📊Actuarial Mathematics Unit 7 – Loss Models & Severity Distributions

Key Concepts

• Loss models quantify and analyze the financial impact of uncertain events (insurance claims, natural disasters)
• Severity distributions describe the probability distribution of the magnitude of individual losses
  • Common severity distributions include exponential, gamma, Pareto, and lognormal
• Frequency distributions model the number of losses occurring in a given time period (Poisson, negative binomial)
• Aggregate loss models combine severity and frequency distributions to estimate total losses over a specified period
• Parameter estimation techniques (method of moments, maximum likelihood estimation) fit distributions to historical loss data
• Goodness-of-fit tests (chi-square, Kolmogorov-Smirnov) assess the appropriateness of a chosen model
• Loss models inform pricing, reserving, and risk management decisions in insurance and finance

Types of Loss Models

• Individual loss models focus on the severity distribution of single losses without considering frequency
• Collective risk models incorporate both severity and frequency distributions to analyze aggregate losses
  • Compound Poisson model assumes losses follow a Poisson process with independently and identically distributed severities
• Credibility theory models blend individual and collective approaches based on the credibility of available data
• Extreme value theory models focus on the tail behavior of loss distributions to capture rare, high-impact events
• Copula models capture dependence structures between multiple risk factors or lines of business
• Bayesian models incorporate prior information and update estimates as new data becomes available
• Simulation models generate realistic loss scenarios by sampling from underlying distributions

Severity Distributions

• Exponential distribution has a constant hazard rate and is memoryless, often used for modeling small to medium-sized losses
• Gamma distribution is flexible and can model a wide range of shapes, making it suitable for various types of losses
  • Special cases include the exponential (shape parameter = 1) and Erlang (shape parameter is an integer) distributions
• Pareto distribution has a heavy tail and is used to model large losses and extreme events
  • Generalized Pareto distribution extends the standard Pareto to allow for more flexible tail behavior
• Lognormal distribution is positively skewed and has a heavy right tail, often used for modeling claim sizes
• Weibull distribution can model increasing, decreasing, or constant hazard rates depending on its shape parameter
• Burr distribution is a flexible three-parameter distribution that can capture a wide range of skewness and kurtosis levels
• Mixed distributions (mixture of exponentials, composite models) combine multiple distributions to better fit complex loss data

Probability Theory Foundations

• Probability spaces consist of a sample space, a set of events, and a probability measure satisfying certain axioms
• Random variables are functions that map outcomes in a sample space to real numbers
  • Discrete random variables have countable outcomes, while continuous random variables have uncountable outcomes
• Probability density functions (PDFs) and cumulative distribution functions (CDFs) characterize the behavior of continuous random variables
  • PDF: $f_X(x) = \frac{d}{dx}F_X(x)$, CDF: $F_X(x) = P(X \leq x)$
• Moment generating functions (MGFs) and probability generating functions (PGFs) uniquely determine a distribution and facilitate calculations
  • MGF: $M_X(t) = E[e^{tX}]$, PGF: $G_X(z) = E[z^X]$
• Convolutions and compound distributions describe the distribution of sums of independent random variables
• Central Limit Theorem states that the sum of many independent random variables converges to a normal distribution under certain conditions

Parameter Estimation

• Method of moments estimates parameters by equating sample moments to population moments
  • For a parameter $\theta$, solve $\frac{1}{n}\sum_{i=1}^n X_i^k = E[X^k]$ for $k=1,2,\ldots$
• Maximum likelihood estimation (MLE) finds parameters that maximize the likelihood of observing the given data
  • Likelihood function: $L(\theta) = \prod_{i=1}^n f(x_i; \theta)$, solve $\frac{d}{d\theta}\log L(\theta) = 0$
• Bayesian estimation incorporates prior beliefs and updates estimates using Bayes' theorem as new data arrives
  • Posterior distribution: $f(\theta|x) \propto f(x|\theta)f(\theta)$, where $f(\theta)$ 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: $-2\log(\frac{L(\theta_0)}{L(\theta_1)}) \sim \chi^2_k$, 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: $-2\log L(\hat{\theta}) + 2k$, BIC: $-2\log L(\hat{\theta}) + k\log(n)$, 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