Insurance Ratemaking & Reserving

Bühlmann credibility · Chain-ladder reserving

Family 09ActuarialRegression

Insurance pricing combines a priori class tariffs (from industry data or theory) with a posteriori individual experience. Bühlmann (1967) formalised this as a credibility problem: the premium $P_{\text{cred}} = Z\, \bar{X} + (1-Z)\, \mu$ mixes observed claims $\bar{X}$ and the portfolio mean $\mu$ with a weight $Z \in [0,1]$ that increases with observation count and with between-risk heterogeneity. Loss reserving — estimating ultimate liabilities from partially developed claims — uses the chain-ladder method (Mack 1993) on loss-development triangles.

Educational Purpose

Educational demonstration of core actuarial techniques. Not insurance, reserving, or regulatory advice. Real ratemaking must satisfy jurisdiction-specific laws (non-discrimination, gender-neutral pricing in the EU, etc.) and regulatory filing requirements.

The models

OR family: Actuarial Solver class: Credibility / regression Measure: Physical $\mathbb{P}$ Realism: ★★★ Exact

Bühlmann credibility premium

Suppose risk $i$ has underlying mean $\Theta_i$ drawn from a prior with mean $\mu = \mathbb{E}[\Theta]$ and between-risk variance $a = \text{Var}[\Theta]$. For risk $i$, $n$ observations of claim amounts $X_{i,j}$ have within-risk variance $s^2 = \mathbb{E}[\text{Var}(X_{i,j} | \Theta_i)]$. The Bayesian-linear estimator of $\Theta_i$ (equivalently, the credibility premium) is:

Bühlmann credibility (1967) $$ P_{\text{cred}}(i) \;=\; Z\, \bar{X}_i \;+\; (1 - Z)\, \mu $$ $$ Z \;=\; \frac{n}{n + s^2/a} $$

$Z$ grows to 1 as $n \to \infty$ (full credibility on own experience); shrinks toward 0 when within-risk variance $s^2$ dominates between-risk variance $a$ (little information in individual experience).

Interpretation

The credibility factor $Z$ answers: how much weight should I put on this risk’s own experience vs the portfolio average? A new policyholder ($n = 0$) gets $Z = 0$ and pays the class tariff. A long-tenured one with many observations gets $Z \to 1$ and pays their own empirical frequency. The crossover point $n = s^2/a$ is where the two sources of information carry equal weight.

Chain-ladder reserving (Mack 1993)

Let $C_{i,j}$ = cumulative paid claims for accident year $i$ at development year $j$. Triangular data: $C_{i,j}$ observed for $i + j \le I$. Estimate future development factors:

Chain-ladder development factors $$ \hat{f}_j \;=\; \frac{\sum_{i=1}^{I-j} C_{i,j+1}}{\sum_{i=1}^{I-j} C_{i,j}}, \qquad j = 0, 1, \ldots, J-1 $$ $$ \hat{C}_{i,j+1} \;=\; C_{i,j}\, \hat{f}_j \quad (\text{for unobserved } i+j \ge I) $$

Project each incomplete row forward using successive $\hat{f}$’s to estimate ultimate claims $\hat{C}_{i,J}$. Reserve = ultimate − paid to date. Mack (1993) supplied standard errors; Mack-Bornhuetter-Ferguson combines with a priori estimates.

Collective risk model

Aggregate loss $S = \sum_{k=1}^{N} X_k$ with $N$ = claim count, $X_k$ = severities. Panjer (1981) recursion computes the compound distribution of $S$ efficiently when $N$ is in the $(a, b, 0)$ class (Poisson, binomial, negative binomial). Ruin theory (Cramér-Lundberg) gives bankruptcy probability under premium-income vs aggregate-claim dynamics.

Interactive credibility

Bühlmann credibility calculator

Portfolio mean $\mu$

$500

Between-risk var $a$

5000

Within-risk var $s^2$

80000

Observation count $n$

Observed mean $\bar{X}_i$

$800

Credibility $Z$

—

Credibility premium

—

Crossover $n^*$

—

$Z$ at $n = 10$

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$Z$ at $n = 100$

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Gold curve: credibility factor $Z(n) = n/(n + s^2/a)$. Current $n$ marked with a red vertical line. Higher $a/s^2$ steepens the curve — individual experience becomes informative faster.

Extensions & limitations

Extensions

Bühlmann-Straub (1970): weights observations by exposure size; extends to unequal risk volumes.
Hierarchical credibility (Jewell 1975): multi-level models with geographic / line-of-business tiers.
Generalised linear models (GLM): Poisson-frequency + Gamma-severity GLMs, now the industry standard for personal-lines ratemaking.
GBMs and ML for pricing: increasingly used with careful regulatory oversight and fairness auditing.
Panjer recursion, ruin theory: for aggregate-loss distribution and solvency.
Catastrophe (cat) bonds, ILS: transfer tail risk to capital markets; see systemic-risk page for network context.

Limitations of this demo

Single-risk credibility only; real tariffs use multi-class GLM structure with interaction effects.
Regulatory constraints (non-discrimination, rate-filing rules) are ignored.
Chain-ladder projects future development deterministically; Mack (1993) gave the standard-error formulas.
No treatment of reinsurance, IBNR vs IBNER distinction, or tail-factor selection.

Key references

Bühlmann, H. (1967).

Experience Rating and Credibility.

ASTIN Bulletin, 4(3), 199–207. doi:10.1017/S0515036100008989

Bühlmann, H. & Straub, E. (1970).

Glaubwürdigkeit für Schadensätze.

Mitteilungen VSVM, 70, 111–133.

Jewell, W. S. (1975).

The Use of Collateral Data in Credibility Theory: A Hierarchical Model.

Giornale dell’Istituto Italiano degli Attuari, 38, 1–16.

Mack, T. (1993).

Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates.

ASTIN Bulletin, 23(2), 213–225. doi:10.2143/AST.23.2.2005092

Panjer, H. H. (1981).

Recursive Evaluation of a Family of Compound Distributions.

ASTIN Bulletin, 12(1), 22–26. doi:10.1017/S0515036100006796

Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2019).

Loss Models: From Data to Decisions, 5th ed.

Wiley. ISBN 978-1-119-52378-9.

Werner, G. & Modlin, C. (2016).

Basic Ratemaking, 5th ed.

Casualty Actuarial Society.

Bühlmann, H. & Gisler, A. (2005).

A Course in Credibility Theory and its Applications.

Springer. ISBN 978-3-540-25753-0.