Econ. & Mat. Enrique Navarrete Palisade Risk Conference “Advanced Topics in Finance and Engineering: Extreme Value Theory (EVT), Risk Management, and Applications” Econ. & Mat. Enrique Navarrete Palisade Risk Conference Rio de Janeiro 2009

Extreme Value Theory TOPICS: Introduction and motivation; Use of the Gumbel distribution (Extreme Value Distribution); Use of the Generalized Extreme Value Distribution (GEV); Parameter estimation by Maximum Likelihood (MLE); Identification of the tail parmeter (Hill’s method); Estimation of extreme loss percentiles; Examples ®Scalar Consulting, 2009

INTRODUCTION TO EXTREME VALUE THEORY (EVT)

What is the “maximimun” claims level we can expect ? Extreme Value Theory Motivation: Maximum insurance claims (monthly maxima, N = 90 months) What is the “maximimun” claims level we can expect ? By simulation methods, could we expect to get a number larger than the historical maximum? ®Scalar Consulting, 2009

Extreme Value Theory Motivation: = RiskWeibull(1,2171;172469;RiskShift(144825)) ®Scalar Consulting, 2009

Extreme Value Theory Motivation: = RiskWeibull(1,2171;172469;RiskShift(144825)) For monthly data, how often should we expect to see the values at the 99,5 % and 99,9 % levels ? ®Scalar Consulting, 2009

Extreme Value Theory Related Question: If the chance of volcanic eruption today is 0,006 %, how do we interpret this small probability ? ®Scalar Consulting, 2009

Extreme Value Theory Related Question: If * N then: N = 1/ (0,006 %) = 16,666 days = 45,6 years N (number of days) * Daily probability = 1 event N (time window to see an event) = 1 / Probability ®Scalar Consulting, 2009

Extreme Value Theory Back to Problem: The percentiles we have calculated indicate possible claim values that can actually occur, therefore these are the minimum monthly reserves to be held to cover possible claims at these confidence levels VAR ®Scalar Consulting, 2009

Extreme Value Theory Back to Problem: Now these confidence levels have failure rates: Example: By setting up a monthly reserve of $ 823,000 (VAR 99,5 %), we would expect to cover all claims approximately 199/200 months (= 99,5 %) and will not be able to cover claims approx. 1 every 200 months ®Scalar Consulting, 2009

Extreme Value Theory Application: How do we set an appropriate level of monthly reserves that fails (falls short of claims) approximately once every 2 years ? ®Scalar Consulting, 2009

Extreme Value Theory Application: How do we set a appropriate level of monthly reserves that fail (fall short of claims) approximately once every 2 years ? Failure rate = (1/24 ) months = 4,2 % Confidence level = (1 - 1/24) = 95,8% VAR 95,8% = $ 590,000. ®Scalar Consulting, 2009

Extreme Value Theory More Applications: How high should we build a dam that fails (allows flooding) once every 40 years ? How strong to build homes to support hurricanes and collapse every 80 years ? How resistant to build antennae in presence of very strong winds ? How strong to build materials in general? ®Scalar Consulting, 2009

EXTREME VALUE THEORY AND APPLICATIONS

Extreme Value Theory Generalized Extreme Value Distribution (GEV): Under certain conditions, the GEV distribution is the limit distribution of sequences of independent and identically distributed random variables. = location parameter; = scale parameter = shape (tail) parameter ®Scalar Consulting, 2009

< 0 (Reversed Weibull) Extreme Value Theory Fisher-Tippett-Gnedenko Theorem: Only 3 possible families of distributions for the maximumm depending on the parameter ! Probability > 0 (Fréchet) = 0 (Gumbel) Loss Distribution $ < 0 (Reversed Weibull) ®Scalar Consulting, 2009

Extreme Value Theory Generalized Extreme Value Distribution (GEV): For modeling maxima, the case < 0 is not interesting (“thin tails”); For the case (Gumbel), we can take shortcuts and avoid estimating the tail parameter; use Gumbel (Extreme Value Distribution); For the case > 0 (“fat tails”), we have to use the GEV Distribution and estimate the tail parameter (Hill’s Plot). ®Scalar Consulting, 2009

(Extreme Value Distribution) Gumbel Distribution (Extreme Value Distribution) = 0

Formulas apply to Gumbel distribution Extreme Value Theory Location and Scale parameters (MOM): Obtain sample mean ( ) and sample standard deviation (s) from the series of maxima; 2) We are assuming initially that the distribution is Gumbel ( = 0); 3) Estimate location ( ) and scale parameters ( ) using formulas from Method of Moments (MOM); Formulas apply to Gumbel distribution ®Scalar Consulting, 2009

Extreme Value Theory Location and Scale parameters (MOM): where = Euler´s Constant : . Limiting difference between the harmonic series and the natural logarithm ®Scalar Consulting, 2009

Extreme Value Theory Example 1: (MOM) Maximum losses (monthly, N = 60) ®Scalar Consulting, 2009

Extreme Value Theory Location and Scale parameters (MLE): As an alternative to MOM, we can calculate the location and scale parameters by Maximum Likelihod Estimation ie. and that maximize the function: ®Scalar Consulting, 2009

Extreme Value Theory Example 1: (MLE) Maximum losses (monthly, N = 60) ®Scalar Consulting, 2009

Extreme Value Theory Example 1: @RISK: =RiskExtvalue(46170;37285) ®Scalar Consulting, 2009

Extreme Value Theory Example 1: When distribution is Gumbel ( = 0), we can use the @RISK Extreme Value distribution: @RISK: =RiskExtvalue(46170;37285) ®Scalar Consulting, 2009

Generalized Extreme Value Distribution (GEV) > 0

Extreme Value Theory Generalized Extreme Value Distribution (GEV): Since in general , we need to estimate this parameter by Hill’s Method. Graph of: ®Scalar Consulting, 2009

To get the loss percentiles we need to estimate the shape parameter Extreme Value Theory Example 2: (MLE) Maximum losses (monthly, N = 60) Location and scale parameters are very different, suggesting distribution is not Gumbel To get the loss percentiles we need to estimate the shape parameter ®Scalar Consulting, 2009

Extreme Value Theory Example 2: Hill’s Diagram = 0,4 ®Scalar Consulting, 2009

Extreme Value Theory Example 2: (MLE) Maximum losses (monthly, N = 60) We obtain very different GEV percentiles since the distribution is not Gumbel ( ). ®Scalar Consulting, 2009

Extreme Value Theory Example 2: Since , we cannot use the Gumbel distribution; either estimate and use EVT or use a @RISK distribution, (not the Extreme value Distribution ie.Gumbel) since it will stay short. ®Scalar Consulting, 2009

Extreme Value Theory Example 2: @RISK: =RiskPearson5(2,2926;124899;RiskShift(-12413)) ®Scalar Consulting, 2009

= 0.06 (ie. for all practical purposes the distribution is Gumbel) Extreme Value Theory Example 1 (Gumbel): Hill’s Plot = 0.06 (ie. for all practical purposes the distribution is Gumbel) ®Scalar Consulting, 2009

Extreme Value Theory Example 3: Hill’s Plot = 0.01 (Gumbel) ®Scalar Consulting, 2009

Extreme Value Theory Example 4: Hill’s Plot = 0.38 (not Gumbel, use GEV) ®Scalar Consulting, 2009

Enrique Navarrete , Scalar Consulting www.grupoescalar.com enrique.navarrete.p@gmail.com MSc. University of Chicago BS. Economics, BS. Mathematics, MIT Risk Software, Consulting and Auditing Risk courses offered jointly with Universidad Iberoamericana, several countries