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Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 E s t i m a t i o n Es st ti im ma at ti io on n E o f t h e of f t th he e o m a r g i n a l e x p e c t e d s h o r

  1. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 E s t i m a t i o n Es st ti im ma at ti io on n E o f t h e of f t th he e o m a r g i n a l e x p e c t e d s h o r t f a l l ma ar rg gi in na al l e ex xp pe ec ct te ed d s sh ho or rt tf fa al ll l m Juan Juan Cai John H. J. Einmahl Tilburg University, NL Tilburg University, NL Laurens de Haan Chen Zhou Erasmus University Rotterdam, NL De Nederlandsche Bank University of Lisbon, PT Erasmus University Rotterdam, NL 1

  2. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 Expected shortfall of an asset X at probability level p is ( ) ( ) − ≤ − ← E X X F p X ( ) { } = ≤ where and : F x P X x X ← the inverse function of . F F X X 2

  3. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 = ∑ � A bank holds a portfolio R y R i i i � Expected shortfall at probability level p ( ) − < − E R R VaR p � Can be decomposed as ( ) ∑ − < − i y E R R VaR i i p � The sensitivity to the i-th asset is ( ) − < − E R R VaR i p (is marginal expected shortfall in this case) 3

  4. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 More generally: Consider a random vector ( ) X Y , Marginal expected shortfall (MES) of X at level p is ( ) ( ) > − ← E X Y F 1 p Y (these are losses hence “ Y big” is bad). All these are risk measures i.e. characteristics that are indicative of the risk a bank occurs under stress conditions. 4

  5. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 We are interested in MES under exceptional stress conditions of the kind that have occurred very rarely or even not at all. This is the kind of situation where extreme value can help. ( ) ( ) > − ← We want to estimate E X Y F 1 p Y for small p on the basis of i.i.d. observations ( ) ( ) ( ) … X Y , , X Y , , , X Y , 1 1 2 2 n n and we want to prove that the estimator has good properties. 5

  6. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 When we say that we want to study a situation that has hardly ever occurred, this means 1 ≤ i.e., that we need to consider the case p n when a non-parametric estimator is impossible, since we need to extrapolate. On the other hand we want to obtain a limit result, as n (the number of observations) goes to infinity. 1 ≤ is essential, we then have Since the inequality p n ( ) = = as n → ∞ . to assume and p p np O 1 n n 6

  7. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 Note that a parametric model in this situation is also not realistic: The model is generally chosen to fit well in the central part of the distribution but we are interested in the (far) tail where the model may not be valid. Hence it is better to “let the tail speak for itself”. This is the semi-parametric approach of extreme- value theory. 7

  8. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 = 1 Notation : ( t big and p small, ) t p ← ⎛ ⎞ 1 ( ) = − ⎜ ⎟ U t : F 1 ⎝ ⎠ 1 X t ← ⎛ ⎞ 1 ( ) = − ⎜ ⎟ U t : F 1 ⎝ ⎠ 2 Y t ⎛ ⎞ ⎛ ⎞ 1 θ = > MES : E X Y U ⎜ ⎟ ⎜ ⎟ p 2 ⎝ ⎠ ⎝ ⎠ p 8

  9. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 ⎛ ⎞ ⎛ ⎞ ∞ 1 > > ∫ ⎜ ⎟ P X ⎜ x Y , U ⎟ dx ⎛ ⎞ ⎛ ⎞ 2 ⎝ ⎠ ⎝ ⎠ 1 p θ = > = 0 ⎜ ⎟ E X Y ⎜ U ⎟ ⎧ ⎫ ⎛ ⎞ p 2 ⎝ ⎠ ⎝ ⎠ p 1 > ⎨ ⎬ ⎜ ⎟ P Y U 2 ⎝ ⎠ ⎩ ⎭ p ⎧ ⎫ ⎛ ⎞ ∞ 1 1 = > > ∫ ⎨ ⎬ ⎜ ⎟ P X x Y , U dx 2 ⎝ ⎠ ⎩ ⎭ p p 0 ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ∞ 1 1 1 1 = > > ∫ ⎨ ⎬ i.e., ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ U P X xU , Y U dx 1 ⎝ ⎠ 1 ⎝ ⎠ 2 ⎝ ⎠ ⎩ ⎭ p p p p 0 θ ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ∞ 1 1 1 = > > ∫ ⎨ ⎬ p ⎜ ⎟ ⎜ ⎟ P X xU , Y U dx ⎛ ⎞ 1 ⎝ ⎠ 2 ⎝ ⎠ ⎩ ⎭ 1 p p p 0 U ⎜ ⎟ 1 ⎝ ⎠ p 9

  10. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 p ↓ We consider the limit of this as . 0 x = Conditions (1) : First note (take upstairs) 1 ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ 1 1 > > ⎨ ⎬ P X U ⎜ ⎟ , Y U ⎜ ⎟ 1 ⎝ ⎠ 2 ⎝ ⎠ ⎩ ⎭ p p { } ( ) ( ) = − < − < 1 , 1 P F X p F Y p 1 2 where F and F are the distribution functions of X 1 2 and Y . This is a copula. 10

  11. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 p ↓ : We impose conditions on the copula as 0 Suppose there exists a positive function ( ) R x y (the , dependence function in the tail) such that for all ≤ ≤ ∞ ∨ > ∧ < ∞ 0 x y , , x y 0, x y ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ 1 x y ( ) > > = ⎨ ⎬ i.e., ⎜ ⎟ ⎜ ⎟ lim P X U , Y U R x y , 1 ⎝ ⎠ 2 ⎝ ⎠ ⎩ ⎭ ↓ p p p p 0 ⎧ ⎫ 1 p p ( ) ( ) ( ) − < − < = ⎨ ⎬ . lim P 1 F X , 1 F Y R x y , ⎩ 1 2 ⎭ ↓ p x y p 0 11

  12. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 This condition indicates and specifies dependence specifically in the tail. ( usual condition in extreme value theory ) θ we have (2) : Compare: in the definition of p ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ 1 1 > > ⎨ ⎬ P X x U ⎜ ⎟ , Y U ⎜ ⎟ 1 ⎝ ⎠ 2 ⎝ ⎠ ⎩ ⎭ p p y = ) and in the condition we have (for 1 ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ x 1 > > ⎨ ⎬ . ⎜ ⎟ , ⎜ ⎟ P X U Y U 1 ⎝ ⎠ 2 ⎝ ⎠ ⎩ ⎭ p p 12

  13. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 In order to connect the two we impose a second x > condition, on the tail of X : for 0 { } > P X tx − = 1 . γ lim x 1 { } > →∞ P X t t γ is a positive parameter. Where 1 This second condition implies a similar condition ← ⎛ ⎞ 1 ( ) = − for the quantile function ⎜ ⎟ ⎠ namely U t F 1 ⎝ 1 t ( ) U tx ( ) = x > γ . 1 lim x 0 1 ( ) →∞ U t t 1 13

  14. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 { } > is “regularly varying at We say that P X t ( ) − 1 γ ∈ infinity” with index and U is also RV . − γ 1 1 1 1 γ . regularly varying, with index 1 (usual condition is extreme value theory) These two conditions are the basic conditions of one-dimensional extreme value theory. 14

  15. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 Examples: Student distribution, Cauchy distribution. It is quite generally accepted that most financial data satisfy this condition. ( ) − − = + lower order 1 Sufficient condition: γ 1 F t ct 1 powers. Under these conditions we get the first result: 15

  16. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 ⎛ ⎞ ⎛ ⎞ 1 > ⎜ ⎟ E X Y ⎜ U ⎟ θ ( ) 2 ⎝ ⎠ ⎝ ⎠ ∞ p − = = 1 ∫ p γ lim lim R x ,1 dx 1 ⎛ ⎞ ⎛ ⎞ ↓ ↓ 1 1 p 0 p 0 0 U ⎜ ⎟ U ⎜ ⎟ 1 ⎝ ⎠ 1 ⎝ ⎠ p p θ goes to infinity as p ↓ at the same rate Hence 0 p ⎛ ⎞ 1 as ⎠ , the value-at-risk for X . U ⎜ ⎟ 1 ⎝ p Now we go to statistics and look at how to θ . estimate p 16

  17. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 We do that in stages: ( ) → ∞ , ( ) θ where = n → First we estimate k k n k n 0 k n as n → ∞ . θ non-parametrically (it is Clearly we can estimate k n just inside the sample). θ The second stage will be the extrapolation from k n θ with ≤ to . p 1 n p For the time being we suppose that X is a positive random variable. 17

  18. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 ⎛ ⎞ ⎛ ⎞ n θ = > Recall ⎠ ⎜ ⎟ ⎜ ⎟ E X Y U ⎝ ⎠ ⎝ k 2 k n ( ) First step : replace quantile n k by U 2 Y − ( k − th order corresponding sample quantile n k n , statistic from above). θ is then The obvious estimator of k n 1 n ∑ X 1 { } 1 > � i n Y Y ∑ n θ = − = { } i n k n , = . i 1 : X 1 k { } ⎛ ⎞ > i n Y Y n k − , i n k n = > i 1 ⎜ ⎟ P Y U ⎝ ⎠ 2 k 18

  19. Estimation of the marginal expected shortfall Laurens de Haan, Poitiers, 2012 First result : Under some strengthening of our conditions (relating to R and to the sequence ( ) k n ) ⎛ ⎞ � k θ d − → Θ k ⎜ ⎟ , 1 n ⎜ ⎟ θ ⎝ ⎠ k n a normal random variable that we describe now. 19

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