A Bayesian methodology for systemic risk assessment in financial networks
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1 A Bayesan methodology for systemc rsk assessment n fnancal networks Lutgard A. M. Veraart London School of Economcs and Poltcal Scence September 2015 Jont work wth Axel Gandy (Imperal College London) 7th General AMaMeF and Swssquote Conference, EPFL Preprnt avalable at SSRN: Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
2 The problem Consder nterbank market as network: Nodes consst of n banks wth ndces n N = {1,..., n}. Edges L j represent nomnal nterbank lablty of bank to bank j. Stress tests: Suppose some banks default on ther labltes. How do losses spread along the edges? What f edges are not observable? A matrx L = (L j ) R n n s a labltes matrx f L j 0, L = 0, j Total nomnal nterbank labltes of bank : r (L) := m j=1 L j. Total nomnal nterbank assets of bank : c (L) := m j=1 L j. In practce, L j not fully observable, but r (L), c (L) are. How to fll n the mssng data? Implcatons for stress testng? Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
3 Man contrbutons Development of Bayesan framework (Gbbs sampler) for samplng from dstrbuton of labltes matrx condtonal on ts row and column sums. Applcaton to systemc rsk assessment: Can gve probabltes for outcomes of stress tests. Results show lmtatons of classcal approach. Show that general monotoncty arguments relatng severty of systemc rsk to number of edges do not hold n general. Code s avalable as R-package (systemcrsk) on CRAN. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
4 Admssble labltes matrx Exstence of admssble labltes matrx Theorem (Exstence of an admssble labltes matrx) Consder two vectors a, l [0, ) n satsfyng n =1 a = n =1 l. Then there exsts a matrx L [0, ) n n wth dag(l) = 0, c(l) = a, r(l) = l f and only f a n l j N. j=1 j Proof contans algorthm gvng explct constructon. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
5 The Bayesan framework The Bayesan framework Constructs adjacency matrx A = (A j ); attaches labltes L j. Model: Parameters: For, j N : A j Bernoull(p j ) L j {A j = 1} Exponental(λ j ) L j = 0 f A j = 0. p [0, 1] n n, dag(p) = 0; p j probablty of exstence of drected edge from to j, λ R n n, governs dstrbuton of weghts gven that edge exsts. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
6 The Bayesan framework The Bayesan framework Constructs adjacency matrx A = (A j ); attaches labltes L j. Model: Parameters: For, j N : A j Bernoull(p j ) L j {A j = 1} Exponental(λ j ) L j = 0 f A j = 0. p [0, 1] n n, dag(p) = 0; p j probablty of exstence of drected edge from to j, λ R n n, governs dstrbuton of weghts gven that edge exsts. Observatons: c(l) = a R n, r(l) = l R n. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
7 The Bayesan framework The Bayesan framework Constructs adjacency matrx A = (A j ); attaches labltes L j. Model: Parameters: For, j N : A j Bernoull(p j ) L j {A j = 1} Exponental(λ j ) L j = 0 f A j = 0. p [0, 1] n n, dag(p) = 0; p j probablty of exstence of drected edge from to j, λ R n n, governs dstrbuton of weghts gven that edge exsts. Observatons: c(l) = a R n, r(l) = l R n. Man nterest: Dstrbuton of h(l) a, l. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
8 The Bayesan framework Gbbs samplng for L a, l Markov Chan Monte Carlo (MCMC): Interested n samplng from a gven dstrbuton. Construct a Markov chan wth ths statonary dstrbuton. Run chan. Chan converges to statonary dstrbuton. Key dea of Gbbs sampler: a step of the chan updates one or several components of the entre parameter vector by samplng them from ther jont condtonal dstrbuton gven the remander of the parameter vector. Here parameter vector s matrx L: Intalse chan wth matrx L that satsfes r(l) = l, c(l) = a. MCMC sampler produce a sequence of matrces L 1, L 2,.... Quantty of nterest: E[h(L) l, a] 1 N N =1 h(lδ+b ), N number of samples, b burn-n perod, δ N thnnng parameter. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
9 The Bayesan framework Updatng components of L Need to decde whch elements of L need to be updated. Need to determne how the new values wll be chosen,.e., need to determne ther dstrbuton condtonal on remander of elements of L. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
10 The Bayesan framework Illustraton of updatng submatrces L 1 j 1 L 1 j 2 L 1 j 1 L 1 j 2 L 2 j 2 L 2 j 3 L 1 j 1 L 1 j 2 L 2 j 3 L 2 j 2 L 3 j 4 L 3 j 3 L 2 j 1 L 2 j 2 L 3 j 3 L 3 j 1 L 4 j 4 L 4 j 1 Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
11 The Bayesan framework Updatng - Illustraton Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
12 Applcatons to systemc rsk assessment Stress testng Balance sheets and fundamental defaults Balance sheet of bank : Assets Labltes external assets a (e) external labltes l (e) nterbank assets a := c (L) nterbank labltes l := r (L) net worth w := w (L, a (e), l (e) ) w := w (L, a (e), l (e) ) := a (e) + c (L) l (e) r (L). Stress tests: apply proportonal shock s [0, 1] n to external assets; shocked external assets are s a (e). Fundamental defaults: { w (L, s a (e), l (e) ) < 0} Fundamental defaults can be checked from balance sheet aggregates wthout needng to know the whole matrx L! To check for contagous defaults we need to know L. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
13 Applcatons to systemc rsk assessment Emprcal example Emprcal example - data Balance sheet data (n mllon Euros) from banks n the EBA 2011 stress test: Bank code Bank a (e) + a a w DE017 DEUTSCHE BANK AG 1,905,630 47,102 30,361 DE018 COMMERZBANK AG 771,201 49,871 26,728 DE019 LANDESBANK BADEN-WURTTEMBERG 374,413 91,201 9,838 DE020 DZ BANK AG 323, ,099 7,299 DE021 BAYERISCHE LANDESBANK 316,354 66,535 11,501 DE022 NORDDEUTSCHE LANDESBANK -GZ- 228,586 54,921 3,974 DE023 HYPO REAL ESTATE HOLDING AG 328,119 7,956 5,539 DE024 WESTLB AG, DUSSELDORF 191,523 24,007 4,218 DE025 HSH NORDBANK AG, HAMBURG 150,930 4,645 4,434 DE027 LANDESBANK BERLIN AG 133,861 27,707 5,162 DE028 DEKABANK DEUTSCHE GIROZENTRALE 130,304 30,937 3,359 Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
14 Applcatons to systemc rsk assessment Emprcal example Stress testng We apply a determnstc shock to external assets of all 11 banks n the network by consderng the shocked external assets s a (e) wth s = 0.97 N. Shock causes fundamental default of 4 banks: DE017, DE022, DE023, DE024. We apply the clearng approach by Esenberg & Noe (2001) and [Rogers & V. (2013)] to determne whch banks suffer contagous defaults. Gbbs sampler allows us to derve posteror default probabltes for remanng 7 banks. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
15 Applcatons to systemc rsk assessment Emprcal example Default probabltes of banks as a functon of p DE DE020 DE p Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
16 Applcatons to systemc rsk assessment Emprcal example Default probabltes for clearng wth default costs DE020 DE019 DE025 DE028 DE021 DE027 DE p Lutgard A. M. Veraart (LSE) (a) AClearng Bayesan approach wth α to= systemc 1, β = rsk0.7 September / 17
17 Applcatons to systemc rsk assessment Emprcal example Mean out-degree of banks,.e., E[ j A j a, l], for dfferent p ER n the Erdős-Rény network l DE DE DE DE DE DE DE DE DE DE DE Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
18 Summary Summary Development of Bayesan framework (Gbbs sampler) for samplng from dstrbuton of labltes matrx condtonal on ts row and column sums. Can be used for stress tests usng emprcal data. Can be used as a smulaton tool to analyse heterogeneous networks. Can ncorporate addtonal nformaton such as expert vews etc. on the network structure. R package (systemcrsk) avalable from CRAN. Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
19 Summary References I Esenberg, L. & Noe, T. H. (2001). Systemc rsk n fnancal systems. Management Scence 47, Gandy, A. & Veraart, L. A. M. (2015). A Bayesan methodology for systemc rsk assessment n fnancal networks. Avalable at SSRN: Rogers, L. C. G. & Veraart, L. A. M. (2013). Falure and rescue n an nterbank network. Management Scence 59, Lutgard A. M. Veraart (LSE) A Bayesan approach to systemc rsk September / 17
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