A simple model to account for diversification in credit …risklab.es/es/jornadas/2006/Juan Antonio...

VI Jornada de Riesgos Financieros Risklab-Madrid19 de octubre de 2006

A simple model to account for diversification in credit risk.

Application to a bank’s portfolio model.

Juan Antonio de Juan HerreroMetodologías de Riesgo Corporativo. BBVA

2


Introduction (I)

Minimum credit capital requirements under the new Basel II Capital Accord are based on the estimation of the 99,9% n.c. systematic credit risk for a large portfolio under a one-factor Merton-type credit model.

The model results in a closed-form solution that provides additive risk contributions and it is easy to implement.

Two key limitations:

It measures only systematic credit risk.It might not recognize the full impact of diversification.

Both issues can be effectively addressed within a multifactor setting in a simulation-based credit portfolio framework.

The use of simulation-based credit portfolio models is now widespread. However, there are benefits for seeking analytical, closed-form models, both for regulatory applications and for credit portfolio management.

3


Introduction (II)

Examples of analytical approaches:

The granularity adjustment (Gordy 2004;Martin and Wilde, 2002).

Multifactor adjustment (Pykhtin 2004).

The model showed here focuses on the second issue and consists in an adjustment (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 modelIts parameterizationExample of application

For more details: A simple multifactor “factor adjustment” for the treatment of credit capital diversification. García Céspedes, J.C., de Juan Herrero, J.A., Kreinin, A. and Rosen, D. . To appear in the Journal of Credit Risk.

4


The Model (I)

Consider a single-step model with K sectors. For each obligor j in a given sector k, the credit losses at the end of the horizon are driven by a single-factor Merton model.

The creditworthiness of obligor j in sector k is driven by a single systematic factor:

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.

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−⋅⋅=

−

∈∑ j

k

kkj

kSectorjjjk PD

zPDNNLGDEADEC

ρρ

1)( %9,991

jkkkk ZY ερρ −+= 1

K

Systematic factor Idiosyncratic component

∑=

=k

ksf ECEC

1

5


The Model (II)

Let’s introduce the concept of a diversification factor, DF, defined as:

We seek 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.

sf

mf

ECECDF =

∑=

⋅≈K

kk

mf ECparparDFEC1

21 ),,( L

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

1

11

21

221

112

L

MOMM

L

L

KK

K

K

qq

qqqq

6


The Model (III)

Both sources can be summarized in two parameters:

The average correlation.

The “Capital Diversification Index” (its inverse can be interpreted as the effective number of sectors).

In both cases, the weights wi=ECi /ECsf 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:

∑∑

∑∑

∑∑

∑∑

= ≠

= ≠

= ≠

= ≠

⋅

⋅⋅=

⋅

⋅⋅= K

i ijji

K

i ijjiij

K

i ijji

K

i ijjiij

ww

wwq

ECEC

ECECq

1

1

1

1β

( ) ∑∑

=

= ==K

iksf

K

kk

wEC

ECCDI

1

22

1

2

∑=

⋅≈K

kk

mf ECCDIDFEC1

),( β

7


The Model: Capital allocation and risk contributions

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 ECk’s, the function

is a homogeneous function of degree one in the ECk’s.

Applying Euler’s theorem leads to the additive marginal capital decomposition

∑=

⋅=K

kkK

mf ECCDIDFECECEC1

1 ),(),,( βK

∑=

⋅∂∂

=K

kk

k

mfmf EC

ECECEC

1

8


The Model: Capital allocation and risk contributions

Applying the chain rule one can obtain a closed-form expression for the kth marginal diversification factor:

where

is the average correlation of factor k 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.

corrsizek DFDFDFDF Δ+Δ+=

[ ]ββ

−⋅−

−⋅

∂∂

+⎥⎦⎤

⎢⎣⎡ −⋅

∂∂

+=∂∂

= k

sfk

sfk

k

mf

k QCDIECEC

DFCDIECEC

CDIDFDF

ECECDF

1

122

Sector concentration Sector correlationOverall

diversification

∑∑

≠

≠⋅

=kj j

kj jkjk EC

ECqQ

9


DF surface parameterization (I)

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, K.

PDk ∈[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, ECk, the single-factor capital for the portfolio, ECsf, and the CDI.

Compute the “true” ECmf from a multifactor model and the empirical DF.

Estimate de function DF(CDI, β) by fitting a parametric function through the points.

10


DF surface parameterization (II)

22.000 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, Pn(CDI, β), with the constraints:

1),1(,1)1,( == βnn PCDIP

These constraints suggest a polynomial of the form:

The specification uses the second-order approximation (in each variable).

There are alternative specifications. For example, by M. Pykhtin’s suggestion, we

tested the functional form obtaining similar results.

ji

jijin CDIaCDIP )1()1(1),(

1,, −−+= ∑

≥

ββ

( ) aba CDICDIDF βββ +⋅−= 1),(

11


DF surface parameterization (III)

Estimated parametric DF polynomial model.

R2=99.4% and the volatility of errors (in capital (%)) is 11 bp, with a mean error of 4 bp.

Coefficients Standard error t-stat Lower 95% Upper 95%

A11 -0,852 0,0009 -911 -0,854 -0,850A21 0,426 0,0019 225 0,422 0,430A12 0A22 -0,481 0,0023 -205 -0,486 -0,449

0%

2%

4%

6%

8%

10%

12%

14%

16%

0% 2% 4% 6% 8% 10% 12% 14% 16%

Actual Capital (%)

Estim

ated

Cap

ital (

%) (

DF

Mod

el)

12


DF surface parameterization (IV)

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 Zk is determined by

The structure of dependence depends on K parameters, being the entries of the correlation matrix

Sample of 22.000 new portfolios with up to 10 sectors where, in addition, the variables βk 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).

kkkk ZZ ηββ ⋅−+⋅= 1

jiq jiij ≠⋅= ββ

13


Application to a bank’s portfolio model (I)

The portfolio model is based on Monte Carlo simulation and covers systematic risk, idiosyncratic risk to specific positions and a non-deterministic LGD.

The systematic credit losses of the bank’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, LGDand 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%

14


Application to a bank’s portfolio model (II)

Marginal sector diversification factors:

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.

DFk DF Sector size DF Sector correlationSector 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%

MC based model vs. DF model

40,0%

50,0%

60,0%

70,0%

80,0%

90,0%

100,0%

40,0% 50,0% 60,0% 70,0% 80,0% 90,0% 100,0%

MC based marginal diversification factors

DF

mod

el b

ased

mar

gina

dive

rsifi

catio

n fa

ctor

s

15


Application to a bank’s portfolio model (III)

Sensitivity to the “average” sector correlation:

At this level of CDI, EC behaves almost linearly in β.

4612,0=∂∂β

DF

The model allows us to compute the sensitivity of the multifactor capital to different

parameters (EAD, LGD, PD,asset correlation): ktors

kkktor

s

mf

parECDF

parEC

secsec ∂∂

⋅=∂∂

E.g.: Marginal sensitivity to EADs for the different

sub-portfolios in each sector.

(%)sec

jiosubportfolkktorjiosubportfol

mf

ECDFEAD

EC⋅=

∂∂

∈

-8%

-6%

-4%

-2%

0%

2%

4%

6%

8%

45% 50% 55% 60% 65%

"Average" correlation

Port

folio

cap

ital (

%ch

ange

)

MC based model vs. DF model

0,0%

5,0%

10,0%

15,0%

0,0% 5,0% 10,0% 15,0%

MC based marginal sensitivities to EAD's

DF

mod

el b

ased

mar

gina

sens

itivi

ties

to E

AD

's

16


Application to a bank’s portfolio model (IV)

Stress testing in sector 1.

Stress in EAD

Stress in PD

MF capital vs SF capital

-50%

-30%

-10%

10%

30%

50%

-100% -75% -50% -25% 0% 25% 50% 75% 100%

% Variation of Exposure (sector 1)

Port

folio

cap

ital (

% c

hang

e

MF capital SF capital


-30%

-20%

-10%

0%

10%

20%

30%

-60% -40% -20% 0% 20% 40% 60%% Variation of PD (sector 1)

Port

folio

cap

ital (

% c

hang

e



0% 20% 40% 60% 80% 100%% Variation of PD (sector 1)

EADsfEC

ΔΔ

EADmfEC

ΔΔ


10% 60% 110% 160% 210% 260%% Variation of PD (sector 1)


PDsfEC

ΔΔ

PDmfEC

ΔΔ

17


Application to a bank’s portfolio model (V)

Stress testing in sector 1.

Stress in asset correlation


10% 60% 110% 160% 210% 260% 310% 360%

% Variation of asset correlation (sector 1)


ρΔΔ mfEC

ρΔΔ sfEC


-40%-30%-20%-10%

0%10%20%30%40%

-60% -40% -20% 0% 20% 40% 60%

% Variation of asset correlation (sector 1)

Port

folio

cap

ital (

% c

hang

e)


18


Conclusions

The presented DF model consists in a simple factor adjustment 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 risk 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 risk management tool for:

Understanding concentration risk and capital allocation.

Identifying capital sensitivities

Stress testing

“Real-time” marginal risk contributions for new deals or portfolios

Potential use to extend the Basel II regulatory framework to a general multifactor setting.

19


References

García Céspedes J.C., de Juan Herrero J.A., Kreinin A. and Rosen D., A simple

multifactor “factor adjustment” for the treatment of credit capital diversification. To

appear in the Journal of Credit Risk.

Gordy, M. (2004). Granularity adjustment in portfolio credit risk measurement. In Risk

Measures for the 21st Century (G.Szegö, ed.), Wiley.

Martin, R., and Wilde, T. (2002). Unsystematic Credit Risk. Risk, November, 123-8.

Pykhtin, M., (2004). Multi-factor adjustment. Risk, March, 85-90.

A simple model to account for diversification in credit …risklab.es/es/jornadas/2006/Juan Antonio...

Documents

Transcript of A simple model to account for diversification in credit …risklab.es/es/jornadas/2006/Juan Antonio...