Exam STAM Formula Sheet: Key Distributions and Formulas
A comprehensive reference of essential distributions, formulas, and relationships for Exam STAM preparation.
Key Severity Distributions
The severity distributions on Exam STAM form the foundation for pricing, reserving, and risk measurement. Each distribution has distinctive properties that make it suitable for specific types of insurance losses. Mastering their formulas and relationships is essential for exam success.
Exponential: f(x) = (1/theta)*exp(-x/theta) for x > 0. Mean E[X] = theta, variance Var(X) = theta^2, CV = 1. The CDF is F(x) = 1 - exp(-x/theta) and the survival function is S(x) = exp(-x/theta). The hazard rate h(x) = 1/theta is constant, giving the memoryless property. Limited expected value: E[X wedge d] = theta*(1 - exp(-d/theta)). The exponential is the simplest severity model and serves as a building block for more complex distributions.
Pareto (single parameter form): f(x) = alpha*theta^alpha / (x+theta)^(alpha+1) for x > 0. Mean E[X] = theta/(alpha-1) for alpha > 1. Variance Var(X) = alpha*theta^2/((alpha-1)^2*(alpha-2)) for alpha > 2. The survival function S(x) = (theta/(x+theta))^alpha has a polynomial (heavy) tail, making the Pareto ideal for modeling large liability losses. Limited expected value: E[X wedge d] = theta/(alpha-1) * [1 - (theta/(theta+d))^(alpha-1)]. The mean excess function e(d) = (d+theta)/(alpha-1) is linear and increasing, a distinctive diagnostic feature.
More Severity Distributions
Lognormal: if Y = ln(X) follows N(mu, sigma^2), then X is lognormal. Mean E[X] = exp(mu + sigma^2/2). Variance Var(X) = exp(2*mu + sigma^2)*(exp(sigma^2) - 1). The CDF and limited expected values involve the standard normal CDF Phi: F(x) = Phi((ln(x) - mu)/sigma) and E[X wedge d] = exp(mu + sigma^2/2)*Phi((ln(d) - mu - sigma^2)/sigma) + d*(1 - Phi((ln(d) - mu)/sigma)). The lognormal is versatile and widely used when losses span several orders of magnitude.
Gamma with shape alpha and scale theta: E[X] = alpha*theta, Var(X) = alpha*theta^2. The PDF is f(x) = x^(alpha-1)*exp(-x/theta) / (theta^alpha * Gamma(alpha)). When alpha = 1, the gamma reduces to the exponential. When alpha is a positive integer, the gamma is the distribution of the sum of alpha independent exponentials. The gamma is commonly used in GLMs for severity modeling because its variance is proportional to the square of the mean.
Weibull with shape tau and scale theta: S(x) = exp(-(x/theta)^tau). When tau = 1, the Weibull reduces to the exponential. When tau > 1, the hazard rate increases (suitable for wear-out phenomena). When tau < 1, the hazard rate decreases. The mean is E[X] = theta*Gamma(1 + 1/tau).
Key Frequency Distributions
Poisson with parameter lambda: P(N=n) = exp(-lambda)*lambda^n / n!. Mean and variance are both lambda, so the variance-to-mean ratio equals 1. The Poisson is the default claim count model and is used when there is no evidence of overdispersion. Its probability generating function is P(z) = exp(lambda*(z-1)).
Negative Binomial with parameters r and beta: Mean E[N] = r*beta, Var(N) = r*beta*(1+beta). The variance exceeds the mean by a factor of (1+beta), making the negative binomial suitable for overdispersed count data. It arises as a gamma mixture of Poissons: if Lambda ~ Gamma(r, beta) and N|Lambda ~ Poisson(Lambda), then N ~ Negative Binomial. PGF: P(z) = (1/(1+beta*(1-z)))^r.
Binomial with parameters m and q: Mean E[N] = m*q, Var(N) = m*q*(1-q). The variance is less than the mean (underdispersion), which occurs when there is a fixed maximum number of possible claims. PGF: P(z) = (1-q+qz)^m.
Aggregate Loss Formulas
For the compound (collective) model S = X_1 + ... + X_N with N independent of the i.i.d. X_i: E[S] = E[N]*E[X]. Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. The MGF of S is M_S(t) = P_N(M_X(t)), where P_N is the PGF of N evaluated at M_X(t). For compound Poisson specifically: M_S(t) = exp(lambda*(M_X(t) - 1)) and Var(S) = lambda*E[X^2].
The normal approximation: P(S <= s) is approximately Phi((s - E[S])/sqrt(Var(S))). This is justified by the central limit theorem for large portfolios. The continuity correction improves the approximation for discrete aggregate distributions: use P(S <= s) approximately equals Phi((s + 0.5 - E[S])/sqrt(Var(S))).
Credibility and Reserving Formulas
Buhlmann credibility: Z = n/(n+k), where k = v/a. The credibility premium is Z*X_bar + (1-Z)*mu. Here v = E[Var(X|theta)] is the expected process variance and a = Var(E[X|theta]) is the variance of hypothetical means. The Buhlmann-Straub extension uses total weight W instead of n: Z = W/(W+k).
Full credibility standard for frequency: n_0 = (z_{(1+p)/2}/k)^2. For aggregate losses: n_0 = (z_{(1+p)/2}/k)^2 * (1 + CV^2), where CV is the coefficient of variation of severity. Partial credibility: Z = sqrt(n/n_0).
Chain ladder: age-to-age factor f_j = (sum of losses at age j+1)/(sum of losses at age j). CDF from age j to ultimate = product of all subsequent age-to-age factors. Reserve for accident year = reported losses * (CDF - 1). Bornhuetter-Ferguson reserve = expected ultimate * (1 - 1/CDF).
Risk Measure Formulas
VaR_p(X) = F^{-1}(p). For exponential: VaR_p = -theta*ln(1-p). For Pareto: VaR_p = theta*((1-p)^{-1/alpha} - 1). For lognormal: VaR_p = exp(mu + sigma*z_p).
TVaR_p(X) = E[X | X > VaR_p] = (1/(1-p))*integral from VaR_p to infinity of x*f(x)dx. For exponential: TVaR_p = VaR_p + theta (by the memoryless property). For Pareto with alpha > 1: TVaR_p = VaR_p * alpha/(alpha-1) + theta/(alpha-1). TVaR is a coherent risk measure; VaR is not (it fails subadditivity).
Policy Modification Formulas
With ordinary deductible d: expected cost per loss = E[X] - E[X wedge d]. Expected cost per payment = (E[X] - E[X wedge d])/S(d). With policy limit u: expected cost per loss = E[X wedge u]. With both deductible d and limit u: expected cost per loss = E[X wedge (d+u)] - E[X wedge d].
Loss elimination ratio: LER(d) = E[X wedge d]/E[X]. Inflation at rate r transforms severity parameter theta to theta*(1+r) for scale distributions (exponential, Pareto, gamma) while shape parameters remain unchanged.
Study Tips
Create a formula sheet organized by topic (severity, frequency, aggregate, credibility, reserving, risk measures, policy modifications) and review it regularly. For each distribution, know the PDF, CDF, survival function, mean, variance, limited expected value, and any special properties. Practice computing each formula from memory, then verify against your sheet. During the exam, having these formulas at your fingertips saves significant time and reduces errors. Focus especially on limited expected values for the exponential and Pareto, as these appear most frequently in Exam STAM problems involving deductibles, limits, and reinsurance pricing.