SOA ASTAMAdvanced Short-Term Actuarial Mathematics

Aggregate losses $S$ take values 0, 100 and 200 with probabilities 0.5, 0.3 and 0.2. Calculate the stop-loss premium with deductible 50.

Individual losses are exponential with mean 400. An excess-of-loss treaty attaches at 500. Calculate the expected reinsurance recovery per loss.

Cumulative reported losses: AY1: 400, 600, 660; AY2: 500, 800; AY3: 450. Using volume-weighted development factors and no tail, calculate the chain-ladder ultimate for AY3.

Losses above a threshold of 10,000 follow a generalized Pareto distribution with $\xi=0.5$ and $\beta=4{,}000$. Calculate the probability an exceedance is greater than 8,000.

Annual claim severity for a class is lognormal. A Bayesian analyst assumes the log-severity has a normal likelihood with known variance $\sigma^2=0.25$ and a normal prior on the mean $\mu$ with mean 8 and variance 0.04. Ten claims have mean log-severity 8.4. Derive the posterior distribution of $\mu$ and the Bühlmann credibility factor implied by it.

Two risks have $m_1=10$ and $m_2=20$ exposure units with sample means 100 and 120. EPV = 8,000 and VHM = 50. Calculate the Bühlmann–Straub credibility estimate for risk 1's mean.

A Mack reserve estimate has estimated process variance 5,000 and parameter variance 3,000 for total unpaid claims. Calculate the standard error of the reserve estimate.

An insurer cedes 40% of every loss under a quota-share treaty. Calculate the ratio of the variance of retained losses to the variance of gross losses.

Claim counts are Poisson with $\lambda=2$; severity takes values 1 and 2 with probabilities 0.6 and 0.4. Use the Panjer $(a,b,0)$ recursion ($a=0,b=\lambda$ for Poisson) to compute $f_S(0), f_S(1)$ and $f_S(2)$ for aggregate losses $S$.

Explain when Bühlmann–Straub credibility should be used in place of plain Bühlmann credibility, and why unequal exposures matter for the estimate of the variance of hypothetical means (VHM).