MAS-II cheat sheet

For frequency-severity models, aggregate loss S = sum of N i.i.d. severities; E[S] = E[N]·E[X] and Var(S) = E[N]·Var(X) + Var(N)·E[X]^2 (compound variance formula).

Collective risk model: if N is Poisson(λ) independent of X, S is compound Poisson with mgf M_S(t) = exp(λ(M_X(t) − 1)); Var(S) = λ·E[X^2].

Panjer recursion: for N in the (a,b,0) class, f_S(x) = [1/(1 − a·p_0)]·Σ_{y=1}^{x} (a + b·y/x)·f_X(y)·f_S(x − y), with f_S(0) = M_N(f_X(0)); Poisson has a = 0, b = λ.

Credibility: Bühlmann–Straub estimate = Z·(individual mean) + (1 − Z)·(overall mean), where Z = n/(n + k) and k = E[v]/E[a] (process variance over between-group variance).

Bühlmann (no exposure weighting) uses Z = n/(n + EPV/VHM); as n → ∞, Z → 1 and the estimate converges to the group's own experience.

Limited fluctuation credibility: choose n such that P(|sample mean − μ| ≤ r·μ) ≥ p; for normal approximation n ≈ (z_p/r)^2·(CV)^2, where CV = σ/μ.

Bayesian credibility: with a conjugate normal prior, the posterior mean is a credibility-weighted average of prior mean and sample mean, with Z = n/(n + σ^2/τ^2).

GLM link functions: log link for Poisson/gamma (log-linear mean), logit link for binomial; canonical links give mean = b'(θ) and variance = φ·b''(θ).

Poisson deviance residual: d_i = sign(y_i − μ̂_i)·sqrt(2[y_i·ln(y_i/μ̂_i) − (y_i − μ̂_i)]); gamma and binomial deviances follow the same scaled-likelihood pattern.

MLE: set score function to zero; asymptotic variance of θ̂ is the inverse Fisher information, so SE(θ̂) = sqrt(1/I(θ̂)); Wald CI uses θ̂ ± z·SE.

Likelihood ratio test: 2[ℓ(full) − ℓ(reduced)] ~ χ² with df = difference in parameters; AIC = 2k − 2ℓ and BIC = k·ln(n) − 2ℓ penalize complexity.

Tweedie distribution: Var(Y) = φ·μ^p with 1 < p < 2; it is a compound Poisson–gamma, so it models aggregate losses with a point mass at zero.

Chain-ladder development factor: f_j = Σ_i C_{i,j+1} / Σ_i C_{i,j} over the same accident years; ultimate for the latest diagonal is C_{i,I}·Π_{j≥I} f_j.

Mack chain-ladder standard error of the ultimate combines process variance and estimation variance across development periods; prediction intervals use the normal or lognormal approximation around the estimate.

Bornhuetter–Ferguson: ultimate = reported + (1 − 1/Πf)·(expected ultimate), where expected ultimate = premium × expected loss ratio; the complement of the reported proportion is the unreported share.

Cape Cod (Stanard–Bühlmann): estimate the expected loss ratio from the pooled data, ELR = Σ reported / Σ (premium × reported proportion), then apply Bornhuetter–Ferguson with that ELR.

Over-dispersed Poisson GLM reproduces chain-ladder ultimates when accident year and development year are both factors and the log link is used; the dispersion φ scales the variance.

Bootstrap or simulation-based reserving resamples Pearson residuals, refits the GLM, and generates a predictive distribution of reserves to obtain the reserve risk (percentile) estimate.

Bühlmann–Straub assumptions: E[X_ij | θ_i] = μ(θ_i), 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(θ)]/Var(μ(θ)).

Credibility premium formula: P = Z·(weighted mean of the group's observations) + (1 − Z)·(collective mean μ), where Z depends on total exposure w_i·.

For a normal-normal hierarchical model, the posterior mean is the credibility estimate and the posterior variance is v·(1 − Z); this gives the predictive distribution for the next period.

GLM estimation: the IRLS algorithm iterates β^(t+1) = (X'WX)^(−1)X'Wz with working weights W and working response z; convergence yields the MLE and the Fisher information X'WX.

Scale parameter estimates: Poisson φ = 1 (or Pearson χ²/df if over-dispersed), binomial φ = 1, gamma and normal φ estimated from the Pearson or deviance statistic.

For a GLM with log link, the fitted mean is exp(x'β); a one-unit increase in x_j multiplies the mean by exp(β_j), so coefficients are multiplicative on the mean scale.

Reserving diagnostics: plot Pearson residuals against fitted values and development age; a funnel shape or trend signals model misspecification (wrong link, missing interaction, or over-dispersion).

The chain-ladder estimate is the MLE under an over-dispersed Poisson GLM with a saturated development-year factor and accident-year factor; the GLM framework allows formal tests and prediction intervals.

IBNR = ultimate − reported (paid or incurred to date); IBNER refers to the change in the estimate of reported-but-not-fully-developed claims, and the total reserve includes both IBNR and case reserves.

Frequency-severity decomposition: model N with Poisson or negative binomial and X with gamma, lognormal, or Weibull; the aggregate mean is E[N]·E[X] and the variance follows the compound variance formula.

Negative binomial as a Poisson–gamma mixture: Var(N) = μ + μ^2/r; the over-dispersion parameter r controls the extra-Poisson variability and the variance exceeds the mean.

Zero-inflated Poisson: P(N = 0) = π + (1 − π)·exp(−λ); the mean is (1 − π)·λ and the variance is (1 − π)·λ·(1 + π·λ).

Severity distributions: gamma has mean αθ and variance αθ^2; lognormal has mean exp(μ + σ^2/2) and variance exp(2μ + σ^2)(exp(σ^2) − 1); Weibull has mean λ·Γ(1 + 1/k).

Deductibles: for an ordinary deductible d, the expected payment per loss is E[(X − d)_+] = E[X] − E[min(X, d)]; for a franchise deductible, the full loss is paid once X > d.

Policy limits: expected payment with limit u is E[min(X, u)]; combining a deductible d and limit u gives E[min(X, u) − min(X, d)]_+.

Stop-loss reinsurance with attachment a: E[(S − a)_+] = E[S] − E[min(S, a)]; the stop-loss premium can be computed from the survival function: ∫_a^∞ S_S(x) dx.

Simulation: use inverse-transform sampling with the generalized inverse F^(−1)(U); for the aggregate loss, simulate N then N severities and sum, repeating many times to obtain the predictive distribution.

Variance reduction: antithetic variates use U and 1 − U to reduce variance; control variates use a correlated variable with a known mean to adjust the estimator.

Method of moments: equate sample moments to population moments and solve; for a two-parameter distribution, use the first two moments (mean and variance).

MLE invariance: if θ̂ is the MLE of θ, then g(θ̂) is the MLE of g(θ); this applies to functions such as the mean, variance, and tail probabilities.

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

Confidence interval for a proportion or mean uses the Wald, Wilson, or exact (Clopper–Pearson) interval; the exact interval is preferred for small samples or extreme proportions.

Goodness-of-fit: Pearson χ² statistic = Σ (O − E)^2/E; the Kolmogorov–Smirnov test compares the empirical CDF to the fitted CDF.

Model selection: compare nested models with the likelihood ratio test; compare non-nested models with AIC or BIC; lower AIC/BIC indicates a better trade-off between fit and complexity.

Cross-validation: k-fold CV estimates out-of-sample prediction error; for time-series or reserving data, use leave-one-out or forward-chaining to respect the data structure.

Residual analysis: standardized deviance residuals should be approximately normal; plot them against fitted values, covariates, and time to detect patterns.

Bayesian estimation: the posterior is proportional to the likelihood times the prior; the posterior predictive distribution integrates over the posterior and is used for forecasting future losses.

Bühlmann credibility with a Poisson frequency model: the credibility factor for a group with exposure n is Z = n/(n + k), where k = E[Var(N|θ)]/Var(E[N|θ]) = 1/λ (for Poisson–gamma).

For a Poisson–gamma model, the posterior for θ given total counts is gamma with updated parameters, and the posterior mean is the credibility premium; the predictive distribution is negative binomial.

For a normal–normal model, the posterior is normal with mean Z·x̄ + (1 − Z)·μ and variance σ^2·(1 − Z); the predictive distribution for a new observation has variance σ^2·(1 + 1/n) plus the prior variance term.

Credibility for loss ratios: the observed loss ratio is weighted with the expected loss ratio; the credibility factor depends on the variance of the loss ratio and the number of exposure units.

Chain-ladder development factors are volume-weighted averages of individual development ratios; they are sensitive to large claims and to the latest diagonal.

The expected loss ratio method (loss ratio method) sets ultimate = premium × expected loss ratio; it is used when the data are too immature for chain-ladder.

The Bornhuetter–Ferguson method blends the expected loss ratio with the reported losses via the reported proportion; it is more stable than chain-ladder for immature accident years.

The reserve risk distribution can be obtained by simulating the development factors or by bootstrapping the GLM; the 99.5th percentile is commonly used for solvency capital.

Back-testing: compare actual development to predicted development one year later; a pattern of adverse or favorable development indicates model bias.

Over-dispersion in a Poisson GLM: estimate φ = Pearson χ²/(n − p); if φ > 1, use a quasi-Poisson or negative binomial model, and inflate standard errors by sqrt(φ).

Quasi-likelihood: the estimating equations use the mean and variance functions only; the scale parameter φ is estimated from the Pearson statistic and the standard errors are multiplied by sqrt(φ).

Negative binomial GLM: Var(Y) = μ + μ^2/θ; the dispersion parameter θ is estimated by MLE, and the model reduces to Poisson as θ → ∞.

Tweedie GLM: the variance power p is estimated by profile likelihood; p = 1 is Poisson, p = 2 is gamma, and 1 < p < 2 gives a compound Poisson–gamma for aggregate losses.

Offset in a GLM: include an offset term such as log(exposure) in the linear predictor so that the mean is proportional to exposure; the coefficient on the offset is fixed at 1.

Hazard and survival models: the Cox proportional hazards model has hazard h(t|x) = h_0(t)·exp(x'β); the baseline hazard is unspecified and the partial likelihood estimates β.

Survival function and hazard: S(t) = exp(−H(t)) where H(t) = ∫_0^t h(s) ds; for a constant hazard λ, S(t) = exp(−λt) and the mean is 1/λ.

Copulas: a copula couples marginal distributions into a joint distribution; the Gaussian copula has no tail dependence, while the t-copula has symmetric tail dependence.

Dependence measures: Kendall's tau and Spearman's rho are rank-based and invariant to monotone transformations; for the Gaussian copula, tau = (2/π)·arcsin(ρ).