Actuarial Mathematics - Core Modelling and Risk Content (CS2)
This domain carries the modelling spine of CS2: survival and multi-state models, ARIMA time series, aggregate-loss and ruin theory, dependence/extremes, and Monte Carlo simulation. Questions rarely ask for a definition alone - you are expected to specify a model, derive a quantity, and then interpret it in actuarial terms, so method marks usually exceed the final-answer mark.
Survival and multi-state modelling
- The Kaplan–Meier estimator is the non-parametric survivor curve; Nelson–Aalen gives the cumulative hazard. Both handle right-censoring, not left-truncation.
- Cox proportional hazards writes h(t|x) = h₀(t)·exp(β′x): the baseline hazard is left unspecified and β is estimated by partial likelihood. Proportional hazards is the assumption to test (e.g. Schoenfeld residuals), not an automatic property.
- In Markov multi-state models, transition intensities q_ij drive the Kolmogorov forward equations; an A–E (actual vs expected) comparison is the standard goodness-of-fit diagnostic.
Time series
Stationarity underpins everything. AR(p) has a PACF that cuts off after lag p; MA(q) has an ACF that cuts off after lag q; ARMA has both tailing off. Box–Jenkins follows identification → estimation → residual checking (ACF of residuals, Ljung–Box) → forecast, and the residual check is where marks are usually lost.
Aggregate losses and ruin
For the collective risk model, E[S] = E[N]E[X] and Var[S] = E[N]Var[X] + Var[N]E[X]². Panjer recursion applies to the (a,b,0) claim-count class with severity on the non-negative integers. In the Cramér–Lundberg process, the adjustment coefficient R solves the Lundberg equation λ + cR = λM_X(R), giving ψ(u) ≤ e^{−Ru}.
Worked example
Poisson claims, λ = 1, exponential severity with mean 1, premium rate c = 1.5. Then 1 + 1.5R = 1/(1 − R), so (1 + 1.5R)(1 − R) = 1, giving 0.5R − 1.5R² = 0 and hence R = 1/3 (reject R = 0). The Lundberg bound is ψ(u) ≤ e^{−u/3}; at u = 10 the ruin probability is at most 0.036.
Traps that cost marks
Forgetting premium income when setting up the surplus process; applying Panjer to a continuous severity; quoting ACF/PACF cutoffs the wrong way round; treating censored observations as deaths; and reporting a VaR or TVaR figure without stating the confidence level and time horizon.