Severity, Frequency, and Aggregate Loss

Transcription

1 Part I Severity, Frequency, and Aggregate Loss 1 Basic Probability Raw moments Central moments µ n = E[X n ] µ n = E[(X µ) n ] Skewness γ 1 = µ 3 /σ 3 Kurtosis γ 2 = µ 4 /σ 4 Coefficient of variation CV = σ/µ Covariance Cov[X, Y ] = E[(X µ X )(Y µ Y )] = E[XY ] E[X] E[Y ] Correlation ρ XY = Cov[X, Y ]/(σ X σ Y ) MGF M X (t) = E[e tx ] PGF P X (t) = E[t X ] = M X (log t) Moments via MGF M (n) X (0) = E[Xn ] Moments via PGF P (n) X (1) = E[X(X 1)... (X n + 1)] Conditional mean E[X] = E Y [E X [X Y ]] 2 Variance Sample variance Var[ X] = Var Mixtures F (x) = [ 1 n ] n X i = Var[X] n n w i F Xi (x), w 1 + w w n = 1 Bernoulli shortcut Var[Y ] = (a b) 2 q(1 q) 3 Conditional Variance Conditional variance Var[X] = Var[E[X I]] + E[Var[X I]] 1

2 4 Expected Values Payment per loss with deductible E[(X d) + ] = d d (x d)f(x) dx = Payment per loss with E[X d] = xf(x) dx + ds(d) = claims limit/limited expected value 0 Decomposition relation E[X] = E[(X d) + ] + E[X d] Payment per payment event/mean residual life LEV higher moments E[(X d) k ] = Deductible + Limit (max. payment = u d) d d e(d) = E[X d X > d] = E[(X d) +] 1 F (d) d 0 kx k 1 S(x) dx E[Y L ] = E[X u] E[X d] 5 Parametric Distributions 0 S(x) dx S(x) dx Tail weight: 1. Compare moments More moments = less tail weight 2. Density ratios Low ratio = numerator has less tail weight 3. Hazard rate Increasing hazard rate = less tail weight 4. Mean residual life Decreasing MRL = less tail weight 6 Lognormal Distribution Continuously compounded growth rate α Continuously compounded dividend return δ Volatility σ v Lognormal parameters µ = (α δ 1 2 σ2 v)t σ = σ v t Asset price at time t S t = S 0 exp(µ + Zσ) Strike price K European call (option to buy) C = max{0, S T K} European put (option to sell) C = max{0, K S T } American options exercise at any time up to T Black-Scholes ˆd 1 = (log(k/s 0 ) µ σ 2 )/σ ˆd 2 = (log(k/s 0 ) µ)/σ Cumulative distribution Pr[S t < K] = Φ( ˆd 2 ) Limited expected value E[S t S t < K] = S 0 e (α δ)t Φ( ˆd 1 ) Φ( ˆd 2 ) E[S t S t > K] = S 0 e (α δ)t Φ( ˆd 1 ) Φ( ˆd 2 ) 2

3 7 Deductibles, LER, Inflation Loss elimination ratio Inflation if Y = (1 + r)x LER(d) = E[X d]/ [ E[X] E[Y d] = (1 + r) E X ] d 1+r 8 Other Coverage Modifications With deductible d, maximum covered loss u, coinsurance α, policy limit/maximum payment L = α(u d): Payment per loss E[Y L ] = α(e[x u] E[X d]) Second moment of per-loss E[(Y L ) 2 ] = E[(X u) 2 ] E[(X d) 2 ] 2d E[Y L ] 9 Bonuses With earned premium P, losses X, proportion of premiums r: Bonus B = c(rp X) + 10 Discrete Distributions p k (a, b, 0) recursion = a + b p k 1 k zero-truncated relation p T 0 = 0, p T n = p n 1 p 0 zero-modified relation p M n = 1 pm 0 1 p 0 p n 11 Poisson/Gamma If S = X 1 + X X N, where X Gamma(α, θ), N Poisson(λ) where λ varies by X, then S NegBinomial(r = α, β = θ). 12 Frequency Distributions Exposure and Coverage Modifications Model Original Exposure mod. Coverage mod. Exposure n 1 n 2 n 1 Pr[X > 0] 1 1 υ Poisson λ (n 2 /n 1 )λ υλ Binomial m, q (n 2 /n 1 )m, q m, υq Neg. Binomial r, β (n 2 /n 1 )r, β r, υβ 3

4 13 Aggregate Loss Models: Approximating Distribution Compound variance Var[S] = Var[X] E[N] + E[X] 2 Var[N] 14 Aggregate Loss Models: Recursive Formula Frequency p n = Pr[N = n] Severity f n = Pr[X = n] Aggregate loss g n = Pr[S = n] = f S (n) 1 k ( (a, b, 0) recursion g k = a + bj ) f j g k j 1 af 0 k j=1 15 Aggregate Losses Aggregate Deductible d/h 1 E[S d] = d(1 F (d)) + hjg hj j=0 16 Aggregate Losses Misc. Topics If X Exponential(θ), N Geometric(β), then F S (x) = 1 β 1+β F X (x), where X Exponential(θ(1 + β)). If S = X X n, then S Gamma(α = n, θ). Method of rounding: p k = F X (k + 1/2) F X (k 1/2) 17 Ruin Theory Ruin probability, discrete, finite horizon ψ(u, t) Survival probability, continuous, infinite horizon φ(u) 1+β [x = 0] + 4

5 Part II Empirical Models 18 Review of Mathematical Statistics Bias Consistency Mean square error Sample variance s 2 = 1 n 1 Biasˆθ(θ) = E[ˆθ θ θ] lim Pr[ ˆθ n θ < δ] = 1, δ > 0 n MSEˆθ(θ) = E[(ˆθ θ) 2 θ] n (x i x) 2 k=1 Variance of s 2 σ 2 /n MSE/Bias relation MSEˆθ(θ) = Var[ˆθ] + Biasˆθ(θ) 2 19 Empirical Distribution for Complete Data Total number of observations n Observations in j-th interval n j Width of j-th interval c j c j 1 n j Empirical density f n (x) = n(c j c j 1 ) 20 Variance of Empirical Estimators with Complete Data Empirical variance Var[Sn (x)] = S n (x)(1 S n (x))/n = n x (n n x )/n 3 21 Kaplan-Meier and Nelson Åalen Estimators Risk set at time y j r j Loss events at time y j s j Kaplan-Meier product-limit estimator j 1 S n (t) = j 1 Nelson-Åalen cumulative hazard Ĥ(t) = ( 1 s ) i, y j 1 t < y j r i s i r i, 22 Estimation of Related Quantities y j 1 t < y j Exponential extrapolation: fit S n (y k ) = exp( y k /θ), and solve for the parameter θ. 5

6 23 Variance of Kaplan-Meier and Nelson-Åalen Estimators Greenwood s approximation for KM Greenwood s approximation for NÅ 100(1 α)% log-transformed confidence interval for KM 100(1 α)% log-transformed confidence interval for NÅ Var[S n (y j )] = S n (y j ) 2 Var[Ĥ(y j)] = j s i r 2 i j s i r i (r i s i ) (S n (t) 1/U, S n (t) U ), U = exp z α/2 Var[S n (t)] S n (t) log S n (t) (Ĥ(t)/U, Ĥ(t)U), U = exp z Var[Ĥ(t)] α/2 Ĥ(t) 24 Kernel Smoothing Uniform kernel density k y (x) = 1 2b, y b x y + b 0, x < y b x (y b) Uniform kernel CDF K y (x) = 2b, y b x y = b 1, y + b < x Triangular kernel density height = 1/b, base = 2b Empirical probability at y i p n (y i ) n Fitted density ˆf(x) = p n (y i )k yi (x) n Fitted distribution ˆF (x) = p n (y i )K yi (x) Use conditional expectation formulas to find moments of kernel-smoothed distributions; condition on the empirical distribution. 25 Approximations for Large Data Sets Right/upper endpoint of j-th interval c j Number of new entrants in [c j, c j+1 ) d j Number of withdrawals in (c j, c j+1 ] u j Number of events in (c j, c j+1 ] s j Risk set for the interval (c j, c j+1 ] r j Conditional mortality rate in (c j, c j+1 ] q j j 1 Population at time c j P j = d i u i s i Generalized relation r j = P j + αd j βu j UD of entrants/withdrawals α = β = 1/2 i=0 6

7 Part III Parametric Models 26 Method of Moments For a k-parameter distribution, match the first k empirical moments to the fitted distribution: E[X m ] = n (x i x) m 27 Percentile Matching Interpolated k-th order statistic x k+w = (1 w)x k + wx k+1, 0 < w < 1 Smoothed empirical 100p-th π p = x p(n+1) percentile 28 Maximum Likelihood Estimators Likelihood function L( θ) = Loglikelihood l = log L n Pr[X X i θ] X i are the observed events each is a subset of the sample space. Maximize l by finding θ such that l θ i = 0 for each parameter in the fitted distribution. 29 MLEs Special Techniques Exponential Gamma (fixed α) Normal Poisson Neg. Binomial Lognormal MLE = sample mean MLE = method of moments MLE(µ) = sample mean, MLE(σ 2 ) = population variance MLE = sample mean MLE(rβ) = sample mean take logs of sample, then use Normal shortcut Censored exponential MLE Take each observation (including censored ones) and subtract the deductible; sum the result and divide by the number of uncensored observations. 7

8 30 Estimating Parameters of a Lognormal Distribution 31 Variance of MLEs For n estimated parameters θ = (θ 1, θ 2,..., θ n ), the estimated variance of a function of MLEs is computed using the delta method: σ1 2 σ 12 σ 1n Covariance matrix Σ( θ) σ 21 σ2 2 σ 2n = σ n1 σ n2 σn 2 ( ) Delta method Var[g( θ)] g ( ) = ( θ) θ g [ θ 2 l( θ) Fisher s information I(θ rs ) = E ] θ s θ r Covariance-information relation Σ( θ)i( θ) = I n 32 Fitting Discrete Distributions To choose which (a, b, 0) distribution to fit to a set of data, compute the empirical mean and variance. Then note Binomial Poisson Negative Binomial E[N] > Var[N] E[N] = Var[N] E[N] < Var[N] 33 Cox Proportional Hazards Model Hazard class i/covariate z i logarithm of proportionality constant for class i β i Proportionality constant/relative risk c = exp(β 1 z 1 + β 2 z β n z n ) Baseline hazard H 0 (t) Hazard relation H(t z 1,..., z n ) = H 0 (t)c 34 Cox Proportional Hazards Model: Partial Likelihood Number at risk at time y k Proportionality constants of members at risk {c 1, c 2,..., c k } Failures at time y {j 1, j 2,..., j d } ( d k Breslow s partial likelihood exp(β ji ) exp(β i ) ) d 8

9 35 Cox Proportional Hazards Model: Estimating Baseline Survival Risk set at time y j R(y j ) Proportionality constants for the c i members of risk set R(y j ) Baseline hazard rate Ĥ 0 (t) = y j t s j i R(y j) c i 36 The Generalized Linear Model 37 Hypothesis Tests: Graphic Comparison Adjusted fitted density f (x) = f(x) 1 F (d) Adjusted fitted distribution F F (x) F (d) (x) = 1 F (d) D(x) plot D(x) = F n (x) F (x) empirical observations x 1, x 2,..., x n p-p plot (F n (x j ), F (x j )) Normal probability plot (x j, F 1 (F n (x))) Where the p-p plot has slope > 1, then the fitted distribution has more weight than the empirical distribution; where the slope < 1, the fitted distribution has less weight. 38 Hypothesis Tests: Kolmogorov-Smirnov Komolgorov-Smirnov statistic D = max F n (x) F (x; ˆθ) KS-statistic is the largest absolute difference between the fitted and empirical distribution. Should be used on individual data, but bounds on KS can be established with grouped data. Fitted distribution must be continuous. Uniform weight across distribution. Lower critical value for fitted parameters and for more samples. 39 Hypothesis Tests: Anderson-Darling Anderson-Darling statistic u A 2 (F n (x) F (x)) 2 = n t F (x)(1 F (x)) f (x) dx AS-statistic used only on individual data. Heavier weight on tails of distribution. Critical value independent of sample size, but decreases for fitted parameters. 9

10 40 Hypothesis Tests: Chi-square Total number of observations Hypothetical probability X is in j-th group Number of observations in j-th group n p j n j Chi-square statistic Q = k (n j E j ) 2 E j=1 j E j = n j p j Degrees of freedom df = total number of groups, minus number of estimated parameters, minus 1 if n is predetermined 41 Likelihood Ratio Algorithm, Schwarz Bayesian Criterion Likelihood Ratio compute loglikelihood for each parametric model. Twice the difference of the loglikelihoods must be greater than 100(1 α)% percentile of chi-square with df = difference in the number of parameters between the compared models. Schwarz Bayesian Criterion Compute loglikelihoods and subtract r 2 log n, where r is the number of estimated parameters in the model and n is the sample size of each model. Part IV Credibility 42 Limited Fluctuation Credibility Poisson Frequency Poisson frequency of claims λ Margin of acceptable fluctuation k Confidence of fluctuation being within k P Severity CV CVs 2 = σs/µ 2 2 s ( ( ) Φ 1 1+P 2 n 0 n 0 = k Credibility for Frequency Severity Aggregate n 0 n 0 Exposure units e F λ λ CV s 2 n 0 ( ) 1 + CV 2 λ s Number of claims n F n 0 n 0 CV 2 s n 0 (1 + CV 2 Aggregate losses s F n 0 µ s n 0 µ s CV 2 s n 0 µ s (1 + CV 2 ) 2 s ) s ) 10

11 43 Limited Fluctuation Credibility: Non-Poisson Frequency Credibility for Frequency Severity Aggregate ( ) σf 2 σs 2 n 0 σ 2 f Exposure units e F n 0 µ 2 n 0 f µ f µ 2 + σ2 s s µ f µ f µ 2 s) Number of claims n F Aggregate losses s F n 0 σ 2 f µ f n 0 σ 2 s µ 2 s n 0 µ s σ 2 f µ f n 0 σ 2 s µ s n 0 µ s ( σ 2 f n 0 + σ2 s µ f µ 2 s ( ) σ 2 f + σ2 s µ f µ 2 s Poisson group frequency is the special case µ f = σf 2 = λ. If a compound Poisson frequency model is used, you cannot use the Poisson formula you must use the mixed distribution (e.g., Poisson/Gamma mixture is Negative Binomial). 44 Limited Fluctuation Credibility: Partial Credibility Credibility factor Manual premium (presumed value before observations) Observed premium Credibility premium Z = n/n F = e/e F = s/s F M X P C = Z X + (1 Z)M 45 Bayesian Estimation and Credibility Discrete Prior Constructing a table: First row is the prior probability, the chance of membership in a particular class before any observations are made. Second row is the likelihood function of the observation(s) given the hypothesis of membership in that particular class. Third row is the joint probability, the product of Rows 1 and 2. Row 4 is the posterior probability, which is Row 3 divided by the sum of Row 3. Row 5 is the hypothetical mean or conditional probability, the expectation or probability of the desired outcome given that the observations belong that that class. Row 6 is the expectation, Bayesian premium, or expected probability, the desired result given the observations, and is the sum of the products of Rows 4 and 5. 11

12 46 Bayesian Estimation and Credibility Continuous Prior Observations x = (x 1, x 2,..., x n ) Prior density π(θ) Model density f( x θ) Joint density f( x, θ) = f( x θ)π(θ) Unconditional density f( x) = f( x, θ) dθ f( x, θ) Posterior density π(θ x 1,..., x n ) = f( x) Predictive density f(x n+1 x) = f(x n+1 θ)π(θ x) dθ Loss function minimizing MSE Loss function minimizing absolute error Zero-one loss function posterior mean E[Θ x] posterior median posterior mode A conjugate prior is the prior distribution when the prior and posterior distributions belong to the same parametric family. 47 Bayesian Credibility: Poisson/Gamma With Poisson frequency with mean λ, where λ is Gamma distributed with parameters α, θ, Number of claims x Number of exposures n Average claims per exposure x = x/n Conjugate prior parameters α, γ = 1/θ Posterior parameters α = α + x γ = γ + n Credibility premium P C = α /γ 48 Bayesian Credibility: Normal/Normal With Normal frequency with mean θ and fixed variance v, where θ is Normal with mean µ and variance a, vµ + na x Posterior parameters µ = v + na Posterior variance a = va v + na n Credibility factor Z = n + v/a Credibility premium µ 12

13 49 Bayesian Credibility: Binomial/Beta With Binomial frequency with parameters M, q, where q is Beta with parameters a, b, Number of trials m Number of claims in m trials k Posterior parameters a = a + k b = b + m k Credibility premium P C = a /(a + b ) 50 Bayesian Credibility: Exponential/Inverse Gamma With exponential severity with mean Θ, where Θ is inverse Gamma with parameters α, β, Posterior parameters α = α + n β = β + n x 51 Bühlmann Credibility: Basics Expected hypothetical mean µ = E[E[X θ]] Variance of hypothetical mean (VHM) a = Var[E[X θ]] Expected value of process variance (EPV) v = E[Var[X θ]] Bühlmann s k k = v/a Bühlmann credibility factor Z = n/(n + k) Bühlmann credibility premium P C = Z X + (1 Z)µ 52 Bühlmann Credibility: Discrete Prior No additional formulas 53 Bühlmann Credibility: Continuous Prior No additional formulas 54 Bühlmann-Straub Credibility 55 Exact Credibility Bühlmann equals Bayesian credibility when the model distribution is a member of the linear exponential family and the conjugate prior is used. 13

14 Frequency/Severity Poisson/Gamma Normal/Normal Binomial/Beta Bühlmann s k k = 1/θ = γ k = v/a k = a + b 56 Bühlmann As Least Squares Estimate of Bayes Variance of observations Var[X] = p i Xi 2 X 2 Bayesian estimates Covariance Y i Cov[X, Y ] = p i X i Y i XȲ Mean relationship E[X] = E[Y ] = E[Ŷ ] regression slope/bühlmann credibility estimate b = Z = Cov[X, Y ] Var[X] regression intercept a = (1 Z) E[X] Bühlmann predictions Ŷ i = a + bx i 57 Empirical Bayes Non-Parametric Methods For uniform exposures, Number of exposures/years data n Number of classes/groups r Observation of of group i, year j x ij Unbiased manual premium ˆµ = x = 1 r n x ij rn j=1 Unbiased EPV ˆv = 1 r 1 n (x ij x i ) 2 r n 1 j=1 Unbiased MHV â = 1 r ( x i x) 2 ˆv r 1 n Bühlmann credibility factor Z = n n + ˆk 58 Empirical Bayes Semi-Parametric Methods When the model is Poisson, v = µ, and we have EPV ˆv = µ VHM â = Var[S] ˆv Sample variance Var[S] = σ 2 = (x i x) 2 n 1 14

15 Part V Simulation 59 Simulation Inversion Method Random number u [0, 1] Inversion relationship Pr[F 1 (u) x] = Pr[F (u) F (x)] = F (x) Method x i = F 1 (u) Simply take the generated uniform random number u and compute the inverse CDF of u to obtain the corresponding simulated x i. 15