MAS-II flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Compound variance formula

For aggregate loss S = sum of N i.i.d. severities X, Var(S) = E[N]·Var(X) + Var(N)·E[X]^2, and E[S] = E[N]·E[X].

Panjer recursion

A recursive formula for the aggregate loss distribution when the frequency belongs to the (a,b,0) class, using f_S(0) = M_N(f_X(0)) and a weighted sum of prior probabilities.

Bühlmann credibility factor

Z = n/(n + k), where k = expected process variance divided by the variance of hypothetical means; as exposure n grows, Z approaches 1.

Bühlmann–Straub model

A credibility model with exposure weights w_ij, where Var(X_ij|θ_i) = v(θ_i)/w_ij; the credibility factor is Z_i = w_i·/(w_i· + k) with k = E[v]/Var(μ).

Limited fluctuation credibility

Choose the exposure n so that the probability the sample mean is within a tolerance r of the true mean is at least p; for normal data n ≈ (z_p/r)^2·CV^2.

Bayesian credibility

With a conjugate normal prior, the posterior mean is Z·sample mean + (1 − Z)·prior mean, with Z = n/(n + σ^2/τ^2).

Poisson–gamma posterior

For Poisson counts with a gamma prior on the mean, the posterior is gamma with updated shape and rate; the posterior mean is the credibility premium.

Negative binomial variance

A Poisson–gamma mixture with Var(N) = μ + μ^2/r; the over-dispersion parameter r controls the extra-Poisson variability.

Zero-inflated Poisson mean

For a zero-inflated Poisson with inflation probability π and Poisson mean λ, the mean is (1 − π)·λ and the variance is (1 − π)·λ·(1 + π·λ).

Gamma severity moments

A gamma distribution with shape α and scale θ has mean αθ and variance αθ^2; the coefficient of variation is 1/sqrt(α).

Lognormal severity moments

A lognormal with parameters μ and σ has mean exp(μ + σ^2/2) and variance exp(2μ + σ^2)·(exp(σ^2) − 1).

Ordinary deductible payment

For an ordinary deductible d, the expected payment per loss is E[(X − d)_+] = E[X] − E[min(X, d)].

Policy limit payment

For a policy limit u, the expected payment per loss is E[min(X, u)]; combining a deductible d and limit u gives E[min(X, u) − min(X, d)]_+.

Stop-loss premium

For stop-loss reinsurance with attachment a, the expected payment is E[(S − a)_+] = ∫_a^∞ S_S(x) dx, where S_S is the survival function of aggregate loss.

Inverse-transform sampling

Generate U uniform on (0,1) and set X = F^(−1)(U); this produces a draw from the distribution with CDF F.

Antithetic variates

A variance-reduction technique that uses U and 1 − U as paired inputs; negative correlation between the paired outputs reduces the variance of the average.

Poisson deviance residual

d_i = sign(y_i − μ̂_i)·sqrt(2[y_i·ln(y_i/μ̂_i) − (y_i − μ̂_i)]); the sum of squared deviance residuals is the deviance.

GLM log link

The mean is exp(x'β); a one-unit increase in a covariate multiplies the mean by exp(β_j), so coefficients are multiplicative on the mean scale.

GLM scale parameter

The dispersion φ multiplies the variance function; Poisson and binomial have φ = 1, while gamma and normal estimate φ from the Pearson or deviance statistic.

Over-dispersed Poisson GLM

A Poisson GLM with an estimated dispersion φ = Pearson χ²/(n − p); standard errors are multiplied by sqrt(φ) to account for the extra variability.

Tweedie variance power

Var(Y) = φ·μ^p with 1 < p < 2; the distribution is a compound Poisson–gamma with a point mass at zero, used for aggregate losses.

Offset in a GLM

A term such as log(exposure) added to the linear predictor with a fixed coefficient of 1, so that the mean is proportional to exposure.

AIC and BIC

AIC = 2k − 2ℓ and BIC = k·ln(n) − 2ℓ; lower values indicate a better trade-off between fit and complexity, with BIC penalizing parameters more heavily.

Likelihood ratio test

2[ℓ(full) − ℓ(reduced)] is compared to a χ² distribution with degrees of freedom equal to the difference in the number of parameters.

Delta method

Var(g(θ̂)) ≈ [g'(θ̂)]^2·Var(θ̂); for a vector parameter, use the gradient and the covariance matrix of θ̂.

Chain-ladder development factor

f_j = Σ_i C_{i,j+1} / Σ_i C_{i,j} over accident years with data in both periods; the ultimate is the latest diagonal multiplied by the product of future factors.

Bornhuetter–Ferguson method

Ultimate = reported + (1 − 1/Πf)·(expected ultimate), where expected ultimate = premium × expected loss ratio and (1 − 1/Πf) is the unreported proportion.

Cape Cod method

Estimate the expected loss ratio as Σ reported / Σ (premium × reported proportion), then apply Bornhuetter–Ferguson using that pooled loss ratio.

Mack chain-ladder standard error

Combines process variance and estimation variance across development periods to produce a prediction error for the ultimate; intervals use a normal or lognormal approximation.

IBNR reserve

Incurred but not reported reserve equals ultimate minus reported (paid or incurred) to date; it covers claims not yet reported and development on reported claims.

Cox proportional hazards

The hazard is h(t|x) = h_0(t)·exp(x'β); the baseline hazard is unspecified and β is estimated by partial likelihood.

Gaussian copula tail dependence

The Gaussian copula has zero tail dependence, while the t-copula has symmetric positive tail dependence; Kendall's tau equals (2/π)·arcsin(ρ) for the Gaussian copula.