MAS-I cheat sheet

Principle of indemnity: recovery is limited to the actual financial loss; valued policies and replacement-cost coverage are the recognized exceptions.

Insurable interest must exist at policy inception for life insurance and at the time of loss for property insurance.

For ground-up loss X and deductible d, insurer payment per loss is (X − d)+ with E[(X − d)+] = E[X] − E[X∧d]; a per-payment deductible is applied per claim rather than per loss.

Coinsurance clause pays (carried limit / required limit) × loss when the insured carries less insurance than the coinsurance percentage requires.

Loss ratio = incurred losses / earned premium; combined ratio = loss ratio + expense ratio; a combined ratio below 1.00 indicates an underwriting profit.

Adverse selection is the tendency of higher-risk insureds to buy or retain more coverage at a given rate; controls include underwriting, waiting periods, and experience rating.

Moral hazard is post-contract behavior change that raises expected claim cost; controls include deductibles, coinsurance, policy limits, and exclusions.

Rate filing methods: prior approval (approval required before use), file-and-use (use upon filing), use-and-file (file shortly after use); prior approval is the most restrictive.

Risk-based capital compares total adjusted capital to the authorized control level computed from asset, credit, underwriting, and other risk charges; falling below prescribed ratios triggers escalating regulatory action levels.

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B); the denominator equals Σ over the partition of the prior times the likelihood.

For a nonnegative random variable, E[X] = ∫₀^∞ S(x) dx and E[X∧d] = ∫₀^d S(x) dx.

Var(X) = E[X²] − (E[X])²; Var(aX + b) = a² Var(X); for independent X, Y, Var(X + Y) = Var(X) + Var(Y).

Moment generating function: M_X(t) = E[e^{tX}]; the k-th derivative at 0 gives E[X^k], and existence in a neighborhood of 0 uniquely determines the distribution.

Law of total variance: Var(X) = E[Var(X|Y)] + Var(E[X|Y]); the tower property gives E[X] = E[E[X|Y]].

Exponential distribution is memoryless: P(X > s + t | X > s) = P(X > t); the geometric distribution is the discrete counterpart.

Poisson(λ) has mean and variance both equal to λ, and sums of independent Poisson variables are Poisson with summed rates.

Sufficiency via factorization: T(X) is sufficient for θ if and only if the joint density equals g(T(x), θ)·h(x).

Cramér-Rao bound: any unbiased estimator satisfies Var(θ̂) ≥ 1/(n·I₁(θ)), where I₁(θ) is the Fisher information from one observation; the delta method gives Var(g(θ̂)) ≈ [g'(θ)]²Var(θ̂).

v = 1/(1 + i); the nominal rate conversion is i = (1 + i^(m)/m)^m − 1, and the discount rate satisfies d = i/(1 + i) = 1 − v.

Annuity-immediate a_n = (1 − v^n)/i; annuity-due ä_n = (1 − v^n)/d = (1 + i)·a_n.

Force of interest δ_t satisfies δ = ln(1 + i); the accumulation function is a(t) = exp(∫₀^t δ_s ds).

NPV = Σ CF_t·v^t at the valuation rate; the IRR is the rate at which NPV equals zero, so accept a project when NPV > 0 at the required return.

Macaulay duration D = Σ t·CF_t·v^t / P; modified duration D* = D/(1 + i), and for small yield changes ΔP/P ≈ −D*·Δi.

Convexity adjustment: ΔP/P ≈ −D*·Δi + ½·C·(Δi)², where C = Σ t(t+1)·CF_t·v^(t+2) / P.

Redington immunization requires PV(assets) = PV(liabilities), asset duration equal to liability duration, and asset convexity at least liability convexity.

Forward rate implied by annual spot rates: (1 + s₂)² = (1 + s₁)(1 + f₁,₂), so f₁,₂ = (1 + s₂)²/(1 + s₁) − 1.

Price of a bond at yield i: coupon·a_n + redemption·v^n, where n is the number of remaining coupon periods.

Collective risk model S = X₁ + ⋯ + X_N with N independent of i.i.d. severities: E[S] = E[N]·E[X] and Var(S) = E[N]·Var(X) + Var(N)·(E[X])².

Compound Poisson case: E[S] = λ·E[X] and Var(S) = λ·E[X²] = λ(Var(X) + (E[X])²).

Panjer recursion for severities in the (a, b, 0) class: p_n = Σ_{k=1}^{n} (a + b·k/n)·f_k·p_{n−k}, starting from p₀.

Surplus process U(t) = u + ct − S(t); the net profit condition c > λ·E[X] gives positive drift and ultimate ruin probability strictly less than 1.

The adjustment coefficient R is the positive solution of λ + c·R = λ·M_X(R); Lundberg's inequality bounds ψ(u) ≤ e^{−Ru}.

Bühlmann credibility: Ẑ = Z·X̄ + (1 − Z)·μ with Z = n/(n + k) and k = E[Var(X|θ)] / Var(E[X|θ]).

Deductible and limit pricing rely on E[X∧d] = ∫₀^d S(x) dx and E[(X − d)+] = E[X] − E[X∧d]; the loss elimination ratio is LER = E[X∧d]/E[X].

Aggregate stop-loss with attachment a costs E[(S − a)+], computed from the aggregate loss distribution rather than from individual severities.

Individual risk model S = Σ X_i assumes independent but not necessarily identically distributed losses, so Var(S) = Σ Var(X_i).