📊Actuarial Mathematics Unit 7 – Loss Models & Severity Distributions
Loss models are crucial tools in actuarial science, quantifying the financial impact of uncertain events like insurance claims and natural disasters. These models use severity distributions to describe individual loss magnitudes and frequency distributions to model loss occurrences over time.
Actuaries employ various types of loss models, including individual, collective risk, and extreme value models. They utilize common severity distributions like exponential, gamma, and Pareto to fit historical data. Parameter estimation techniques and goodness-of-fit tests ensure accurate model selection and calibration for pricing, reserving, and risk management decisions.
Bayesian estimation incorporates prior beliefs and updates estimates using Bayes' theorem as new data arrives
Posterior distribution: f(θ∣x)∝f(x∣θ)f(θ), where f(θ) is the prior distribution
Empirical Bayes estimation uses data to estimate prior distribution parameters, combining frequentist and Bayesian approaches
Kernel density estimation is a non-parametric method that estimates the PDF of a random variable using a smoothing function
Confidence intervals quantify the uncertainty in parameter estimates based on the sampling distribution of the estimator
Model Selection and Goodness-of-Fit
Goodness-of-fit tests assess the compatibility of a fitted model with observed data
Chi-square test compares observed and expected frequencies in discrete categories
Kolmogorov-Smirnov test measures the maximum distance between the empirical and fitted CDFs
Likelihood ratio tests compare the goodness-of-fit of nested models
Test statistic: −2log(L(θ1)L(θ0))∼χk2, where k is the difference in the number of parameters
Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) balance model fit and complexity
AIC: −2logL(θ^)+2k, BIC: −2logL(θ^)+klog(n), where k is the number of parameters and n is the sample size
Cross-validation assesses model performance by partitioning data into training and validation sets
Residual analysis examines the differences between observed and fitted values to detect model inadequacies
Graphical methods (Q-Q plots, P-P plots) visually compare the fitted distribution to the empirical distribution
Applications in Insurance
Pricing and ratemaking: Loss models help determine premiums that cover expected losses and expenses while providing a fair return
Pure premium: E[S]=E[N]E[X], where S is aggregate loss, N is claim frequency, and X is claim severity
Reserving: Loss models estimate future claim liabilities for claims that have occurred but are not yet fully settled (IBNR, IBNER)
Chain ladder method uses historical claim development patterns to project ultimate losses
Reinsurance: Loss models help design and price reinsurance contracts that transfer risk from primary insurers to reinsurers
Excess-of-loss reinsurance covers losses above a specified retention level
Quota share reinsurance covers a fixed percentage of losses
Capital allocation: Loss models inform the allocation of capital to different lines of business or risk categories based on their risk profiles
Solvency and risk management: Loss models assess an insurer's ability to meet its obligations and guide risk mitigation strategies
Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) measure the potential magnitude of extreme losses
Advanced Topics and Extensions
Generalized linear models (GLMs) extend linear regression to accommodate non-normal response variables and link functions
Exponential dispersion family includes Poisson, gamma, and Tweedie distributions
Copulas model the dependence structure between multiple risk factors or lines of business
Common copulas: Gaussian, t, Clayton, Frank, Gumbel
Extreme value theory (EVT) focuses on modeling the tail behavior of loss distributions
Block maxima approach fits a generalized extreme value (GEV) distribution to the maximum losses in fixed time intervals
Peaks-over-threshold (POT) approach fits a generalized Pareto distribution (GPD) to losses exceeding a high threshold
Bayesian hierarchical models allow for the incorporation of multiple sources of uncertainty and the borrowing of information across related groups
Machine learning techniques (neural networks, gradient boosting, random forests) can enhance loss modeling by capturing complex, non-linear relationships
Spatial and temporal dependence models capture the correlation structure of losses across different geographic regions or time periods
Catastrophe modeling simulates the impact of natural disasters (hurricanes, earthquakes) on insured losses using physical and statistical models