Asymptotic Risk Factor Model with Volatility Factors

Transcription

1 Asymptotic Risk Factor Model with Volatility Factors Abdoul Aziz Bah 1 Christian Gourieroux 2 André Tiomo 1 1 Credit Agricole Group 2 CREST and University of Toronto March 27, 2017 The views expressed are solely those of the authors and do not necessarily reflect the views of the Credit Agricole Group. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 1 / 40

2 1. INTRODUCTION Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 2 / 40

3 Introduction Three categories of models : 1. The portfolio credit Value-at-Risk (VaR) models. 2. The reduced form models. 3. The structural models. The Asymptotic Single Risk Factor Model is the basis for : The analysis of credit risk. Performing the stress tests (including the possibility to account for a rating scale). Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 3 / 40

4 Motivation The standard ASRF Model V k,i,t = α k ρz t + 1 ρu k,i,t, k = 1, 2, where EZ t = 0, VZ t = 1, u s are i.i.d. standard normal, and ρ is interpreted as a correlation. A single factor model with linear effect of the factor. This basic model has been extended in the literature to include more than a single linear factor [see e.g. Gagliardini, Gourieroux (2005)]. But this extension does not include nonlinear factor such as volatility The idea behind this paper is that nonlinear effects can be appropriately captured by introducing time varying volatility factors Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 4 / 40

5 Objective of the paper In this paper we... Extend the standard Asymptotic Single Risk Factor Model to a Risk Factor Model with both drift and volatility factors. Provide a calibration step based on the asymptotic principle (granularity theory) to derive consistent estimators of the parameters and consistent smoothed factor values. Discuss the consequences of using misspecified ASRF model, i.e. of neglecting volatility factors. Provide an application of the standard ASRF model and the Risk Factor Model with both drift and volatility factors in a stress testing framework for corporate credit portfolio by using comprehensive database managed within Credit Agricole Group. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 5 / 40

6 2. THE RISK FACTOR MODEL Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 6 / 40

7 The Model Model defined by means of latent variables interpretable as log asset/liability ratios : Y. The distribution of these latent variables depends on the class of risk at the previous period. k = 1 : investment grade Y 1,i,t k = 2 : speculative grade Y 2,j,t k = 3 : default (absorbing state) We assume : { Y 1,i,t = α 1 β 1Z t + γ 1tu 1,i,t, i = 1,..., n 1,t 1, Y 2,j,t = α 2 β 2Z t + γ 2tu 2,j,t, j = 1,..., n 2,t 1, where u 1,i,t, u 2,j,t are i.i.d. standard normal. 3 types of factors : i) Z t, linear factor ii) γ 1t, γ 2t nonlinear stochastic volatility factors. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 7 / 40

8 The Model The conditional distribution of Yk,i,t is : Yk,i,t Z t, γ 1t, γ 2t N(α k β k Z t, γkt), 2 k = 1, 2. (1) The log asset/liability ratios are not directly observed. The observed variables are the new ratings : Y k,i,t = 1, investment grade if Yk,i,t > c 1, Y k,i,t = 2, speculative grade if c 1 > Yk,i,t > c 2, Y k,i,t = 3, default if c 2 > Yk,i,t, where c 1, c 2 are unknown thresholds. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 8 / 40

9 The Model Conditional Migration Probabilities From rating k to default : Π k,3,t = P(Y k,i,t < c 2) = Φ From rating k to investment grade : Π k,1,t = P(Y k,i,t > c 1) = 1 Φ From rating k to speculative grade : ( ) c2 α k + β k Z t, k = 1, 2; γ kt Π k,2,t = 1 Π k,1,t Π k,3,t. ( ) c1 α k + β k Z t, k = 1, 2; γ kt Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 9 / 40

10 The Model Conditional Migration Probabilities These formulas can be used to see how the transformed probit migration probabilities depend on the factors : Φ 1 (Π k,3,t ) = c2 α k + β k Z t γ kt, k = 1, 2, t = 1,..., T, Φ 1 (1 Π k,1,t ) = c1 α k + β k Z t γ kt, k = 1, 2, t = 1,..., T. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 10 / 40

11 The Model Identification Restrictions By appropriate drift and scaling of factors and parameters, we assume without loss of generality: β 1 = 1, c 2 = α 1, α 1 α 2 = 1, and c = c 1 - c 2 > 0, β = β 2 Φ 1 (Π 1,3,t) = Zt γ 1t, Φ 1 (1 Π 1,1,t) = c + Zt γ 1t. Φ 1 (Π 2,3,t) = 1 + βzt γ 2t, Φ 1 (1 Π 2,1,t) = c βzt γ 2t. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 11 / 40

12 The Model The expressions of migration probabilities extend the standard ASRF formulas based on the assumption : γ 1t = γ 2t = γ, and the latent variables are defined with another identification restriction : EZ t = 0, VZ t = 1. ( ) c2 α k ρ Π k,3,t = Φ + Z 1 ρ 1 ρ t ( ) c1 α k ρ Π k,1,t = 1 Φ + Z 1 ρ 1 ρ t, Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 12 / 40

13 The Model The new model increases the number of factors and allows for different reasons of entering into the default state. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 13 / 40

14 3. CALIBRATION Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 14 / 40

15 Calibration of RFM Asymptotic Principle When the numbers of firms n k,t are large k, t : ˆΠ k,l,t Π k,l,t. Then for given thresholds c, β, we can use the closed form expressions : 1 = Φ 1 (1 Π 1,1,t) Φ 1 (Π 1,3,t) 0, γ 1t c 1 = Φ 1 (1 Π 2,1,t) Φ 1 (Π 2,3,t) 0, γ 2t c Z t = cφ 1 (Π 1,3,t) ( Z1t(c, β)) Φ 1 (1 Π 1,1,t) Φ 1 (Π 1,3,t) = cφ 1 (Π 2,3,t) 1/β[ 1]( Z2t(c, β)). Φ 1 (1 Π 2,1,t) Φ 1 (Π 2,3,t) Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 15 / 40

16 Calibration of RFM The calibration is as follows : Plug in for ˆΠ Ẑ1t(c, β), Ẑ2t(c, β) Estimation of the thresholds : Smoothed factor values: (ĉ, ˆβ) = arg min c>0 T [Ẑ1t(c, β) Ẑ2t(c, β)] 2. t=1 Then plug in for ˆγ 1t, ˆγ 2t. Ẑ t = 1 2 [Ẑ1t(ĉ, ˆβ) + Ẑ 2t(ĉ, ˆβ) ] ; Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 16 / 40

17 Calibration of Standard RFM The closed form expressions in the standard RFM : Π k,3,t = Φ 1 Π k,1,t = Φ ( Φ 1 ) (Π k,3 ) ρ + Z 1 ρ 1 ρ t, ( Φ 1 ) (1 Π k,1 ) ρ + Z 1 ρ 1 ρ t. (2) k = 1, 2 Ẑ k,3,t (ρ) = 1 ρ ρ Φ 1 (ˆΠ k,3,t ) Φ 1 (ˆΠ k,3 ) ρ, Ẑ k,1,t (ρ) = 1 ρ ρ Φ 1 (1 ˆΠ k,1,t ) Φ 1 (1 ˆΠ k,1 ) ρ. (3) Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 17 / 40

18 Calibration of Standard RFM The calibration steps : step 1 : : Estimate ρ ˆρ = arg min ρ T 2 t=1 k=1 l=1,3 Ẑ k,l,t(ρ) k=1 l=1,3 step 2 : The smoothed factor value is deduced by : Ẑ t = k=1 l=1,3 Ẑ k,l,t(ˆρ). 2 Ẑk,l,t(ρ). Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 18 / 40

19 Misspecified ASRFM What are the consequences of implementing an ASRF model, when the true model is a RFM with volatility factors? Intuitively : Ẑ t a 0 + a 1 1 γ 1t + a 2 1 γ 2t + a 3 Z t γ 1t + a 4 Z t γ 2t. A pseudo factor linear combination of 1 1,, Z t, Z t. γ 1t γ 2t γ 1t γ 2t Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 19 / 40

20 4. ILLUSTRATION Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 20 / 40

21 The Data Set Corporate loans granted by the Credit Agricole Group. From December 2007 to December The internal ratings have been aggregated to get the 3 rating classes. Quarterly observations providing 35 quarterly migration matrices. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 21 / 40

22 A Complete Migration Matrix Figure: PIT Migration Matrix Figure: TTC Migration Matrix Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 22 / 40

23 The Simplified Migration Matrix IG SG D IG 93,79% 5,90% 0,31% SG 2,21% 96,73% 1,06% D 0,00% 0,00% 100% PIT Migration Matrix IG SG D IG 70,91% 28,69% 0,39% SG 8,23% 89,26% 2,51% D 0,00% 0,00% 100% TTC Migration Matrix Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 23 / 40

24 Estimation Results Two models are estimated : 1. The standard Single Risk Factor Model Estimated ρ : ˆρ = ρ estimated by Basel formula : ˆρ = The model with volatility factors Estimated c : ĉ = 0.11 Estimated β : ˆβ = 7.5 Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 24 / 40

25 Summary statistics for the factors Zt Z t γˆ 1t γˆ 2t Mean STD Table: Descriptive Statistics For each volatility factor γ 1t, γ 2t is computed the standard deviation/mean ratio to check the hypothesis of constant volatility. For γ 1t : 0.15 For γ 2t : 0.03 Both are significantly different from constant. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 25 / 40

26 Summary statistics for the factors It is also interesting to study the (unconditional) link between the 3 factors: Ẑ t, ˆγ 1t, ˆγ 2t. Ẑ t γˆ 1t γˆ 2t Ẑ t γˆ 1t γˆ 2t 1 Table: Correlation Matrix Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 26 / 40

27 Bivariate Plots Figure: Bi-variate plots Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 27 / 40

28 Graphs of factors (Annualized) Graphics Figure: Evolution of the volatility factors Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 28 / 40

29 Graphs of factors (Annualized) Graphics Figure: Evolution of the linear factor Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 29 / 40

30 Discussion on Misspecification How to interpret the factor Z t in the single risk factor model? Regress Z ˆ t on 1ˆ, γ 1t 1, γˆ 2t Ẑ t, γˆ 1t Ẑ t. γˆ 2t Zˆ t = γ 1ˆ 1ˆ Ẑt 0.24 Ẑt, 1t γ 2t γˆ 1t γˆ 2t R 2 adjusted = 0.87 (4) Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 30 / 40

31 5. STRESS TESTS Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 31 / 40

32 Stress Testing Approach As usual the stress tests are performed along the following steps : i) Select a set of macro-variables to be stressed and define the directions and magnitude of the stresses (i.e. of the shocks). ii) Estimate a dynamic model to relate the evolution of the factors to the macro-variables : Zt for the Single Risk Factor model, Z t, γ 1t, γ 2t for the model with volatility factors. iii) Then apply the shocks to the macro-variables and look at their impact on first the factor, second the migration matrices. iv) Use these stressed matrices to deduce the term structure of ratings. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 32 / 40

33 The macro-variables We consider a list of macro-variables associated with the scenarios proposed by the EBA. This macro-variables are : The French GDP growth rate The French inflation rate The change in French unemployment rate A long run interest rate (the 10 year OAT rate) The market index return (computed from the CAC40 index). The change in real estate index is in the EBA scenarios, but is not introduced, since it is more relevant for mortgages than for corporate loans. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 33 / 40

34 Link between the macro-variables and the factors : Dynamic Model The linear factor ˆ Z t = ˆ Z t GDPg t R 2 adjusted = 0.68 (5) Ẑ t = Ẑ t INFL t 0.28 Euribor t, R 2 adjusted = 0.56 (6) Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 34 / 40

35 Link between the macro-variables and the factors : Dynamic Model The volatility factors { log γ1,t ˆ = log ˆγ 1,t INFL t Euribor t R 2 adjusted = 0.70 (7) { log γ2,t ˆ = log ˆγ 2,t Euribor t R 2 adjusted = 0.32 (8) Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 35 / 40

36 Scenarios and Stress Tests We consider the scenarios proposed in EBA(2016). Macro-variables Baseline (%) Adverse (%) OAT 10 year 1,3 1,4 2 2 GDP Growth 1,7 1,6-1,1 0,6 Inflation 1,3 1,6 0,5 1 Unemployment rate 10,2 10,1 10,6 11,1 Residential property prices 1,5 2,3-4,3-1,5 Stock price shocks - - EBA Stress Test Scenarios Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 36 / 40

37 Scenarios and Stress Tests For each scenario we deduce : The stressed factor values of Z* in the ASRF model and of Z and volatility factors in the second RFM for the futures years, from the complete regression models. The stressed future migration matrices or ratings. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 37 / 40

38 6. CONCLUSION Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 38 / 40

39 Conclusion We propose a RFM with both drift and volatility factors : Extend the standard ASRF Introduce volatility factors i.e. nonlinear factors. Provide an estimation and calibration method which is valid for large cross-sections and even for small number of observation dates. We illustrate the models [RFM with volatility factors and Standard ASRF Model] in a stress testing exercise for a corporate credit portfolio. Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 39 / 40

40 THANK YOU Bah, Gouriéroux, Tiomo Asymptotic Risk Factor Model with Volatility Factors Paris 40 / 40