Simulation Techniques: Monte Carlo Methods for Actuaries

Monte Carlo Fundamentals

Monte Carlo simulation uses random sampling to estimate quantities that are analytically intractable. The basic algorithm: (1) generate random variates from specified distributions using the inverse transform or acceptance-rejection methods, (2) compute the quantity of interest for each realization, (3) average the results. The law of large numbers guarantees convergence, and the CLT provides confidence intervals. The standard error of the estimate decreases as 1/sqrt(n). For actuarial applications, Monte Carlo estimates aggregate loss distributions, ruin probabilities, and option values.

Variance Reduction

Variance reduction techniques improve simulation efficiency. Antithetic variates pair each random draw U with 1-U, creating negative correlation that reduces variance. Control variates use a correlated quantity with a known expected value to adjust estimates. Stratified sampling partitions the probability space and samples from each stratum. Importance sampling concentrates samples in regions that contribute most to the estimate. Exam MAS-I tests the inverse transform method, basic simulation algorithms, estimation of simulation error, and conceptual understanding of variance reduction techniques.