Risk-Consistent Conditional Systemic Risk Measures

Transcription

1 Risk-Consistent Conditional Systemic Risk Measures Thilo Meyer-Brandis University of Munich joint with: H. Hoffmann and G. Svindland, University of Munich Workshop on Systemic Risk and Financial Networks IPAM, UCLA March 24, 2015

2 Motivation Let X = (X 1,..., X d ) L d (F) represent (monetary) risk factors associated to a system of d interacting financial institutions.

3 Motivation Let X = (X 1,..., X d ) L d (F) represent (monetary) risk factors associated to a system of d interacting financial institutions. Traditional approach to risk management: Measuring stand-alone risk of each institution η (X i ) for some univariate risk measure η : L (F) R.

4 Motivation Axiomatic characterization of (univariate) risk measures: A map η : L (F) R is a monetary risk measure, if it is Monotone: X 1 X 2 η(x 1 ) η(x 2 ). Cash-invariant:] η(x + m) = η(x ) m for all m R. (Constant-on-constants: η(m) = m for all m R.) A monetary risk measure is called convex, if it is Convex: η(λx 1 + (1 λ)x 2 ) λη(x 1 ) + (1 λ)η(x 2 ) for λ [0, 1]. (Quasi-convex: η(λx 1 + (1 λ)x 2 ) max{η(x 1 ), η(x 2 )}.) A convex risk measure is called coherent, if it is Pos. homogeneity: η(λx ) = λη(x ) for λ 0.

5 Motivation However, Financial crisis: traditional approach to regulation and risk management insufficiently captures Systemic risk: risk that in case of an adverse (local) shock substantial parts of the system default. Question: Given the system X = (X 1,..., X d ), what is an appropriate risk measure of systemic risk? ρ : L d (F) R

6 Motivation Most proposals of systemic risk measures in the post-crisis literature are of the form ρ(x ) := η (Λ(X )) for some univariate risk measure η : L (F) R and some aggregation function Λ : R d R.

7 Motivation Some examples: Appropriate aggregation to reflect systemic risk? Systemic Expected Shortfall [Acharya et al., 2011] ( d ) ρ(x ) := ES q X i i=1 where ES q is the univariate Expected Shortfall at level q [0, 1].

8 Motivation Deposit Insurance [Lehar, 2005], [Huang, Zhou & Zhu, 2011] ρ(x ) := E ( d i=1 X i ) SystRisk [Brunnermeier & Cheridito, 2013] ρ(x ) := η SystRisk ( d i=1 α i X i + β i (X + i v i ) ) where η SystRisk is some utility-based univariate risk measure.

9 Motivation Contagion model Π ji denotes the proportion of the total liabilities L j of bank j which it owes to bank i. X i represents the capital endowment of bank i. If a bank i defaults, i.e. X i < 0, this generates further losses in the system by contagion. Systemic risk concerns the total loss in the network generated by a profit/loss profile X = (X 1,..., X d ).

10 Motivation [Chen, Iyengar & Moallemi, 2013], [Eisenberg, Noe, 2001] Total loss in the system induced by some initial loss profile x R d : Λ(x) := min y,b R d d y i γb i i=1 subject to y i = x i b i + d Π ji y j Bank i decreases its liabilities to the remaining banks by y i. This decreases the equity values of its creditors, which can result in a further default. j=1 A regulator injects b i (weighted by γ > 1) into bank i. i

11 Motivation Various extensions of the Eisenberg & Noe framework to include further channels of contagion, see e.g. [Amini, Filipovic & Minca, 2013] [Awiszus & Weber, 2015] [Cifuentes, Ferrucci & Shin, 2005] [Gai & Kapadia, 2010] [Rogers & Veraart, 2013]

12 Motivation Conditional risk measuring interesting in systemic risk identification of systemic relevant structures CoVar and CoES [Adrian, Brunnermeier, 2011] ρ(x ) := VaR q ( d i=1 X i X j VaR q (X j ) ) [Acharya et al., 2011] ES q (X j ) d d X i VaR q ( X i ) i=1 i=1

13 Motivation conditional risk measure η : L (F) L (G) for some sub-σ-algebra G F Broad literature on conditional risk measures in a dynamic setup (e.g. [Detlefsen & Scandolo, 2005], [Föllmer & Schied, 2011], [Frittelli & Maggis, 2011], [Tutsch, 2007]). In the context of systemic risk, see also [Föllmer, 2014] and [Föllmer & Klüppelberg, 2014]:

14 Motivation But also conditional aggregation Λ(x, ω) appears naturally; f.ex.: The way of aggregation might depend on macroeconomic factors: Countercyclical regulation: more severe when economy is in good shape than in times of financial distress Stochastic discounting: Λ(x, ω) = Λ(x)D(ω), where D is some stochastic discount factor. Liability matrix Π = Π(ω) in contagion model above might be stochastic (derivative exposures between banks)

15 Motivation Objective of this presentation: Structural analysis of conditional systemic risk measures ρ G : L d (F) L (G), G F, of the form ρ G (X ) := η G (Λ G (X )) where η G is some univariate conditional risk measure and Λ G some conditional aggregation function.

16 Motivation Structure of remaining presentation: 1. Axiomatic characterization of conditional systemic risk measures of this type 2. Examples and simulation study 3. Strong consistency and risk-consistency

17 . AXIOMATIC CHARACTERIZATION OF RISK-CONSISTENT CONDITIONAL SYSTEMIC RISK MEASURES

18 Aim: Axiomatic characterization of conditional systemic risk measures of the form η (Λ(X )) in terms of properties on constants and risk-consistent properties. Risk-consistent properties ensure a consistency between local - that is ω-wise - risk assessment and the measured global risk. For deterministic risk measures [Chen, Iyengar & Moallemi, 2013], f. ex.: Risk-monotonicity: if for given risk vectors X and Y we have ρ(x (ω)) ρ(y (ω)) in a.a. states ω, then ρ(x ) ρ(y ).

19 In the deterministic case: [Chen, Iyengar & Moallemi, 2013] on finite probability space [Kromer, Overbeck & Zilch, 2013] general probability space Our contribution: Conditional setting More comprehensive structural analysis More flexible aggregation and axiomatic setting

20 Notation: Given (Ω, F, P) and a sub-σ-algebra G F: A realization of a function ρ G : L d (F) L (G) is a function ρ G (, ) : L d (F) Ω R such that ρ G (X, ) ρ G (X ) for all X L d (F). A realization ρ G (, ) has continuous path if ρ G (, ω) : R d R is continuous for all ω Ω. We denote the constant random vector X ω, ω Ω, by X ω (ω) := X ( ω) ω Ω.

21 Definition: A function ρ G : L d (F) L (G) is called risk-consistent conditional systemic risk measure (CSRM), if it is Monotone on constants: x, y R d with x y ρ G (x) ρ G (y) and if there exists a realization ρ G (, ) with continuous paths such that ρ G is Risk-monotone: For all X, Y L d (F) ρ G (X ω, ω) ρ G (Y ω, ω) a.s. ρ G (X, ω) ρ G (Y, ω) a.s. (Risk-) regular: ρ G (X, ω) = ρ G (X ω, ω) a.s. X L d (G)

22 Further risk-consistent properties: We say that ρ G is Risk-convex: If for X, Y, Z L d (F), α L (G) with 0 α 1 ρ G (Z ω, ω) = α(ω)ρ G (X ω, ω)+ ( 1 α(ω) ) ρ G (Y ω, ω) a.s. ρ G (Z, ω) α(ω)ρ G (X, ω)+(1 α(ω))ρ G (Y, ω) a.s. Risk-quasiconvex: If for X, Y, Z L d (F), α L (G), 0 α 1 ρ G (Z ω, ω) = α(ω)ρ G (X ω, ω)+ ( 1 α(ω) ) ρ G (Y ω, ω) a.s. ρ G (Z, ω) ρ G (X, ω) ρ G (Y, ω) a.s. Risk-pos.-homogeneous: If for X, Y L d (F), 0 α L (G) ρ G (Y ω, ω) = α(ω)ρ G (X ω, ω) a.s. ρ G (Y, ω) α(ω)ρ G (X, ω) a.s.

23 Further properties on constants: We say that ρ G is Convex on constants: If for all x, y R d and λ [0, 1] ρ G (λx + (1 λ)y) λρ G (x) + (1 λ)ρ G (y) Positively homogeneous on constants: If for all x R d and λ 0 ρ G (λx) = λρ G (x)

24 Proposition: Let ρ G : L d (F) L (G) be a function which has a realization with continuous paths. Further suppose that ρ G (x) = s a i (x)i Ai, x R d, i=1 where a i (x) R and A i are pairwise disjoint sets such that Ω = s i=1 A i for s N { }. Define k : Ω N; ω i such that ω A i. Then ρ G is risk-monotone if and only if ρ G (X ω )I Ak(ω) ρ G (Y ω )I Ak(ω) for a.a. ω ρ G (X ) ρ G (Y ). Also the remaining risk-consistent properties can be expressed in a similar way without requiring a particular realization of ρ G.

25 Proposition: The following holds for a risk-consistent CSRM ρ G : If ρ G is risk-monotone and monotone on constants, then ρ G is monotone. If ρ G is risk-quasiconvex and convex on constants, then ρ G is quasiconvex. If ρ G is risk-convex and convex on constants, then ρ G is convex; ρ G is risk positively homogeneous and positively homogeneous on constants iff ρ G is positively homogeneous;

26 Definition: A function Λ G : R d Ω R is a conditional aggregation function (CAF), if 1. Λ G (x, ) L (G) for all x R d. 2. Λ G (, ω) is continuous for all ω Ω. 3. Λ G (, ω) is monotone (increasing) for all ω Ω. Furthermore, Λ G is called concave (positively homogeneous) if Λ G (, ω) is concave (positively homogeneous) for all ω Ω.

27 Given a CAF Λ G, we extend the aggregation from deterministic to random vectors in the following way: Λ G : L d (F) L (F) X (ω) Λ G (X (ω), ω) Notice that the aggregation of random vectors is ω-wise in the sense that given a certain state ω Ω, in that state we aggregate the sure payoff X (ω).

28 Definition: Let F, G L (F). A function η G : L (F) L (G) is a conditional base risk measure (CBRM), if it is Monotone: F G η G (F ) η G (G). Constant on G-constants: η G (α) = α α L (G).

29 Additional properties of CBRMs: Convexity: For all α L (G) with 0 α 1 η G (αf + (1 α)g) αη G (G) + (1 α)η G (G) Quasiconvexity: For all α L (G) with 0 α 1 η G (αf + (1 α)g) η G (F ) η G (G) Positive homogeneity: For all α L (G) with α 0 η G (αf ) = αη G (F )

30 Theorem: (Decomposition of conditional systemic risk measures) A map ρ G : L d (F) L (G) is a risk-consistent CSRM if and only if there exists a CBRM η G : L (F) L (G) and a CAF Λ G such that ) ρ G (X ) = η G ( ΛG (X ) X L d (F), where Λ G (X ) := Λ G (X (ω), ω).

31 Theorem cont : Furthermore, the following equivalences hold: and ρ G is risk-convex iff η G is convex; ρ G is risk-quasiconvex iff η G is quasiconvex; ρ G is risk-positive homogeneous iff η G is positive homogeneous; ρ G is convex on constants iff Λ G is concave; ρ G is positive homogeneous on constants iff Λ G is positive homogeneous.

32 . EXAMPLES AND SIMULATION STUDY

33 Examples CoVar (analogue CoES) [Adrian & Brunnermeier, 2011]: Let { ( η G (F ) := VaR λ (F G) := essinf P F G G ) > λ }, G L (G) where F L (F), λ L (G), and 0 < λ < 1. For a fixed j {1,..., d} and q (0, 1), define the event A := {X j VaR q (X j )} and let G := σ(a). Define Λ(x) := x i, x R. Then the CoVar is represented by η G (Λ(X )), which is a positively homogeneous CSRM.

34 Examples Example: Extended contagion model Π ji denotes the proportion of the total interbank liabilities L j of bank j which it owes to bank i. X i represents the capital endowment of bank i. If a bank i defaults, i.e. X i < 0, this generates further losses in the system by contagion. Systemic risk concerns the total loss in the network generated by a profit/loss profile X = (X 1,..., X d ).

35 Examples More realistic contagion aggregation: Total aggregated loss in the system for loss profile x R d : subject to Λ(x) := min y,b R d d ( ( ) ) x i + b i + Π y + γbi i d x i + b i + Π ji y j, L i i i=1 y i = max and y 0, b 0. Bank i decreases its liabilities to the remaining banks by y i. This decreases the equity values of its creditors, which can result in a further default. j=1 A regulator injects b i (weighted by γ > 1) into bank i.

36 Examples Conditional contagion aggregation: Let the proportional liability matrix Π ji (ω) L (G) be stochastic. Then the corresponding CAF becomes d ( ( ) ) Λ(x, ω) := min x i + b i + Π (ω)y + γbi y,b R d i i=1 d subject to y i = max y i + b i x i + Π ji (ω)y j, L i i and y 0, b 0. j=1

37 Examples Numerical case study d = 10 financial institutions; One realization of an Erdös-Rényi graph with success probability p = 0.35 and with half-normal distributed weights/liabilities; Initial capital endowment proportional to the total interbank assets; 0.5 correlated normally distributed shocks on this initial capital;

38 31 (0.10) Examples FI (0.90) 61 (0.41) 6 (0.10) FI FI FI 6 45 (0.26) 75 (0.44) FI (0.35) FI (0.19) 78 (0.53) 45 (0.26) 6 (0.04) 125 (0.39) 25 (0.10) 34 (1.00) 69 (0.27) 59 (0.23) 8 (0.05) 82 (0.32) FI (0.05) 60 (0.24) 90 (0.35) 13 (0.05) 67 (0.30) 51 (0.20) 118 (0.52) 49 (0.19) FI (1.00) FI (1.00) FI 5 52 (0.16)

39 Examples Statistics for Λ for shock scenarios: γ Mean % Quantile Standard Deviation bi Initially defaulted banks 2.78 Defaulted banks without regulator 3.83 Defaulted banks with regulator

40 Examples Systemic ranking by CoVar:

41 . STRONG CONSISTENCY AND RISK-CONSISTENCY

42 Strong consistency and risk-consistency We now consider CSRMs ρ G : L d (F) L (G) fulfilling: Monotonicity: X Y = ρ G (X ) ρ G (Y ) Strong Sensitivity: X Y and P(X > Y ) > 0 = P ( ρ G (X ) < ρ G (Y ) ) > 0 Locality: ρ G (X I A + Y I A C ) = ρ G (X )I A + ρ G (Y )I A C A G Lebesgue property: For any uniformly bounded sequence (X n ) n N in L d (F) such that X n X P-a.s. ρ G (X ) = lim n ρ G(X n ) P-a.s.

43 Strong consistency and risk-consistency On (Ω, F, P) let E be a family of sub-σ-algebras of F such that {, Ω}, F E. Definition: A family ( ρ G )G E measures (CSRM) of conditional systemic risk ρ G : L d (F) L (G) is (strongly) consistent if for all G, H E with G H and X, Y L d (F) ρ H (X ) ρ H (Y ) a.s. = ρ G (X ) ρ G (Y ) a.s..

44 Strong consistency and risk-consistency Remark: In case ρ G and ρ H are univariate monetary risk measures that are constant-on-constants, strong consistency is equivalent to ρ G (X ) = ρ G ( ρh (X ) ).

45 Strong consistency and risk-consistency Definition: Given a CSRM ρ G we define f ρg : L (G) L (G); α ρ G (α1 d ) and its corresponding inverse function (which is well-defined) f 1 ρ G : Im f ρg L (G).

46 Strong consistency and risk-consistency Lemma: f ρg and fρ 1 G are antitone, strongly sensitive, local, and fulfill the Lebesgue property. Further ρ G (L d (F)) = f ρ G (L (G)).

47 Strong consistency and risk-consistency Lemma: The following statements are equivalent 1. (ρ G ) G E is strongly consistent; 2. ρ G (X ) = ρ G (f ρ 1 ( ρh H (X ) ) ) 1 d X L d (F), G H

48 Strong consistency and risk-consistency Lemma: The following statements are equivalent 1. (ρ G ) G E is strongly consistent; 2. ρ G (X ) = ρ G (f ρ 1 ( ρh H (X ) ) ) 1 d X L d (F), G H Remark: In case ρ G and ρ H are univariate risk measures that are constant-on-constants, fρ 1 H = id and 2. reduces to ( 2. ρ G (X ) = ρ G ρh (X ) )

49 Strong consistency and risk-consistency Remark: Let (ρ G ) G E be a family of CRMs and define the normalized family ( ρ G ) G E := ( f 1 ρ G ρ G ) G E. Then ( ρ G ) G E is a family of CRMs which is consistent iff (ρ G ) G E is consistent. Further, f ρg = id which can be considered as a vector generalization of the constant-on-constants property for univariate risk measures.

50 Strong consistency and risk-consistency Theorem: Let (ρ G ) G E be strongly consistent. Assume there exists a continuous realizations ρ G (, ) for all G E and that f 1 ρ F ρ F (x) R x R d. Then for all G E the risk measure ρ G is risk-consistent, i.e. there exists an CAF Λ G and a CBRM η G such that ρ G = η G Λ G.

51 Strong consistency and risk-consistency Theorem cont : Further, the CAF are strongly consistent in the sense that for all G H Λ H (X ) Λ H (Y ) = Λ G (X ) Λ G (Y ) (X, Y L d (F)), which is equivalent to f 1 Λ G ( ΛG (X ) ) = f 1 Λ F ( ΛF (X ) ), for all X L d (F).

52 Strong consistency and risk-consistency Law-invariant, consistent systemic risk measures: Definition: A CSRM is called conditional law-invariant if µ X ( G) = µ Y ( G) = ρ G (X ) = ρ G (Y ), where µ X ( G) and µ Y ( G) are the G-conditional distributions of X, Y L d (F) resp. Assumption: There exists H E such that (Ω, H, P) is atomless, and (Ω, F, P) is conditionally atomless given H.

53 Strong consistency and risk-consistency Theorem [Föllmer, 2014]: Let (ρ G ) G E be a family of univariate, conditionally law-invariant, monetary risk measures. Then (ρ G ) G E is strongly consistent iff the ρ G are certainty equivalents of the form ρ G (F ) = u 1( E P (u(f ) G) ), F L, (F) where u : R R is strictly increasing and continuous.

54 Strong consistency and risk-consistency Theorem: Let (ρ G ) G E be a family of conditionally law-invariant CSRMs. Then (ρ G ) G E is strongly consistent iff each ρ G is of the form ( ( ρ G (X ) = g G f 1 u EP (u(x ) G) )), X L d (F), where u : R d R is strictly increasing and continuous : Im f u R is the unique inverse function of f u : R R; x u(x1 d ) f 1 u g G : L (G) L (G) is antitone, strongly sensitive, local, and fulfills the Lebesgue property In particular, g G = f ρg for all G E.

55 Strong consistency and risk-consistency Corollary: Let (ρ G ) G E be a family of conditionally law-invariant, strongly consistent CSRMs. Under the assumptions from above, the risk-consistent decomposition ρ G = η G Λ G is given by a stochastic certainty equivalent η G of the form η G (F ) = U 1 G (E P (U G (F ) G)), F L (F), where U G := f u g 1 G : L (F) L (F) and g G is a certain extension of g G from L (G) to L (F), and the aggregation Λ G := g G f 1 u u. Further, the CBRM η G and the aggregation Λ G are related to each other by U G (Λ G (x)) = u(x) x R d, G E.

56 Strong consistency and risk-consistency Corollary: Let CBRMs (η G ) G E and CAFs (Λ G ) G E be given such that (η G ) G E are strongly consistent and (ρ G := η G Λ G ) G E are conditionally law-invariant, strongly consistent CSRMs. Then the CBRMs (η G ) G E must be certainty equivalents of the form η G (F ) := u 1( E P (u(f ) G) ), F L (F), where u : R R is strictly increasing and continuous, and the CAFs (Λ G ) G E must be of the form Λ G = f (α G Λ(x) + β G ), for some deterministic AF Λ, G-constants α G, β G L (G), and some strictly increasing and continuous f : R R.

57 References V. V. Acharya, L. H. Pedersen, T. Philippon, and M. Richardson. Measuring systemic risk. Working paper, March Adrian, T. and M. K. Brunnermeier (2011). Covar. Technical report, National Bureau of Economic Research. Amini, H., D. Filipovic & A. Minca (2013), Systemic risk with central counterparty clearing, Swiss Finance Institute Research Paper No , Swiss Finance Institute Awiszus, K. & Weber, S. (2015): The Joint Impact of Bankruptcy Costs, Cross-Holdings and Fire Sales on Systemic Risk in Financial Networks, preprint Artzner, Delbaen, Eber, Heath: Coherent measures of risk, Math. Fin., 1999 Brunnermeier, M. K. and P. Cheridito (2013). Measuring and allocating systemic risk. Available at SSRN

58 References Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., and Montrucchio, L., Riskmeasures: rationality and diversication, Mathematical Finance, 21, (2011) Chen, C., G. Iyengar, and C. C. Moallemi (2013). An axiomatic approach to systemic risk. Management Science 59(6), Cifuentes, R., G. Ferrucci & H. S. Shin (2005), Liquidity risk and contagion, Journal of the European Economic Association 3 (2-3), ] Detlefsen, K. and G. Scandolo (2005). Conditional and dynamic convex risk measures. Finance and Stochastics 9(4), Föllmer, H., Spatial Risk Measures and their Local Specication: The Locally Law-Invariant Case, Statistics & Risk Modeling 31(1), 79103, Föllmer, H. & A. Schied (2011). Stochastic Finance: An introduction in discrete time (3rd ed.). De Gruyter.

59 References Föllmer, H., Klüppelberg, C., Spatial risk measures: local specification and boundary risk. In: Crisan, D., Hambly, B. and Zariphopoulou, T.: Stochastic Analysis and Applications In Honour of Terry Lyons. Springer, 2014, Frittelli, M. & M. Maggis (2011). Conditional certainty equivalent. International Journal of Theoretical and Applied Finance 14 (01), 4159 Gai, P. & Kapadia, S. (2010), Contagion in financial networks, Bank of England Working Papers 383, Bank of England X. Huang, H. Zhou, and H. Zhu. A framework for assessing the systemic risk of major financial institutions. Journal of Banking & Finance, 33(11): , Kromer E., Overbeck, L., Zilch, K.. Systemic risk measures on general probability spaces, preprint, 2013.

60 References A. Lehar. Measuring systemic risk: A risk management approach. Journal of Banking & Finance, 29(10): , 2005 Rogers, L. C. G. & L. A. M. Veraart (2013), Failure and rescue in an interbank network, Management Science 59(4), Tutsch, S. (2007). Konsistente und konsequente dynamische Risikomae und das Problem der Aktualisierung. Ph. D. thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II.