Credit and Systemic Risks in the Financial Services Sector

Transcription

1 Credit and Systemic Risks in the Financial Services Sector Measurement and Control of Systemic Risk Workshop Montréal Jean-François Bégin (Stat & Actuarial Sciences, Simon Fraser) Mathieu Boudreault ( Mathematics and Statistics, UQAM) Delia Doljanu (M.Sc. Financial Engineering, HEC Montreal) Geneviève Gauthier (Decision Sciences, HEC Montreal) September, 2017

2 In this presentation... I 1 The rst objective is to construct and estimate a rm-speci c credit risk model. 1 Includes default risk, recovery risk and regime switches. 2 multivariate version: a measure of systemic risk. 2 According to the existing literature (Elton et al. 2001, Collin-Dufresne et al. 2001, Lando & Skødeberg (2002), Driessen (2005), Chen et al. (2009), Dionne et al. (2010), and Huang & Huang (2012) among others), conclusions are sensitive to 1 the model s characteristics, 2 the data, and 3 the estimation procedure.

3 In this presentation... II 3 The model is constructed carefully to replicate the desired empirical facts. 1 A regime-switching framework captures the behavior changes during the crisis. 2 Negative relationship between recovery rates and default probabilities to reproduce empirical facts. 4 Pricing is performed numerically (no closed-form solution). 5 The estimation 1 ltering approach based on CDS data. 2 the lter must be tailored for the two latent variables (the leverage ratio and the regime) 6 Empirical studies

4 The model CDS premium Evolution of the mean 5-year CDS premiums (bps) (across rms) and of the di erence between mean 10-year CDS premiums and 1-year CDS premiums (across rms) for both CDX.NA.IG.21.V1 and CDS.NA.HY.21.V1 portfolios, between January 2005 and December The gray surface corresponds to the nancial crisis (July 2007 to March 2009)

5 Leverage Structural framework I fa t : t 0g is the value of the rm s assets. fl t : t 0g is the liabilities process: it represents company s obligations toward creditors. The unobserved leverage ratio X t = L t A t. (1) The unobserved regime h t prevailing on (t 1, t] is modeled as a time-homogenous Markov chain with transition probabilities p ij = P(h t = jjh t 1 = i), i, j 2 f1, 2g. (2)

6 Leverage Structural framework II The latent regime s dynamics is 8 < 11 U p U >p11 if h t = 1 h t+1 = : 11 U p U >p21 if h t = 2 (3) where U is a uniform random variable and the log-leverage ratio s process satis es 8 >< ln X t + µ P σ 2 p t + σ 1 t ε P t+1 if h t+1 = 1 ln X t+1 = (4) >: ln X t + µ P σ 2 p t + σ 2 t ε P t+1 if h t+1 = 2 where ε P t iid N(0,1). t=1

7 Default intensity The default intensity I The survival probability is " P t [τ > T jτ > t ] = E t!# T 1 exp t H s s=t (5) where the default intensity process satis es α Xs H s = β + (6) θ α > 0 ) default intensity increases with leverage. α > 1 ) default intensity is a convex function of leverage. θ is the default threshold: the default intensity is large when leverage is greater than θ and small otherwise. β = 0 and α! : the basic structural model underlying the leverage ratio process X.

8 Recovery rate Recovery rate Literature In the vast majority of credit risk models, an exogenous speci cation of the recovery rate R is required. A constant based on empirical studies, such as those by Carty & Lieberman (1996) and Altman & Kishore (1996). In CreditMetrics (1997), a beta-distributed random recovery rate, independent of the default process, is used. However, Altman et al. (2004) and Altman (2006) both argue the importance of having a recovery rate structure that is inversely related to the default probability.

9 Recovery rate The recovery process The liquidation and legal fees represent a fraction κ of the market value of assets at default. A proxy for the recovery rate is R τ = min ((1 κ) A τ; L τ ) L τ = min (1 κ) 1 ; 1. (7) X τ The leverage ratio process impacts both the moment of default (through the default intensity) and the recovery rate at the moment of default, inducing a negative correlation between the two quantities. Di erent under P and Q.

10 Risk-neutral measures Pricing measures The market model is incomplete, implying that there are an in nitely many pricing measures. Among these measures, we restrict the choices to those preserving the model structure: 8 >< ln X t+1 = >: n o where ε Q t variables. t=1 ln X t + µ Q 1 ln X t + µ Q 2 Transition probabilities : σ t + σ 1 p t ε Q t+1 if h t+1 = 1 σ t + σ 2 p t ε Q t+1 if h t+1 = 2 are independent standard normal random q ij = Q(h t = jjh t 1 = i), i, j 2 f1, 2,..., K g. (8)

11 Pricing Pricing 1 No closed-form solution for corporate bond prices and CDS premiums. 2 Numerical scheme: extensions of Schönbucher (2002) tree and Yuen and Yang (2010) s trinomial lattice.

12 Data Data I rms of the CDX North American IG and HY indices (Markit Group, Sept. 2013). 1 Multiple credit ratings and sectors. 2 Weekly term structure of CDS premiums over (provided by Markit) for a maximum of 469 weeks. (2013 is keep for out-of-sample analysis) 1 Prices for maturities of 1, 2, 3, 5, 7, and 10 years are available for most rms (125 IG and 100 HY) rms (4 IG and 10 HY) were removed from the sample since there were not enough observations ,796 observations in our nal sample.

13 Filtering Filtering I The state equation is 8 >< ln X t + ln X t+1 = >: µ P 1 ln X t + µ P 2 The measurement equation is σ t + σ 1 p t ε P t+1 if h t+1 = 1, σ t + σ 2 p t ε P t+1 if h t+1 = 2. (9) ln CDS obs t,i = ln CDS model t,i + N (0, δ i ) {z } (10) fct(x t,h t,θ Q,i) i = 1, 2, 3, 5, 7, 10. Both equations are nonlinear transformations of the state variables. A single set of P and Q parameters is estimated at once for each rm.

14 Filtering Filtering II Both state and observation equations are nonlinear 1 An extension of Tugnait (1982) s detection-estimation algorithm (DEA): instead of running M Kalman lters in parallel, we use M unscented Kalman lters (UKF). 2 The short rate is not modeled explicitly and it remains xed during the pricing for a given day. However, the rate changes from one week to another and is set to the three-month constant maturity Treasury rate.

15 In-sample In-sample t I IG HY

16 In-sample Random recovery rate I 1 Monte Carlo simulation. For each rm, 1 For each date t, using the ltered leverage bx t and the ltered regime bh t as initial point, Monte Carlo simulation produces 10 6 weekly paths for the leverage ratio under P for the next 10 years. 2 Default are derived from the corresponding simulated intensity. 3 The recovery rate is recorded at default time. 2 For each rm and each date, we obtain a term structure of (1) expected recovery rates, (2) standard deviation, (3) skewness and kurtosis. 1 This can never be empirically observed (not enough default observations for a single rm).

17 In-sample Random recovery rate II Moody s stated that the average recovery rate (across rms) is 52% and the standard deviation (across rms) is 26%. With our sample of rms, the average (across rms) expected recovery rate is between 47% to 52% (depending on the time of default) while the standard deviation (across rms) of the expected recovery rate is about 27% to 29%.

18 In-sample Credit spread pricing Credit spreads are de ned as the di erence between risky and riskless zero-coupon yields: CS (t, s) = log EQ t (LGD τ 1 τt ) T t h i = log E Q t (LGD τ ) E Q t (1 τt ) + Cov Q t (LGD τ, 1 τt ) T t (11) (12) 1 If the recovery rate is random but independent of the other risk factors, h i CS (t, s) = log E Q t (LGD) E Q t (1 τt ) (13) T t then there is no e ect of the ramdomness in the credit spreads. 2 If the recovery rate is constant coming from empirical data, then it is like replacing E Q t (LGD) with E P t (LGD).

19 In-sample Model implied credit spread curves Pre-crisis Crisis Post-crisis

20 4L Billio et al. (2012) : there are four major determinants of nancial crises 1 Leverage 2 Losses 3 Linkage 4 Liquidity We take advantage of our framework that captures the three rst "L" to measure systemic risk.

21 The model Correlation I The leverage ratio X (i) and the regime h (i) for rm i is 8 ln X (i) t + µ (i) σ (i) 2 1 t + σ (i) p 1 t ε (i) t+1 if h (i) t+1 = 1, ln X (i) t+1 = >< >: ln X t + µ (i) σ (i) 2 2 t + σ (i) p 2 t ε (i) t+1 if h (i) t+1 = 2. Cov P t h i ε (i) t+1, ε(j) t+1 h (i) t+1, h(i) t+1 h (i) t+1, h(i) t+1 = ρ (i,j), h (i) t+1,h(i) t+1 2 f(1, 1), (1, 2), (2, 1), (2, 2)g

22 Estimation Estimation 1 However, estimating all rms simultaneously is not numerically feasible. The estimation is thus broken down into two stages. 1 First, the rm-speci c parameters are estimated. 2 The second stage then focuses on the interrelation between rms while keepingthe rm-speci c parameters xed. 2 This approach is similar to the Inference Function for Margin (IFM) estimator proposed by Joe and Xu (1996).

23 Data Data I 1 Results based on 35 rms of the nancial sector : 16 insurance companies and 19 banking rms. 1 Weekly term structure of CDS premiums over for a maximum of 417 weeks. 2 Premiums of maturities of 1, 2, 3, 5, 7 and 10 years.

24 Results Proportion of rms in high volatility regime A The credit crunch begins (August 1, 2007). B The Federal Reserve Board approves the nancing arrangement between JPM and BSC (March 14, 2008). C LEH les for Chapter 11 bankruptcy protection. MER is taken over by the BACORP. AIG almost defaulted the next day (September 15, 2008). D Three large U.S. life insurance companies seek TARP funding : LNC, HIG and GNWTH (November 17, 2008). E U.S. Treasury Department, Fed, and FDIC announce a package of guarantees, liquidity access, and capital for BACORP (January 16, 2009).

25 Results Default probability

26 Results Recovery rates I The average recovery rate calculated in this study is consistent with the literature. 1 Altman et al. (2005) nd an average recovery at default of 53% and 35% for senior secured and unsecured bonds, respectively. 2 In Vazza and Gunter (2012), senior secured and unsecured bonds have an average discounted recovery rate of 56.4% and 42.9%, respectively, during the period.

27 Results Correlation estimates I

28 Results Correlation estimates II

29 Systematic risk Systematic Risk I The systemic risk measure is de ned as the expected value of a loss over a period of three months given that it is higher than the 99th percentile of the loss distribution. 1 Generate a sample of losses for each rm L (i) t,t+3 months = N(i) I τ (i) t+3 and compute the total losses across the rms : L t,t+3 months = n L (i) t,t+3 months i=1 2 Compute the systemic risk measure (SR) for the nancial service sector as the sample average of L t,t+3 months I Lt,t+3 months >VaR 0.99 (L t,t+3 months )

30 Systematic risk Systematic Risk II 3 Calculate the systemic risk contribution SR (so-called nominal price, in millions) for each subsector : L (i) t,t+3 months I L t,t+3 months >VaR 0.99(L t,t+3 months), SS 2 fbanks, insurancesg i2ss 4 The relative contributions RSR are scaled versions of the nominal price measures. We simply divide the contributions by the sum of the total liabilities for each respective subsector. 5 The RSR measure is forward-looking as it is based on CDS data and does not require a large sample of rms.

31 Systematic risk Systematic Risk III

32 Systematic risk Systematic Risk IV 1 Test whether a subsector s contribution could be used to forecast the other s systemic risk. 2 Granger-type (1969) causality tests adapted for GARCH e ect are employed. 3 The banking subsector s systemic risk only Granger-causes the insurance subsector s systemic risk when the lag length is equal to one (at a con dence level of 95%). 4 However, the insurer s systemic risk does not Granger-cause the bank s systemic risk for any lag length.

33 Conclusion I 1 Model includes recovery risk, negative dependence between default probabilities and recovery rates, and regime switching. 2 A rm-by- rm estimation procedure based on a ltering procedure deals with latent variables and microstructure noises. 1 Estimation based on CDS data of 200 rms. 2 The in-sample performance of the model reveals that it is exible enough to adjust to various rms and nancial cycles 3 An out-of-sample study concludes that the model is reliable and outperforms other considered benchmarks.

34 Conclusion II 3 Credit spread 1 The recovery uncertainty and its negative relationship with the default probability have a major impact on mid- and long-term credit spreads. 4 Recovery risk 1 Firm-speci c recovery term structure 5 Systemic risk 1 Multivariate version of the model 2 Our results indicate an increase in correlation during the high-volatility regime in comparison with the stable regime for 33 out of the 35 rms within the portfolio 3 Correlations are stronger during nancial turmoil 4 Systemic Risk measures