Application of Extreme Value Theory to Risk Capital Estimation
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Application of Extreme Value Theory to Risk Capital Estimation

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Presented at Life Conference 2010.

Presented at Life Conference 2010.

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Application of Extreme Value Theory to Risk Capital Estimation Application of Extreme Value Theory to Risk Capital Estimation Presentation Transcript

  • Life conference and exhibition 2010Steven Morrison and Alex McNeil Application of Extreme Value Theory to Risk Capital Estimation 7-9 November 2010© 2010 The Actuarial Profession  www.actuaries.org.uk
  • Application of EVT to risk capital estimation:Agenda• Motivation• Background theory• VaR case study• Summary• Questions or comments?© 2010 The Actuarial Profession  www.actuaries.org.uk 1
  • Application of extreme value theory to risk capital estimationSteven Morrison and Alex McNeil Motivation© 2010 The Actuarial Profession  www.actuaries.org.uk
  • Motivation• Measures of risk capital are based on the (extreme) tail of a distribution – Value at Risk (VaR) – Conditional Value at Risk (CVaR) / Expected Shortfall• In particular, Solvency II SCR is defined as a 99.5% VaR over a one year horizon• Generally needs to be estimated using simulation 1. Generate real-world economic scenarios for all risk drivers affecting the balance sheet over one year 2. Revalue the balance sheet under each real-world scenario – e.g. Monte Carlo („nested stochastic), Replicating Formula, Replicating Portfolio 3. Estimate the statistics of interest© 2010 The Actuarial Profession  www.actuaries.org.uk 3
  • Motivation• An insurer who has gone through such a simulation exercise states – “Our Solvency Capital Requirement is £77.5m”• How confident can we be in this number?• Many sources of uncertainty – Choice of economic scenario generator (ESG) models and their calibration – Liability model assumptions e.g. dynamic lapse rules – Choice of scenarios sampled i.e. choice of real-world ESG random number seed© 2010 The Actuarial Profession  www.actuaries.org.uk 4
  • Motivation• The same insurer re-runs their internal model using a different random number seed (but all other assumptions are unchanged) – “Our Solvency Capital Requirement is now £82.8m”• So, simulation-based capital estimates are subject to statistical uncertainty – Can we estimate this statistical uncertainty? – How can we reduce the amount of statistical uncertainty?• In this presentation, we will address these questions using a statistical technique known as Extreme Value Theory (EVT)© 2010 The Actuarial Profession  www.actuaries.org.uk 5
  • Application of extreme value theory to risk capital estimationSteven Morrison and Alex McNeil Background theory© 2010 The Actuarial Profession  www.actuaries.org.uk
  • Application of extreme value theory to risk capital estimationSteven Morrison and Alex McNeil VaR case study© 2010 The Actuarial Profession  www.actuaries.org.uk
  • VaR Case studyLiability book• UK-style with profits – Management actions, dynamic EBR, dynamic bonus rates, regular premiumsValuation methodology• Nested stochastic – 1,000 real-world outer scenarios – 1,000 risk-neutral inner scenarios per outer scenario© 2010 The Actuarial Profession  www.actuaries.org.uk 8
  • Estimated distribution of liability value at end of year(empirical quantile method)• Estimate „empiricalquantiles‟ by ranking1,000 scenarios• Estimated 99.5% VaR =£77.5m(995th worst-casescenario)• Note that estimateddistribution is „lumpy‟,particularly as we gofurther out in the tail© 2010 The Actuarial Profession  www.actuaries.org.uk 9
  • Estimated distributions using different scenario sets(empirical quantile method)Initial set of 1,000 real-world scenarios• 99.5% VaR = £77.5mSecond set of 1,000real-world scenarios• 99.5% VaR = £82.8m© 2010 The Actuarial Profession  www.actuaries.org.uk 10
  • What is the ‘true’ VaR?• Two different estimates for VaR – £77.5m – £82.8m – Which is „correct‟?• Both use same (subjective) modelling assumptions – Same economic scenario generator and calibration – Same liability model assumptions e.g. dynamic lapse rules• Difference is purely due to different random number streams used to generate the economic scenarios© 2010 The Actuarial Profession  www.actuaries.org.uk 11
  • Two important questions1. Can we reduce the sensitivity of the estimate to the choice of random numbers? – Run more scenarios – May not be feasible because of model run-time – Find a „better‟ estimator than the empirical quantile2. Given a particular estimate of the 99.5% VaR, can we estimate the uncertainty around this?© 2010 The Actuarial Profession  www.actuaries.org.uk 12
  • Application of Extreme Value Theory• Recall that Extreme Value Theory tells us something about the shape of the distribution in the tail – Distribution of liability value beyond some threshold is (approximately) Generalised Pareto – Parameterised by 2 parameters• Estimate the tail of the distribution by: 1. Picking a threshold 2. Fitting the 2 parameters of the Generalised Pareto Distribution to values in excess of the threshold© 2010 The Actuarial Profession  www.actuaries.org.uk 13
  • Choice of threshold 20 • Choice of threshold is subjective – But examination of „mean excess Mean Excess (£m) 15 function‟ helps identify a suitable choice 10 5 • We have judged that a threshold of 20 40 60 80 100 Threshold (£m) £40m is suitable for this particular 20 case study Mean Excess (£m) – Approximately 26% of scenarios 15 exceed the threshold 10 5 20 40 60 80 100 Threshold (£m)© 2010 The Actuarial Profession  www.actuaries.org.uk 14
  • Estimated distributions using different scenario sets(Extreme Value Theory method)Initial set of 1,000 real-worldscenarios• 99.5% VaR = £75.9m• 95% confidence interval =[71.1m, 84.3m]Second set of 1,000 real-world scenarios• 99.5% VaR = £77.8m• 95% confidence interval =[72.2m, 87.7m]© 2010 The Actuarial Profession  www.actuaries.org.uk 15
  • Application of extreme value theory to risk capital estimationSteven Morrison and Alex McNeil Summary© 2010 The Actuarial Profession  www.actuaries.org.uk
  • Summary• Simulation-based measures of risk capital, e.g. VaR, are subject to statistical uncertainty• Extreme Value Theory provides a robust method for estimating VaR – Allows statistical uncertainty to be estimated – Statistical uncertainty lower than „naïve‟ quantile estimation – Provides an estimate of entire tail of distribution, allowing estimate of more extreme VaR, CVaR etc.© 2010 The Actuarial Profession  www.actuaries.org.uk 17
  • Questions or comments?Expressions of individual views bymembers of The Actuarial Professionand its staff are encouraged.The views expressed in this presentationare those of the presenter.© 2010 The Actuarial Profession  www.actuaries.org.uk 18