Actuarial Mathematics and Risk Theory
Actuarial Mathematics and Risk Theory is the quantitative backbone of the CAA Fellow syllabus. The exam tests whether you can move fluently between life-table notation, multiple-state models and aggregate-loss theory, and whether you can justify the assumptions behind each formula rather than simply compute.
Life-contingent mathematics
- Core identities: d_x = l_x − l_{x+1}; p_x = 1 − q_x; μ_{x+t} = −d/dt ln l_{x+t}.
- Fractional-age assumptions and their consequences: constant force gives l_{x+t} = l_x · p_x^t with μ constant between integer ages; UDD makes deaths uniform across the year and μ_{x+t} = q_x/(1 − t·q_x); Balducci makes 1/l_{x+t} linear.
- Commutation functions collapse net-premium calculations to one line: A_x = M_x/D_x, ä_x = N_x/D_x, hence P = M_x/N_x.
- Multiple decrement and Markov models: separate dependent rates q_x^{(j)} from the associated single-decrement rates that would apply if only cause j operated. Withdrawal is normally a multiple decrement, not a competing risk inside the life table.
Aggregate loss and ruin
- Compound Poisson aggregate loss: frequency Poisson(λ), severity X, aggregate S = ΣX_i over N claims. Panjer recursion gives an exact distribution when severity is in the (a,b,0) class; otherwise use translated-gamma or normal-power approximations with caution.
- Premium principles: net, expected-value, variance, standard-deviation, exponential and Esscher. Be ready to check monotonicity, additivity and the no-rip-off constraint.
- Cramér–Lundberg: the adjustment coefficient R solves λ + cR = λ·E[e^{RX}], and the Lundberg bound is ψ(u) ≤ e^{−Ru}.
Worked example
Claim counts are Poisson with λ = 400 per year and claim sizes are exponential with mean 25 (万元). Premium income is c = 11,000 per year, so the loading is positive and R exists. For exponential severity, α = 1/25 = 0.04 and the equation λ + cR = λα/(α − R) rearranges to R = α − λ/c = 0.04 − 400/11,000 = 0.003636 per 万元. With initial surplus u = 2,000, Lundberg gives ψ(u) ≤ exp(−0.003636 × 2,000) = exp(−7.27) ≈ 0.0007. Marks come from noting that R > 0 requires c > λ·E[X]; without that, the bound is vacuous.
Traps that cost marks
- Using independent rates where dependent rates are required in a decrement table.
- Forgetting that under UDD the force of mortality rises within the year, so μ_{x+0.5} exceeds q_x.
- Applying a normal approximation to a small, skewed portfolio.
- Quoting ψ(u) = e^{−Ru} as exact; it is exact only in special severity cases.