Empirical Analysis of Bank Capital and New Regulatory Requirements for Risks in Trading Portfolios

Empirical Analysis of Bank Capital and New Regulatory Requirements for Risks in

Trading Portfolios

Michael Jacobs, Ph.D., CFASenior Manager

Deloitte & Touche LLPAudit and Enterprise Risk Services

June 2012

The views expressed herein are those of the authors and do not necessarily represent the views of the Deloitte and Touche LLP or of the Board of Governors of the Federal Reserve System

Outline• Background and Motivation

• Introduction and Conclusions

• Review of the Literature

• Theory and Empirical Methodology

• Data and Summary Statistics

• Empirical Results and Discussion

• Directions for Future Research

Background and Motivation• The 2007-2009 financial crisis was the impetus behind a new set of

financial regulations known as “Basel 2.5” and “Basel III”• While credit risk played a central role, not widely known that the

epicenter of these losses was in institutions’ trading portfolios• Supervisors require sufficient capital to remain a going concern in a

downturn, but trading book losses far exceeded minimums • BCBS introduced 2 metrics: incremental risk charge (IRC) for credit

risk in trading books & stressed Value-at-Risk (S-VaR) for estimated mark-to-market losses in a downturn period

• IRC measures credit risk in a trading portfolio over capital formation period of 1 year termed the capital horizon (CH)

• Period in which unable to rebalance a trading portfolio is the liquidity horizon (LH) set by regulators to be at least 3 months

• Constant level of risk (CLR) portfolio dynamics: a certain amount of risk in crises of generating sufficient income to cover losses

Background and Motivation (continued)• CLR implies migration (default) risk only within the LH (CH) so

rebalancing causes the risk profile realigned within the year• This influences measured annual default rates upon which

probability of default (“PD”) estimates & risk measures are based• Through-the-cycle (“TTC”) ratings quantify credit risk over a business

cycle: stable measures not a function of temporary fluctuations in the economy but of obligor fundamentals– Due to this dynamic should observe cyclicality of transitions across the cycle – Ratings agencies or some banks estimate these & are generally favored by

supervisors in that they result in less cyclicality in capital measures• Point-in-time (“PIT”) ratings quantify the credit risk over the short

term & includes the effects of systematic factors – Observe a greater (less) frequency of transition across (within) ratings– Also used banks concerned about an obligor’s ability to make payments– Examples include vendor models for PD (Moody’s KMV “EDFs”)

Background and Motivation (continued)• A spike in the VIX

not seen in 2 decades coincided with massive bank losses

• Clearly, the rise in investor risk aversion affected traded credit adversely

• Reproduced from: Jacobs, Michael, 2012, Stress testing credit risk portfolios, Working paper (presented to Risk-Incisive Media Training Conference: Credit Risk Management, New York, NY, March 20th, 2012 )

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VIX Volatility Index and C&I Charge-off Rates

VIX Volatility Index C&I Charegoff Rates

Introduction and Conclusions• We measure the impact of new regulations on firms’ regulatory

capital & reach conclusions different and more nuanced as compared to regulatory impact or academic studies

• We find evidence that IRC capital may be substantially more than expected by regulatory impact or academic investigations

• We find this to vary significantly by portfolio dynamics (CLR, CR), rating system (TTC, PIT), risk model (single vs. multi-factor), sector / asset class

• We develop a general theoretical credit risk framework subsuming the structural models of Basel II IRB (B2-IRB) and CreditMetrics (CM)– We implement a version of the industry standard multi-factor generalization of

CM in a proprietary R software package that is tractable– We extend the basic B2-IRB and CM models in a Bayesian analysis

• In the empirical implementation we utilize extensive data-sets from Datastream (bond and CDS indices in various sectors)

Introduction and Conclusions (continued)• Capital charges are found higher for either PIT / CP ratings /

dynamics vs. either TTC or CLR, accentuated for financial or sovereign as compared to industrial sectors

• Regulatory capital is found to be larger than economic capital for industrial but not for the financial or sovereign sectors

• A comparison of the single to a multi-factor CM model shows that capital estimates larger in the latter and that results do not differ as much across sectors and rating systems– However, for the financial / sovereign by orders of magnitude vs. industrial or

the Basel II asymptotic single risk factor (“ASRF”) • We perform a Bayesian implementation of the Basel II asymptotic

single risk factor ASRF and the basic CM – Treating correlation matrices and default rates as random rather than as fixed

parameters– Illustrates that the new framework potentially adds substantial uncertainty to

the capital estimates

Review of the Literature• Gordy (2003): proves that marginal EC calculated using a portfolio

model of C-VaR depends on the properties of the portfolio vs. portfolio invariance property of Basel 2 IRB ASRF model

• Frey and McNeil (2003) analyze the mathematical structure of portfolio credit risk models, modeling of dependence exploring the role of copulas in latent variable frameworks in popular models

• Jimenez-Martin et al (2009) conclude concept of VaR is intended to capture outcomes on a typical day & in market panic sets in & justifies an additional S-VaR measure to correct for the shortfalls

• Acharya and Schnabl (2009): later was likely the case because the pre-crisis period was characterized by unusually low volatility

• Perignon et al (2010a,b) conclude C-VaR is often inaccurate & has little ability to predict future losses, pre-(post-) crisis data banks systematically over-(under-)estimate trading book capital needs

• Berg (2010) calibrates the term structure of risk premia before & in financial crisis finding risk premium term structure flat (increasing) before (during) but long-run mean was of the same magnitude

Review of the Literature (continued)• BCBS (2009a): IRC as a requirement on banks that model

specific risk to measure and hold capital against default risk that is incremental to any risk captured in banks’ credit VaR models – Implies IRC should be based CLR at 1-year capital horizon to account for

both liquidity risk as well as concentration risk– Market risk is measured as in the current 10 day VaR plus an S-VaR with

99.9th percentile to remain consistent with A-IRB– Still specific risk but only when C-VaR model is only for common factors

• BCBS (2009b, 2010a): extensive survey banks internationally concludes that new requirements will result in material increase of regulatory capital (IRC & S-VaR of 51.7% & 28.8%, resp.)

• BCBS (2010c): focus on the macroeconomic impact finds a positive long-term effect as justification of the current regime– Higher regulatory capital & depressed lending outweighed by a stronger

banking system less vulnerable to financial crises

Review of the Literature (continued)• Kashyap, Stein and Hanson (2010) are in line BCBS (2010c) finding a

modest increase in 10% bank capital & 25 BPs in lending rates– But also highlight the flip-side: small changes in banking sector borrowing

costs may imply large flows of funds to the “shadow banking system”• Demirguc-Kunt et al (2010) study the financial crisis era equity price

reactions of banks with respect to capital ratios & find the ones given more emphasis in Basel III are those most sensitive to equity returns

• Varatto (2011): quantitative analysis of the impact of the Basel 2.35 for trading portfolios utilizing various U.S. bond indices – Finds that capital requirements could potentially be much more than

estimated by the impact studies conducted by the regulators – Ascribes the diminished capital impact reported by the banks an artifice of an

assumed risk reduction due to hedging which in crisis may be questionable• Yavin et al (2011) investigate IRC by constructing alternative TPMs for

unsecuritized credit products in the trading book– Argue that banks may need to make discretionary choices in methodology

Theory and Empirical Methodology• Assume a single-period (i.e., fixed time horizon T) model of a credit

portfolio consisting of m obligors & an integer-valued state variable:

0,1,...,ijS n 0,1,...,i m• Sij measure of the creditworthiness of each obligor i such that better

state is associated with an increasing value & default is coded by :0 1ij iS y 0 0ij iS y

• Assume that t = 0 all m obligors in some non-default state & focus on binary outcomes of default and non-default. Define random vector of default indicators for the credit portfolio with joint distribution:

1 1 1,.., ~ ,..., 0,1T m

m m mY Y p P Y y Y y Y y y

• Denote the marginal default probabilities (or “long-run” PD) by: 1 1,...,i ip P Y i m

• Count the number of defaulted obligors at time T with the R.V.:

1

m

ii

M Y

Y

Theory and Empirical Methodology (continued)

• …

• As realized loss if obligor i defaults is LGDi the random exposure is EADi, we may define the random overall loss as:

1

m

i i ii

L Y LGD EAD

• We may always posit alternative credit risk models with the same

multivariate distribution of Y or S and we will term equivalent any set of models with equality in distribution (E.I.D.)amongst these:

:d d

EID Y Y S S

• The simplifying assumption of the exchangeability with respect to default, the mathematical formalism that corresponds to notion of homogenous segments commonly implemented in credit risk models:

1 11 1 ,.., ,.., ,.., ,..,d d

m mm mS S S S Y Y Y Y

• Implies that for any k ϵ {1,..,m-1} all the possible m!/(k!*(m-k)!)) k dimensional marginal distributions of S are identical


• …

• Now consider latent variables models, in terms of general structure and there relation to copulas, with an m-dimensional random vector and a sequence of cut-off points with the following boundaries:

• The state vector and cutoff points are related by:

• It follows that this collection (often interpreted as an obligor’s asset and liability values at the horizon) constitutes a latent variable model (LVM) with respect to the underlying state vector S:

• In this framework, the marginal distribution function of X and the marginal default probability of obligor i are given by :

1,.., mX XX 1 1,...,i ind d i m 0

id 1

ind

1 1,..., , 1,...,i ii j i jS j d X d i m j n

i iF x P X x 1ii ip F d

11 1, : , . . . ,..,

Tii j mj n i m

LVM X d w r t S S

X d S S


• …

• We propose a simple criterion for the equivalence of two LVMs in terms of the marginal distributions of the state vector & the copula of X, which facilitates the comparison of various industry risk models:

• The first condition is that that the X’s admit the same copula, the second that marginal distributions of the state vectors coincide:

• If we are in default mode (i.e., binary sate vector) these reduce to the condition that the marginal PDs are identical in both models

• This is only a sufficient condition: in general the converse of this does not hold that the equivalence does not imply the same copulae

• JP Morgan’s CreditMetrics (“CM”): assume that the state vector is distributed as a multivariate Gaussian random variable & interpreted as a (standardized) asset value change process over horizon T

0 01 1, : , . . . , , : , . . .i i

i j i jj n j ni m i mLVM X d w r t LVM X d w r t

X d S S X d S S

1,.., , 1,..,i ii j i jP X d P X d j n i m


• …

• In CM default threshold is set such that obligor i’s PD is equal to the empirical default frequency of in credit quality & segmentation by some kind of PD rating system, either internal by the rating agencies

• The dependency structure of the random vector X may estimated in a factor model framework, which means that we express components:

• p<m dimensional random vector Z represents systematic factors, a standardized independent Gaussian random vector ε idiosyncratic factors, constant matrices A and B contain factor loadings on these

• In empirical implementation Z is usually interpreted as observable global, country and industry effects, while ε is an unobserved regression residual in the calibration of the factor weight matrix

• In the empirical tests we consider versions of the CM model for both univariate (p=1) or “Base Cm” and multivariate cases (p>1)

X AZ+Bε +μ

2 21 ,..,

T TmE E E diag

Σ X X X X AΩA

~ ,ppR N p m Z 0 Ω ~ ,m

m MR Nε 0 I

Data and Summary Statistics • Estimate the IRC for various test portfolios with different credit rating, maturity sectoral characteristics, a sample portfolio of bond indices using the basic single-factor and multi-factor version of CM• Utilize rating transition matrices of two kinds, TTC from Moody’s Investors Services, and PIT from Kamakura Corporation’s PD model for three sectors (industrial, financial and sovereign) and at two accumulation frequencies, quarterly and

annually, in order to test the effect of the CLR vs. the CP assumption• Meant to be portfolios representative of various sectors, daily bond indices compiled by Bank of America-Merrill Lynch and sourced from Datastream spanning the period 1/2/97 to 12/19/11• The industrial Portfolio 1 has 4 rating groups (Aa-Aaa, Baa-A, B-Ba and C-CCC), the financial Portfolio 3 rating groups (Aa-Aaa, A and Baa), and the sovereign Portfolio 3 has 4 rating groups (Aaa, Aa, A and Baa), spanning all maturities

Data and Summary Statistics (continued)

• … Aaa Aa A Baa Ba B Caa-C Default Aaa Aa A Baa Ba B Caa-C Default

Aaa 88.96% 9.69% 1.31% 0.04% 0.01% 0.00% 0.00% 0.00% Aaa 30.29% 15.36% 24.51% 18.48% 6.31% 2.23% 1.24% 1.58%

Aa 0.66% 89.23% 9.48% 0.48% 0.10% 0.05% 0.00% 0.00% Aa 18.86% 12.49% 26.73% 25.32% 9.33% 3.23% 1.71% 2.32%

A 0.04% 1.34% 91.15% 6.53% 0.69% 0.20% 0.04% 0.01% A 6.93% 7.52% 24.18% 32.65% 14.96% 5.83% 3.33% 4.60%

Baa 0.00% 0.17% 3.59% 88.42% 6.45% 1.14% 0.10% 0.09% Baa 1.52% 2.82% 14.26% 32.30% 20.97% 10.24% 6.95% 10.94%

Ba 0.00% 0.04% 0.32% 4.30% 83.82% 9.82% 0.65% 1.00% Ba 0.32% 0.89% 6.72% 23.30% 21.71% 13.69% 11.48% 21.89%

B 0.00% 0.04% 0.12% 0.37% 4.70% 84.52% 5.13% 5.13% B 0.08% 0.27% 2.78% 13.62% 17.44% 13.72% 13.94% 38.14%

Caa-C 0.00% 0.00% 0.02% 0.95% 1.30% 8.63% 65.16% 23.94% Caa-C 0.02% 0.07% 0.88% 5.45% 9.07% 8.68% 10.43% 65.40%

Aaa Aa A Baa Ba B Caa-C Default Aaa Aa A Baa Ba B Caa-C Default

Aaa 87.27% 12.23% 0.53% 0.01% 0.00% 0.00% 0.00% 0.00% Aaa 30.32% 15.32% 24.22% 18.27% 6.28% 2.27% 1.33% 2.01%

Aa 0.92% 89.88% 8.85% 0.33% 0.02% 0.00% 0.00% 0.00% Aa 18.87% 12.40% 26.26% 24.93% 9.24% 3.25% 1.81% 3.24%

A 0.09% 3.38% 89.88% 6.01% 0.49% 0.04% 0.03% 0.05% A 6.76% 7.25% 23.19% 31.61% 14.60% 5.81% 3.51% 7.27%

Baa 0.08% 0.78% 8.94% 82.74% 6.19% 0.99% 0.11% 0.21% Baa 1.50% 2.76% 14.00% 32.06% 21.03% 10.58% 7.70% 10.37%

Ba 0.00% 0.15% 1.12% 8.07% 81.04% 7.60% 0.87% 1.15% Ba 0.31% 0.87% 6.60% 23.15% 22.05% 14.57% 13.39% 19.05%

B 0.00% 0.12% 0.41% 0.93% 9.79% 76.57% 6.83% 5.39% B 0.08% 0.28% 2.90% 14.51% 19.39% 16.28% 18.38% 28.18%

Caa-C 0.00% 0.01% 0.04% 0.43% 1.05% 12.96% 47.37% 38.14% Caa-C 0.03% 0.08% 1.03% 6.66% 11.85% 12.30% 16.54% 51.52%

Aaa Aa A Baa Ba B Caa-C Default Aaa Aa A Baa Ba B Caa-C Default

Aaa 88.96% 9.69% 1.31% 0.04% 0.01% 0.00% 0.00% 0.00% Aaa 30.32% 15.32% 24.22% 18.27% 6.28% 2.27% 1.33% 2.01%

Aa 0.66% 89.23% 9.48% 0.48% 0.10% 0.08% 0.00% 0.00% Aa 18.87% 12.40% 26.26% 24.93% 9.24% 3.25% 1.81% 3.24%

A 0.04% 1.34% 91.15% 6.53% 0.69% 0.20% 0.04% 0.01% A 6.76% 7.25% 23.19% 31.61% 14.60% 5.81% 3.51% 7.27%

Baa 0.00% 0.17% 3.59% 88.42% 6.45% 1.14% 0.10% 0.09% Baa 1.50% 2.76% 14.00% 32.06% 21.03% 10.58% 7.70% 10.37%

Ba 0.00% 0.04% 0.32% 4.30% 83.82% 9.82% 0.65% 1.00% Ba 0.31% 0.87% 6.60% 23.15% 22.05% 14.57% 13.39% 19.05%

B 0.00% 0.04% 0.12% 0.37% 4.70% 84.52% 5.13% 5.13% B 0.08% 0.28% 2.90% 14.51% 19.39% 16.28% 18.38% 28.18%

Caa-C 0.00% 0.00% 0.02% 0.95% 1.30% 8.63% 65.16% 23.94% Caa-C 0.03% 0.08% 1.03% 6.66% 11.85% 12.30% 16.54% 51.52%

Table 3.3: Sovereign Sector, Through-the-Cycle Table 3.6: Sovereign Sector, Point-in-Time

Table 3: Point-in-Time (Kamakura KRIS 1990-201) vs. Through-the-Cycle (Moody's DRS 1980-2011) Transition Matrices - Annual Capital Horizon and Quarterly Liquidity Horizon (Constant Level of Risk Assumption)

Table 3.1: Industrial Sector, Through-the-Cycle Table 3.4: Industrial Sector, Point-in-Time

Table 3.2: Financial Sector, Through-the-Cycle Table 3.5: Financial Sector, Point-in-Time


• …Sector Rating Mean Median

Standard Deviation

Coefficient of

Variation Minimum Maximum

Correlations between Systematic Factors

Aa-Aaa 0.0377% 0.0274% 0.7167% 18.99 -12.3977% 11.6545% 100.00% 36.07% 8.84% 8.26% 32.92% 55.05%

Baa-A 0.0433% 0.0331% 0.5247% 12.11 -11.5403% 7.4375% 100.00% 8.68% 16.46% 66.26% 48.96%

B-Ba 0.0372% 0.0418% 0.5308% 14.27 -6.0864% 10.8899% 100.00% 78.83% 24.00% 51.39%C-Caa 0.0194% 0.0425% 0.4478% 23.12 -4.7283% 8.3753% 100.00% 29.40% 60.87%

Aa-Aaa 0.0286% 0.0185% 0.3977% 13.90 -2.9119% 8.0429% 100.00% 46.23% 14.20% - 88.56% 83.12%

A 0.0411% 0.0097% 0.3375% 8.22 -2.5594% 4.5207% 100.00% 24.17% - 92.23% 80.79%

Baa 0.0380% 0.0000% 0.7499% 19.74 -3.9653% 10.7130% 100.00% - 51.09% 95.84%

Aaa 0.0334% 0.0097% 0.5006% 14.97 -4.0076% 7.9405% 100.00% 38.39% -32.57% -33.72% 68.09% 88.05%

Aa 0.0164% 0.0147% 0.0964% 5.87 -0.8925% 0.9980% 100.00% -80.28% -68.16% 91.26% 68.36%

A -0.0259% 0.0000% 2.9434% -113.85 -17.2285% 20.3193% 100.00% 76.78% 75.88% 49.29%

Baa -0.0165% -0.0205% 1.2872% -78.19 -8.9365% 7.4074% 100.00% 74.02% 60.84%

Table 4: Bank Of America Merrill Lynch United States Bond Indices Logarithmic Daily Returns 1/2/97 to 12/19/11 (Source: Datastream )

Loadings on Systematic Factor

60.09%

75.10%

50.61%

Portfolio 3 - Datastream Sovereign Indices

Correlations

Portfolio 1 -Datastream Industrials Indices

Portfolio 2 - Datastream Financials Indices

Datastream Total Bond M

arket Index

Datastream

Sovereigns Index

Datastream

Financials Index

Datastream

Industrials Index

• High variation in returns relative to the mean & no clear pattern in CV• Correlations vary significantly between sectors and patterns with respect to nearness in rating not always intuitive• Factor loadings for 2-factor CM show that better ratings and financial / sovereign vs. industrial more sensitive to common factors


• …

Empirical Results and Discussion

• …Expected Loss - Credit Metrics

Credit VaR - Base Credit Metrics

Credit Capital - Base Credit Metrics

Credit VaR - Mutlifactor Credit Metrics

Credit Capital - Mutlifactor Credit Metrics

Expected Loss - Basel 2

Credit VaR - Basel 2 IRB

Credit Capital - Basel 2 IRB

Industrial Sector & Constant Risk 2.60% 6.65% 4.05% 16.45% 13.85% 2.60% 11.05% 8.45%Financial Sector & Constant Risk 0.00% 0.79% 0.79% 70.50% 70.50% 0.01% 0.34% 0.33%Sovereign Sector & Constant Risk 0.00% 0.47% 0.47% 66.49% 66.49% 0.01% 0.15% 0.14%Industrial Sector & Constant Portfolio 2.32% 6.39% 4.07% 16.81% 14.49% 2.33% 9.95% 7.62%Financial Sector & Constant Portfolio 0.08% 15.11% 15.04% 69.31% 69.24% 0.08% 1.47% 1.39%Sovereign Sector & Constant Portfolio 0.04% 12.09% 12.05% 67.58% 67.54% 0.04% 1.76% 1.72%Industrial Sector & Constant Risk 2.63% 7.17% 4.54% 16.30% 13.68% 2.63% 9.29% 6.66%Financial Sector & Constant Risk 2.17% 31.93% 29.76% 71.53% 69.36% 2.17% 11.34% 9.17%Sovereign Sector & Constant Risk 1.44% 28.74% 27.29% 66.76% 65.32% 1.45% 12.47% 11.03%Industrial Sector & Constant Portfolio 3.71% 9.76% 6.05% 16.15% 12.44% 0.93% 6.46% 5.53%Financial Sector & Constant Portfolio 2.17% 33.53% 31.36% 71.09% 68.92% 2.17% 10.12% 7.95%Sovereign Sector & Constant Portfolio 1.09% 26.82% 25.74% 66.98% 65.90% 1.09% 12.15% 11.06%

Thro

ugh-

the-

Cycl

e Ra

ting

sPo

int-

in-T

ime

Rati

ngs

Table 5: Incremental Risk Charge (IRC) for CreditMetrics vs. Basel II IRB Models

• Generally PIT & CP higher IRC vs. PIT & CR• EC-BCM financial / sovereign EC much smaller only for TTC and CR, else higher by

an order of magnitude • Universally EC-MCM higher order of magnitude vs. EC-BCM or RC-B2• Less differences across ratings or portfolio dynamics in the EC-MCM than in the EC-

BCM: factor loadings outweighs correlation matrix inputs

Empirical Results and Discussion (continued)

• …

Multifactor CreditMetrics Simulated Incremental Risk Charge (IRC) Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix

Datastream Industrial Bond Indices as of 4Q11 (Constant Risk - Moody's Quarterly Transitions 1982-2011)Credit Losses

Pro

bab

ility

-0.20 -0.15 -0.10 -0.05 0.00

05

10

15

20

B2-cVar999=11.05%CM-cVar999=16.45% EL=2.50%

CreditMetrics Simulated Incremental Risk Charge (IRC) Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix

Datastream Financial Bond Indices as of 4Q11 (Constant Risk - Moody's Quarterly Transitions 1982-2011)Credit Losses

Pro

ba

bility

-0.010 -0.008 -0.006 -0.004 -0.002 0.000

02

00

04

00

06

00

08

00

01

00

00

B2-cVar999=0.34%

CM-cVar999=0.79%

EL=0.00%

Multifactor CreditMetrics Simulated Incremental Risk Charge (IRC) Credit Loss Distribution: Through-the-Cycle Rating Migration Matrix

Datastream Financial Bond Indices as of 4Q11 (Constant Risk - Moody's Quarterly Transitions 1982-2011)Credit Losses

Pro

ba

bili

ty

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

01

23

45

B2-cVar999=0.34%CM-cVar999=70.50%

EL=0.00%

• This show the different loss distributions comparisons between EC and RC across industrial & financial sectors, for TTC ratings and CR

Empirical Results and Discussion: Bayesian Estimation

• …

• In order to implement this we need prior distributions for correlation and default rate parameters – shown here are the ones for industrial sector and through-the cycle ratings

• We implement a non-parametric bootstrap of the bond index history to derive prior distributions for the correlation

• In the case of default rates, we fit the history time series to Vasicek distributions, and use those as prior

Prior Distribution of Correlation: Rating Aa & A

Industrial Bond Return Indices: Datastream 1997-2011(mean=0.361,stdev=0.014)Correlation

Pro

babi

lity

Den

sity

0.2 0.3 0.4 0.5 0.6 0.7

02

46

8

Prior Distribution of Correlation: Rating Aa & Ba


Pro

babi

lity

Den

sity

0.0 0.1 0.2 0.3 0.4

02

46

Prior Distribution of Correlation: Rating Aa & Caa


Pro

babi

lity

Den

sity

-0.1 0.0 0.1 0.2 0.3 0.4

02

46

810

Prior Distribution of Correlation: Rating A & Ba


Pro

babi

lity

Den

sity

-0.1 0.0 0.1 0.2 0.3 0.4

02

46

8

Prior Distribution of Correlation: Rating A & Caa


Pro

babi

lity

Den

sity

0.0 0.1 0.2 0.3 0.4 0.5

02

46

8

Prior Distribution of Correlation: Rating Ba & Caa


Pro

babi

lity

Den

sity

0.6 0.7 0.8 0.9 1.0

02

46

8

0e+00 2e-04 4e-04 6e-04 8e-04 1e-03

0e+0

04e

+05

8e+0

5

Prior Distribution of Default Rate: Rating Aaa

Through-the-Cycle / Industrial / Quarterly:Moody's 1982-2011(theta=0.0001,rho=0.010)Default Rate

Pro

babi

lity

Den

sity

0e+00 2e-04 4e-04 6e-04 8e-04 1e-03

0e+0

04e

+05

8e+0

5

Prior Distribution of Default Rate: Rating Aa


Pro

babi

lity

Den

sity

0.0000 0.0005 0.0010 0.0015 0.0020

0e+0

02e

+05

4e+0

5

Prior Distribution of Default Rate: Rating A


Pro

babi

lity

Den

sity

0.0000 0.0005 0.0010 0.0015 0.0020

0e+0

02e

+05

4e+0

5

Prior Distribution of Default Rate: Rating Baa


Pro

babi

lity

Den

sity

0.000 0.005 0.010 0.015 0.020

050

0015

000

Prior Distribution of Default Rate: Rating Ba


Pro

babi

lity

Den

sity

0.00 0.01 0.02 0.03 0.04

020

0040

0060

00

Prior Distribution of Default Rate: Rating B


Pro

babi

lity

Den

sity

0.00 0.05 0.10 0.15

050

010

0015

00

Prior Distribution of Default Rate: Rating Caa


Pro

babi

lity

Den

sity

Empirical Results and Discussion: Bayesian Estimation (continued)

• …

• Numerical coefficients of variation in the posterior distribution of the IRC are almost all higher in the CM-EC than the B2-RC model

• This pattern is accentuated for the financial / sovereign sector & PIT ratings, but otherwise no clear pattern

• Industrials have generally higher CV than financial / sovereign in the case of EC-CM, but this is reversed for B2RC

• Overall, in both the CM and regulatory model, there is a high degree of uncertainty in the estimates, more so for TTC vs. PIT ratings

CreditMetrics Credit VaR - Point Estimate

CreditMetrics Credit VaR - Posterior Mean

CreditMetrics Credit VaR - Posterior Standard Error

CreditMetrics Credit VaR - 25th Percentile

CreditMetrics Credit VaR - 95th Percentile

CreditMetrics Credit VaR - Numerical Coeffi cient of Variation

Basel II IRB Credit VaR - Point Estimate

Basel II IRB Credit VaR - Posterior Mean

Basel II IRB Credit VaR - Posterior Standard Error

Basel II IRB Credit VaR - 25th Percentile

Basel II IRB Credit VaR - 95th Percentile

Basel II IRB Credit VaR - Numerical Coeffi cient of Variation

Industrial Sector & Constant Risk 6.65% 7.56% 10.68% 0.34% 33.32% 4.3645 11.05% 11.07% 6.82% 0.34% 28.18% 2.7982Financial Sector & Constant Risk 0.79% 0.80% 1.60% 0.0005% 2.93% 3.6801 0.34% 0.34% 0.71% 0.0005% 2.14% 6.2132Sovereign Sector & Constant Risk 0.47% 0.47% 0.73% 0.0020% 1.66% 3.5327 0.15% 0.15% 0.37% 1.96E-05 1.04% 7.0416Industrial Sector & Constant Portfolio 6.39% 7.63% 10.85% 0.33% 33.69% 4.3721 9.95% 9.93% 6.33% 0.33% 25.95% 2.4247Financial Sector & Constant Portfolio 15.11% 16.11% 20.95% 0.02% 91.13% 5.6569 1.47% 1.47% 1.91% 0.02% 1.47% 4.5507Sovereign Sector & Constant Portfolio 12.09% 12.90% 18.04% 0.01% 73.65% 5.7077 1.76% 1.76% 2.16% 0.01% 7.69% 4.3491Industrial Sector & Constant Risk 7.17% 8.08% 11.46% 0.34% 36.81% 4.5120 9.29% 9.31% 6.09% 0.34% 24.63% 2.4703Financial Sector & Constant Risk 31.93% 35.52% 32.25% 0.57% 100.00% 2.7989 11.34% 11.39% 6.96% 0.57% 28.65% 2.3323Sovereign Sector & Constant Risk 28.74% 31.85% 30.81% 0.37% 100.00% 3.1280 12.47% 12.44% 7.38% 0.37% 30.62% 2.2489Industrial Sector & Constant Portfolio 9.76% 8.04% 11.33% 0.33% 36.50% 4.4969 6.46% 6.45% 4.78% 0.33% 18.88% 2.7833Financial Sector & Constant Portfolio 33.53% 35.44% 32.18% 0.54% 100.00% 2.8065 10.12% 10.13% 6.45% 0.54% 26.24% 2.4084Sovereign Sector & Constant Portfolio 26.82% 29.79% 29.92% 0.28% 100.00% 3.3479 12.15% 12.12% 7.25% 0.28% 29.94% 2.2643

Table 6: Incremental Risk Charge (IRC) for CreditMetrics vs. Basel II IRB Models- Bayesian Analysis

Thro

ugh-

the-

Cycl

e Ra

ting

sPo

int-

in-T

ime

Rati

ngs

Empirical Results and Discussion: Bayesian Estimation (continued)

• …

Bayesian Posterior of CreditMetrics Incremental Risk Charge (IRC) Credit Value-at-Risk:

C-VaR

Pro

ba

bilit

y

0.0 0.1 0.2 0.3 0.4 0.5 0.6

05

10

15

20

25

30

CM.cVar999=6.64%

CM.cVar999.mu.pst=7.56%

CM.cVar999.q025.pst=0.34%

CM.cVar999.q0975.pst=33.32%

CM.cVar999.CV.pst=4.36, CM.cVar999.StdDev.pst=11.08%

Bayesian Posterior of Basel II IRB Incremental Risk Charge (IRC) Credit Value-at-Risk

Datastream Industrial Bond Indices as of 4Q11 (Constant Risk: Through-the-Cycle Rating Migration Matrix - Moody's Quarterly Transitions 1982-2011)C-VaR

Pro

ba

bilit

y

0.0 0.1 0.2 0.3 0.4 0.5

02

46

81

0

B2.cVar999=11.05%B2.Var999.mu.pst=11.08%

B2.cVar999.q025.pst=0.34%

B2.cVar999.q0975.pst=28.32%

B2.cVar999.CV.pst=2.36, B2.cVar999.StdDev.pst=6.82%

• For industrials and TTC ratings, this shows the much longer tail in the posterior distribution of the C-VaR for EC vs. RC

• Note also the very different shapes: EC-CM has a mode at 0 and rapidly drops off, whereas for RC-B2 it has a mode further right

Directions for Future Research • Implementing a multifactor CM for a realistic, large portfolio of

equities/bonds held by a bank comparable to reported to regulators• Estimating the models herein for indices of different instruments

such as credit default swaps (CDS)• Extension of the Bayesian estimation framework: expert elicitation of

priors and to multifactor CM• Extension of CM to different copula models for the vector of

systematic factors (e.g., Student-T, Archimedean) • Consideration of challenger / benchmark credit risk models (e.g.,

actuarial: CreditRisk+, reduced-form: Jarrow-Chava)• Modeling of stressed VaR by various means (historical simulation vs.

extreme value theory-EVT)• Development of a theoretical framework that would endogenously

give rise to a process for the measure of risk in a trading portfolio

Empirical Analysis of Bank Capital and New Regulatory Requirements for Risks in Trading Portfolios

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