Link Prediction in Dynamic Financial Networks. José Luis Molina-Borboa Serafín Martínez-Jaramillo

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1 Link Prediction in Dynamic Financial Networks José Luis Molina-Borboa Serafín Martínez-Jaramillo

2 Outline Introduction Data Method and estimator Results Conclusions, improvements and further work Link Prediction in Dynamic Financial Networks 2 / 19

3 Introduction In stress testing and systemic risk, different frameworks have taken into account network effects either on the solvency side, liquidity or both. There is an urgent need to model bank behaviour in the context of networks. This would allow to discover the most likely network under specific circumstances. In Mexico, we don t have data gaps on the interaction between banks across many different markets. Therefore, we decided to use a statistical approach to take advantage of our data and test how can systemic risk measurement and stress tests benefit from it. Link prediction is useful for extracting missing information, identifying relationships and evaluating evolving mechanisms in networks (Lü & Zhou, 2010). It has been widely used in social and biological networks. More recently, the dynamical aspects of time-changing networks have been incorporated into this task (da Silva Bastos,2012; Tylenda et. al., 2009). Link Prediction in Dynamic Financial Networks 3 / 19

4 Objectives and contribution This work: First known approach to link prediction in financial networks using empirical data Different problem than the classic LP problem (missing data, social networks, etc.) Robust statistical estimation to capture banks reactions in the network given economic environment Uses a nonparametric link prediction technique for large scale dynamic networks, introduced by Sarkar et. al. (2012). The estimator is consistent and captures local neighbourhood evolution, as well as evolving characteristics of the involved nodes. Link Prediction in Dynamic Financial Networks 4 / 19

Data To test the predictor, we used daily bilateral exposures on 43 banks (during the first semester of 2013) arising from: Cross-holding of securities between banks (including repo transactions,

5 Data To test the predictor, we used daily bilateral exposures on 43 banks (during the first semester of 2013) arising from: Cross-holding of securities between banks (including repo transactions, securities lending and securities used as collateral). Interbank deposits and loans in local and foreign currency, credit lines extended for settlement purposes. Valuation of derivatives transactions (swaps, forwards and options), valuation of repo transactions and securities trading. FX related exposures. Link Prediction in Dynamic Financial Networks 5 / 19

6 The method Following Sarkar et. al. (2012), let G = {G 1,..., G t} be the observed sequence of exposure networks (directed) and let N t,p = {N t(i),..., N t p+1(i)} denote the last p neighbourhoods of i. Think of useful metrics for link prediction (e.g. (cn ij(t), ll i,j(t), d + i (t))) and define a cube S Z 3 + with all the possible combinations of the three metrics, with each s S representing a combination of the metrics values. For the pair (i, j) in G t, calculate their metrics and locate them in the cube S, call this cell s t(i, j). Now, define the set d t(i) = {η it(s), η + it (s) s S} where: η it(s) counts the number of pairs in N t 1(i) located in s S η + it (s) counts the number of such pairs connected at the next time step Link Prediction in Dynamic Financial Networks 6 / 19

7 The method Now, define the set d t(i) = {η it(s), η + it (s) s S} where: η it(s) counts the number of pairs in N t 1(i) located in s S η + it (s) counts the number of such pairs connected at the next time step Link Prediction in Dynamic Financial Networks 7 / 19

8 Overview of the method Figure : by Sarkar et. al. (2012) Link Prediction in Dynamic Financial Networks 8 / 19

9 The method Let Y t (i, j) = 1 {i j exists in Gt }, then Y t+1(i, j) G Bernoulli(g(ψ t(i, j))) ψ t(i, j) = {s t(i, j), d t(i)} g( ) [0, 1] depends on pair-specific characteristics s t(i, j) and the local neighbourhood characteristics d t(i). Since the elements of ψ are functions of N t,p(i), Y t+1(i, j) is then independent from G given N t,p(i). Link Prediction in Dynamic Financial Networks 9 / 19

10 The estimator The estimator for g( ) at time t is given by: i g(ψ t(i, j)) = j t Γ(ψ t(i, j), ψ t (i, j )) Y t +1(i, j ). i j t Γ(ψ t(i, j), ψ t (i, j )) Factorize Γ into pair and neighbourhood characteristics: Γ(ψ t(i, j), ψ t (i, j )) = K(d t(i), d t (i )) ξ(s t(i, j), s t (i, j )), where ξ(s t(i, j), s t (i, j )) = 1 {s t (i,j )=s t (i,j)} + ζ t1 {d1 (s,s )=1}. (1) 1 + ζ tn(s t(i, j)) Here, d 1 is the L 1 distance, n(s) is the set of cells at L 1 distance one from s and ζ T is a bandwidth parameter. K(d t(i), d t (i )) is a discrete analogous for a kernel function. Link Prediction in Dynamic Financial Networks 10 / 19

11 The estimator After some manipulation, we get g t(ψ t(i, j)) = K(d t(i), d t (i )) ξ(s t(i, j), s t (i, j )) Y t +1(i, j ) i j t, K(d t(i), d t (i )) ξ(s t(i, j), s t (i, j )) i j t and after some more: ( K(d t(i), d t (i )) η + i g t(ψ t(i, j)) = t i,t +1 (st(i, j)) + ζt ( K(d t(i), d t (i )) η i,t +1(s t(i, j)) + ζ t i t s n(s t (i,j)) s n(s t (i,j)) η + i,t +1 (s) ) η i,t +1(s) ). Link Prediction in Dynamic Financial Networks 11 / 19

12 The estimator In our case, we let ζ t = 0 to (finally) obtain K(d t(i), d t (i )) η + i g(ψ t(i, j)) = t i,t +1 (st(i, j)) K(d t(i), d t (i )) η i,t +1(s t(i, j)) i t Where K(d t(i), d t (i )) = e D(d t(i),d t (i ))/b T D(d t(i), d t (i )) = TotVar(X, Y ) s S X Beta ( η it(s), + η it(s) η it(s) + ) Y Beta ( η + i t (s), η i t (s) η+ i t (s)) Link Prediction in Dynamic Financial Networks 12 / 19

13 Results Link Prediction in Dynamic Financial Networks 13 / 19

14 Results Classification performance: Observed 1 0 Predicted ,516 Link Prediction in Dynamic Financial Networks 14 / 19

15 Results Metric Value Sensibility (TP rate) Specificity (1- FP rate) AUC ACC RMSE SAR Link Prediction in Dynamic Financial Networks 15 / 19

16 Contagion risk prediction Methodology similar to Batiz et. al. (2013) was used to assign weights to the predicted links, preserving the observed history of preference indices (Cocco et. al., 2009). Classic idiosincratic contagion risk was then calculated. July 2014 Final statistics Predicted Observed Initial losses (milliards of MXN) Final losses (milliards of MXN) Number of failed banks Banks with capital ratio < 4% Total losses (% of system s assets) 9.12% 10.78% Link Prediction in Dynamic Financial Networks 16 / 19

17 Strengths and weaknesses Strengths: Ability to accurately predict link appearance and detecting changes in the network based on local evolution. The absence of a link was easier to predict than the creation of a new link. Weaknesses: Underestimation of links corresponding to the most connected nodes (correctly classified, but underestimated probability). Conversely, overestimation of the probability of links from isolated nodes to central nodes. This is due to the enormous denominator for large banks (really big neighbourhoods). Link Prediction in Dynamic Financial Networks 17 / 19

18 Improvements and further work This model could be improved by: Designing the cube S so it reflects the economic status of a bank (capital ratio, LCR, NPL, etc.) and include a variable measuring the stress in the system. Assesing the multiplex structure of the exposures network and predict each layer individually and add up those predictions. Modify the kernel function to consider changes further away in time in order to incorporate predictions with macroeconomic models. Incorporate a priori knowledge on the behaviour of some banks (given their characteristics) and the observed persistence of some links (Molina et. al., Journal of Network Theory in Finance, in press). As further work, we would like to use gaussian graphical models, such as the Bayesian autologistic model, because of their ability to capture and probabilistically model the spatial relation between variables. They could also provide with a greater explanatory power. Link Prediction in Dynamic Financial Networks 18 / 19

19 Thank you Thank you very much. Link Prediction in Dynamic Financial Networks 19 / 19

Filling in the Blanks: Network Structure and Systemic Risk

Filling in the Blanks: Network Structure and Systemic Risk Kartik Anand, Bank of Canada Ben Craig, Federal Reserve Bank of Cleveland & Deutsche Bundesbank Goetz von Peter, Bank for International Settlements