A simple model to account for diversification in credit risk. Application to a bank s portfolio model.

Transcription

1 A simple model to account for diversification in credit ris. Application to a ban s portfolio model. Juan Antonio de Juan Herrero Metodologías de Riesgo Corporativo. BBVA VI Jornada de Riesgos Financieros Rislab-Madrid 19 de octubre de 2006

2 Introduction (I) 2 Minimum credit capital requirements under the new Basel II Capital Accord are based on the estimation of the 99,9% n.c. systematic credit ris for a large portfolio under a one-factor Merton-type credit model. The model results in a closed-form solution that provides additive ris contributions and it is easy to implement. Two ey limitations: It measures only systematic credit ris. It might not recognize the full impact of diversification. Both issues can be effectively addressed within a multifactor setting in a simulationbased credit portfolio framewor. The use of simulation-based credit portfolio models is now widespread. However, there are benefits for seeing analytical, closed-form models, both for regulatory applications and for credit portfolio management. A simple model to account for diversification in credit ris.

3 Introduction (II) 3 Examples of analytical approaches: The granularity adustment (Gordy 2004;Martin and Wilde, 2002). Multifactor adustment (Pyhtin 2004). The model showed here focuses on the second issue and consists in an adustment (based on a few and intuitive parameters) to the single-factor credit capital model which recognizes the diversification in systematic credit losses from a multifactor setting. This presentation shows an overview of: The model Its parameterization Example of application For more details: A simple multifactor factor adustment for the treatment of credit capital diversification. García Céspedes, J.C., de Juan Herrero, J.A., reinin, A. and Rosen, D.. To appear in the Journal of Credit Ris. A simple model to account for diversification in credit ris.

4 The Model (I) 4 Consider a single-step model with sectors. For each obligor in a given sector, the credit losses at the end of the horizon are driven by a single-factor Merton model. The creditworthiness of obligor in sector is driven by a single systematic factor: Y = ρ Z + 1 ρ ε Systematic factor Idiosyncratic component For asymptotically fine-grained sector portfolios, the stand alone capital (for a confidence level of 99,9%) for each sector Under the Basel II single-factor model, or equivalently assuming perfect correlation between all the sectors, the overall capital is simply the sum of the stand-alone capital. = Sector EAD LGD N N sf = 1 = 1 ( PD ) 1 ρ ρ z 99,9% PD A simple model to account for diversification in credit ris.

5 The Model (II) 5 Let s introduce the concept of a diversification factor, DF, defined as: DF = sf We see to approximate it by a scalar function of a small number of intuitive parameters s.t. In order to specify the parameters, one has to notice that there are two main sources of diversification: The correlation between the systematic sector factors. The sector concentration or distribution of relative sizes of the various sector portfolios. DF( par1, par2, L) = 1 1 q M q 21 1 q q 12 1 M 2 L L O L q1 q2 M 1 A simple model to account for diversification in credit ris.

6 The Model (III) 6 Both sources can be summarized in two parameters: The average correlation. β q i i i= 1 i i= 1 i = = i= 1 i i i= 1 i The Capital Diversification Index (its inverse can be interpreted as the effective number of sectors). CDI = 1 sf ( ) = = 2 In both cases, the weights w i = i / sf are determined by the one-factor capitals, accounting for the size of sector exposures as well as for their credit characteristics. Therefore, it seems natural to use a two-factor parameterization: i= 1 DF( CDI, β ) 2 w = 1 q 2 i w i i w w w A simple model to account for diversification in credit ris.

7 The Model: Capital allocation and ris contributions 7 In a single-factor credit model, capital allocation is straightforward: the capital attributed to a sector is the same as its stand-alone capital. Under a multifactor model it is necessary to obtain contributions on a marginal basis. One of the advantages of having an analytical model is that it provides tractable solutions for capital contributions. Given that DF only depends on CDI and β, and both are homogeneous functions of degree zero in the s, the function (,, ) = DF( CDI, β 1 ) is a homogeneous function of degree one in the s. Applying Euler s theorem leads to the additive marginal capital decomposition = 1 = = 1 A simple model to account for diversification in credit ris.

8 The Model: Capital allocation and ris contributions 8 Applying the chain rule one can obtain a closed-form expression for the th marginal diversification factor: 1 DF sf DF DF = = DF + 2 CDI + 2 sf CDI β 1 CDI where Overall Sector concentration Sector correlation diversification is the average correlation of factor to the rest of the systematic factors in the portfolio. The above expression shows that the marginal capital allocation resulting from the DF model leads to an intuitive decomposition of diversification effects into three components. DF Q = = DF + ΔDF q size + ΔDF corr [ Q β ] A simple model to account for diversification in credit ris.

9 DF surface parameterization (I) 9 The general parameterization methodology can be summarized as follows: Simulate a large number of portfolios, each of them consisting of a set of fine-grained homogeneous sectors with the same PD and EAD. In each simulation, sample independently The number of sectors,. PD [0,10%]. Asset correlation is given as a function of the PDs from Basel II formula for wholesale exposures. The average factor correlation β [0,100%] For each portfolio, Compute the stand-alone capital for each sector,, the single-factor capital for the portfolio, sf, and the CDI. Compute the true from a multifactor model and the empirical DF. Estimate de function DF(CDI, β) by fitting a parametric function through the points. A simple model to account for diversification in credit ris.

10 DF surface parameterization (II) randomly simulated portfolios with up to 10 sectors (CDI [0,1-1]). The figure shows the DF for the simulated data. The DF surface is estimated as a polynomial function of both the CDI and β, P n (CDI, β), with the constraints: P ( CDI,1) = 1, (1, β ) = 1 n P n These constraints suggest a polynomial of the form: i P n( CDI, β ) 1+ ai, (1 β ) (1 CDI) i, 1 = The specification uses the second-order approximation (in each variable). There are alternative specifications. For example, by M. Pyhtin s suggestion, we tested the functional form a b a ( β ) CDI β DF( CDI, β ) = 1 + obtaining similar results. A simple model to account for diversification in credit ris.

11 DF surface parameterization (III) 11 Estimated parametric DF polynomial model. Coefficients Standard error t-stat Lower 95% Upper 95% A11-0,852 0, ,854-0,850 A21 0,426 0, ,422 0,430 A12 0 A22-0,481 0, ,486-0,449 16% Estimated Capital (%) (DF Model) 14% 12% 10% 8% 6% 4% 2% 0% 0% 2% 4% 6% 8% 10% 12% 14% 16% Actual Capital (%) R 2 =99.4% and the volatility of errors (in capital (%)) is 11 bp, with a mean error of 4 bp. A simple model to account for diversification in credit ris.

12 DF surface parameterization (IV) 12 The parameterization has been tested against a richer structure of correlations between the systematic factors (instead of assuming a structure given by the average beta). The dependence structure between the systematic factors Z is determined by Z = β Z + 1 β η The structure of dependence depends on parameters, being the entries of the correlation matrix q = β β i i i Sample of new portfolios with up to 10 sectors where, in addition, the variables β have been simulated randomly as independent uniform variables in the interval [0-1]. The performance of the model under this structure is generally as good as in the case of a single correlation, with a volatility of the errors of only 14 bp (with a mean error of 4 bp). A simple model to account for diversification in credit ris.

13 Application to a ban s portfolio model (I) 13 The portfolio model is based on Monte Carlo simulation and covers systematic ris, idiosyncratic ris to specific positions and a non-deterministic LGD. The systematic credit losses of the ban s credit portfolio are determined by 6 different factors. The portfolio is divided into portfolios (sectors) each of them exposed to one systematic factor. Each sector is composed of fine-grained sub-portfolios with different PD, LGD and intra-portfolio asset correlation. The DF model provides an acceptable approximation to the multifactor systematic capital obtained by Monte Carlo. Average Beta 55,30% CDI 37,65% Theoretical DF 77,81% Monte Carlo based DF 75,77% Relative error 2,69% A simple model to account for diversification in credit ris.

14 Application to a ban s portfolio model (II) 14 Marginal sector diversification factors: DF DF Sector size DF Sector correlation Sector 1 86,29% 9,27% -0,79% Sector 2 60,52% -25,72% 8,43% Sector 3 77,65% -1,16% 1,00% Sector 4 42,97% -28,04% -6,80% Sector 5 41,74% -29,59% -6,48% Sector 6 42,19% -29,45% -6,16% The size component of the sector diversification factor increases contribution for the biggest sector (sector 1). The rest of the sectors benefit from their relative size, lower than the average. Sector 2 and 3 are penalized by being their sector factors more correlated than average to the rest of the factors. MC based model vs. DF model DF model based margina diversification factors 100,0% 90,0% 80,0% 70,0% 60,0% 50,0% 40,0% 40,0% 50,0% 60,0% 70,0% 80,0% 90,0% 100,0% MC based marginal diversification factors A simple model to account for diversification in credit ris.

15 Application to a ban s portfolio model (III) 15 Sensitivity to the average sector correlation: DF = 0,4612 β At this level of CDI, behaves almost linearly in β. Portfolio capital (%change) 8% 6% 4% 2% 0% -2% -4% -6% -8% 45% 50% 55% 60% 65% "Average" correlation The model allows us to compute the sensitivity of the multifactor capital to different parameters (EAD, LGD, PD,asset correlation): par sector s = DF par sector s MC based model vs. DF model E.g.: Marginal sensitivity to EADs for the different sub-portfolios in each sector. EAD subportfolio sector = DF subportfolio (%) DF model based margina sensitivities to EAD's 15,0% 10,0% 5,0% 0,0% 0,0% 5,0% 10,0% 15,0% MC based marginal sensitivities to EAD's A simple model to account for diversification in credit ris.

16 Application to a ban s portfolio model (IV) 16 Stress testing in sector 1. Stress in EAD MF capital vs SF capital MF capital vs SF capital Portfolio capital (% change 50% 30% 10% -10% -30% -50% -100% -75% -50% -25% 0% 25% 50% 75% 100% % Variation of Exposure (sector 1) MF capital SF capital Δ sf ΔEAD Δ ΔEAD 0% 20% 40% 60% 80% 100% % Variation of PD (sector 1) Stress in PD Portfolio capital (% change MF capital vs SF capital 30% 20% 10% 0% -10% -20% -30% -60% -40% -20% 0% 20% 40% 60% % Variation of PD (sector 1) Δ sf ΔPD Δ Δ PD MF capital vs SF capital 10% 60% 110% 160% 210% 260% % Variation of PD (sector 1) MF capital SF capital MF capital SF capital A simple model to account for diversification in credit ris.

17 Application to a ban s portfolio model (V) 17 Stress testing in sector 1. Stress in asset correlation Portfolio capital (% change) MF capital vs SF capital 40% 30% 20% 10% 0% -10% -20% -30% -40% -60% -40% -20% 0% 20% 40% 60% % Variation of asset correlation (sector 1) MF capital vs SF capital Δ sf Δρ Δ Δρ 10% 60% 110% 160% 210% 260% 310% 360% % Variation of asset correlation (sector 1) MF capital SF capital MF capital SF capital A simple model to account for diversification in credit ris.

18 Conclusions 18 The presented DF model consists in a simple factor adustment to the single-factor systematic credit capital model recognizing diversification from a multifactor setting. It depends on two intuitive parameters, the average correlation between sectors and the distribution of the ris between sectors (capital diversification index). Once parameterized, it provides a simple complement to Monte Carlo simulation for the computation and analysis of portfolio economic capital. It can be used as a ris management tool for: Understanding concentration ris and capital allocation. Identifying capital sensitivities Stress testing Real-time marginal ris contributions for new deals or portfolios Potential use to extend the Basel II regulatory framewor to a general multifactor setting. A simple model to account for diversification in credit ris.

19 References 19 García Céspedes J.C., de Juan Herrero J.A., reinin A. and Rosen D., A simple multifactor factor adustment for the treatment of credit capital diversification. To appear in the Journal of Credit Ris. Gordy, M. (2004). Granularity adustment in portfolio credit ris measurement. In Ris Measures for the 21 st Century (G.Szegö, ed.), Wiley. Martin, R., and Wilde, T. (2002). Unsystematic Credit Ris. Ris, November, Pyhtin, M., (2004). Multi-factor adustment. Ris, March, A simple model to account for diversification in credit ris.