Understand, interpret, and apply the following statistical models and assumptions used for outstanding claims reserves: Mack's model, Poisson model, and Overdispersed Poisson model.

Chain ladder gives you a single reserve number. Mack, Poisson, and Overdispersed Poisson supply the statistical scaffolding that lets you attach a standard error to it.

Mack (1993) rebuilds chain ladder as a stochastic model without assuming a parametric distribution. Let $C_{i,k}$ denote cumulative paid claims for accident year $i$ at development period $k$, with $i = 1, \dots, I$ and $k = 1, \dots, I$. Three assumptions:

Assumption 1 (mean). Development factors $f_1, \dots, f_{I-1}$ exist with $E[C_{i,k+1} \mid C_{i,1}, \dots, C_{i,k}] = f_k \, C_{i,k}$. Each accident year follows the same age-to-age multiplier in expectation, conditional on its own history.

Assumption 2 (variance). Parameters $\sigma_k^2$ exist with $\text{Var}[C_{i,k+1} \mid C_{i,1}, \dots, C_{i,k}] = \sigma_k^2 \, C_{i,k}$. Conditional variance scales linearly with the running cumulative.

Assumption 3 (independence). Different accident years are independent.

KEY: Assumption 2 is the load-bearing one. Variance proportional to the current cumulative is what makes the volume-weighted average the right estimator for $f_k$.

The chain-ladder estimators emerge directly from the assumptions:

Common mistakes

• Quoting $\hat f_k$ as a simple average of link ratios. Mack's estimator is the volume-weighted average $\sum C_{i,k+1} / \sum C_{i,k}$, not the arithmetic mean.
• Using $I - k$ degrees of freedom in $\hat\sigma_k^2$. The correct divisor is $I - k - 1$.
• Treating Mack's three assumptions as a distributional claim. Mack pins down only mean and variance, no full distribution.

Bottom line

• Mack's model is distribution-free: three assumptions (mean factor $f_k$, variance proportional to $C_{i,k}$, AY independence) deliver the chain-ladder estimate plus a closed-form MSE.
• Mack's development factor is the volume-weighted average $\hat f_k = \sum C_{i,k+1} / \sum C_{i,k}$, with $\hat\sigma_k^2$ using divisor $I-k-1$, not a simple average of link ratios.
• Poisson model assumes incremental claims $X_{ij} \sim \text{Poisson}(\alpha_i \beta_j)$; its MLE reproduces the chain-ladder reserve exactly, with constraint $\sum_j \beta_j = 1$ for identifiability.
• Overdispersed Poisson (ODP) keeps the multiplicative mean but inflates variance: $\text{Var}(X_{ij}) = \phi \, \alpha_i \beta_j$ ($\phi > 1$ typical), with $\phi$ from squared Pearson residuals, denominator $n - p$, $p = 2I-1$.

Exam shortcut

For Mack, sum the columns first and divide once. Never average link ratios row-by-row; that gives the wrong $\hat f_k$. For ODP dispersion, $\hat\phi = (\text{sum of squared Pearson residuals}) / (n - p)$ with $n$ observed cells and $p = 2I - 1$ parameters. When asked to compare Mack vs ODP standard errors, recall that point reserves match across the three models. Anchor the answer to variance and $\phi$, not the mean.

The full lesson (about 2,360 words, 16 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 6b

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