Empirical Bayes methods and credibility premiums are key tools in actuarial science for estimating future losses. They combine individual risk experience with collective data to create more accurate predictions, balancing the reliability of each source.

These techniques use statistical models to determine how much weight to give individual versus group data. The resulting credibility premiums help insurers set fair rates that reflect both overall trends and specific policyholder history.

Fundamentals of credibility theory

• Credibility theory provides a framework for combining individual risk experience with collective risk experience to estimate future losses or premiums
• Credibility theory assigns weights to individual and collective experience based on the credibility of each source of information
• Key concepts in credibility theory include credibility premium, credibility factor, and full vs. partial credibility

Classical credibility premium

Bühlmann model for experience rating

• Bühlmann model assumes a portfolio of risks with unknown risk parameters $\Theta$ and observable risk characteristics $X$
• Goal is to estimate the pure premium $\mu(\Theta)$ for each risk based on its individual experience $X$ and the collective experience of the portfolio
• Bühlmann model derives the credibility premium as a weighted average of individual and collective experience

Least squares criteria for credibility premium

• Credibility premium is derived by minimizing the expected squared error loss function $E[(\hat{\mu}(X) - \mu(\Theta))^2]$
• Optimal credibility premium is a linear function of the individual experience $X$: $\hat{\mu}(X) = \alpha + \beta X$
• Coefficients $\alpha$ and $\beta$ are determined by the least squares criteria

Full credibility vs. partial credibility

• Full credibility is assigned to individual experience when it is sufficiently large and stable to be fully reliable (e.g., large number of exposures or claims)
• Partial credibility is assigned when individual experience is not fully credible, and collective experience is given some weight
• Credibility factor $Z$ determines the weight given to individual experience, with $1-Z$ assigned to collective experience

Greatest accuracy credibility

Conditional distribution of risk parameters

• Greatest accuracy credibility assumes a conditional distribution of risk parameters $\Theta$ given the observable risk characteristics $X$
• Common choices for the conditional distribution include the normal distribution for continuous parameters and the Poisson distribution for claim counts
• Hyperparameters of the conditional distribution are estimated from the collective experience of the portfolio

Derivation of credibility premium formula

• Credibility premium is derived as the expected value of the pure premium $\mu(\Theta)$ given the individual experience $X$: $\hat{\mu}(X) = E[\mu(\Theta) | X]$
• Credibility premium formula depends on the choice of conditional distribution and the estimated hyperparameters
• For the normal conditional distribution, the credibility premium is a linear function of $X$ with credibility factor $Z = \frac{n}{n + k}$, where $n$ is the number of exposures and $k$ is a constant

Interpretation of credibility factor Z

• Credibility factor $Z$ represents the weight given to individual experience, with $1-Z$ assigned to collective experience
• $Z$ increases with the number of exposures or claims in the individual experience, reflecting greater credibility of larger datasets
• $Z$ approaches 1 for fully credible individual experience and 0 for non-credible individual experience

Bühlmann-Straub model

Assumptions and notation

• Bühlmann-Straub model generalizes the Bühlmann model to allow for varying risk volumes and exposure periods
• Assumes a portfolio of risks with unknown risk parameters $\Theta_j$ and observable risk characteristics $X_{ij}$ for risk $i$ in exposure period $j$
• Risk volumes $P_{ij}$ measure the size or exposure of each risk in each period

Derivation of credibility premium formula

• Credibility premium is derived by minimizing the expected squared error loss function, weighted by the risk volumes $P_{ij}$
• Optimal credibility premium is a weighted average of individual and collective experience, with weights proportional to $P_{ij}$
• Credibility factor $Z_j$ for each exposure period $j$ depends on the risk volumes and the variance components of the model

Estimation of model parameters

• Bühlmann-Straub model requires estimation of the variance components $\sigma^2$ (within-risk variance) and $\tau^2$ (between-risk variance)
• Variance components are estimated from the observed data using methods such as analysis of variance (ANOVA) or maximum likelihood estimation (MLE)
• Estimated variance components are used to calculate the credibility factors $Z_j$ and the credibility premiums for each risk

Empirical Bayes credibility

Relationship to greatest accuracy credibility

• Empirical Bayes credibility is a special case of greatest accuracy credibility, where the prior distribution of risk parameters is estimated from the data
• Prior distribution represents the collective experience of the portfolio, while the likelihood function represents the individual experience
• Credibility premium is derived as the posterior mean of the risk parameter, given the individual experience

Nonparametric estimation of prior distribution

• Nonparametric methods estimate the prior distribution without assuming a specific parametric form
• Kernel density estimation (KDE) is a common nonparametric method that estimates the prior density as a weighted average of kernel functions centered at each data point
• Bandwidth parameter controls the smoothness of the estimated density and is selected by cross-validation or other data-driven methods

Semiparametric estimation of prior distribution

• Semiparametric methods combine parametric and nonparametric techniques to estimate the prior distribution
• Parametric component captures the main features of the prior, while the nonparametric component captures deviations from the parametric form
• Example: Gaussian mixture models with unknown number of components, estimated using the EM algorithm or Bayesian methods

Hierarchical credibility models

Motivation for hierarchical models

• Hierarchical models account for multiple levels of variation and dependence in the data, such as risks nested within classes or territories
• Allows for borrowing strength across different levels of the hierarchy, improving the precision of credibility estimates
• Particularly useful when data is sparse or unbalanced at some levels of the hierarchy

Model assumptions and notation

• Assumes a hierarchical structure of risks, with unknown risk parameters $\Theta_{ij}$ at each level of the hierarchy
• Observable risk characteristics $X_{ijk}$ for risk $i$ in class $j$ and exposure period $k$
• Hyperparameters at each level of the hierarchy capture the variability and dependence of risk parameters across levels

Derivation of credibility premium formula

• Credibility premium is derived by minimizing the expected squared error loss function, weighted by the risk volumes at each level of the hierarchy
• Optimal credibility premium is a weighted average of individual, class, and overall experience, with weights determined by the credibility factors at each level
• Credibility factors depend on the variance components and the amount of data at each level of the hierarchy

Applications of credibility theory

Experience rating in property/casualty insurance

• Credibility theory is used to adjust premiums based on the past claims experience of individual policyholders or groups
• Credibility premiums balance the individual experience with the class or manual rates, providing a more accurate estimate of future losses
• Bühlmann-Straub model is commonly used for experience rating, with risk volumes measured by payroll, sales, or other exposure bases

Prospective premium calculation in life insurance

• Credibility theory is used to estimate mortality rates for individual life insurance policies, based on the experience of the insured and the overall portfolio
• Hierarchical credibility models can account for variation across age, gender, smoking status, and other risk factors
• Credibility estimates are used to set premiums that are adequate, equitable, and competitive

Credibility for excess loss coverages

• Excess loss coverages, such as stop-loss or umbrella policies, protect against large or catastrophic losses
• Credibility theory is used to estimate the frequency and severity of excess losses, which are often rare and highly variable
• Empirical Bayes methods are particularly useful for excess loss credibility, as they can capture the heavy-tailed nature of the loss distribution

Bayesian credibility models

Conjugate prior distributions

• Conjugate prior distributions are chosen to match the likelihood function, resulting in a posterior distribution of the same form as the prior
• Common conjugate priors include the normal distribution for normal likelihoods, the gamma distribution for Poisson likelihoods, and the beta distribution for binomial likelihoods
• Conjugate priors simplify the calculation of the posterior distribution and the credibility premium

Posterior distribution of risk parameters

• Posterior distribution of risk parameters combines the prior distribution (collective experience) with the likelihood function (individual experience) using Bayes' theorem
• Posterior distribution represents the updated beliefs about the risk parameters after observing the individual experience
• Posterior mean, mode, or median can be used as point estimates of the risk parameters

Credibility premium as posterior mean

• Credibility premium is the posterior mean of the pure premium, given the individual experience and the prior distribution
• For conjugate prior distributions, the credibility premium is a weighted average of the prior mean and the individual experience, with weights determined by the credibility factor
• Credibility factor depends on the relative precision of the prior distribution and the individual experience

Evaluation of credibility estimates

Bias vs. variance trade-off

• Credibility estimates aim to balance the bias (systematic error) and variance (random error) of the premium estimates
• Bias arises from the use of simplified models or assumptions, such as the linearity of the credibility premium or the choice of prior distribution
• Variance arises from the limited sample size of individual experience and the estimation of model parameters

Mean squared error of credibility premium

• Mean squared error (MSE) measures the average squared deviation of the credibility premium from the true pure premium
• MSE can be decomposed into bias and variance components, providing insights into the sources of error in the credibility estimates
• Optimal credibility factor minimizes the MSE, striking a balance between bias and variance

Empirical evaluation using holdout samples

• Holdout samples are used to evaluate the performance of credibility models on data not used in the model development
• Data is split into training and holdout samples, with the model fitted on the training data and evaluated on the holdout data
• Common evaluation metrics include the MSE, the mean absolute error (MAE), and the predictive log-likelihood
• Cross-validation techniques, such as k-fold or leave-one-out cross-validation, provide more robust estimates of model performance by averaging over multiple holdout samples