Checking Aggregate Distribution Calculations

Why Aggregate Tables Need More Than a Sum Check

A compound aggregate-loss calculation combines uncertainty in the number of claims with uncertainty in the amount of each claim. A table can look plausible and even sum to one while still assigning probability to an impossible aggregate amount or combining claim outcomes incorrectly. A reliable audit therefore uses several independent checks: total probability, the zero-loss probability, feasible support, and moments.

Let \(N\) be the number of claims, let \(X_1,X_2,\ldots\) be independent and identically distributed claim amounts, and assume the severities are independent of \(N\). The aggregate loss is \(S=\sum_{j=1}^{N}X_j\), with \(S=0\) when \(N=0\). For a discrete model, probabilities at each possible value of \(S\) can be found by conditioning on \(N\) and convolving the severity distribution for each claim count.

No single check is sufficient. A total of one verifies normalization but not whether probability was put in the right rows. A moment agreement is a useful cross-check, but distinct distributions can have the same first two moments. The strongest routine audit combines all of these checks with a reconstruction of the count-conditioned probabilities.

A Systematic Audit

Moment Checks and Their Justification

Suppose \(N\) and the severities satisfy the independence assumptions above, and the relevant means and variances are finite. Write \(m=E[X_1]\) and \(\sigma_X^2=\operatorname{Var}(X_1)\). Conditioning on the claim count gives the mean and variance of \(S\):

The mean follows because, conditional on \(N=n\), the sum has mean \(nm\). For the variance, the conditional variance is \(n\sigma_X^2\), and the conditional mean is \(nm\). Applying the law of total variance gives the result:

For a direct table check, calculate \(E[S]=\sum_s sP(S=s)\) and \(E[S^2]=\sum_s s^2P(S=s)\), then use \(\operatorname{Var}(S)=E[S^2]-(E[S])^2\). The table should agree with both the frequency-severity formulas and the assumptions used to construct it. These formulas do not apply in this form if the severities are dependent on \(N\), or if claims are not independent and identically distributed; in those cases, derive moments under the actual model.

Support and Convolution Must Agree

A support check asks whether an aggregate amount can be formed at all. If claim amounts belong to a finite set and the claim count is bounded, list the sums for each possible count before accepting a table. When the count is unbounded, the support may also be unbounded, but each listed value must still be attainable under the severity model. If severities are strictly positive, a positive count cannot produce zero; if zero severities are possible, that conclusion changes.

Convolution is the arithmetic of combining independent claim amounts. For two claims with discrete severity probabilities \(p_x=P(X=x)\), the probability of a sum \(s\), conditional on two claims, is \(\sum_x p_xp_{s-x}\), over severity values \(x\) for which both terms are possible. In particular, distinct ordered pairs that give the same sum must all be included. The next example checks a table whose probabilities sum to one but whose support and convolution entries are wrong.

When the Zero Atom Is a Special Case

The shortcut \(P(S=0)=P(N=0)\) requires strictly positive claim amounts. More generally, conditional on \(N=n\), the aggregate is zero when all \(n\) claim amounts are zero, provided claim amounts are nonnegative. If \(P(X=0)=r\), independence gives \(P(S=0\mid N=n)=r^n\) for \(n\geq 1\), while \(P(S=0\mid N=0)=1\). Thus:

where \(r^0=1\), including when \(r=0\). This reduces to \(P(N=0)\) when \(r=0\). If negative severities or offsets are allowed, cancellation can also produce zero; the displayed formula then does not apply. State the severity assumptions before using a zero-atom check.

A further useful diagnostic is the smallest and largest possible aggregate amounts conditional on a given claim count. For strictly positive severities with minimum \(x_{\min}\) and maximum \(x_{\max}\), exactly \(n\) claims imply \(nx_{\min}\leq S\leq nx_{\max}\). If the claim count itself has a finite maximum \(n_{\max}\), then \(S\leq n_{\max}x_{\max}\). These bounds do not identify every feasible discrete sum, so exact support enumeration may still be needed.

Common Audit Traps

• Checking only that probabilities sum to one. A misplaced probability can preserve the total. Check each entry against count-conditioned outcomes.
• Forgetting to combine counts. The same aggregate amount may result from different claim counts; add all such unconditional contributions.
• Counting unordered pairs incorrectly. For two independent claims, \(x+y\) and \(y+x\) are distinct outcomes when \(x\ne y\), so their probabilities both contribute to the sum \(x+y\).
• Using the zero-atom shortcut without checking severity support. Positive claim amounts make zero equivalent to no claims; severities that can be zero do not.
• Treating a moment match as proof. Matching a mean and variance is a consistency check, not a complete verification of every table entry.

An efficient audit writes down the model first, including the count probabilities, severity support, and independence assumptions. It then checks normalization and the zero atom, enumerates feasible sums, reconstructs any questionable rows by count, and compares moments. This sequence catches both missing probability mass and errors that a sum-to-one check cannot reveal. These verification methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.