Use convolution and recursive formulas to derive probability and distribution functions for aggregate claims distributions with (a,b,0) or (a,b,1) frequency, and with discrete severity distributions.

The aggregate loss $S = X_1 + \cdots + X_N$ sits between a frequency $N$ and i.i.d. severities $X_i$. Direct convolution is honest but slow. Panjer recursion collapses the work into a single loop once $N$ lives in the (a,b,0) or (a,b,1) class.

Compound model. $S = \sum_{i=1}^{N} X_i$ with $N$ independent of i.i.d. $X_i$. The pgf stacks: $P_S(z) = P_N(P_X(z))$. Moments: $E[S] = E[N]\,E[X]$ and $\operatorname{Var}(S) = E[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,E[X]^{2}$. Use both as table-top sanity checks on any computed pmf.

Convolution. The n-fold convolution of $X$ at $x$ is the pmf of $X_1 + \cdots + X_n$:

$$f_X^{*n}(x) = \sum_{y=0}^{x} f_X^{*(n-1)}(y)\,f_X(x-y),\qquad f_X^{*0}(x) = \mathbf{1}\{x=0\}.$$

The aggregate pmf is the mixture:

$$f_S(x) = \sum_{n=0}^{\infty} p_n\,f_X^{*n}(x).$$

Rigorous, no assumptions on $N$. But cost is $O(N_{\max}\,x_{\max}^{2})$. Use it when the frequency support is small (truncated, deterministic, or a handful of terms), or as a check against Panjer on the same problem.

(a,b,0) class. Defined by $p_k = (a + b/k)\,p_{k-1}$ for $k \ge 1$. Only three families satisfy this:

Common mistakes

• Applying Panjer with the wrong $(a, b)$. For binomial with $m = 10, q = 0.2$, the correct values are $a = -0.25$ and $b = 2.75$, not $a = q,\, b = mq$. Always derive from the ratio formula.
• Forgetting to rescale (a,b,1) probabilities. Setting $p_0^{M} = 0.30$ without bumping the other masses leaves a pmf that sums to less than one. Multiply non-zero $p_k$ by $(1-p_0^{M})/(1-p_0)$.
• Dropping the correction term in (a,b,1) Panjer. Skipping $(p_1^{M} - (a+b)p_0^{M})f_X(x)$ gives ordinary Panjer on a modified frequency, which is wrong at every $x \ge 1$.

Bottom line

• Aggregate pmf by conditioning on count: $f_S(x) = \sum_{n} p_n\,f_X^{*n}(x)$; the pgf identity $P_S(z)=P_N(P_X(z))$ yields $E[S]=E[N]E[X]$ and the compound variance.
• (a,b,0) class: ratio $p_k/p_{k-1} = a + b/k$ for $k \ge 1$, exactly Poisson, binomial, and negative binomial, identified by the sign of $a$.
• (a,b,1) class: same ratio for $k \ge 2$ with $p_0$ and $p_1$ free; includes zero-truncated and zero-modified versions.
• Panjer (a,b,0), severity on positives: $f_S(x) = \sum_{y=1}^{x}(a + by/x)\,f_X(y)\,f_S(x-y)$, $f_S(0) = p_0$.

Exam shortcut

Identify the family fast: $a = 0$ Poisson, $a < 0$ binomial, $a > 0$ negative binomial. Then read $b$ off the formulas. Geometric is the NB with $b = 0$. For (a,b,1), compute the rescale factor $(1 - p_0^{M})/(1 - p_0)$ first, multiply original $p_1$ by it to get $p_1^{M}$, then plug into the correction term once and reuse across all $x$. Spot-check $f_S(1) = p_1^{M} f_X(1)$ for (a,b,1) or $p_1 f_X(1)$ for (a,b,0).

The full lesson (about 4,453 words, 30 min read) adds 6 worked examples, all 12 common mistakes, a self-check, free in the app.

Learning objectives

• 2a

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