Loss Distributions for Actuarial Models

Outline loss distributions for actuarial models, introduce concepts and fit with R basics for exams; cover statistical properties, excesses, retention limits, parameter estimation, and goodness-of-fit with R exam walkthrough.

Learn to install R and RStudio on macOS, including downloading from the R project site, installing XQuartz, and running basic R code with package installation.

Learn R basics for actuaries: install packages like e1071, create and manipulate vectors, use sequences and repeats, and apply loops, if statements, and functions to compute summary statistics.

Learn core R basics for actuaries part 2, including vectors, matrices, lists, and data frames, then apply linear models, distributions, and plotting to practical data.

Introduce loss distributions as the actual losses on general insurance contracts and their use in risk modeling, pricing, reserves, and ruin theory.

Loss distributions are treated as continuous with positive skew and fat tails; log-normal and exponential distributions model losses well, while normal or uniform may be unsuitable.

Explore excess and retention limits as risk management tools, illustrating how excess shifts the first loss to the insured and retention caps losses, and discuss insurer and policyholder benefits.

Explore reinsurance basics, including proportional treaties with premium and loss sharing, excess of loss for each loss or total losses, retention limits, and maximum likelihood estimation.

Explore parameter estimation for loss distributions, comparing method of moments and maximum likelihood under truncation, with an exponential model and a practical excess-of-loss example.

Collect loss data, arrange and plot it, then fit a distribution by selecting a suitable family and estimating parameters. Apply a chi-square goodness-of-fit test to validate the fit.

Introducing compound distributions and their role in ruin theory and general insurance, this course outline presents their mean, variance, and moment generating function for Poisson and negative binomial models.

Explore compound distributions in actuarial models, linking loss frequency and severity to model total losses. Learn how this approach informs pricing, reserves, and ruin probability in general insurance.

Derive the mean of compound distributions by computing the expected total loss S = sum_{i=1}^N X_i, with N random and X_i iid. Conclude E[S] = E[N] E[X].

Explore the variance of a compound distribution for total losses given the number of losses, guided by the orange book and preparing for the moment generating function.

Examine the compound distribution and express the total losses' moment generating function in terms of the MGF of loss size and the number of losses.

Derive the compound Poisson moment generating function for total losses by combining the Poisson frequency with the distribution of individual losses, linking to the number of claims.

Derive the moment generating function for a binomial distribution with parameters M and Q in the context of compound distributions, then simplify to a practical loss model.