Extreme Value Theory in Risk Management See McNeil, Extreme Value - PowerPoint PPT Presentation

ST 810-006 Statistics and Financial Risk Extreme Value Theory in Risk Management See McNeil, Extreme Value Theory for Risk Managers Risk management is essentially about tail events. Probabilists have developed a theory specific to extreme, or tail, events. The generalized Pareto distribution (GPD) plays a central role. 1 / 13

ST 810-006 Statistics and Financial Risk Framework X 1 , X 2 , . . . are identically distributed random variables with continuous distribution function F ( x ) = P ( X i ≤ x ) . These X ’s represent losses , so we shall focus on the upper tail of F ( · ). For a level q such as 0 . 95 or 0 . 99, VaR q ( F ) = F − 1 ( q ) and ES q ( F ) = E [ X | X > VaR q ( F )] . 2 / 13

ST 810-006 Statistics and Financial Risk Generalized Pareto Distribution The GPD has distribution function  � � − 1  1 + ξ x ξ 1 − ξ � = 0 β � � G ξ,β ( x ) =  − x 1 − exp ξ = 0 , β where β > 0 is a scale parameter, ξ is a shape parameter, and: x ≥ 0 when ξ ≥ 0; 0 ≤ x ≤ − β/ξ when ξ < 0. The GPD is heavy-tailed when ξ > 0. With ξ > 0, E ( X k ) < ∞ only for k < 1 /ξ. 3 / 13

ST 810-006 Statistics and Financial Risk Excess Loss Distribution The conditional excess distribution function is F u ( y ) = P ( X ≤ u + y | X > u ) , y ≥ 0 . That is, F u ( y ) = F ( y + u ) − F ( u ) . 1 − F ( u ) Key result: for “nice” F ( · ) and for large u , F u ( y ) ≈ G ξ,β ( u ) ( y ) , where ξ and β ( · ) depend on F ( · ). 4 / 13

ST 810-006 Statistics and Financial Risk That is, for a high threshold u , losses over that threshold are approximately GPD. Proposal: given data X 1 , X 2 , . . . , X n , choose a “sensible” u , and estimate ξ and β = β ( u ) by fitting the GPD (e.g. by Maximum Likelihood) to the N u observed excesses { X i : X i ≥ u } . Note that F ( x ) = [1 − F ( u )] F u ( x − u ) + F ( u ) ≈ [1 − F ( u )] G ξ,β ( u ) ( x − u ) + F ( u ) and F ( u ) can be estimated by ( n − N u ) / n . 5 / 13

ST 810-006 Statistics and Financial Risk So for x ≥ u , F ( x ) can be estimated by � N u � � n − N u � ˆ F ( x ) = G ˆ β ( x − u ) + ξ, ˆ n n � � x − u �� − 1 / ˆ ξ = 1 − N u 1 + ˆ ξ ˆ n β 6 / 13

ST 810-006 Statistics and Financial Risk Value at Risk The approximation ˆ F ( x ) yields, for q > N u / n , �� n � � − ˆ ˆ ξ β � VaR q = u + (1 − q ) − 1 . ˆ N u ξ 7 / 13

ST 810-006 Statistics and Financial Risk Expected Shortfall For the approximating GPD ES q 1 − ξ + β ( u ) − ξ u 1 = . VaR q (1 − ξ )VaR q So we can estimate ES q by � β − ˆ ˆ VaR q ξ u � ES q = + 1 − ˆ (1 − ˆ ξ ξ ) 8 / 13

ST 810-006 Statistics and Financial Risk EVT and GARCH Above, losses have been modeled as the independent and identically distributed X 1 , X 2 , . . . . This ignores volatility dynamics. An EVT-GARCH approach models losses by Y 1 , Y 2 , . . . , where Y t = σ t X t , and σ t = σ ( Y t − 1 , Y t − 2 , . . . ) satisfies a GARCH recursion. Then for one step ahead, VaR or ES is calculated conditionally on σ t : VaR q ( Y t | σ t ) = σ t � � VaR q ( X ) 9 / 13

ST 810-006 Statistics and Financial Risk For k > 1 steps ahead, it is necessary to simulate Y t , Y t +1 , Y t + k − 1 . We have a model only for the upper tail of X . Proposal: use the empirical distribution for the rest of the distribution (i.e., historical simulation, or bootstrap ). 10 / 13

ST 810-006 Statistics and Financial Risk Extreme Value Distribution Classic extreme value theory deals with the distribution of M n = max( X 1 , X 2 , . . . , X n ) . We look for constants a n > 0 and b n such that M n − b n a n converges in distribution to some limit H ( · ). Fisher-Tippett-Gnedenko Theorem: H ( · ) must be the Generalized Extreme Value Distribution; for some ξ , � exp[ − (1 + ξ x ) − 1 /ξ ] ξ � = 0 H ( x ) = exp ( − e − x ) ξ = 0 . 11 / 13

ST 810-006 Statistics and Financial Risk If constants a n and b n can be found, F ( · ) belongs to the (maximum) domain of attraction of H ( · ). The three cases are: ξ > 0: the Fr´ echet distribution; ξ = 0: the Gumbel distribution; ξ < 0: the Weibull distribution. Pickands-Balkema-de Haan Theorem: the distribution function F ( · ) is “nice” if and only if it belongs to the domain of attraction of H ( · ) for some ξ . The parameter ξ has the same value in the GEVD as in the GPD. 12 / 13

ST 810-006 Statistics and Financial Risk A necessary and sufficient condition for F ( · ) to belong to the domain of attraction of the Fr´ echet distribution (GEVD with ξ > 0) is 1 − F ( tx ) 1 − F ( t ) → x − 1 /ξ as t → ∞ . That is, 1 − F ( x ) is regularly varying as x → ∞ . The same condition is therefore also necessary and sufficient for convergence of F u ( y ) to the GPD. 13 / 13