Elicitability and Identifjability of Measures of Systemic Risk - PowerPoint PPT Presentation

Elicitability and Identifjability of Measures of Systemic Risk Tobias Fissler Imperial College London based on joint work with Jana Hlavinová and Birgit Rudlofg IMS–FIPS London 10 – 11 September 2018 T. Fissler (Imperial College London) Measures of Systemic Risk 13 April 2018 1 / 29

(Law-invariance) If X d Y then Briefjng: Risk Measures Monotonicity If X 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) Y . X Y . X Y X (Sub-additivity) Y . X Y a.s. then X . Let the random variable Y model the gains and losses of a fjnancial c position. money one has to add to Y in order to make it acceptable. That is Properties Let Y X be random variables. Cash-invariance For any m X m X m . . Homogeneity For any c cX 2 / 29 A risk measure ρ maps Y to the real value ρ ( Y ) P R which stands for the ρ ( Y + ρ ( Y )) = 0 .

(Law-invariance) If X d Y then Briefjng: Risk Measures X 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) Y . X Y . X Y X (Sub-additivity) Y . Y a.s. then Let the random variable Y model the gains and losses of a fjnancial Monotonicity If X X . c cX Homogeneity For any c Properties money one has to add to Y in order to make it acceptable. That is position. 2 / 29 A risk measure ρ maps Y to the real value ρ ( Y ) P R which stands for the ρ ( Y + ρ ( Y )) = 0 . Let Y , X be random variables. Cash-invariance For any m P R ρ ( X + m ) = ρ ( X ) ´ m . ⇝ ρ (0) = 0 .

(Law-invariance) If X d Y then Briefjng: Risk Measures Y . 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) Y . X Y . X Y X (Sub-additivity) X Let the random variable Y model the gains and losses of a fjnancial Y a.s. then Monotonicity If X Properties money one has to add to Y in order to make it acceptable. That is position. 2 / 29 A risk measure ρ maps Y to the real value ρ ( Y ) P R which stands for the ρ ( Y + ρ ( Y )) = 0 . Let Y , X be random variables. Cash-invariance For any m P R ρ ( X + m ) = ρ ( X ) ´ m . ⇝ ρ (0) = 0 . Homogeneity For any c ą 0 ρ ( cX ) = c ρ ( X ) .

(Law-invariance) If X d Y then Briefjng: Risk Measures X 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) Y . X Y . X Y (Sub-additivity) Properties Let the random variable Y model the gains and losses of a fjnancial money one has to add to Y in order to make it acceptable. That is position. 2 / 29 A risk measure ρ maps Y to the real value ρ ( Y ) P R which stands for the ρ ( Y + ρ ( Y )) = 0 . Let Y , X be random variables. Cash-invariance For any m P R ρ ( X + m ) = ρ ( X ) ´ m . ⇝ ρ (0) = 0 . Homogeneity For any c ą 0 ρ ( cX ) = c ρ ( X ) . Monotonicity If X ď Y a.s. then ρ ( X ) ě ρ ( Y ) .

(Law-invariance) If X d Y then Briefjng: Risk Measures Let the random variable Y model the gains and losses of a fjnancial 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) Y . X 2 / 29 Properties money one has to add to Y in order to make it acceptable. That is position. A risk measure ρ maps Y to the real value ρ ( Y ) P R which stands for the ρ ( Y + ρ ( Y )) = 0 . Let Y , X be random variables. Cash-invariance For any m P R ρ ( X + m ) = ρ ( X ) ´ m . ⇝ ρ (0) = 0 . Homogeneity For any c ą 0 ρ ( cX ) = c ρ ( X ) . Monotonicity If X ď Y a.s. then ρ ( X ) ě ρ ( Y ) . (Sub-additivity) ρ ( X + Y ) ď ρ ( X ) + ρ ( Y ) .

Briefjng: Risk Measures Let the random variable Y model the gains and losses of a fjnancial 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) 2 / 29 Properties money one has to add to Y in order to make it acceptable. That is position. A risk measure ρ maps Y to the real value ρ ( Y ) P R which stands for the ρ ( Y + ρ ( Y )) = 0 . Let Y , X be random variables. Cash-invariance For any m P R ρ ( X + m ) = ρ ( X ) ´ m . ⇝ ρ (0) = 0 . Homogeneity For any c ą 0 ρ ( cX ) = c ρ ( X ) . Monotonicity If X ď Y a.s. then ρ ( X ) ě ρ ( Y ) . (Sub-additivity) ρ ( X + Y ) ď ρ ( X ) + ρ ( Y ) . = Y then ρ ( X ) = ρ ( Y ) . (Law-invariance) If X d

E F Y Y Briefjng: Risk Measures (Examples) Y 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) F q Y d VaR ES Value-at-Risk ) F (close to 0). Then (if F q F and Let Y Expected Shortfall 3 / 29 Let Y „ F and α P (0 , 1) (close to 0). Then VaR α ( Y ) = ´ q ´ α ( F ) = ´ inf t x P R | F ( x ) ě α u .

Briefjng: Risk Measures (Examples) Value-at-Risk 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) 3 / 29 Expected Shortfall Let Y „ F and α P (0 , 1) (close to 0). Then VaR α ( Y ) = ´ q ´ α ( F ) = ´ inf t x P R | F ( x ) ě α u . Let Y „ F and α P (0 , 1) (close to 0). Then (if F ( q ´ α ( F )) = α ) ż α ES α ( Y ) = 1 = ´ E F [ Y | Y ď q ´ ( ) VaR β ( Y ) d β α ( F )] . α 0

Measures of Systemic Risk Aggregate the system with some monotone aggregation function 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) allocations and thus ignores transaction costs. Bail-out costs. This is insensitive with respect to capital Y . Measure the risk via n be set-valued). Suppose you have some fjnancial system consisting of n fjrms. The Use some kind of generalisation of quantiles to replace VaR (this will in the tails!) Caveat: Ignores the dependence structure! (Usually high correlation Y n Y Y Apply some scalar risk measure to each component: How to measure the risk of the entire system Y ? 4 / 29 system can be represented as a random vector Y = ( Y 1 , . . . , Y n ) .

Measures of Systemic Risk n 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) allocations and thus ignores transaction costs. Bail-out costs. This is insensitive with respect to capital Y . Measure the risk via Aggregate the system with some monotone aggregation function Suppose you have some fjnancial system consisting of n fjrms. The be set-valued). Use some kind of generalisation of quantiles to replace VaR (this will in the tails!) Caveat: Ignores the dependence structure! (Usually high correlation Apply some scalar risk measure to each component: How to measure the risk of the entire system Y ? 4 / 29 system can be represented as a random vector Y = ( Y 1 , . . . , Y n ) . ρ ( Y ) = ( ρ ( Y 1 ) , . . . , ρ ( Y n ))

Measures of Systemic Risk n 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) allocations and thus ignores transaction costs. Bail-out costs. This is insensitive with respect to capital Y . Measure the risk via Aggregate the system with some monotone aggregation function Suppose you have some fjnancial system consisting of n fjrms. The be set-valued). Use some kind of generalisation of quantiles to replace VaR (this will in the tails!) Caveat: Ignores the dependence structure! (Usually high correlation Apply some scalar risk measure to each component: How to measure the risk of the entire system Y ? 4 / 29 system can be represented as a random vector Y = ( Y 1 , . . . , Y n ) . ρ ( Y ) = ( ρ ( Y 1 ) , . . . , ρ ( Y n ))

Measures of Systemic Risk Suppose you have some fjnancial system consisting of n fjrms. The 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) allocations and thus ignores transaction costs. Aggregate the system with some monotone aggregation function be set-valued). Use some kind of generalisation of quantiles to replace VaR (this will in the tails!) Caveat: Ignores the dependence structure! (Usually high correlation Apply some scalar risk measure to each component: How to measure the risk of the entire system Y ? 4 / 29 system can be represented as a random vector Y = ( Y 1 , . . . , Y n ) . ρ ( Y ) = ( ρ ( Y 1 ) , . . . , ρ ( Y n )) Λ: R n Ñ R . Measure the risk via ρ (Λ( Y )) . ⇝ Bail-out costs. This is insensitive with respect to capital

i x i i x i Measures of Systemic Risk x 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) x i exp i n x v i v i i n x i Feinstein, Rudlofg, Weber (2017) i n x x i i n x n Examples for the aggregation Example 1 Take an ex ante point of view: How do we need to allocate additional 5 / 29 money k P R n in order to make the aggregate system Λ( Y + k ) acceptable under ρ ? R ( Y ) = t k P R n | ρ (Λ( Y + k )) ď 0 u .

Measures of Systemic Risk n 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) n n Feinstein, Rudlofg, Weber (2017) 5 / 29 n Example 1 Take an ex ante point of view: How do we need to allocate additional money k P R n in order to make the aggregate system Λ( Y + k ) acceptable under ρ ? R ( Y ) = t k P R n | ρ (Λ( Y + k )) ď 0 u . Examples for the aggregation Λ: R n Ñ R ÿ ÿ ´ x ´ Λ( x ) = x i , Λ( x ) = i , i =1 i =1 [ α i ( x i ´ v i ) + ´ β i ( x i ´ v i ) ´ ] , ÿ ÿ [1 ´ exp (2 x ´ Λ( x ) = Λ( x ) = i )] . i =1 i =1

, they are upper sets. So is continuous, then R Y is closed. convex, then the set R Y is convex. Measures of Systemic Risk 13 April 2018 Measures of Systemic Risk T. Fissler (Imperial College London) is concave and If If n R Y R Y Due to the monotonicity of 6 / 29 Properties I of R ( Y ) = t k P R n | ρ (Λ( Y + k )) ď 0 u The values of R are subsets of R n .