Model insurance claim frequency and severity.

A pricing actuary needs two pieces to set a premium: how often claims happen and how big they are when they do. Frequency and severity each have their own distributional toolbox, and MAS-I tests both fluently.

Frequency distributions. Claim counts are non-negative integers, so the candidate distributions are discrete.

Poisson($\lambda$): $P(N=k)=e^{-\lambda }{\lambda }^k/k!$, with mean and variance both equal to $\lambda$. Use when arrivals are independent and the portfolio is homogeneous.

Negative binomial($r,\beta$): mean $r\beta$, variance $r\beta (1+\beta )$. Variance exceeds the mean by factor $(1+\beta )$. It arises as a gamma-mixed Poisson, so it is the default when policyholders vary in underlying risk.

Binomial($m,q$): mean $mq$, variance $mq(1-q)$. Variance is below the mean. Appropriate when the count is bounded (claims per policy capped at $m$).

KEY: Compare sample mean and variance first. $\text{Var}/\text{Mean}<1$ suggests binomial, $\approx 1$ suggests Poisson, $>1$ suggests negative binomial.

The $(a,b,0)$ class. All three satisfy $p_k/p_{k-1}=a+b/k$ for constants $a,b$. Poisson has $a=0,b=\lambda$; negative binomial has $a=\beta /(1+\beta ),b=(r-1)\beta /(1+\beta )$; binomial has $a=-q/(1-q),b=(m+1)q/(1-q)$.

Common mistakes

• Forgetting that compound-Poisson variance uses $E[X^2]$, not $\text{Var}(X)$. Plugging variance alone systematically understates the answer.
• Confusing negative binomial parameterizations. Some texts use $(r,p)$ with $p=1/(1+\beta )$. Mean is then $r(1-p)/p$. Always confirm which form the question uses.
• Fitting Pareto when $\alpha \le 1$. The mean does not exist. Reported sample mean is meaningless as an estimator; use the median or a different family.

Bottom line

• Frequency: Poisson ($E=\text{Var}$), Negative Binomial (over-dispersed), Binomial (under-dispersed). The $(a,b,0)$ class covers all three, identified from a plot of $k{p}_k/p_{k-1}$.
• Variance-to-mean ratio classifies the family: below 1 binomial, near 1 Poisson, above 1 negative binomial.
• Negative binomial is the gamma-mixed Poisson model for heterogeneous portfolios, inflating variance above the mean.
• Severity: Exponential, Gamma, Weibull, Lognormal, Pareto. Choose by tail weight: lognormal and Pareto for heavy tails, gamma and Weibull for lighter tails.

Exam shortcut

When the question gives a sample mean and variance for counts, compute the ratio first. The answer family is decided before you reach for a formula. For aggregate-loss problems, write $E[S]$ and $\text{Var}(S)$ using the general two-term formula, then simplify only if $N$ is Poisson.

The full lesson (about 1,569 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B5

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