Statistical information risk
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Statistical Information Risk. Gary G. Venter Guy Carpenter Instrat. Statistical Information Risk. Estimation risk Uncertainty related to estimating parameters of distributions from data Projection risk Uncertainty arising from the possibility of future changes in the distributions

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Statistical Information Risk

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Statistical information risk

Statistical Information Risk

Gary G. Venter

Guy Carpenter Instrat


Statistical information risk1

Statistical Information Risk

  • Estimation risk

    • Uncertainty related to estimating parameters of distributions from data

  • Projection risk

    • Uncertainty arising from the possibility of future changes in the distributions

  • Both quantifiable to some extent


Topics

Topics

  • Impact of information risk

  • Estimation risk 1 – asymptotic theory

  • Estimation risk 2 – Bayesian updating

  • Estimation risk 3 – both together

  • Projection risk


Impact of information risk

Impact of Information Risk

  • Collective risk theory

    • L = Si=1N Xi

    • E(L) = E(N)E(X)

    • Var(L) = E(N)Var(X) + E(X)2Var(N)

    • CV(L)2 = Var(X)/[E(N)E(X)2] + Var(N)/E(N)2

    • = [CV(X)2 + VM(N)] / E(N)

    • = [ 49 + 1 ] / E(N)

    • Goes to zero as frequency gets large

  • Add projection risk

    • K = JL, E(J) = 1, so E(K) = E(L)

    • CV(K)2 = [1+CV(J)2]CV(L)2 + CV(J)2

    • Lower limit is CV(J)2


Impact of projection risk on aggregate cv cv x 7 vm n 1

Impact of Projection Risk on Aggregate CVCV(X)=7, VM(N)=1


Percentiles of lognormal loss ratio cv x 7 vm n 1 e lr 65 3 e n s

Percentiles of Lognormal Loss RatioCV(X)=7, VM(N)=1, E(LR)=65, 3 E(N)’s


Estimation risk 1 asymptotic theory

Estimation Risk 1 – Asymptotic Theory

  • At its maximum, all partial derivatives of log-likelihood function with respect to parameters are zero

  • 2nd partials are negative

  • Matrix of negative of expected value of 2nd partials called “information matrix”

  • Usually evaluated by plugging maximizing values into formulas for 2nd partials

  • Matrix inverse of this is covariance matrix of parameters

  • Distribution of parameters is asymptotically multivariate normal with this covariance


Estimation risk 1 asymptotic theory1

Estimation Risk 1 – Asymptotic Theory

  • See Loss Models p. 63 for discussion

  • Simulation proceeds by simulating parameters from normal, then simu-lating losses from the parameters

  • See Loss Models p. 613 for how to simulate multivariate normals


Information matrix pareto example

Information Matrix – Pareto Example


Estimation risk 2 bayesian updating

Estimation Risk 2 – Bayesian Updating

  • Bayes Theorem:

    • f(x|y)f(y) = f(x,y) = f(y|x)f(x)

    • f(x|y) = f(y|x)f(x)/f(y)

    • f(x|y)  f(y|x)f(x)

  • Diffuse priors

    • f(x)  1

    • f(x)  1/x


Estimation risk 2 bayesian updating frequency example

Estimation Risk 2 – Bayesian UpdatingFrequency Example

  • Poisson distribution with diffuse prior f(l) 1/l

    • f(k| l) = e- llk/k!

    • f(l|k)  f(k| l)/l e- llk-1

    • Gamma distribution in k, 1

    • That’s posterior; predictive distribution of next observation is mixture of Poisson by that gamma, which is negative binomial

    • After n years of observation, predictive distribution is negative binomial with same mean as sample and variance = mean*(1+1/n)

    • Parameter distribution for l is gamma with mean of sample and variance = mean/n

    • Can simulate from negative binomial or from gamma then Poisson


Estimation risk 3 bayes information

Estimation Risk 3 – Bayes + Information

  • Tests developed by Rodney Kreps

  • Likelihood function is proportional to f(data|parameters)

  • Assume prior for parameters is proportional to 1

  • Then likelihood function is proportional to f(parameters|data)


Likelihood function for small sample pareto fit scaled to max 1 log scale

Likelihood Function for Small Sample Pareto Fit Scaled to Max = 1, Log Scale


Testing assumptions of parameter distribution

Testing Assumptions of Parameter Distribution

  • Use information matrix to get covariance matrix of parameters

  • This is quadratic term of expansion of likelihood function around max

  • Compare bivariate normal and bivariate lognormal parameter distributions to scaled likelihood function


Probability contours of likelihood

Probability Contours of Likelihood


Normal approximation

Normal Approximation


Lognormal approximation contours

Lognormal Approximation Contours


95 th percentile comparison

95th Percentile Comparison


33 rd percentile comparison

33rd Percentile Comparison


Quantifying projection risk

Quantifying Projection Risk

  • Regression 100q% confidence intervals

    • t(q/2;N-2) is the upper 100(q/2)% point from a Student “T” distribution with N-2 degrees of freedom


Projection risk example

Projection Risk Example


Finis

finis


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