Advanced Reserving Methods in Insurance

A statistical framework used to represent the distribution of random variables in general insurance reserving.

The mean of the random variable Y.

The variance of the random variable Y.

The function that relates the mean of the distribution to the parameter θ.

The function that scales the variance of the distribution based on the parameter ϕ.

The function that represents the likelihood of observing a particular value y.

A common distribution used to model claim frequency, often referred to as the counting process.

A common distribution used to model claim severity, often used for continuous values.

The proportion of losses incurred in a specific development period that is attributable to a particular accident year. It represents the relative size of losses from each accident year across different development periods.

The proportion of losses incurred in a specific development period that is attributable to a particular accident year. It represents the relative size of losses from each accident year across different development periods.

The scaling factor used to adjust the cumulative losses for each accident year to the corresponding cumulative losses from the chain ladder method. It accounts for differences in the overall development pattern.

The scaling factor used to adjust the cumulative losses for each accident year to the corresponding cumulative losses from the chain ladder method. It accounts for differences in the overall development pattern.

A method of reserving that uses a combination of the chain ladder method and a cross-classified model to estimate ultimate losses. It considers both the development pattern and the accident year effects.

A method of reserving that uses a combination of the chain ladder method and a cross-classified model to estimate ultimate losses. It considers both the development pattern and the accident year effects.

The calculation of the α parameter, which represents the scaling factor used to ensure consistency with the chain ladder method, involves subtracting the sum of β parameters for all previous development periods from 1.

The calculation of the α parameter, which represents the scaling factor used to ensure consistency with the chain ladder method, involves subtracting the sum of β parameters for all previous development periods from 1.

A statistical model used in actuarial science for estimating future claims payments. It assumes a specific distribution, often an over-dispersed Poisson, for the claims counts. This distribution incorporates a parameter representing the spread or dispersion of the claims data.

A measure of variation in expected claims beyond the usual Poisson distribution. It captures the fact that claims often exhibit more variability than expected by a simple Poisson model.

A reserving method where the loss development factors (LDFs) are calculated using the cumulative losses across all accident periods. This method assumes that the pattern of loss development is consistent across different accident years.

A reserving method where the loss development factors are calculated using the cross-classified data, i.e., the combined data of accident period and development period. It takes into account potential differences in the loss development pattern across various accident years.

The loss development factor is a factor that is used to estimate the future development of claims. It is calculated as the ratio of cumulative claims at a later development period to the cumulative claims at an earlier development period.

The estimates of future incremental losses and reserves derived from the ODP model are equal to those obtained using the conventional Chain Ladder method when the dispersion parameter is constant for all cells of the data.

When the ODP model results are adjusted for bias, the estimates of future incremental losses and reserves become the Minimum Variance Unbiased Estimators (MVUEs). This implies that the adjusted estimates are the most accurate and efficient among all unbiased estimators.

When the ODP Mack and ODP Cross-Classified models are adjusted for bias, they produce identical estimates of reserves. This signifies their equivalence in terms of the final reserve estimates after accounting for bias.

A method used to estimate ultimate claims by projecting past development patterns into the future.

The amount of loss incurred in a specific accident period and development period. It is derived by subtracting the cumulative loss up to the previous development period from the cumulative loss at the current development period.

A parameter representing the age-to-age factor for a specific accident period (k). It indicates the expected growth factor in cumulative losses from one development period to the next for that accident year.

A parameter representing the development factor for a specific development period (j). It indicates the expected growth factor in cumulative losses from one development period to the next.

A statistical modeling framework that uses a design matrix (X) with covariates to represent the relationships between the observed loss data and the parameters.

Variables that represent the effects of accident period and development period on the loss development triangle. These variables are included as entries in the design matrix of a GLM.

The difference between the loglikelihood of a saturated model and a nested model. Used to measure the goodness of fit of the nested model compared to the saturated model.

A statistical model where each observation (claim) has its own parameter. In other words, it perfectly fits the data.

A statistical model that is a simplified version of the saturated model. It has fewer parameters and might not perfectly fit the data.

The logarithm of the likelihood of observing a specific set of data given a statistical model. It reflects how well the model explains the observed data.

The fitted cumulative losses for each year. In the context of the example, each entry in the table represents the sum of losses up to that year.

The incremental losses for each year. They represent the difference between cumulative losses in consecutive years.

The formula for calculating the loglikelihood of a saturated model in an overdispersed model. It takes into account the dispersion parameter (phi) and the fitted incremental losses (µ).

The formula for calculating the loglikelihood of a nested model in an overdispersed model. It takes into account the dispersion parameter (phi) and the fitted incremental losses (µ).

Used to check if an insurance model is correctly accounting for the spread (variance) of losses in different time periods.

A statistical family that describes how random variables are distributed, used in insurance reserving to model claim frequency and severity.

This function captures how the variance of a random variable changes as the mean changes within the EDF.

Used to solve for heteroscedasticity by adjusting the scale parameter 'ϕ' in the EDF.

A special case of the EDF that is particularly useful for modeling claim costs.