IFoA CS2 cheat sheet

CS2 Actuarial Mathematics: Core Modelling and Risk

• Survival function and force of mortality: S(x) = exp(-∫₀ˣ μ(t)dt), f(x) = S(x)μ(x); curtate expectation eₓ = Σₖ≥₁ ₖpₓ. These identities underpin every life-contingent cash flow model in CS2.

• Life-table recursion: ₜ₊₁pₓ = ₜpₓ · pₓ₊ₜ and ₜqₓ = 1 − ₜpₓ. Fractional-age assumptions (UDD, constant force, Balducci) give different ₜpₓ for 0<t<1; UDD implies ₜqₓ = t·qₓ and μₓ₊ₜ = qₓ/(1−t·qₓ).

• Expected present value of a life annuity: äₓ = Σₖ≥₀ vᵏ ₖpₓ; insurance Aₓ = Σₖ≥₀ vᵏ⁺¹ ₖpₓ qₓ₊ₖ. Relationship äₓ = (1 − Aₓ)/d with d = 1 − v, and Aₓ = 1 − d·äₓ, are standard identities.

• Thiele's differential equation for a policy reserve: d/dt V = δ·V + P − b·μ − (S − V)·μ, where b is the sum at risk on death and S is the survival benefit. This is the continuous-time analogue of the recursive reserve formula.

• Recursive reserve formula (discrete): (V + P)(1+i) = qₓ₊ₜ · S + pₓ₊ₜ · ₜ₊₁V, where S is the death benefit. Solve forward from terminal reserve ₀V at maturity to obtain premium or reserve.

• Multiple-decrement and multiple-state models: independent rates qᵢ relate to dependent rates via qᵢᵈ = qᵢ · (1 − ½ Σⱼ≠ᵢ qⱼ) under constant-force/UDD assumptions. Markov models use transition intensities μᵢⱼ and Kolmogorov forward equations.

• Profit testing: model cash flows including premiums, expenses, claims, reserves and interest; profit signature Πₜ = profit at t · ₜ₋₁p (survival to start of year). NPV at risk discount rate r; IRR solves NPV = 0. Use emerging surplus to set premiums or assess profitability.

• Mortality projection and selection: use a parametric law (e.g., Gompertz μₓ = Bcˣ, Makeham adds constant) or the Lee-Carter model ln m(x,t) = aₓ + bₓ kₜ + ε. Selection effects mean select mortality q₍ₓ₎ < ultimate qₓ₊ₜ for several years after underwriting.

• Risk measures: VaR at level α is the α-quantile of loss; TVaR/ES = E[L | L > VaR_α]. Coherent risk measures satisfy monotonicity, translation invariance, positive homogeneity, and subadditivity; VaR fails subadditivity in general, TVaR does not.