IAI CS1 cheat sheet

Actuarial Statistics (CS1)

• Survival function: S(x) = P(X > x) = 1 - F(x); it is non-increasing, right-continuous, with S(0) = 1 and S(∞) = 0.

• Force of mortality: μ(x) = f(x)/S(x) = -d/dx ln S(x); hence S(x) = exp(-∫₀ˣ μ(t)dt) and f(x) = μ(x)S(x).

• Curtate future lifetime: K = floor(T), with P(K = k) = ₖpₓ - ₖ₊₁pₓ = ₖpₓ qₓ₊ₖ; the complete expectation e̊ₓ = ∫₀^∞ ₜpₓ dt.

• Constant force of mortality μ over an interval gives ₜpₓ = e^(-μt); under uniform distribution of deaths, ₜqₓ = t·qₓ for 0 ≤ t ≤ 1 and μ(x + t) = qₓ/(1 - t·qₓ).

• Whole life insurance: Āₓ = ∫₀^∞ vᵗ ₜpₓ μ(x+t) dt; under constant force μ and δ, Āₓ = μ/(μ + δ).

• Annuity relations: äₓ = (1 - Āₓ)/d with d = 1 - v, and 1 = d·äₓ + Āₓ; the variance of the present value is ²Āₓ - (Āₓ)² evaluated at δ replaced by 2δ.

• Premium principle: net premium P satisfies P·äₓ = Āₓ for whole life, so P = Āₓ/äₓ; gross premiums add expense loadings to the equivalence equation.

• Recursions: Āₓ = v·qₓ + v·pₓ·Āₓ₊₁ and äₓ = 1 + v·pₓ·äₓ₊₁; these support recursive and select-and-ultimate life table calculations.

• Thiele's differential equation: d/dt (ₜV) = δ·ₜV + P - μ(x+t)(1 - ₜV) for a unit sum assured, linking reserves to premiums and mortality.