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Cattolica del Sacro Cuore statistical and actuarial sciences Curriculum actuarial sciences for insurance 2021
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Data Analysis for Insurance focuses on statistical methods to model and predict risk, crucial for premium calculation and financial stability. It covers identifying underlying stochastic mechanisms through data analysis, hypothesis testing with distributions like Poisson (for accident frequency) and Lognormal (for accident costs), and estimating parameters using the likelihood principle. The process involves generating random numbers for simulation and applying linear models (simple and multiple) to understand variable relationships. Key concepts include diagnostics, the Central Limit Theorem for portfolio risk assessment, and using dummy variables for categorical data, enabling personalized premium strategies based on customer profiles.
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- Core Objective: Understand and model stochastic mechanisms generating insurance data (e.g., number/cost of accidents) to facilitate ex-ante risk assessment and premium computation.
- Data Analysis Process: Utilizes past information from databases, computes descriptive statistics, visualizes data through histograms, and makes hypotheses about underlying distributions.
- Key Distributions:
- Poisson Distribution: Ideal for modeling the number of accidents, characterized by a single parameter (lambda) representing both mean and variance, under assumptions of constant, independent, and rare occurrences.
- Lognormal Distribution: Suited for describing the costs of accidents, derived from a normally distributed random variable.
- Gamma Distribution: Another flexible distribution used in insurance with shape and scale parameters.
- Gaussian distribution is generally avoided due to its unlimited domain and symmetry, which may not fit insurance losses.
- Parameter Estimation: The Likelihood Principle is the primary method to identify unknown distribution parameters by maximizing the probability of observing the collected data (Maximum Likelihood Estimation).
- Model Validation and Simulation:
- After hypothesizing a distribution, diagnostic checks (e.g., comparing mean and variance for Poisson, Q-plots) validate its fit to empirical data.
- Once validated, the stochastic process can be used to simulate future events, providing insights for scenario planning.
- Premium Calculation:
- Crucial for an insurance company's financial stability.
- The Fair Premium is typically the expected value of losses, E(X).
- Modern approaches involve Personalization of Premium, using customer profiles and covariates to precisely assess individual risk.
- Random Number Generation: Utilizes the inverse transform sampling method, where random numbers from a Uniform(0,1) distribution are transformed using the inverse of a target distribution's CDF to simulate outcomes (e.g., future accidents).
- Linear Models:
- Simple Linear Model (Y = a + bX + ε): Explores relationships between a dependent variable (Y) and a single independent variable (X), accounting for randomness (ε). Parameters (a, b) are estimated using Ordinary Least Squares (OLS).
- Multiple Linear Model (Y = Xβ + ε): Extends to multiple explanatory variables using matrix notation, where X includes a column of ones for the intercept.
- Fit vs. Predict: "Fitting" involves using known data to model existing risks, while "Predicting" uses a trained model on new data to forecast behavior or risk for new customers.
- Central Limit Theorem (CLT): Essential for understanding portfolio-level risk, stating that the distribution of sample means for a large number of independent risks tends towards a normal distribution, regardless of the individual risk distributions.
- Dummy Variables: Transforms categorical data (e.g., smoker/non-smoker) into binary (0/1) numerical representations, enabling their inclusion in regression models as "factors" or "levels."
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