IAI CM1 cheat sheet

Whole life insurance APV: A_x = Σ_{k=0}^∞ v^{k+1} · {}k p_x · q{x+k} = Σ v^{k+1} · {}_k|q_x; under constant force μ, A_x = μ/(μ+δ).

Term insurance APV: A^1_{x:n|} = Σ_{k=0}^{n-1} v^{k+1} · {}k p_x · q{x+k}; endowment adds pure endowment A_{x:n|} = v^n · {}n p_x, so A{x:n|} = A^1_{x:n|} + v^n {}_n p_x.

Whole life annuity-due: ä_x = (1 − A_x)/d with d = i/(1+i) = 1 − v; immediate annuity satisfies a_x = (1 − A_x)/i and ä_x = 1 + a_x.

Premium equation: P · ä_x = A_x for whole life, so P = A_x/ä_x = d·A_x/(1 − A_x); for n-pay policies use the appropriate annuity-due denominator ä_{x:n|}.

Reserves by recursion: (V + P)(1+i) = q_{x+k} + p_{x+k} · _{k+1}V, so prospective reserve equals APV of future benefits minus APV of future net premiums.

Gross premium: G · ä = A + APV(expenses); expense-loaded reserves discount future expenses and premiums at the valuation basis, not the pricing basis.

Survival models: force of mortality μ_{x+t} = −d/dt ln({}_t p_x); S(x) = exp(−∫_0^x μ_s ds) and e_x = ∫_0^∞ {}_t p_x dt.

Fractional ages: under UDD, {}t q_x = t·q_x for 0≤t<1 and μ{x+t} = q_x/(1 − t·q_x); under constant force, {}_t p_x = (p_x)^t.

Multiple decrements: independent rates q' relate to dependent rates via q^{(j)} = ∫_0^1 {}_t p_x^{(τ)} μ_x^{(j)} dt; under UDD in the multiple decrement table, q^{(j)} = (q'^{(j)}/q'^{(τ)})·q^{(τ)}.

MLE: log-likelihood ℓ(θ) = Σ ln f(x_i;θ); solve ∂ℓ/∂θ = 0 and confirm via second derivative; invariance gives MLE of g(θ) as g(θ̂).

Cramér–Rao lower bound: Var(θ̂) ≥ 1/(n·I(θ)) with I(θ) = −E[∂²ℓ/∂θ²]; an estimator achieving the bound is efficient.

Confidence interval for a mean: x̄ ± z_{1−α/2}·σ/√n when σ known; use t_{n−1} when σ is estimated by s and the sample is normal.

Hypothesis tests: reject H0 when the test statistic exceeds the critical value; p-value is the probability, under H0, of a result at least as extreme as observed.

Central limit theorem: for i.i.d. variables with mean μ and variance σ², (X̄ − μ)/(σ/√n) converges in distribution to N(0,1).

Distribution facts: Poisson mean equals variance λ; exponential mean 1/λ with memoryless property; Gamma(α,λ) has mean α/λ and variance α/λ².

Method of moments sets sample moments equal to population moments; it is consistent but generally less efficient than MLE.

Bayes' theorem: posterior ∝ likelihood × prior; a conjugate prior family keeps the posterior in the same family, e.g. Beta prior with binomial data.