Exam FAM formulas, and when to use each

Coverage modifications

• Limited expected value: E[X ∧ u] is the integral of S(x) from 0 to u. Exponential with mean θ: θ(1 − e^(−u/θ)).
• Ordinary deductible d: the payment per loss has mean E[X] − E[X ∧ d]; per payment, divide by S(d).
• Deductible d, maximum covered loss u and coinsurance α: α(E[X ∧ u] − E[X ∧ d]) per loss.
• Loss elimination ratio: E[X ∧ d] ÷ E[X]. Exponential: 1 − e^(−d/θ).
• Inflation r with the deductible fixed: for the new loss Y = (1 + r)X, E[Y ∧ d] = (1 + r)E[X ∧ d ÷ (1 + r)].

Frequency and aggregate losses

Using the (a, b, 0) class

• (a, b, 0): P(N = k) ÷ P(N = k − 1) = a + b ÷ k for k = 1, 2, 3, ...; the (a, b, 1) class starts the same ratio at k = 2.
• Compound: E[S] = E[N]E[X] and Var(S) = E[N]Var(X) + Var(N)E[X]². Compound Poisson: Var(S) = λE[X²].
• Stop-loss: E[(S − d)+] = E[S] − E[S ∧ d].
• Exponential with mean θ: VaR at p is −θ ln(1 − p), and TVaR is VaR + θ.

Credibility

• Full standard in expected claims: n₀ = (y ÷ k)², where y is the normal value with P(−y < Z < y) = p. With p = 0.90 and k = 0.05, n₀ = 1,082.
• Poisson claim counts: frequency needs n₀ expected claims, severity n₀ × CV², and aggregate losses n₀(1 + CV²), where CV is the claim size's coefficient of variation.
• Partial credibility: Z = √(n ÷ full standard), capped at 1. Estimate = Z × observed + (1 − Z) × prior.

Reserving and ratemaking

• Age-to-ultimate factor: the product of every age-to-age factor still to come.
• Chain-ladder: ultimate = latest value × age-to-ultimate factor; unpaid = ultimate − paid to date.
• Bornhuetter-Ferguson: ultimate = reported + expected ultimate × (1 − 1 ÷ age-to-ultimate factor).
• Expected loss ratio method: ultimate = earned premium × expected loss ratio.
• Loss cost method: indicated rate = (pure premium + fixed expense per exposure) ÷ (1 − variable expense ratio − profit provision).

Options

• Put-call parity, no dividends: C − P = S₀ − Ke^(−rT).
• Binomial tree: q = (e^(rh) − d) ÷ (u − d), and value = e^(−rh)(q × up value + (1 − q) × down value).
• Delta in one period: (up value − down value) ÷ (S₀u − S₀d).
• Black-Scholes call: S₀N(d₁) − Ke^(−rT)N(d₂), with d₁ = (ln(S₀ ÷ K) + (r + σ² ÷ 2)T) ÷ (σ√T) and d₂ = d₁ − σ√T.

Survival models

• Survival over t years: tpx = l(x + t) ÷ l(x), and tpx = exp(−∫ μ) over those t years.
• A death between ages x + u and x + u + n: (l(x + u) − l(x + u + n)) ÷ l(x).
• Curtate expectation: e(x) = p(x)(1 + e(x + 1)).
• Fractional ages: under UDD, sqx = s × qx; under a constant force, spx = (px)^s.

Insurance, annuities, premiums and policy values

• Second moments use twice the force of interest: Var(Z) = ²A(x) − A(x)².
• A(x) = 1 − d·ä(x), and the pure endowment nEx = vⁿ × npx.
• Endowment insurance = term insurance + pure endowment.
• Equivalence principle, whole life of 1 with level annual premiums: P = A(x) ÷ ä(x).
• Policy value recursion: (tV + P − e)(1 + i) = q × (S + E) + p × (t+1)V, with q and p at age x + t, S the benefit and E its claim expense.
• Net premium policy value for whole life of 1: tV = 1 − ä(x + t) ÷ ä(x).