, \ldots\]with $f_S(0) = p_N(0)$ if $f_X(0) = 0$, or more generally:
\[f_S(0) = P_N(f_X(0))\]where $P_N$ is the probability generating function of $N$. The Panjer recursion is $O(n^2)$ in the number of discretization points — far more efficient than direct convolution or Fast Fourier Transform for moderate-sized loss ranges, though FFT wins for very large grids.
Stop-Loss Premiums and the Limited Expected Value
The stop-loss premium with retention $d$ (also called the net stop-loss premium) is the pure premium for excess-of-loss reinsurance:
\[\pi(d) = E[\max(S - d, 0)] = E[(S-d)_+] = \int_d^{\infty} (s-d)\,f_S(s)\,ds\]It can be rewritten as:
\[\pi(d) = E[S] - d + \int_0^d F_S(s)\,ds = E[S] - d + \int_0^d F_S(s)\,ds\]or equivalently $\pi(d) = E[S] - E[\min(S, d)]$ where $E[\min(S,d)]$ is the limited expected value:
\[E[X \wedge d] = \int_0^d [1 - F_X(x)]\,dx = E[X] - \pi(d)\]For the Pareto($\alpha$, $\theta$) severity:
\[E[X \wedge d] = \frac{\theta}{\alpha-1}\left[1 - \left(\frac{\theta}{\theta+d}\right)^{\alpha-1}\right], \quad \alpha > 1\]The excess loss variable (residual life) given the loss exceeds $d$ is:
\[Y = X - d \mid X > d, \quad E[Y] = \frac{E[(X-d)_+]}{1 - F_X(d)} = e(d)\]For exponential $X \sim \text{Exp}(\theta)$, $e(d) = \theta$ for all $d$ (memoryless property); for Pareto, $e(d) = (\theta+d)/(\alpha-1)$, linear and increasing.
Risk Measures for Insurance
The Value-at-Risk at confidence level $p$ is the $p$-quantile of aggregate losses:
\[\text{VaR}_p(S) = F_S^{-1}(p) = \inf\{s : F_S(s) \ge p\}\]For Solvency II, $p = 99.5\%$ over a one-year horizon defines the Solvency Capital Requirement (SCR).
The Conditional Value-at-Risk (CVaR, also called TVaR or ES) averages losses above VaR:
\[\text{CVaR}_p(S) = E[S \mid S > \text{VaR}_p(S)] = \frac{1}{1-p}\int_p^1 \text{VaR}_u(S)\,du\]For a continuous distribution:
\[\text{CVaR}_p(S) = \text{VaR}_p(S) + \frac{\pi(\text{VaR}_p(S))}{1-p}\]where $\pi(\cdot)$ is the stop-loss premium. CVaR is coherent (satisfies subadditivity, monotonicity, translation invariance, positive homogeneity) while VaR is not in general.
| Risk measure | Formula | Coherent? | Solvency II use |
|---|---|---|---|
| VaR$_{99.5\%}$ | $F_S^{-1}(0.995)$ | No | SCR standard formula |
| CVaR$_{99\%}$ | $E[S\mid S > \text{VaR}_{99\%}]$ | Yes | Internal models |
| Standard deviation | $E[S] + \lambda\,\text{SD}(S)$ | No | Simple pricing loads |