5.1 Individual and collective risk models

Risk models are essential tools in actuarial science, helping quantify and manage financial risks. Individual models focus on each insured unit, providing detailed insights but requiring more data. Collective models treat portfolios as a whole, offering simplicity at the cost of less granular risk assessment.

These models form the foundation for insurance pricing, reserving, and solvency calculations. By combining frequency and severity distributions, actuaries can estimate aggregate claims and make informed decisions about risk management strategies. Understanding these models is crucial for navigating the complex world of insurance and financial risk.

Types of risk models

• Risk models quantify and assess potential financial losses in actuarial applications such as insurance pricing, reserving, and capital requirements
• Two main categories of risk models are individual models and collective models, each with different assumptions and approaches to modeling claims

Individual vs collective models

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• Individual models focus on modeling the claims experience of each insured unit separately, considering their specific characteristics and risk factors
• Collective models treat the entire portfolio as a whole, modeling the aggregate claims arising from the group without distinguishing between individual risks
• Individual models provide more granular insights but require more data and computations, while collective models offer simplicity and tractability at the expense of less detailed risk assessment

Assumptions and limitations

• Risk models rely on assumptions about the underlying claim processes, such as independence between claims, stationarity of claim distributions over time, and homogeneity of risk within the portfolio
• Models are simplifications of reality and may not capture all aspects of the real-world claims experience, leading to potential model risk and uncertainty
• Limitations arise from data quality, parameter estimation, and the inherent randomness of claims, requiring careful model selection, validation, and sensitivity analysis

Individual risk models

• Individual risk models assess the claims experience of each insured unit separately, considering their specific risk characteristics and exposure
• Key components include claim frequency, claim severity, and risk factors such as age, gender, occupation, and policy features

Structure of individual models

• Individual models typically consist of two main components: a frequency model for the number of claims and a severity model for the claim amounts
• Frequency models often use discrete probability distributions such as Poisson, negative binomial, or binomial, depending on the nature of the claims process
• Severity models use continuous probability distributions such as gamma, lognormal, or Pareto to represent the size of individual claim amounts

Key components and variables

• Claim frequency: the number of claims occurring within a specified time period, modeled using discrete probability distributions
• Claim severity: the size or amount of each individual claim, modeled using continuous probability distributions
• Risk factors: policyholder characteristics (age, gender, occupation) and policy features (deductibles, limits) that influence the likelihood and size of claims
• Exposure: the measure of risk associated with each insured unit, such as the number of policies, the sum insured, or the duration of coverage

Modeling individual claim amounts

• Claim severity models aim to capture the distribution of individual claim sizes, which often exhibit right-skewness and heavy tails
• Common distributions for modeling claim amounts include gamma, lognormal, Weibull, and Pareto, chosen based on goodness-of-fit tests and domain knowledge
• Parameter estimation techniques such as maximum likelihood estimation (MLE) or method of moments (MoM) are used to fit the chosen distribution to historical claim data
• Tail risk measures like value-at-risk (VaR) and expected shortfall (ES) quantify the potential for large claims and inform risk management decisions

Aggregate claims distribution

• The aggregate claims distribution combines the frequency and severity models to determine the total claims for the portfolio over a given time period
• Convolution techniques or Monte Carlo simulation are used to derive the aggregate claims distribution from the individual frequency and severity distributions
• Key risk measures such as expected value, variance, and quantiles of the aggregate claims provide insights into the overall risk profile and inform pricing and reserving decisions
• Aggregate loss distributions enable the calculation of risk premiums, stop-loss premiums, and the allocation of capital to ensure solvency and profitability

Collective risk models

• Collective risk models focus on the aggregate claims arising from a portfolio of risks, treating the portfolio as a whole without distinguishing between individual risks
• Key components include claim frequency, claim severity, and the resulting aggregate claims distribution

Structure of collective models

• Collective models consist of two main components: a frequency model for the number of claims and a severity model for the claim amounts
• The frequency and severity models are combined using compound distributions to obtain the aggregate claims distribution for the entire portfolio
• Common frequency distributions include Poisson, negative binomial, and binomial, while severity distributions include gamma, lognormal, and Pareto

Key components and variables

• Claim frequency: the number of claims occurring within a specified time period for the entire portfolio, modeled using discrete probability distributions
• Claim severity: the size or amount of each individual claim, modeled using continuous probability distributions
• Exposure: the measure of risk associated with the portfolio, such as the total number of policies, the aggregate sum insured, or the total premium income
• Risk parameters: the characteristics of the frequency and severity distributions, such as the mean and variance, estimated from historical claims data

Claim frequency distributions

• Poisson distribution: models the number of claims as a rare event process, assuming independence between claims and a constant claim rate over time
• Negative binomial distribution: accommodates overdispersion (variance greater than mean) in claim counts, often arising from heterogeneity in risk exposure or contagion effects
• Binomial distribution: models the number of claims as a series of independent Bernoulli trials, suitable for situations with a fixed number of policies and a constant claim probability

Claim severity distributions

• Gamma distribution: a flexible two-parameter distribution for modeling right-skewed claim amounts, with a shape parameter controlling the skewness and a scale parameter determining the spread
• Lognormal distribution: models claim sizes that are the product of many small multiplicative effects, resulting in a log-transformed normal distribution
• Pareto distribution: captures heavy-tailed claim size distributions, where large claims occur more frequently than in lighter-tailed distributions like gamma or lognormal
• Weibull distribution: a versatile distribution for modeling claim amounts with varying hazard rates, including increasing, decreasing, or constant hazard over time

Aggregate claims distribution

• The aggregate claims distribution is obtained by combining the claim frequency and severity distributions using compound distribution techniques
• Compound Poisson distribution: models the aggregate claims as a sum of a random number of independent and identically distributed (i.i.d.) claim amounts, where the number of claims follows a Poisson distribution
• Compound negative binomial distribution: accommodates overdispersion in the claim frequency while modeling the claim amounts as i.i.d. random variables
• Recursive formulas, Panjer's recursion, or Fast Fourier Transform (FFT) techniques are used to compute the aggregate claims distribution efficiently

Compound distributions

• Compound distributions model the aggregate claims as a sum of a random number of independent and identically distributed (i.i.d.) claim amounts
• The number of claims is modeled by a discrete frequency distribution, while the claim amounts are modeled by a continuous severity distribution

Definition and properties

• Let $N$ be a random variable representing the number of claims, and $X_1, X_2, \ldots, X_N$ be i.i.d. random variables representing the individual claim amounts
• The aggregate claims random variable $S$ is defined as the sum of the individual claim amounts: $S = \sum_{i=1}^N X_i$
• The distribution of $S$ is called a compound distribution, with the frequency distribution of $N$ and the severity distribution of $X_i$ as its building blocks
• Key properties of compound distributions include the expected value $\mathbb{E}[S] = \mathbb{E}[N] \cdot \mathbb{E}[X]$ and the variance $\mathbb{V}ar[S] = \mathbb{E}[N] \cdot \mathbb{V}ar[X] + \mathbb{V}ar[N] \cdot (\mathbb{E}[X])^2$

Poisson compound distribution

• The Poisson compound distribution arises when the claim frequency $N$ follows a Poisson distribution with parameter $\lambda$, and the claim amounts $X_i$ follow a continuous severity distribution
• The probability mass function (PMF) of the aggregate claims $S$ can be computed using Panjer's recursion formula or other numerical methods
• The Poisson compound distribution is widely used in insurance applications due to its simplicity and the memoryless property of the Poisson process

Negative binomial compound distribution

• The negative binomial compound distribution models the claim frequency $N$ using a negative binomial distribution with parameters $r$ and $p$, allowing for overdispersion in the claim counts
• The claim amounts $X_i$ are modeled by a continuous severity distribution, such as gamma or lognormal
• The PMF of the aggregate claims $S$ can be computed using recursive formulas or numerical methods, similar to the Poisson compound distribution

Recursion formulas

• Recursive formulas provide an efficient way to compute the PMF of compound distributions, especially when the claim amount distribution has a simple form
• Panjer's recursion is a general formula applicable to a wide range of frequency and severity distributions, including Poisson, negative binomial, and binomial frequencies
• The recursion formula expresses the probability of a given aggregate claim amount in terms of the probabilities of smaller claim amounts and the parameters of the frequency and severity distributions
• Other recursive methods, such as De Pril's recursion or Hipp's recursion, offer alternative approaches to computing the compound distribution probabilities efficiently

Approximations for aggregate claims

• Approximation methods provide tractable alternatives to exact computation of the aggregate claims distribution, especially when the compound distribution is complex or the portfolio size is large
• Common approximation techniques include the normal approximation, normal power approximation, translated gamma approximation, and simulation methods

Normal approximation

• The normal approximation relies on the Central Limit Theorem (CLT) to approximate the aggregate claims distribution by a normal distribution with matching mean and variance
• The approximation is justified when the number of claims is large and the individual claim amounts are not too heavily tailed
• The normal approximation is simple to implement but may underestimate the probability of large claims in the tail of the distribution

Normal power approximation

• The normal power approximation (NPA) extends the normal approximation by incorporating higher moments (skewness and kurtosis) of the aggregate claims distribution
• NPA uses a polynomial transformation of the standard normal variable to capture the non-normality of the aggregate claims, providing a more accurate approximation than the plain normal approximation
• The approximation is particularly useful when the claim size distribution is moderately skewed and the portfolio size is sufficient for the CLT to hold

Translated gamma approximation

• The translated gamma approximation matches the first three moments (mean, variance, and skewness) of the aggregate claims distribution to a translated gamma distribution
• The translated gamma distribution is a three-parameter distribution that allows for both positive and negative claim amounts, making it suitable for modeling aggregate claims with potential deductibles or reinsurance recoveries
• The approximation is more flexible than the normal approximation and can capture the skewness of the aggregate claims distribution more accurately

Simulation techniques

• Simulation techniques, such as Monte Carlo simulation, provide a flexible and intuitive approach to approximating the aggregate claims distribution
• By generating a large number of scenarios for the claim frequency and severity, the aggregate claims can be simulated and the empirical distribution can be used as an approximation
• Simulation allows for complex dependencies, copulas, and non-standard claim size distributions to be incorporated into the model
• Variance reduction techniques, such as importance sampling or stratified sampling, can be employed to improve the efficiency and accuracy of the simulations

Applications of risk models

• Risk models have wide-ranging applications in the insurance industry, including pricing, reserving, reinsurance, and solvency assessment
• The choice of the appropriate risk model depends on the nature of the portfolio, the available data, and the specific business problem at hand

Insurance pricing and reserving

• Risk models are used to determine the pure premium (expected claims cost) for insurance policies, considering the frequency and severity of claims
• Collective risk models help in setting the overall premium level for a portfolio, while individual risk models allow for risk-based pricing and personalized premiums
• Reserving relies on risk models to estimate the future claims liabilities and ensure adequate funds are set aside to meet the obligations
• Stochastic reserving techniques, such as bootstrapping or Mack's chain ladder method, incorporate the uncertainty of claims development into the reserving process

Reinsurance and risk sharing

• Reinsurance is a risk transfer mechanism where an insurer cedes part of its risk to another insurer (the reinsurer) in exchange for a premium
• Risk models help in designing and pricing reinsurance contracts, such as excess-of-loss or quota share treaties, by quantifying the expected claims and the risk reduction achieved
• Optimal reinsurance strategies can be determined by minimizing the retained risk or maximizing the risk-adjusted profitability, subject to constraints on the reinsurance budget and the risk appetite

Solvency and capital requirements

• Solvency regulations, such as Solvency II in Europe, require insurers to hold sufficient capital to withstand adverse scenarios and ensure policyholder protection
• Risk models are used to assess the capital requirements for different risk categories, such as underwriting risk, market risk, and operational risk
• Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are common risk measures used to quantify the capital needs and ensure the insurer's financial stability
• Stress testing and scenario analysis help in evaluating the resilience of the insurer's balance sheet to extreme events and identifying potential vulnerabilities

Model selection and validation

• Selecting the appropriate risk model is crucial for accurate risk assessment and decision-making
• Model selection involves comparing different candidate models based on their goodness-of-fit, parsimony, and predictive performance
• Information criteria, such as Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC), provide a quantitative basis for model comparison and selection
• Model validation techniques, such as back-testing or out-of-sample testing, assess the model's performance on historical data and its ability to generalize to new data
• Sensitivity analysis helps in understanding the impact of model assumptions and parameter uncertainty on the risk estimates and decision outcomes