Risk Aggregation Inanoglu Jacobs 6 09 V1

Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital

Hulusi Inanoglu and Michael Jacobs, Jr.

Enterprise and Credit Risk Analysis Divisions

June 2009The views expressed herein are those of the authors and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.

Outline• Background and Motivation

• Introduction and Conclusions

• Review of the Literature

• Methodology

• Data and Summary Statistics

• Empirical Results

• Summary and Future Directions

Background and MotivationCentral challenge to enterprise risk measurement and management faced by

diversified financial institutions: a coherent approach to aggregating different risk types.

• Impetus from rapid financial innovation, evolving supervisory standards (Basel 2) and now recent financial crises

• Main risks faced (market, credit and operational) have distinct distributional properties & historically modeled differently

• Extend the scope of the analysis by analyzing A/L mismatch and liquidity risk (Pillar II of IRB framework implications)

• Utilize actual data representative of major banking institutions’ loss experience (call reports)– Explore effect of business mix & inter-risk correlations on total risk

• Apply copula methods for capturing realistic distributional features of & combining different risk types– Compare different copula frameworks (including goodness-of-fit to the data) &

evaluate sensitivity to sampling error

Background and Motivation (continued)

ICAAP: Internal Capital Adequacy Assessment Process • Not a model for economic capital (EC), but a bank’s overall

framework and mechanism for assessing if EC is appropriate • EC may be a quantitative component of ICAAP, but it is not required

of all banks by supervisors (only the largest)• All banks must perform Stress Testing, which includes analysis

around the impact on EC from the following:– Scenario Analysis: extreme broad systematic events (or high quantiles of

underlying risk factors)– Sensitivity Analysis: variation in key parameters due to sampling error or

uncertainty or different specifications of the model

• The contribution of this work is in the latter, as we explore the variability of EC due to underlying statistical noise (sampling error) and to alternative models (specification of copula)

Summary and Conclusions• Estimated loss distributions for 5 largest banks as of 4Q08 (&

Top 200) using quarterly Call Report data 1984-2008 – Proxy for 5 risk types with financials: credit (GCO), operational (ONIE),

market (4QDNTR), liquidity (4QDLGD) & interest income (4QDIG)– Different risk aggregation methodologies: historical bootstrap (empirical

copula), Normal approx., Copulas (Gaussian,Student-t,Archimadean)

• Empirical copula (normal approx.) is found to be most (least) conservative (contrary to asymptotics) & most (least) stable in bootstrap experiment vs. standard copula methods – But EC implies significantly greater proportional diversification benefits

• Document significant differences across banks & aggregation methodologies in absolute risk measures & diversification benefits (ranging 10% to 60%)

• Simple addition over-states risk relative to standard copula formulations by about 30%-20%

Summary and Conclusions (continued)

• Goodness-of-fits tests are mixed across copula models, but in many cases show evidence of poor fit to the data

• Fail to find the effect of business mix to exert a directionally consistent an impact on total integrated diversification benefits

• In a bootstrapping experiment, find the variability of the VaR to be significantly lower (higher) for the empirical & Gaussian copula than other formulations (Normal approximation)

• Find that the contribution of the sampling error in the parameters of the marginal distributions to be an order or magnitude greater than that or the correlation matrices.

• Results constitute a sensitivity analysis that argues for practitioners to err on the side of conservatism in considering a non-parametric EC approach to quantify integrated risk

• .

Review of the Literature• Sklar (1956): mathematical foundation of copula methodology

– Existence of a copula to connect any set of marginal distributions• Embrechts (1999, 2002): first applications to risk management

– Li (2000): credit risk management• Frey & McNeil (2001): copulas as a generalization of dependence according

to linear correlations– Motivation for applying the technique to understanding tail events

• Poon (2001): alternative of a data intensive multivariate extension of extreme value theory (need joint tail events)

• Most finance applications in portfolio risk measurement: Bouye (2001), Longin and Solnik (2001) and Glasserman et al (2002)

• Embrechts et al (2003): reviews & extends recent results on distributional bounds for functions of dependent risks– Main emphasis on Value-at-Risk as a risk measure

• Ward and Lee (2002): joint loss distributions (pair-wise roll-ups Gaussian copula marginal distributions) analytical & numerical

• Kuritzkes et al. (2003): financial conglomerate & Gaussian copula for a large set of diversification results

Review of the Literature (continued)

• Dimakos and Aas (2004): bank with life insurance subsidiary (risk = conditional marginal + unconditional credit risk)– Imposing conditional independence through set of sufficient conditions such that

only pair-wise dependence remains

• Schuermann & Rosenberg (2006): integrated risk management for typical large, internationally active financial institution– Copula approach for aggregating 3 main risk types (market, credit & operational)

where the distributional properties varies widely – Impact of business mix and inter-risk correlations on total risk: former found more

important (“good news” for supervisors)– Compare various simplified approaches applied by practitioners (variance-

covariance approach & regulatory addition approach)

• Aas (2007): incorporates ownership risk from a life insurance subsidiary & combines a base-& top-level aggregation– Risk factors: multivariate GARCH model with Student-t errors

– The model, originally developed DnB Nor is adapted to of Basel II • Genest et al (2009): reviews literature on goodness-of-fit tests for copula

models and proposes a “blanket” test with good size/power properties

Methodology: Value-at-Risk

• Consider a single-valued function (simple sum of losses) of the risk factors (dollar losses: e.g., P&L, credit losses) from time t to t + Δ (Δ = horizon):

• The Value-at-Risk at the confidence level α between times t and t + Δ (Δ is the horizon) is related to the αth quantile of F(π(X)) as and denoted by:

• Vector of K risk factors at time t having a joint distribution function.

1 ,..,t t KtX XX 1 1Pr ,.., K KF X x X x X X

: K , ,1

k

t t t t ii

X

X

,

1

t tVaR F

X

X

• Serious issues with VaR: coherence (Artzner 1997, 1999), loss of information vs. focusing on entire distribution (Diebold et al,1998; Christoffersen and Diebold, 2000; Berkowitz, 2001), possibility for unbounded concentration risk & “gaming” (Embrechts et al. 1999, 2002).

• Therefore, we also look at the expected shortfall (ES), measuring expectation of the risk exposure conditional upon exceeding a VaR threshold:

Et tt t tES VaR

X XX X

, ,

Pr 1 1t t t tt VaR F VaR

X X

XX

Value-at-Risk: The Variance-Covariance Approximation

• Note simply summing losses so no portfolio weights so that standard deviation of horizon losses is the root of the simple quadratic form:

• Interesting & ubiquitous special case (motivated by Markowitz (1959) investment theory), seen in many practical EC frameworks (HSBC, 2008), where risk factors have a valid variance-covariance matrix & are multivariate Gaussian (or risk managers/investors do not care about moments > 2nd) :

• Under the assumptions that minimizing the variance of the total loss is the object, NVaR (N=“normal”) is proportional to the standard deviation of the position according to the quantile of the standard normal distribution:

VT

t t t t t t t t t t

Σ X E X E X X E X

,1

KT

t t K t iti

X

i X 2, , , , ,

1

2K K

Tt t t k t k t i t ij t i t j

i i j

S

i Σ i

1,t t t tNVaR S

X

X

• Case in which the standardized distribution of the positions is the same as that of the total loss yields “Hybrid Value-at-Risk” (HVaR) as follows

, , ,

21 1 1

,1

2t t i t i t j

K K

t ijX X Xi i j

HVaR F F F

X

Value-at-Risk: The Variance-Covariance Approximation (continued)

• The case in which we assume risk factors or losses to be perfectly correlated we call “Perfectly-correlated Value-at-Risk” (PVaR):

• The case in which we assume risk factors or losses to be uncorrelated we call “Uncorrelated Value-at-Risk” (UVaR):

• Obviously that in this framework and in a “mean-variance world”, PVaR (UVaR) forms an upper (lower) bound on the HVaR measure of risk:

, 0,1t t t

KUVaR HVaR PVaR

X X X X

,

21

1t t i

K

Xi

UVaR F

X

,

1

1t t i

K

Xi

PVaR F

X

Methodology: The Method of Copulas

• If the joint distribution is continuously differentiable to the kth degree, that is sufficient for the copula to exist and be unique• Frechet-Hoeffding boundaries for copulas: minimum (maximum) copula, the case of perfect inverse (positive) dependence

amongst random variables:

• Fundamental result (Sklar, 1956): under the appropriate & general mathematical regularity conditions) any joint distribution can be expressed in terms of a copula (or dependence) function & set of marginal distributions. • If we have a K-vector of risk factors, then a copula is a multivariate joint distribution defined on the K-dimensional unit cube, such that each marginal distribution is uniformly distributed on the unit interval: .

• Four technical conditions sufficient for a copula to exist (Nelson, 1999):

:[0,1] [0,1]KC 1 1( ),.., ( )

KX X KF C F x F xX x

0 0iu C u 1 1 1 1,.., , ,.., , 1i i K K i iu u u u u C u i u

1 , 0,1KK

i i iC is K on B x y u

1

( )

,

1 ( ) 0 cardKi i i

N

C K Kx y

V B C where N K z x

z

z

z z

1,..

1

max 1 minK

i jj K

i

W K u C u M

u u u

Methodology: The Method of Copulas (continued)

• Note that P is not necessarily the correlation matrix of X, but in this context the Spearman rank-order correlations of the transformed variables (in cases of other copulas this may a different dependence measure of dependence)

• While for a random vector having a valid joint distribution function the copula will always exist, there is no guarantee that it will be unique.• May always construct a copula for any multivariate distribution according to the method of inversion

– Intuitively: removing the effects of the marginal distributions on dependence relation by substituting in the marginal quantile functions in lieu of the arguments to the original distribution function • If we have a random vector in the kth hyper-unit, them we may write the copula as a function as this as follows:

• Consider a rather common choice of copula function, the Gaussian copula, simply a multivariate standard normal distribution with covariance matrix P:

1,.., 0,1K

Ku u u

1

11 11

2 2

1,.., ; exp

2

Kx xT

K KKx x dx dx

X Xx Ρ x Ρ xΡ

1 1( ),.., ( );X Xi K

G i KC F u F u Xu Ρ

1 1( ),.., ( )X Xi K

i KC F F u F u Xu


• Computationally equivalent to historical simulation method of simply resampling the observed history of joint losses with replacement (or bootstrapping)– Historically, this was on of the standard method for computing VaR for trading positions amongst market risk department practitioners.

• Another commonly employed and closely related choice of copula in the elliptical family is the t-copula with degrees-of-freedom ν:

• Often neglected but fundamental & interesting: empirical copula, a useful tool where there is high uncertainty on the underlying data distribution– Procedure: transform the empirical data distribution into an "empirical copula" by warping such that the marginal distributions become uniform

• Mathematically the empirical copula frequency function has the following representation:

1 1( ),.., ( ); ,X Xi K

T i KC T F u F u u Q

jth

jix is the i order statistic of x

11

1

1,.., # ,.., . .

j

KK

E K j ij

i iC x x s t x x

K K K


• Where the generator function is indexed by a parameter θ, a whole family of copulas may be Archimedean, as in the Clayton copula:

• An important class of copulas: Archimadean family, having simple forms with properties (e.g., associativity) & a variety of dependence structures• Unlike elliptical copulas, most have closed-form solutions and are not derived from the multivariate distribution functions using Sklar’s Theorem• One particularly simple form of k-dimensional Archimadean copula having generator function (satisfying certain conditions):

• Several special cases of note. In the product (independent) copula there is no dependence between variates (i.e., density function is unity everywhere):

:[0,1]

1 0 0

limx

x

0d x

dx

2

20

d x

dx

1

1

( )i

K

A X ii

C F x

x

1

( )i

K

AI X ii

C F x

x lnx x

1

1

( ) 1i

K

AC X ii

C F x

x 1x x

• Where parameter θ=0 we have the case of statistical independence • The Clayton copula exhibits negative tail dependence


• Another commonly employed copulas in the Archimadean family include the Gumbel copula (having the property of positive tail dependence):

• Finally, we consider the Frank copula (having the property of neither positive nor negative tail dependence):

• We may simulate realizations from a multivariate distribution by generating independent random vectors • For example, in the Gaussian case, it is either standard normal and independent random variables that we generate • With knowledge of the marginal distributions of the risk factors (which can be estimated either parametrically or non-parametrically), we can derive a rank-order correlation matrix of the transformed marginal

data• We can make our independent random vectors correlated (by means of a Cholesky decomposition, for instance)

1

1

exp ln ( )i

K

AG X ii

C F x

x lnx x

( )

1

1 1ln 1 exp ln 1

1

X iiF xK

AFi

eC e

e

x 1

ln1

xex

e

Data Description• Quarterly call report data for top 200 banks 1Q84-4Q08

• Corrected for mergers & acquisitions: legacy banks synthetically added into currently surviving banks on pro forma basis

• Proxy for 5 risk types using financial statement data• Credit Risk (CR): gross charge-offs (“GCO”)• Operational Risk (OR): total other non-interest expense

(“ONIE”)• Market Risk (MR): (minus of) net trading revenues deviation

from moving 4 quarter moving average (“NTR-4QD”)• Liquidity Risk (LR): liquidity gap (total loans minus total

deposits) deviation from 4 quarter moving average (“LR-4QD”)• Interest Rate Risk (IR): interest rate gap (interest expense on

deposits minus interest income on loans) deviation from 4 quarter moving average (“IRG-4QD”)

Empirical Results: Summary Statistics (Call Report Data)

Book Value of Total Assets

Book Equity

Book Value of Total Debt

Book Leverage

Ratio2Lending Assets

Percent Lending Assets

Trading Assets

Percent Trading Assets

Total Chargeoffs

Chargeoff

Ratio3

Net Interest Income

Net-Interest Margin

Non-Performing Assets

Non-Performing Assets

Ratio4Trading Revenue

Non-Interst Income

Non-Interest Expense

Other Non-Interest Expense

Aggregate Top 200 Banks 10,758.51 1,007.19 9,751.33 90.64% 5,737.07 53.33% 964.24 8.96% 88.01 1.53% 289.33 5.04% 188.15 3.28% -0.99 189.28 298.53 116.37

JP Morgan Chase 1,849.65 152.69 1,696.96 91.74% 738.44 39.92% 365.71 19.77% 10.75 1.46% 43.38 5.88% 29.23 3.96% 5.02 41.78 46.35 14.40

Bank of America 1,699.71 178.72 1,520.99 89.49% 900.99 53.01% 155.64 9.16% 17.60 1.95% 46.35 5.14% 27.82 3.09% -0.35 29.94 36.13 13.67

Citigroup 1,319.45 101.46 1,217.99 92.31% 620.12 47.00% 200.52 15.20% 15.55 2.51% 35.40 5.71% 28.67 4.62% -4.49 12.28 38.97 17.05

Wells Fargo 1,236.36 105.62 1,130.74 91.46% 792.49 64.10% 52.08 4.21% 7.52 0.95% 35.40 4.47% 25.50 3.22% 0.35 22.16 32.89 12.21

PNC 289.88 25.25 264.62 91.29% 180.79 62.37% 6.09 2.10% 0.62 0.34% 7.22 4.00% 29.23 16.17% -0.13 3.35 9.38 2.85

5th Percentile 2.87 0.24 2.40 83.65% 1.94 39.64% 0.00 0.00% 0.0016 0.04% 0.07 0.67% 0.01 0.36% -0.07 0.00 0.00 0.00

25th Percentile 3.90 0.38 3.44 88.20% 2.66 62.06% 0.00 0.00% 0.01 0.36% 0.12 0.94% 0.05 1.18% 0.00 0.01 0.00 0.01

Average Bank 53.79 5.04 48.06 89.35% 28.69 66.63% 4.82 1.38% 0.44 1.27% 1.45 1.24% 0.94 3.40% 0.00 0.17 0.65 0.16

Median Bank 7.04 0.70 6.35 90.09% 4.38 69.53% 0.00 0.00% 0.04 0.76% 0.20 1.11% 0.11 2.37% 0.00 0.01 0.02 0.02

75th Percentile 15.47 1.65 14.21 91.81% 10.34 75.47% 0.05 0.40% 0.14 1.38% 0.48 1.31% 0.35 3.94% 0.00 0.06 0.10 0.04

95th Percentile 162.91 15.36 152.86 93.83% 92.30 86.12% 6.65 5.25% 1.38 3.93% 3.98 1.85% 2.06 9.75% 0.06 0.70 1.11 0.69

Standard Deviation 218.78 19.64 10.88 4.97% 109.41 15.40% 31.88 5.46% 1.92 1.95% 5.86 0.82% 3.94 4.60% 0.54 0.81 7.99 0.64

Skewness 6.76 6.97 1.99 2.46 6.52 -1.45 9.13 7.28 7.20 6.44 6.59 7.16 6.52 5.61 0.81 8.09 22.03 6.99

Kurtosis 47.29 51.79 52.88 2.99 43.83 3.18 92.04 61.85 55.54 60.24 44.00 64.32 42.57 45.17 64.67 79.00 513.64 53.64

3 - Defined as the ratio of gross-charegeoffs to total lending assets.

4 - Defined as the ratio of non-performing assets to total lending assets.

Table 1.1: Summary Statistics on Characteristics of Top 200 and 5 Largest Banks by Asset Size (Call Report Data As of 20081 )

1 - Dollar amounts expressed in billions.

2 - Defined as the ratio of the book value of total debt to the book value of total assets.

Empirical Results: Summary Statistics (Call Reports & CRSP)

Book Value of Total Assets

Market Value of Equity

Book Equity

Book Value of Total Debt

Quasi-Market Value of

Assets3

Book Leverage

Ratio4

Market Leverage

Ratio5

Market to Book

Ratio6

Aggregate Banks2 9,179.99 644.10 838.58 8,341.40 8,985.50 90.87% 92.83% 97.88%

JP Morgan Chase 1,849.65 117.68 152.69 1,696.96 1,814.64 91.74% 93.51% 98.11%

Bank of America 1,699.71 70.65 178.72 1,520.99 1,591.63 89.49% 95.56% 93.64%

Citigroup 1,319.45 36.57 101.46 1,217.99 1,254.55 92.31% 97.09% 95.08%

Wells Fargo 1,236.36 109.92 105.62 1,130.74 1,240.66 91.46% 91.14% 100.35%

PNC 289.88 17.05 25.25 264.62 281.67 91.29% 93.95% 97.17%

5th Percentile 2.88 0.03 0.26 2.56 2.83 86.03% 81.07% 91.14%

25th Percentile 3.96 0.31 0.38 3.61 4.15 88.43% 86.33% 94.26%

Average Bank 74.63 5.24 6.82 67.82 73.05 90.10% 89.86% 103.04%

Median Bank 8.36 0.73 0.72 7.14 8.07 90.28% 90.74% 98.81%

75th Percentile 16.32 1.86 1.64 14.74 16.76 91.72% 95.52% 104.24%

95th Percentile 205.15 30.09 21.63 189.39 219.72 93.75% 99.33% 112.07%

Standard Deviation 275.91 16.87 24.74 251.42 266.72 2.46% 9.75% 28.99%

Skewness 5.3050 5.1706 5.4962 5.3000 5.2794 -0.4452 -3.8215 7.6426

Kurtosis 28.3161 29.1232 31.4196 28.2416 28.1107 1.0313 21.7520 63.3658

Table 1.2: Summary Statistics on Market Value Characteristics of Banks by Asset

Size (Call Report and CRSP Data As of 20081 )

5 - Defined as the ratio of the book value of total debt to the quasi-market value of assets (defined in 3).6 - Defined as the ratio of the quasi-market value of assets (defined in 3) to book value of total total assets.

1 - Dollar amounts expressed in billions.2- 123 out of the 200 top banks by book value of assets for which we could match to CRSP as of 4Q08.3 - Defined as the market value of equity plus the book value of total debt.4 - Defined as the ratio of the book value of total debt to the book value of total assets.

Empirical Results: Summary Statistics (Call Report Variables)

-5 0 5 10 15 20

x 108

0

0.5

1

1.5

2x 10

-8 Fig. 1.1.1: Book Value of Assets (BVA)

0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15Fig. 1.1.2: Book Leverage Ratio (BLR)

-0.2 0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4Fig. 1.1.3: Percent Lending Assets (PLA)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2Fig. 1.1.4: Percent Trading Assets (PTA)

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

20

40

60

80Fig. 1.1.5: Gross Charge-off Ratio (GCOR)

-0.02 0 0.02 0.04 0.06 0.08 0.10

50

100

150Fig. 1.1.6: Net Interest Margin (NIM)

Figure 1.1; Call Report Variables: 200 Largest Banks by Book Value (As of 4Q08)

Empirical Results: Summary Statistics (Call Report & CRSP)

-5 0 5 10 15 20

x 108

0

1

2

3

4x 10

-8 Fig. 1.2.1: Quasi Market Value of Assets (QMVA)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

2

4

6Fig. 1.2.2: Market Leverage Ratio (MLR)

0.5 1 1.5 2 2.5 3 3.5 40

2

4

6Fig. 1.2.3: Market-to-Book Ratio (MTBR)

Figure 1.2: Call Report & CRSP Variables - 123 of 200 Largest Banks by Book Value (As of 4Q08)

Empirical Results: Summary Statistics (Risk Proxies)

Minimum5th Percentile

25th Percentile Mean Median

75th Percentile

95th Percentile Maxmimum

Standard Deviation Skewness Kurtosis

Gross Chargeoffs2 1.92 3.29 5.24 7.89 6.66 9.88 13.60 31.16 4.51 2.4931 9.0434Non-Interest Expense3 6.20 8.55 12.85 18.47 17.55 24.20 28.64 33.10 6.60 0.1041 -0.9597Net Trading Revenue4 -7.20 -1.79 -0.62 0.01 -0.13 0.25 1.30 16.13 2.14 4.5929 35.3144Liquidity Gap5 -159.68 -112.11 -66.48 -20.10 -20.50 26.51 75.92 375.83 72.07 1.5928 7.8773Interest Rate Gap6

-171.72 -89.86 -57.76 -2.62 7.34 59.23 85.11 153.01 64.80 -0.0682 -0.6063Gross Chargeoffs 0.79 1.07 1.38 1.96 1.74 2.32 3.59 4.53 0.82 1.2623 1.3433Non-Interest Expense 2.15 2.80 3.47 4.29 4.08 5.01 5.96 7.00 1.06 0.3278 -0.3711Net Trading Revenue -1.59 -0.96 -0.32 -0.03 -0.09 0.21 0.87 3.65 0.63 1.9651 11.1842Liquidity Gap -82.87 -26.97 -12.15 2.28 -0.76 13.25 38.46 88.94 24.92 0.4594 2.2183Interest Rate Gap -30.86 -16.66 -8.67 -0.13 0.92 8.46 14.76 25.60 10.92 -0.1630 -0.3518Gross Chargeoffs 0.85 1.21 1.52 2.10 1.91 2.43 3.91 5.81 0.87 1.6790 3.8429Non-Interest Expense 2.81 3.19 3.71 4.14 4.05 4.53 5.20 5.98 0.62 0.3662 0.1090Net Trading Revenue -2.43 -0.36 -0.08 0.00 -0.01 0.05 0.25 4.38 0.61 2.8475 31.4616Liquidity Gap -65.05 -43.17 -20.41 -6.63 -6.85 5.92 35.65 84.95 24.59 0.5063 1.3789Interest Rate Gap -34.99 -17.06 -9.37 -0.42 0.54 9.03 15.35 31.71 12.12 -0.1040 -0.0776Gross Chargeoffs 0.17 0.48 0.91 1.58 1.11 2.19 3.55 4.96 1.01 1.0753 0.2524Non-Interest Expense 0.69 1.67 2.21 3.04 2.58 3.72 5.21 6.43 1.20 0.7202 -0.1864Net Trading Revenue -3.87 -0.39 -0.16 0.03 -0.02 0.06 0.43 9.09 1.04 6.2887 60.3677Liquidity Gap -34.38 -20.14 -10.08 -0.12 -1.93 7.55 20.76 90.05 17.96 2.2629 9.0858Interest Rate Gap -15.34 -11.93 -6.42 -0.58 -0.91 5.83 11.39 14.60 7.32 -0.0244 -0.9156Gross Chargeoffs 0.45 0.60 0.83 1.15 1.00 1.33 2.21 3.50 0.52 1.7393 4.0686Non-Interest Expense 1.94 2.30 2.69 2.99 2.90 3.23 4.03 6.63 0.59 2.6639 14.1990Net Trading Revenue -0.56 -0.12 -0.03 0.00 -0.01 0.02 0.12 0.75 0.13 2.0058 16.6127Liquidity Gap -41.58 -29.25 -12.26 -5.28 -4.91 0.29 19.90 30.69 13.30 0.0705 0.5398Interest Rate Gap -24.80 -12.17 -6.31 -0.24 0.86 6.97 11.49 22.39 8.48 -0.1075 0.0474Gross Chargeoffs 0.10 0.17 0.21 0.33 0.27 0.38 0.68 1.33 0.20 2.4750 7.7287Non-Interest Expense 0.42 0.57 0.75 0.84 0.84 0.93 1.07 1.23 0.15 0.0660 0.4106Net Trading Revenue -0.21 -0.03 -0.01 0.00 0.00 0.00 0.04 0.21 0.04 0.2049 22.0198Liquidity Gap -17.36 -10.17 -5.46 -1.59 -1.85 1.78 8.07 23.62 6.35 0.7620 2.3140Interest Rate Gap -7.58 -3.76 -1.98 -0.02 0.58 2.43 3.52 7.26 2.80 -0.1077 -0.3254

2 - Gross charge-offs (GCO) is our proxy measure credit risk (CR).3 - Other non-interest expense (ONIE) is our proxy measure of operational risk (OR).4 -The deviation to the trailing 4-quarter average in net-trading revenues (NTR-4QD) is our proxy measure of market risk (MR).5 -The deviation to the trailing 4-quarter average of the liquidity gap, defined as total loans minus total deposits, our proxy measure of liquidity risk (LG-4QD).6 -The deviation to the trailing 4-quarter average of the interest rate gap, defined as total interest expense minus total interest income, our proxy measure of interest rate risk (IRG-4QD).

Table 1.3: Summary Statistics on Risk Measures for Top 200 and 5 Largest Banks by Asset Size (Call Report Data 1984-2008)1

Top

200

B

anks

JPM

C

1 - Dollar amounts expressed in billions.

Ban

k of

A

mer

ica

Citi

grou

pW

ells

Far

goP

NC

Historical Quarterly Risk Proxies: Loss Distributions (1984-2008)

-2 0 2 4

x 107

0

1

2x 10

-7 Credit Risk: GCO

-2 0 2 4 6

x 107

0

5x 10

-8 Op Risk: ONIE

-1 0 1 2

x 107

0

0.5

1x 10

-6Market Risk:NTR-4QD

-5 0 5

x 108

0

0.5

1x 10

-8 Liquidity Risk: LG-4QD

-4 -2 0 2 4

x 108

0

5x 10

-9Interest Rate Risk:

IRG-4QD

-5 0 5 10

x 108

0

2

4x 10

-9Total Risk: Sum of

Cr.,Ops.,Mkt.,Liqu.&Int.

Historical Loss Distributions (200 Largest Banks 1984-2008)

-2 0 2 4 6

x 106

0

0.5

1x 10

-6 Credit Risk: GCO

-5 0 5 10

x 106

0

5x 10

-7 Op Risk: ONIE

-5 0 5 10

x 106

0

2

4x 10


-1 0 1 2

x 108

0

2

4x 10


-4 -2 0 2 4

x 107

0

5x 10


IRG-4QD

-1 0 1 2

x 108

0

2

4x 10



Historical Loss Distributions (Citibank 1984-2008)

-2 0 2 4 6

x 106

0

0.5

1x 10

-6 Credit Risk: GCO

0 5 10

x 106

0

2

4x 10

-7Op Risk: ONIE

-4 -2 0 2

x 106

0

0.5

1x 10


-2 -1 0 1 2

x 108

0

1

2x 10


-5 0 5

x 107

0

2

4x 10


IRG-4QD

-2 -1 0 1 2

x 108

0

1

2x 10



Historical Loss Distributions (JP Morgan Chase 1984-2008)

0 2 4 6 8

x 106

0

0.5

1x 10

-6 Credit Risk: GCO

2 4 6 8

x 106

0

0.5

1x 10

-6 Op Risk: ONIE

-5 0 5

x 106

0

2

4x 10

-6 Market Risk:NTR-4QD

-1 0 1 2

x 108

0

1

2x 10


-10 -5 0 5

x 107

0

2

4x 10


IRG-4QD

-2 -1 0 1 2

x 108

0

1

2x 10



Historical Loss Distributions (Bank of America 1984-2008)

-2 0 2 4

x 106

0

1

2x 10

-6 Credit Risk: GCO

0 2 4 6 8

x 106

0

1

2x 10

-6 Op Risk: ONIE

-1 -0.5 0 0.5 1

x 106

0

0.5

1x 10


-10 -5 0 5

x 107

0

2

4x 10

-8Liquidity Risk: LG-4QD

-4 -2 0 2 4

x 107

0

2

4x 10


IRG-4QD

-1 0 1 2

x 108

0

2

4x 10



Historical Loss Distributions (Wells Fargo 1984-2008)

-5 0 5 10 15

x 105

0

5x 10

-6 Credit Risk: GCO

0 5 10 15

x 105

0

2

4x 10

-6 Op Risk: ONIE

-4 -2 0 2 4

x 105

0

5x 10

-5 Market Risk:NTR-4QD

-4 -2 0 2 4

x 107

0

0.5

1x 10


-2 -1 0 1 2

x 107

0

1

2x 10


IRG-4QD

-1 0 1 2

x 108

0

2

4x 10



Historical Loss Distributions (PNC 1984-2008)

Historical Quarterly Risk Proxies: Loss Distributions

(Top 200 Banks 1984-2008)

-2 0 2 4

x 107

0

1

2x 10

-7 Credit Risk: GCO

-2 0 2 4 6

x 107

0

5x 10

-8 Op Risk: ONIE

-1 0 1 2

x 107

0

0.5

1x 10


-5 0 5

x 108

0

0.5

1x 10


-4 -2 0 2 4

x 108

0

5x 10


IRG-4QD

-5 0 5 10

x 108

0

2

4x 10



Historical Loss Distributions (200 Largest Banks 1984-2008)

Historical Quarterly Risk Proxies: Time Series (1984-2008)

0 50 1000

5x 10

6 Credit Risk: GCO

0 50 1000

5

10x 10

6 Op Risk: ONIE

0 50 100-2

0

2x 10

6Market Risk:NTR-4QD

0 50 100-1

0

1x 10

8 Liquidity Risk: LG-4QD

0 50 100-5

0

5x 10

7Interest Rate Risk:

IRG-4QD

0 50 100-1

0

1x 10

8Total Risk: Sum of


Time Series of Quarterly Losses (JP Morgan Chase 1984-2008)

0 50 1000

2

4x 10

7 Credit Risk: GCO

0 50 1000

2

4x 10

7 Op Risk: ONIE

0 50 100-2

0

2x 10

7 Market Risk:TR-4QD

0 50 100-5

0

5x 10

8Liquidity Risk: LG-4QD

0 50 100-2

0

2x 10

8 Interest Rate Risk: IRG-4QD

0 50 100-5

0

5x 10

8Total Risk: Sum of


Time Series of Quarterly Losses (200 Largest Banks 1984-2008)

0 50 1000

5x 10

6 Credit Risk: NCO

0 50 1000

5

10x 10

6 Op Risk: NIE

0 50 100-5

0

5x 10


0 50 100-1

0

1x 10


0 50 100-5

0

5x 10


IRG-4QD

0 50 100-1

0

1x 10

8Total Risk: Sum of


Time Series of Quarterly Losses (Bank of America 1984-2008)

0 50 1000

5x 10

6 Credit Risk: GCO

0 50 1000

5

10x 10

6 Op Risk: ONIE

0 50 100-1

0

1x 10

7 Market Risk:NTR-4QD

0 50 100-1

0

1x 10


0 50 100-2

0

2x 10


0 50 100-1

0

1x 10

8 Total Risk: Sum of Cr.,Ops.,Mkt.,Liqu.&Int.

Time Series of Quarterly Losses (Citibank 1984-2008)

0 50 1000

2

4x 10

6 Credit Risk: GCO

0 50 1000

5

10x 10

6 Op Risk: ONIE

0 50 100-1

0

1x 10

6Market Risk:NTR-4QD

0 50 100-5

0

5x 10


0 50 100-5

0

5x 10


IRG-4QD

0 50 100-5

0

5x 10

7Total Risk: Sum of


Time Series of Quarterly Losses (Wells Fargo 1984-2008)

0 50 1000

1

2x 10

6 Credit Risk: GCO

0 50 1000

1

2x 10

6 Op Risk: ONIE

0 50 100-5

0

5x 10

5 Market Risk:NTR-4QD

0 50 100-5

0

5x 10


0 50 100-1

0

1x 10


0 50 100-5

0

5x 10

7Total Risk: Sum of


Time Series of Quarterly Losses (PNC 1984-2008)

Historical Quarterly Risk Proxies: Time Series

(Top 200 Banks 1984-2008)

0 50 1000

2

4x 10

7 Credit Risk: GCO

0 50 1000

2

4x 10

7 Op Risk: ONIE

0 50 100-2

0

2x 10


0 50 100-5

0

5x 10


0 50 100-2

0

2x 10


0 50 100-5

0

5x 10

8Total Risk: Sum of


Time Series of Quarterly Losses (200 Largest Banks 1984-2008)

Pairwise Correlations: Pearson vs. Spearman (5 Risk Types 1984-2008)

Risk PairType of

Correlation

Aggregate

Banks2JP Morgan Chase

Bank of America Citigroup

Wells Fargo PNC

Pearson 65.17% -5.77% -4.34% 76.65% 10.07% 28.87%

Spearman 60.00% -3.60% -10.00% 78.00% 15.00% 41.00%

Pearson 22.41% 19.73% 5.29% 16.40% 18.42% 9.00%

Spearman -4.90% 15.00% 6.90% 8.10% 19.00% 9.00%

Pearson 53.43% 19.07% 47.87% 31.47% 2.30% 20.85%

Spearman 10.00% -12.00% -17.00% -3.30% -15.00% -15.00%

Pearson -13.28% -7.82% -18.09% -8.78% -14.31% -13.13%

Spearman 33.00% 20.00% 24.00% 33.00% 17.00% 28.00%

Pearson 19.89% 10.92% 12.01% 13.46% -4.28% -9.31%

Spearman 3.00% 10.00% 10.00% 2.70% 1.40% -6.50%

Pearson 15.33% 7.37% -8.55% 11.76% -4.85% -10.22%

Spearman -2.00% -16.00% -24.00% -9.20% -26.00% -18.00%

Pearson -11.74% -14.25% -23.49% -8.79% -15.88% -15.68%

Spearman 7.20% 10.00% -30.00% 12.00% -4.60% -4.20%

Pearson 11.27% 1.56% -18.23% 6.29% -0.94% -3.21%

Spearman 2.30% -36.00% -23.00% -23.00% -25.00% 0.26%

Pearson 24.78% -27.92% -16.70% -19.17% -17.79% 3.38%

Spearman 19.00% -9.10% 8.80% -0.60% 6.80% 3.90%

Pearson 18.97% 19.96% 9.17% 12.38% 9.14% 12.86%

Spearman 13.00% 21.00% 15.00% 26.00% 8.20% 18.00%

Table 1.4: Pairwise Correlations for Top 200 and 5 Largest Banks Risk Proxies (Call Report Data 1984-2008)

Operational and Liquidity Risk

Operational and Market Risk

Credit and Interest Rate Risk

Credit and Liquidity Risk

Credit and Market Risk

Credit and Operational Risk

Interest Rate and Market Risk

Market and Interest Rate Risk

Market and Liquidity Risk

Operational and Interest Rate Risk

Pairwise Correlations, Scatters & Histograms (5 Risk Types 1984-2008)

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph & Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu.

Operat.

Market

Int.Rt.

corr(cr,ops)= 0.6517

corr(mkt,liqu)= 0.1127

corr(int,liqu)= 0.1897

corr(cr,mkt)= 0.2241

corr(ops,liqu)= 0.1533

corr(mkt,int)= 0.2478

corr(cr,liqu)= 0.5343

corr(ops,int)= -0.1174

corr(ops,mkt)= 0.1989

corr(cr,int)= -0.1328

-5 0 5

x 107

-1 0 1

x 108

-5 0 5

x 106

0 5 10

x 106

0 2 4

x 106

-5

0

5

x 107

Pairwise Scattergraph & Pearson Correlations of 5 Risk TypesJP Morgan Chase (Call Report Data 1984-2008)

-1

0

1

x 108

-5

0

5

x 106

0

2

4

x 106

0

5

10

x 106

Credit

Operat.

Liqu.

Int.Rt.

Market

corr(cr,ops)= -0.0577






corr(mkt,int)=-0.2792




-5 0 5

x 107

-1 0 1

x 108

-5 0 5

x 106

2 4 6

x 106

0 5 10

x 106

-5

0

5

x 107

-1

0

1

x 108

-5

0

5

x 106

2

4

6

x 106

0

5

10

x 106

PairPairwise Scatterplot & Pearson Correlations of 5 Risk Types

Bank of America (Call Report Data 1984-2008)

Credit

Liqu.

Int.Rt.

Ops.

Market

corr(cr,ops)= -0.0434


corr(mkt,liqu)= -0.1823



corr(ops,liqu)= -0.0855

corr(mkt,int)=-0.1670




-2 0 2

x 107

-1 0 1

x 108

-1 0 1

x 107

0 5 10

x 106

0 2 4

x 106

-2

0

2

x 107

-1

0

1

x 108

-1

0

1

x 107


Citibank (Call Report Data 1984-2008)

0

2

4

x 106

2

4

6

x 106

Credit

Ops.

Market

Liqu.

Int.Rt.










corr(mkt,int)= -0.1917

-5 0 5

x 107

-5 0 5

x 107

-1 0 1

x 106

0 5 10

x 106

0 2 4

x 106

-5

0

5

x 107

-5

0

5

x 107

-1

0

1

x 106

0

5

10

x 106

0

2

4

x 106


Wells Fargo (Call Report Data 1984-2008)

Ops.

Credit

Market

Liqu.

Int.Rt.


corr(ops,mkt)= -0.0428





corr(mkt,int)= -0.1779




-1 0 1

x 107

-5 0 5

x 107

-5 0 5

x 105

0 1 2

x 106

0 1 2

x 106

-1

0

1

x 107

-5

0

5

x 107

-5

0

5

x 105

0

1

2

x 106

Pairwise Scatterplot & Pearson Correlations of 5 Risk TypesPNC (Call Report Data 1984-2008)

0

1

2

x 106

Credit

Ops.

Market

Liqu.

Int.Rt.


corr(ops,mkt)= -0.0931









Pairwise Correlations, Scatters & Histograms: 5 Risk Types (Top 200 Banks

1984-2008)

-2 0 2

x 108

-5 0 5

x 108

-2 0 2

x 107

0 2 4

x 107

0 2 4

x 107

-2

0

2

x 108

-5

0

5

x 108

-2

0

2

x 107

0

2

4

x 107

Pairwise Scattergraph & Pearson Correlations of 5 Risk TypesTop 200 Banks (Call Report Data 1984-2008)

0

2

4

x 107

Credit

Liqu.

Operat.

Market

Int.Rt.











Spearman Correlations: 5 Risk Types Transformed Data

U1=LNINV(CR)

0.0 0.4 0.8

0.60 -0.049

0.0 0.4 0.8

0.10

0.0

0.4

0.8

0.33

0.0

0.4

0.8

U2=LNINV(OR) 0.03 -0.020 0.072

U2=NINV(MR) 0.0230.

20.

61.

00.19

0.0

0.4

0.8

U4=NINV(LR) 0.13

0.0 0.4 0.8 0.2 0.6 1.0 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations:Top 200 Banks(Transformed Data)

U1=LNINV(CR)

0.0 0.4 0.8

-0.036 0.15

0.0 0.4 0.8

-0.12

0.0

0.4

0.8

0.20

0.0

0.4

0.8

U2=LNINV(OR) 0.10 -0.16 0.10

U2=NINV(MR) -0.36

0.0

0.4

0.8

-0.091

0.0

0.4

0.8

U4=NINV(LR) 0.21

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations:JP Morgan Chase(Transformed Data)

U1=LNINV(CR)

0.0 0.4 0.8

-0.10 0.069

0.0 0.4 0.8

-0.17

0.0

0.4

0.8

0.24

0.0

0.4

0.8

U2=LNINV(OR) 0.10 -0.24 -0.30

U2=NINV(MR) -0.23

0.0

0.4

0.8

0.088

0.0

0.4

0.8

U4=NINV(LR) 0.15

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations: Bank of America (Transformed Data)

U1=LNINV(CR)

0.0 0.4 0.8

0.78 0.081

0.0 0.4 0.8

-0.033

0.0

0.4

0.8

0.33

0.0

0.4

0.8

U2=LNINV(OR) 0.027 -0.092 0.12

U2=NINV(MR) -0.23

0.0

0.4

0.8

-0.006

0.0

0.4

0.8

U4=NINV(LR) 0.26

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations: Citibank (Transformed Data)

U1=LNINV(CR)

0.0 0.4 0.8

0.15 0.19

0.0 0.4 0.8

-0.15

0.0

0.4

0.8

0.17

0.0

0.4

0.8

U2=LNINV(OR) 0.014 -0.26 -0.046

U2=NINV(MR) -0.250.

00.

40.

80.068

0.0

0.4

0.8

U4=NINV(LR) 0.082

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations: Wells Fargo (Transformed Data)

U1=LNINV(CR)

0.0 0.4 0.8

0.41 0.09

0.0 0.4 0.8

-0.15

0.0

0.4

0.8

0.28

0.0

0.4

0.8

U2=LNINV(OR) -0.065 -0.18 -0.042

U2=NINV(MR) 0.0026

0.0

0.4

0.8

0.039

0.0

0.4

0.8

U4=NINV(LR) 0.18

0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations: PNC (Transformed Data)

Spearman Correlations: 5 Risk Types Transformed Data (Top

200 Banks)

U1=LNINV(CR)

0.0 0.4 0.8

0.60 -0.049

0.0 0.4 0.8

0.10

0.0

0.4

0.8

0.33

0.0

0.4

0.8

U2=LNINV(OR) 0.03 -0.020 0.072

U2=NINV(MR) 0.023

0.2

0.6

1.0

0.19

0.0

0.4

0.8

U4=NINV(LR) 0.13

0.0 0.4 0.8 0.2 0.6 1.0 0.0 0.4 0.8

0.0

0.4

0.8

U5=NINV(IR)

Spearman Correlations:Top 200 Banks(Transformed Data)

Dependograms of Multivariate Groupwise Independence Tests

Dependogram of Multivariate Groupwise Independence Tests – Top

200 Banks

Multivariate Groupwise Independence Tests: P-Values

Risk Group

Aggregate

Banks2JP Morgan Chase

Bank of America Citigroup

Wells Fargo PNC

Credit & Operational 0.05% 8.54% 21.53% 0.05% 0.05% 0.05%Credit & Market 8.14% 28.52% 49.40% 27.02% 25.32% 70.68%Credit & Liquidity 26.52% 47.90% 27.22% 73.28% 71.98% 21.63%Credit & Interest Rate 0.05% 2.45% 2.05% 0.05% 0.05% 1.15%Operational & Market 9.74% 3.95% 22.73% 7.44% 8.04% 28.02%Operational & Liquidity 22.43% 4.95% 3.15% 32.62% 35.11% 9.54%Operational & Interest Rate 20.23% 17.43% 0.45% 6.44% 6.04% 17.93%Market & Liquidity 0.35% 0.15% 5.14% 5.94% 6.84% 95.65%Market & Interest Rate 12.54% 35.81% 51.90% 57.29% 60.39% 56.79%Interest Rate & Liquidity 0.05% 1.55% 11.94% 0.75% 0.15% 2.75%Credit, Operational & Market 10.04% 1.25% 58.69% 48.60% 46.30% 91.56%Credit, Operational & Liquidity 38.41% 41.81% 72.68% 62.19% 65.18% 38.91%Credit, Operational & Interest Rate 6.94% 43.61% 69.08% 13.24% 10.94% 23.43%Credit, Market & Liquidity 1.55% 83.47% 43.71% 8.74% 9.44% 56.29%Credit, Market & Interest Rate 99.05% 12.14% 35.91% 25.92% 23.23% 23.03%Credit, Liquidity & Interest Rate 10.94% 94.26% 77.87% 74.88% 76.27% 83.07%Operational, Market & Liquidity 0.45% 0.65% 68.88% 12.84% 12.54% 91.16%Operational, Market & Interest Rate 7.54% 81.87% 46.80% 20.63% 19.23% 5.74%Operational, Liquidity & Interest Rate 22.43% 1.45% 99.25% 46.70% 50.60% 73.88%Market, Liquidity & Interest Rate 13.94% 45.90% 5.34% 14.04% 11.94% 39.81%Credit, Operational, Market & Liquidity 0.25% 53.50% 58.39% 1.45% 1.15% 10.64%Credit, Operational, Market & Interest Rate 5.04% 0.85% 64.19% 10.84% 10.64% 3.75%Credit, Operational, Liquidity & Interest Rate 0.15% 75.47% 89.66% 17.13% 16.03% 58.69%Credit, Market, Liquidity & Interest Rate 12.24% 73.78% 65.58% 49.20% 51.10% 21.93%Operational, Market, Liquidity & Interest Rate 14.74% 60.19% 88.66% 46.00% 47.60% 16.53%Credit, Operational, Market, Liquidity & Interest Rate 0.85% 2.38% 2.09% 4.95% 3.65% 9.56%Global Test 0.05% 2.43% 1.72% 0.05% 0.05% 9.44%

Table 1.5: Genest et al (2004) Mulivariate Groupwise Independence Test P-Values: Top 200 and 5 Largest Banks Risk Proxies (Call Report Data 1984-2008)

Alternative Risk Measures: 99.97th Percentile VaR

Gaussian Copula

Simulation

Gaussian (Variance-

Covariance) Approximation

Historical Bootstrap (Empirical

Copula) Simulation

T-Distribution

Copula Simulation

Archimedean Copula

(Gumbel) Simulation

Archimedean Copula

(Clayton) Simulation

Archimedean Copula (Frank)

Simulation

Magnitude of Risk Measure - Fully Diversified 7.64E+08 6.88E+08 8.59E+08 8.12E+08 9.30E+08 7.28E+08 7.52E+08

Magnitude of Risk Measure - Perfect Correlation 1.09E+09 9.61E+08 2.55E+09 1.07E+09 1.06E+09 1.07E+09 1.06E+09

Diversification Benefit 42.05% 39.60% 196.34% 31.47% 14.51% 46.38% 41.69%

Genest Goodess of Fit Test P-Value 3.55% N/A N/A 6.54% 0.05% 0.25% 1.15%

VaR as a Proportion of Book Value of Assets 7.10% 6.40% 7.98% 7.55% 8.64% 6.76% 6.99%


























Table 2.1: 99.97% Confidence Level Value-at-Risk for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

Top

200

Ban

ksP

NC

JPM

CB

ank

of A

mer

ica

Citi

grou

pW

ells

Far

go

99.97th Percentile Dollar VaR Across Banks and Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Frank Copula

Top

200

JPM

C

Bof

A

Citi

Wel

ls

PN

C

0.00E+00

1.00E+08

2.00E+08

3.00E+08

4.00E+08

5.00E+08

6.00E+08

7.00E+08

8.00E+08

9.00E+08

1.00E+09

$ VaR

Figure 6.1: 99.97th Percentile Value-at-Risk for 5 Risk Types: Credit, Operational, Market, Liquidity and Interest

Rate (200 Largest Banks: Call Report Data 1984-2008)

Gaussian Copula

Normal Approx.

Empirical Copula T-Copula

Gumbel Copula

Clayton Copula

Frank Copula

Top 200 7.64E+08 6.88E+08 8.59E+08 8.12E+08 9.30E+08 7.28E+08 7.52E+08

JPMC 2.30E+08 1.87E+08 3.92E+08 2.38E+08 2.47E+08 2.19E+08 2.32E+08

BofA 1.94E+08 1.82E+08 2.05E+08 2.00E+08 2.07E+08 1.82E+08 2.03E+08

Citi 1.62E+08 1.32E+08 2.77E+08 1.72E+08 2.00E+08 1.49E+08 1.60E+08

Wells 1.63E+08 1.04E+08 1.87E+08 1.71E+08 1.99E+08 1.52E+08 1.58E+08

PNC 4.79E+07 4.66E+07 5.78E+07 5.01E+07 5.23E+07 4.66E+07 4.71E+07

99.97th Percentile VaR/BVA Across Banks and Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Frank CopulaTop200

JPMCBofA

CitiWells

PNC

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

VaR/BVA

Figure 6.2: 99.97th Percentile VaR as a Proportion of BVA for 5 Risk Types: Credit, Operational, Market, Liquidity and

Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Frank Copula

Top 200 7.10% 6.40% 7.98% 7.55% 8.64% 6.76% 6.99%

JPMC 12.43% 10.12% 21.18% 12.88% 13.35% 11.85% 12.54%

BofA 11.43% 10.72% 12.05% 11.75% 12.19% 10.72% 11.93%

Citi 12.28% 10.03% 21.03% 13.07% 15.19% 11.29% 12.13%

Wells 13.18% 8.45% 15.11% 13.86% 16.11% 12.28% 12.81%

PNC 16.52% 16.06% 19.94% 17.30% 18.04% 16.06% 16.25%

99.97th Percentile VaR Diversification Benefit Across Banks and

Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Frank CopulaTop200

JPMC BofACiti

WellsPNC

0.00%

50.00%

100.00%

150.00%

200.00%

250.00%

300.00%

% DivPer

Figure 6.3: 99.97th Percentile VaR Diversification Benefit for 5 Risk Types: Credit, Operational, Market, Liquidity and Interest Rate (200 Largest Banks: Call Report Data

1984-2008) Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Frank Copula

Top 200 42.05% 39.60% 196.34% 31.47% 14.51% 46.38% 41.69%

JPMC 28.27% 31.17% 127.18% 32.60% 20.18% 35.42% 27.74%

BofA 38.31% 36.32% 252.32% 37.50% 25.22% 45.34% 30.13%

Citi 35.23% 37.91% 182.83% 27.53% 11.22% 46.05% 36.41%

Wells 33.97% 41.03% 189.86% 27.82% 10.35% 44.76% 39.28%

PNC 32.16% 31.12% 242.61% 35.21% 21.09% 36.39% 36.09%

Genest et al (2009) Copula Goodness-of-Fit Test P-values Across Banks and

Methodologies

Gaussian Copula

T-Copula

Gumbel Copula

Clayton Copula

Frank Copula

Top200

JPMCBofA

CitiWells

PNC

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

% DivPer

Figure 6.4: Genest et al (2009) Copula GOF Tests for 5 Risk Types: Credit, Operational, Market, Liquidity and Interest

Rate (200 Largest Banks: Call Report Data 1984-2008)Gaussian

Copula T-

CopulaGumbel Copula

Clayton Copula

Frank Copula

Top 200 3.55% 6.54% 0.05% 0.25% 1.15%

JPMC 20.53% 7.24% 0.05% 61.69% 37.23%

BofA 50.10% 6.04% 42.91% 60.79% 39.01%

Citi 32.82% 6.04% 30.12% 12.04% 6.94%

Wells 28.72% 57.19% 30.02% 8.44% 8.14%

PNC 47.70% 35.35% 82.07% 10.94% 6.51%

Discussion of 99.97th Percentile VaR and % Diversification Benefit & GOF

Tests of Model Fit• Dollar VaR (VaR/BVA) increases montonically (generally

decreases) with size of the institution• VCA (ECS or the AGCS) produces consistently the lowest

(highest) VaR; TCS in follows in conservativeness, while the GCS & AFCS (ACCS) is usually toward the middle (low side)

• ECS yields highest PDBs (127%-252%) vs. other models (10%-50%); GCS tends to lie in the middle (41-58%), VCA to the lower end (31-41%) & GCAS is the lowest (10-21%)

• No directionally consistent pattern in PDBs across different business mixes (i.e., higher % trading vs. lending assets)

• GOF tests highly mixed (reject null 14/30 cases), no pattern, not at very high levels of significance->models generally OK?

Alternative Risk Measures: 99th Percentile Expected Shortfall

Gaussian Copula

Simulation

Gaussian (Variance-

Covariance) Approximation

Historical Bootstrap (Empirical

Copula) Simulation

T-Distribution

Copula Simulation

Archimedean Copula

(Gumbel) Simulation

Archimedean Copula

(Clayton) Simulation

Archimedean Copula (Frank)

Simulation














Genest Goodess of Fit Test P-Value 5.01E-01 N/A N/A 6.04E-02 42.91% 60.79% 39.01%

VaR as a Proportion of Book Value of Assets 11.24% 15.56% 11.57% 1.11E-01 12.04% 10.57% 11.53%
















Citi

grou

pW

ells

Far

goP

NC

Table 2.2: 99.9% Confidence Level Expected Shortfall for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate (200 Largest Banks: Call Report Data 1984-2008)

Top

200

Ban

ksJP

MC

Ban

k of

Am

eric

a

Normal Approximation Loss Distributions

-3 -2 -1 0 1 2 3

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

-8Normal Approximation Loss Distribution: 5 Risk Types (Cr.,Ops.,Mkt.,Liqu.Int.)

Wells Fargo: Call Report Data 1984-2008 (Div%=41.03%)

Freq

uenc

y

Losses

VaR99.97%=1.04E+8

-4 -3 -2 -1 0 1 2 3 4 5 6

x 108

0

1

2

3

4

5

6

7

8x 10

-9Normal Approximation Loss Distribution: 5 Risk Types (Cr.,Ops.,Mkt.,Liqu.Int.)

Bank of America: Call Report Data 1984-2008 (Div%=36.32%)

Freq

uenc

yLosses

VaR99.97%=1.82E+8

-5e+08 0e+00 5e+08

0e+0

01e

-09

2e-0

93e

-09

Normal Approximation Annual Loss Distribution

VaR-99.7%=6.88e+8K=$688B,%DivBen=39.26%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

VaR.99=688B

-2e+08 -1e+08 0e+00 1e+08 2e+08

0.0e+

002.0

e-09

4.0e-

096.0

e-09

8.0e-

091.0

e-08

1.2e-

081.4

e-08


VaR-99.7%=1.87e+8K=$187B,%DivBen=31.2%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity

VaR.99=879B

-2e+08 -1e+08 0e+00 1e+08 2e+08

0.0e+

004.0

e-09

6.0e-

098.0

e-09

1.0e-

081.2

e-08

1.4e-

08


VaR-99.7%=6.96e+8K=$182B,%DivBen=36.3%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

VaR.9987=182B

-2e+08 -1e+08 0e+00 1e+08 2e+08

0.0e

+00

2.0e

-09

4.0e

-09

6.0e

-09

8.0e

-09

1.0e

-08

1.2e

-08

1.4e

-08


VaR-99.7%=1.32e+8K=$132B,%DivBen=37.9%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity

VaR.9987=132B

Normal Approximation Loss Distribution: Top 200 Banks

-5e+08 0e+00 5e+08

0e+0

01e

-09

2e-0

93e

-09


VaR-99.7%=6.88e+8K=$688B,%DivBen=39.26%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Den

sity

VaR.99=688B

Empirical Copula Simulated Loss Distributions

-1.5 -1 -0.5 0 0.5 1 1.5

x 108

0

1

2

3

4

5

6

7x 10

-8Empirical Copula Loss Distribution: 5 Risk Types (Cr.,Ops.,Mkt.,Liqu.,Int.)

PNC Call Report Data 1984-2008 (Div%=242.61%)

Fre

quen

cy

Losses

VaR 99.97% = 5.78E+7

-6 -5 -4 -3 -2 -1 0 1 2 3 4

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

-8Empirical Copula Loss Distribution: 5 Risk Types (Cr.,Ops.,Mkt.,Liqu.,Int.)Wells Fargo Call Report Data 1984-2008 (Div%=189.86%)

Fre

quen

cy

Losses

VaR 99.97% = 1.87E+8

Empirical Copula Simulated Annual Loss Distribution

VaR-99.7%=8.59e+8K=$859B,%DivBen=196.3%5 Risk Types (Credit,Market,Oerational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

-5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

2.0e

-09

VaR.99=859B


VaR-99.7%=3.92e+8K=$392B,%DivBen=127.2%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity-2e+08 0e+00 2e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

9

VaR.99=392B


VaR-99.7%=2.02e+8K=$205B,%DivBen=252.3%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

VaR.9997=205B


VaR-99.7%=2.77e+8K=$277B,%DivBen=182.8%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity

-3e+08 -2e+08 -1e+08 0e+00 1e+08 2e+08 3e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

VaR.9997=277B

Empirical Copula Simulated Loss Distribution: Top 200


VaR-99.7%=8.59e+8K=$859B,%DivBen=196.3%5 Risk Types (Credit,Market,Oerational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

-5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

2.0e

-09

VaR.99=859B

Gaussian Copula Simulated Loss Distributions

Gaussian Copula Simulated Annual Loss Distribution

VaR-99.7%=1.63e+8K=$163B,DivPer=33.97%,GOF.Test.PV=32.82%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Wells Fargo (1984-2008)

Dens

ity

-1.5e+08 -1.0e+08 -5.0e+07 0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08


VaR-99.7%=4.79e+7K=$47.9B,DivPer=32.16%,GOF.Test.PV=47.70%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): PNC (1984-2008)

Dens

ity

-6e+07 -4e+07 -2e+07 0e+00 2e+07 4e+07 6e+07

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

2.5e

-08


VaR-99.7%=7.64e-8K=$764B,DivPer=42.05%,GOF.Test.PV=3.55%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

-5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

2.0e

-09

VaR-99.7%=$764B


VaR-99.7%=2.30e-8K=$230B,DivPer=28.27%,GOF.Test.PV=20.53%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JPMC (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

97e

-09

VaR-99.7%=$230B


VaR-99.7%=1.94e+8K=$194B,DivPer=38.31%,GOF.Test.PV=50.10%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

VaR-99.7%=$194B


VaR-99.7%=1.62e+8K=$162B,DivPer=35.23%,GOF.Test.PV=32.82%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity

-1.5e+08 -1.0e+08 -5.0e+07 0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

VaR-99.7%=$162B

Gaussian Copula Simulated Loss Distribution: Top 200 Banks



De

nsi

ty

-5e+08 0e+00 5e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

92

.0e

-09

VaR-99.7%=$764B

Gaussian Copula vs. Normal Approximation & Empirical Copula

Loss Distributions

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

Gaussian Copula Simulated vs. Normal Approximation and Empicial Copula Annual Loss Distribution

GausCopVaR-99.7%=7.64e+8K=$764B,NormApproxVaR-99.7%=6.95e+8K=$695B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

Normal ApproximationGaussian CopulaEmpirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity


2e+08 4e+08 6e+08 8e+08 1e+090.

0e+0

05.

0e-1

01.

0e-0

91.

5e-0

9



Dens

ity


1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

97e

-09

Gaussian Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

GausCopVaR-99.7%=2.30e+8K=$230B,NormApproxVaR-99.7%=1.88e+8K=$188B,EmpCopVaR-99.7%=3.92e+8K=$392B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity

Normal Approximation

Gaussian Copula

Empirical Copula

0e+00 1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

97e

-09


GausCopVaR-99.7%=1.94e+8K=$94B,NormApproxVaR-99.7%=1.82e+8K=$182B,EmpCopVaR-99.7%=2.05e+8K=$205B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity


0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08 2.5e+08 3.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

9


GausCopVaR-99.7%=1.62e+8K=$162B,NormApproxVaR-99.7%=1.32e+8K=$132B,EmpCopVaR-99.7%=2.77e+8K=$277B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity


Gaussian Copula vs. Normal Approximation & Empirical Copula Loss Distribution: Top 200 Banks

2e+08 4e+08 6e+08 8e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9



De

nsi

ty


T-Copula Simulated Loss Distributions

Student-T Copula Simulated Annual Loss Distribution

VaR-99.7%=1.71e+8K=$171B,DivPer=27.82%,GOF.Test.PV=57.19%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Wells Fargo (1984-2008)

Dens

ity

-1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08


VaR-99.7%=5.01e+7K=$501B,DivPer=35.21%,GOF.Test.PV=35.35%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): PNC (1984-2008)

Dens

ity

-6e+07 -4e+07 -2e+07 0e+00 2e+07 4e+07 6e+07

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

2.5e

-08

3.0e

-08



Dens

ity

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

2.0e

-09

VaR-99.7%=$812B


VaR-99.7%=2.38e-8K=$238B,DivPer=32.6%,GOF.Test.PV=7.24%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JPMC (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+080e

+00

2e-0

94e

-09

6e-0

98e

-09

VaR-99.7%=$238B


VaR-99.7%=2.00e+8K=$200B,DivPer=37.5%,GOF.Test.PV=6.04%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08 3e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

9

VaR-99.7%=$200B


VaR-99.7%=1.72e+8K=$172B,DivPer=27.53%,GOF.Test.PV=6.04%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity

-1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

VaR-99.7%=$172B

T-Copula Simulated Loss Distribution: Top 200 Banks



De

nsi

ty

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

92

.0e

-09

VaR-99.7%=$812B

Student-T Copula vs. Normal Approximation & Empirical Copula

Loss Distributions

2e+08 4e+08 6e+08 8e+08 1e+090.

0e+0

05.

0e-1

01.

0e-0

91.

5e-0

9

Student-T Copula Simulated vs. Normal Approximation and Empicial Copula Annual Loss Distribution


Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity

Normal ApproximationStudent-T CopulaEmpirical Copula

1e+08 2e+08 3e+08 4e+08

0e+0

02e

-09

4e-0

96e

-09


TCopVaR-99.7%=2.38e+8K=$238B,NormApproxVaR-99.7%=1.88e+8K=$188B,EmpCopVaR-99.7%=3.92e+8K=$392B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Student-T (1984-2008)

Dens

ity


0e+00 1e+08 2e+08 3e+08 4e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

9

Student-T Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

TCopVaR-99.7%=2.00e+8K=$200B,NormApproxVaR-99.7%=1.82e+8K=$182B,EmpCopVaR-99.7%=2.05e+8K=$205B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

Normal ApproximationStudent-t CopulaEmpirical Copula

0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08 2.5e+08 3.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

Student-T Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

TCopVaR-99.7%=1.72e+8K=$172B,NormApproxVaR-99.7%=1.32e+8K=$132B,EmpCopVaR-99.7%=2.77e+8K=$277B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity


Student-T Copula vs. Normal Approx. & Empirical Copula Loss Distributions:

Top 200 Banks

2e+08 4e+08 6e+08 8e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9



De

nsi

ty


Archimadean (Gumbel) Simulated Loss Distributions

Archimadean (Gumbel) Copula Simulated Annual Loss Distribution

Theta=1.31, VaR-99.7%=9.30e+8K=$930B, DivPerc=14.51%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Den

sity

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09


Theta=1.50, VaR-99.7%=1.99e+8K=$199B, DivPerc=10.35%, GOF.Test.PV=30.02%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Wells Fargo (1984-2008)

Dens

ity

-1e+08 0e+00 1e+08 2e+08

0e+0

02e

-094e

-096e

-098e

-09


Theta=1.31, VaR-99.7%=9.30e+8K=$930B, DivPerc=14.51%, GOF.Test.PV=0.05%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0e+

005.0

e-10

1.0e-0

91.5

e-09 VaR-99.7%=$930B


Theta=1.11, VaR-99.7%=2.47e+8K=$247B, DivPerc=20.18%,GOF.Test.PV=0.05%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08 3e+08

0e+0

02e

-094e

-096e

-09 VaR-99.7%=$247B


Theta=1.09, VaR-99.7%=2.07e+8K=$207B, DivPerc=25.22%, GOF.Test.PV=42.91%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

-3e+08 -2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

01e

-092e

-093e

-094e

-095e

-096e

-097e

-09

VaR-99.7%=$207B


Theta=1.50, VaR-99.7%=2.00e+8K=$200B, DivPerc=11.22%, GOF.Test.PV=30.12%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Dens

ity

-3e+08 -2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

VaR-99.7%=$200B

Archimadean (Gumbel) Simulated Loss Distribution: Top 200

BanksArchimadean (Gumbel) Copula Simulated Annual Loss Distribution


De

nsi

ty

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9

VaR-99.7%=$930B

Gumbel Copula vs. Normal Approximation & Empirical Copula

Loss Distributions

2e+08 4e+08 6e+08 8e+08 1e+090.

0e+0

05.

0e-1

01.

0e-0

91.

5e-0

9



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

Archimadean (Gumbel) Copula Simulated vs. Normal Approximation and Empicial Copula Annual Loss Distribution

GumbCopVaR-99.7%=9.30e+8K=$930B,NormApproxVaR-99.7%=6.95e+8K=$695B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

Normal ApproximationGumbel CopulaEmpirical Copula

1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

97e

-09


GumbCopVaR-99.7%=2.38e+8K=$247B,NormApproxVaR-99.7%=1.88e+8K=$188B,EmpCopVaR-99.7%=3.92e+8K=$392B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity


0e+00 1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

97e

-09

Archimadean (Gumbel) Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

GumbCopVaR-99.7%=2.07e+8K=$207B,NormApproxVaR-99.7%=1.82e+8K=$182B,EmpCopVaR-99.7%=2.05e+8K=$205B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity


0e+00 1e+08 2e+08 3e+08 4e+08

0.0e

+00

2.0e

-09

4.0e

-09

6.0e

-09

8.0e

-09

1.0e

-08

1.2e

-08

Archimadean (Gumbel) Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

GumbCopVaR-99.7%=2.00e+8K=$200B,NormApproxVaR-99.7%=1.32e+8K=$132B,EmpCopVaR-99.7%=2.77e+8K=$277B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity


Gumbel Copula vs. Normal Approx. & Empirical Copula Loss Distributions:

Top 200 Banks

2e+08 4e+08 6e+08 8e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9


GumbCopVaR-99.7%=9.30e+8K=$930B,NormApproxVaR-99.7%=6.95e+8K=$695B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

De

nsity


Archimadean (Clayton) Simulated Loss Distributions

Archimadean (Clayton) Copula Simulated Annual Loss Distribution

Theta=0.12, VaR-99.7%=7.28e+8K=$728B,DivPerc=46.38%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Den

sity

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity

-1e+08 0e+00 1e+08 2e+08

0e+0

02e

-094e

-096e

-098e

-091e

-08



Dens

ity

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0e+

005.0

e-10

1.0e-0

91.5

e-09 VaR-99.7%=$728B



Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

02e

-094e

-096e

-098e

-09

VaR-99.7%=$219B


Theta=-0.0247, VaR-99.7%=1.82e+8K=$182B, DivPerc=45.34%,GOF.Test.PV=60.79%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08 3e+08

0e+0

02e

-09

4e-0

96e

-09

VaR-99.7%=$182B



Dens

ity

-1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

VaR-99.7%=$149B

Archimadean (Clayton) Simulated Loss Distribution: Top 200

BanksArchimadean (Clayton) Copula Simulated Annual Loss Distribution


De

nsi

ty

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9

VaR-99.7%=$728B

Clayton Copula vs. Normal Approximation & Empirical Copula

Loss Distributions

2e+08 4e+08 6e+08 8e+08 1e+090.

0e+0

05.

0e-1

01.

0e-0

91.

5e-0

9



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

Archimadean (Clayton) Copula Simulated vs. Normal Approximation and Empicial Copula Annual Loss Distribution

ClayCopVaR-99.7%=7.28e+8K=$930B,NormApproxVaR-99.7%=6.95e+8K=$728B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

Normal ApproximationClayton CopulaEmpirical Copula

1e+08 2e+08 3e+08 4e+08

0e+0

02e

-09

4e-0

96e

-09


ClayCopVaR-99.7%=2.19e+8K=$247B,NormApproxVaR-99.7%=1.88e+8K=$188B,EmpCopVaR-99.7%=3.92e+8K=$392B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity


0e+00 1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

97e

-09

Archimadean (Clayton) Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

ClayCopVaR-99.7%=1.82e+8K=$207B,NormApproxVaR-99.7%=1.82e+8K=$182B,EmpCopVaR-99.7%=2.05e+8K=$205B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Den

sity


Clayton Copula

Empirical Copula

0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08 2.5e+08 3.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

Archimadean (Clayton) Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

ClayCopVaR-99.7%=1.49e+8K=$149B,NormApproxVaR-99.7%=1.32+8K=$132B,EmpCopVaR-99.7%=2.77e+8K=$277B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity


Clayton Copula vs. Normal Approx. & Empirical Copula Loss Distributions

for Top 200

2e+08 4e+08 6e+08 8e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9


ClayCopVaR-99.7%=7.28e+8K=$930B,NormApproxVaR-99.7%=6.95e+8K=$728B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

De

nsi

ty


Archimadean (Frank) Simulated Loss Distributions

Archimadean (Frank) Copula Simulated Annual Loss Distribution

Theta=1.21, VaR-99.7%=7.52e+8K=$752B,DivPerc=41.64%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Den

sity

-1e+09 -5e+08 0e+00 5e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity

-1e+08 0e+00 1e+08 2e+08

0e+0

02e

-094e

-096e

-098e

-091e

-08



Dens

ity

-5e+08 0e+00 5e+08 1e+09

0.0e+

005.0

e-10

1.0e-0

91.5

e-09 VaR-99.7%=$752B



Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

01e

-092e

-093e

-094e

-095e

-096e

-097e

-09

VaR-99.7%=$232B


Theta=1.21, VaR-99.7%=2.03e+8K=$203B, DivPerc=30.13%, GOF.Test.PV=39.01%5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Dens

ity

-2e+08 -1e+08 0e+00 1e+08 2e+08

0e+0

02e

-09

4e-0

96e

-09

VaR-99.7%=$203B



Dens

ity

-1.5e+08 -1.0e+08 -5.0e+07 0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

VaR-99.7%=$160B

Archimadean (Frank) Simulated Loss Distribution: Top 200

BanksArchimadean (Gumbel) Copula Simulated Annual Loss Distribution


De

nsi

ty

-5e+08 0e+00 5e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9

VaR-99.7%=$752B

Frank Copula vs. Normal Approximation & Empirical Copula

Loss Distributions

2e+08 4e+08 6e+08 8e+08 1e+090.

0e+0

05.

0e-1

01.

0e-0

91.

5e-0

9



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09



Dens

ity


Student-T Copula

Empirical Copula

2e+08 4e+08 6e+08 8e+08 1e+09

0.0e

+00

5.0e

-10

1.0e

-09

1.5e

-09

Archimadean (Frank) Copula Simulated vs. Normal Approximation and Empicial Copula Annual Loss Distribution

FrankCopVaR-99.7%=7.52e+8K=$930B,NormApproxVaR-99.7%=6.95e+8K=$728B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

Dens

ity

Normal ApproximationFrank CopulaEmpirical Copula

1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9


FrankCopVaR-99.7%=2.32e+8K=$232B,NormApproxVaR-99.7%=1.88e+8K=$188B,EmpCopVaR-99.7%=3.92e+8K=$392B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): JP Morgan Chase (1984-2008)

Dens

ity


0e+00 1e+08 2e+08 3e+08 4e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9

Archimadean (Frank) Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

FrankCopVaR-99.7%=2.03e+8K=$207B,NormApproxVaR-99.7%=1.82e+8K=$182B,EmpCopVaR-99.7%=2.05e+8K=$205B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Bank of America (1984-2008)

Den

sity


0.0e+00 5.0e+07 1.0e+08 1.5e+08 2.0e+08 2.5e+08 3.0e+08

0e+0

02e

-09

4e-0

96e

-09

8e-0

91e

-08

Archimadean (Frank) Copula Simulated vs. Normal Approximation and Empirical Copula Annual Loss Distribution

FrankVaR-99.7%=1.60+8K=$160,NormApproxVaR-99.7%=1.32+8K=$132B,EmpCopVaR-99.7%=2.77e+8K=$277B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): Citibank (1984-2008)

Den

sity


Frank Copula vs. Normal Approx. & Empirical Copula Loss Distributions

for Top 200

2e+08 4e+08 6e+08 8e+08 1e+09

0.0

e+

00

5.0

e-1

01

.0e

-09

1.5

e-0

9


FrankCopVaR-99.7%=7.52e+8K=$930B,NormApproxVaR-99.7%=6.95e+8K=$728B,EmpCopVaR-99.7%=8.59e+8K=$859B5 Risk Types (Credit,Market,Operational,Liquidity&InterestRate): 200 Large Banks (1984-2008)

De

nsi

ty


Bootstrap Analysis of Risk Measures: 99.97th Percentile VaR

Normal Approx.

Gaussian Copula T-Copula

Gumbel Copula

Clayton Copul;a


Gumbel Copula

Clayton Copul;a

99.97% VaR in Historical Sample 8.59E+08 6.88E+08 7.64E+08 8.12E+08 7.28E+08 7.28E+08 7.64E+08 8.12E+08 7.28E+08 7.28E+08

95% Confidence Interval 6.96E+07 1.94E+08 5.41E+07 6.90E+07 4.77E+07 4.77E+07 2.70E+08 3.54E+08 1.05E+09 8.68E+08

Numerical Coefficient of Variation 8.10% 28.20% 7.08% 8.49% 6.55% 6.55% 35.37% 43.61% 3.66E+08 3.22E+08



Numerical Coefficient of Variation 6.42% 30.34% 7.78% 7.51% 7.53% 7.00% 41.38% 46.64% 3.35E-01 4.99E-01













Top

200

B

anks

JPM

CB

ank

of

Am

eri

ca

Table 3.1: Bootstrap Analysis of 99.97% Confidence Level Value-at-Risk for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate

(200 Largest Banks: Call Report Data 1984-2008)

Empirical Copula

Resampling of Correlations Resampling of Marginals

Citi

gro

upW

ells

F

argo

PN

C

Bootstrap of Correlations: 99.97th Percentile Dollar VaR Across Banks

and Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top

200

JPM

C

Bof

A

Citi

Wel

ls

PN

C

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

CV

Figure 7.1: Numerical Coefficients of Variation in Bootstrap of Correlations for 99.97th Percentile Value-at-Risk for 5

Risk Types: Credit, Operational, Market, Liquidity and Interest Rate (200 Largest Banks: Call Report Data 1984-

2008) Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top 200 7.08% 28.20% 8.10% 8.49% 10.56% 6.55%

JPMC 7.78% 30.34% 6.42% 7.51% 7.53% 7.00%

BofA 8.85% 27.86% 13.57% 11.09% 10.80% 6.50%

Citi 7.95% 45.26% 9.08% 12.00% 9.57% 6.52%

Wells 8.10% 26.98% 8.91% 16.42% 9.25% 6.00%

PNC 8.96% 32.85% 12.78% 10.89% 9.98% 5.90%

Bootstrap of Margins: 99.97th Percentile Dollar VaR Across Banks

and Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top

200

JPM

C

Bof

A

Citi

Wel

ls

PN

C

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

CV

Figure 7.2: Numerical Coefficients of Variation in Bootstrap of Margins for 99.97th Percentile Value-at-Risk for 5 Risk Types: Credit, Operational, Market, Liquidity and Interest

Rate (200 Largest Banks: Call Report Data 1984-2008)

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top 200 35.37% 28.20% 8.10% 43.61% 39.32% 44.29%

JPMC 41.38% 30.34% 6.42% 46.64% 33.51% 49.89%

BofA 48.21% 27.86% 13.57% 56.14% 37.79% 25.16%

Citi 38.41% 45.26% 9.08% 62.23% 52.66% 54.38%

Wells 40.00% 26.98% 8.91% 45.99% 45.74% 62.14%

PNC 44.82% 32.85% 12.78% 47.44% 44.94% 69.55%

Bootstrap Analysis of Risk Measures: VaR Diversification%

Normal Approx.


Gumbel Copula

Clayton Copul;a


Gumbel Copula

Clayton Copul;a

Diversification % in Historical Sample 1.96E+00 3.96E-01 4.21E-01 3.15E-01 1.45E-01 4.64E-01 4.21E-01 3.15E-01 1.45E-01 4.64E-01

95% Confidence Interval 2.74E-01 4.44E-01 9.88E-02 9.85E-02 1.10E-01 1.11E-01 1.95E-01 1.93E-01 5.34E-02 1.62E-01

Numerical Coefficient of Variation 13.96% 112.15% 23.48% 31.31% 75.46% 23.89% 46.38% 61.37% 36.82% 35.01%
















Table 3.2: Bootstrap Analysis 99.97% Confidence Percent Level Diversification Benefit for 5 Risk Types: Credit, Operational, Market, Liquidity & Interest Rate

(200 Largest Banks: Call Report Data 1984-2008)

Empirical Copula

Resampling of Correlations Resampling of Margins

Top

200

B

anks

PN

CJP

MC

Ban

k of

A

mer

ica

Citi

grou

pW

ells

F

argo

Bootstrap of Correlations: 99.97th Perc. VaR % Diversification Benefit Across

Banks and Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top

200

JPM

C

Bof

A

Citi

Wel

ls

PN

C

0.00%5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

CV

Figure 7.3: Numerical Coefficients of Variation in Bootstrap of Correlations for 99.97th Percentile VaR % Diversification

Benefit for 5 Risk Types: Credit, Operational, Market, Liquidity and Interest Rate (200 Largest Banks: Call Report

Data 1984-2008) Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top 200 23.48% 112.15% 13.96% 31.31% 75.46% 23.89%

JPMC 34.32% 111.16% 12.19% 35.99% 44.12% 30.00%

BofA 24.44% 100.26% 17.59% 39.46% 46.76% 24.93%

Citi 29.02% 158.23% 12.70% 42.37% 88.45% 24.24%

Wells 17.30% 83.43% 15.92% 28.35% 87.93% 22.73%

PNC 24.50% 118.15% 16.58% 22.45% 53.88% 28.27%

Bootstrap of Margins: 99.97th Perc. VaR % Diversification Benefit Across

Banks and Methodologies

Gaussian Copula

Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top

200

JPM

C

Bof

A

Citi

Wel

ls

PN

C

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

CV

Figure 7.4: Numerical Coefficients of Variation in Bootstrap of Margins for 99.97th Percentile VaR % Diversification Benefit for 5 Risk Types: Credit, Operational, Market,

Liquidity and Interest Rate (200 Largest Banks: Call Report Data 1984-2008) Gaussian

Copula Normal Approx.

Empirical Copula

T-Copula

Gumbel Copula

Clayton Copula

Top 200 46.38% 112.15% 13.96% 61.37% 36.82% 35.01%

JPMC 55.64% 111.16% 12.19% 69.44% 38.83% 32.84%

BofA 38.63% 100.26% 17.59% 39.46% 44.47% 26.54%

Citi 45.08% 158.23% 12.70% 83.94% 48.61% 43.72%

Wells 43.85% 17.30% 15.92% 58.20% 53.09% 49.48%

PNC 56.68% 24.50% 16.58% 60.91% 43.03% 19.93%

Discussion: Bootstrap of 99.97th Perc. VaR & % Diver. Benefit Across Banks

and Methodologies • We fail to observe a consistent pattern in the variability of VaR

or PDB across size or types of banks (i.e., business mix). • For either the bootstrap of margins or correlations for VaR or

PDB, ECS (VCA) yields lowest (highest) NCVs • Across models or banks NCVs are an order of magnitude

higher for the resampling of margins vs. correlations• This difference is accentuated for VaR vs. PDB.

• NCVs are higher for the PDB vs. VaR: excluding ECS, NCVs for VaR in bootstrap of correlations (margins) range in 5.9%-32.8% (25.2%-56.1%); respective PDB numbers are 17.3%-158.2% (19.9%-83.9%)

• .,..

Bootstrapping of Value-at-Risk: Normal Approximation

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

x 108

0

1

2

3

4

5

6

7x 10

-8Bootstrap Distribution of Normal Approximation 99.97% VaR (CI=2.82E+7,CV=26.98%)

Wells Fargo: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=1.04E+8

q2.5%=8.94E+7 q97.75%=1.18E+8

2.5 3 3.5 4 4.5

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10


200 Largest Banks: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

q97.75%=7.88E+8

VaR99.97%=6.88E+8

q2.5%=5.92E+8

1 1.5 2 2.5

x 108

0

0.5

1

1.5

2

2.5x 10

-8Bootstrap Distribution of Normal Approximation 99.97% VaR (CI=5.68E+7,CV=30.3%)JP Morgan Chase: 5 Risk Types (Call Report Data 1984-2008)

Value-at-RiskF

requ

ency

VaR99.97%=1.87E+8

q2.5%=1.57E+8 q97.75%=2.14E+8

1.2 1.4 1.6 1.8 2 2.2 2.4

x 108

0

0.5

1

1.5

2

2.5

3

3.5x 10


Bank of America: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=1.82E+8

q97.75%=2.06E+8q2.5%=1.55E+8

0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 108

0

0.5

1

1.5

2

2.5x 10


Citibank: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=1.32E+8

q97.75%=1.62E+8q2.5%=1.02E+8

3 3.5 4 4.5 5 5.5 6 6.5

x 107

0

0.2

0.4

0.6

0.8

1

1.2

1.4x 10


PNC: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=4.66E+7

q2.5%=3.89E+7 q97.75%=5.42E+8

Bootstrapping of Value-at-Risk: Normal Approximation (Top

200 Banks)

2.5 3 3.5 4 4.5

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10



Value-at-Risk

Fre

quen

cy

q97.75%=7.88E+8

VaR99.97%=6.88E+8

q2.5%=5.92E+8

Bootstrapping of Diversification Percentage: Normal Approximation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types: Bank of America Call Report Data 1984-2008 (CI=36.4%,CV=100.3%)

Div%

Fre

quen

cy

DivPer=36.3%

q97.75%=53.9%q2.5%=17.5%

-0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types: Citibank Call Report Data 1984-2008 (CI=59.98%,CV=158.23%)

Div%

Fre

quen

cy

DivPer=37.9%

q2.5%=8.03% q97.75%=68.1%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types: Wells Fargo Call Report Data 1984-2008 (CI=34.2%,CV=83.43%)

Div%

Fre

quen

cy

DivPer=41.0%

q2.5%=23.7% q97.75%=58.0%

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types: PNC Call Report Data 1984-2008 (CI=36.77%,CV=118.15%)

Div%

Fre

quen

cy

DivPer=31.1%

q97.75%=48.6%q2.5%=11.9%

-0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types for 200 Largest Banks: Call Report Data 1984-2008 (CI=44.4%, CV=112.2%)

Div%

Fre

quen

cy

q97.75%=62.6%

DivPer=39.6%

q2.5%=18.2%

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types: JP Morgan Chase Call Report Data 1984-2008 (CI=34.65%,CV=111.2%)

Div%

Fre

quen

cy

DivPer=31.2%

q97.75%=49.1%q2.5%=13.3%

Bootstrapping of Diversification Percentage: Normal Approximation

(Top 200 Banks)

-0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5

4

Bootstrap Distribution of Normal Approximation 99.97% VaR Diversification Percentage5 Risk Types for 200 Largest Banks: Call Report Data 1984-2008 (CI=44.4%, CV=112.2%)

Div%

Fre

quen

cy

q97.75%=62.6%

DivPer=39.6%

q2.5%=18.2%

Bootstrapping of Value-at-Risk: Empirical Copula

7.6 7.8 8 8.2 8.4 8.6 8.8 9 9.2

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

-8Bootstrap Distribution of Empirical Copula 99.97% VaR (CI=7.0E+7,CV=8.1%)


Value-at-Risk

Fre

quen

cy

VaR99.97%=8.59E+8

q2.5%=8.07E+8 q97.75%=8.77E+8

3.6 3.7 3.8 3.9 4 4.1 4.2

x 108

0

1

2

3

4

5

6

7x 10


JP Morgan Chase: 5 Risk Types (Call Report Data 1984-2008)

Value-at-RiskF

requ

ency

VaR99.97%=3.92E+8

q97.75%=4.07E+8q2.5%=3.82E+8

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5

x 108

0

1

2

3

4

5

6

7x 10


Bank of AMerica: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=2.12E+8

q2.5%=1.96E+8 q97.75%=2.24E+8

2.65 2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1

x 108

0

1

2

3

4

5

6

7

8x 10


Citibank: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=2.77E+8

q2.5%=2.74E+8 q97.75%=2.99E+8

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8

x 107

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10


PNC: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=5.78E+7

q97.75%=1.34E+7q2.5%=5.60E+8

1.7 1.75 1.8 1.85 1.9 1.95 2 2.05

x 108

0

1

2

3

4

5

6

7

8

9x 10


Wells Fargo: 5 Risk Types (Call Report Data 1984-2008)

Value-at-Risk

Fre

quen

cy

VaR99.97%=1.87E+8

q2.5%=1.80E+8 q97.75%=1.97E+8

Bootstrapping of Value-at-Risk: Empirical Copula (Top 200 Banks)

7.6 7.8 8 8.2 8.4 8.6 8.8 9 9.2

x 108

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10



Value-at-Risk

Fre

quen

cy

VaR99.97%=8.59E+8

q2.5%=8.07E+8 q97.75%=8.77E+8

Bootstrapping of Diversification Percentage: Empirical Copula

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50

1

2

3

4

5

6

7

8

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for JPMC: Call Report Data 1984-2008 (CI=15.5%,CV=12.2%)

Div%

Fre

quen

cy

DivPer=126.2%

q97.75%=133.3%q2.5%=117.8%

1.7 1.8 1.9 2 2.1 2.2 2.3 2.40

1

2

3

4

5

6

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for 200 Largest Banks: Call Report Data 1984-2008 (CI=27.4%,CV=14.0%)

Div%F

requ

ency

q97.75%=216.9%q2.5%=189.5%

DivPer=196.3%

1.8 2 2.2 2.4 2.6 2.8 30

0.5

1

1.5

2

2.5

3

3.5

4

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for BofA: Call Report Data 1984-2008 (CI=44.4%,CV=17.6%)

Div%

Fre

quen

cy

DivPer=252.3%

q97.75%=259.3%q2.5%=214.9%

1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 2.050

1

2

3

4

5

6

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for Citibank: Call Report Data 1984-2008 (CI=23.22%,CV=12.70%)

Div%

Fre

quen

cy

DivPer=182.8%

q2.5%=167.9% q97.75%=191.2%

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80

0.5

1

1.5

2

2.5

3

3.5

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for PNC: Call Report Data 1984-2008 (CI=40.543%,CV=16.58%)

Div%

Fre

quen

cy

DivPer=242.6%

q97.75%=247.9%q2.5%=207.5%

1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80

0.5

1

1.5

2

2.5

3

3.5

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for PNC: Call Report Data 1984-2008 (CI=40.543%,CV=16.58%)

Div%

Fre

quen

cy

DivPer=242.6%

q97.75%=247.9%q2.5%=207.5%

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for Wells Fargo: Call Report Data 1984-2008 (CI=30.23%,CV=15.92%)

Div%

Fre

quen

cy

q2.5%=167.6% q97.75%=197.9%

DivPer=189.9%

Bootstrapping of Diversification Percentage: Empirical Copula (Top

200 Banks)

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50

1

2

3

4

5

6

7

8

Bootstrap Distribution of Empirical Copula 99.97% VaR Diversification Percentage5 Risk Types for JPMC: Call Report Data 1984-2008 (CI=15.5%,CV=12.2%)

Div%

Fre

quen

cy

DivPer=126.2%

q97.75%=133.3%q2.5%=117.8%

Bootstrapping of Value-at-Risk: Gaussian Copula (Correlations)

Gaussian Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR99.7%=7.64e+8, q2.5%=7.37e+8, q97.5%=7.91e+8, CV=7.0899.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)

De

nsi

ty

7.2e+08 7.4e+08 7.6e+08 7.8e+08 8.0e+08

0.0

e+

00

5.0

e-0

91

.0e

-08

1.5

e-0

82

.0e

-08

2.5

e-0

8


VaR99.7%=2.30e+8, q2.5%=2.21e+8, q97.5%=2.39e+8, CV=7.7899.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JPMC (1984-2008)

Dens

ity

2.15e+08 2.20e+08 2.25e+08 2.30e+08 2.35e+08 2.40e+08 2.45e+08

0e+0

02e

-08

4e-0

86e

-08


VaR99.7%=1.94e+8, q2.5%=1.85e+8, q97.5%=2.02e+8, CV=8.85%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): BofA (1984-2008)

Dens

ity

1.80e+08 1.85e+08 1.90e+08 1.95e+08 2.00e+08 2.05e+08 2.10e+08

0e+0

02e

-08

4e-0

86e

-08

8e-0

8


VaR99.7%=1.62e+8, q2.5%=1.56e+8, q97.5%=1.68e+8, CV=7.95%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Citibank (1984-2008)

Dens

ity

1.55e+08 1.60e+08 1.65e+08 1.70e+08

0e+0

02e

-08

4e-0

86e

-08

8e-0

81e

-07


VaR99.7%=1.63e+8, q2.5%=1.57e+8, q97.5%=1.70e+8, CV=8.10%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Wells Fargo (1984-2008)

Dens

ity

1.50e+08 1.55e+08 1.60e+08 1.65e+08 1.70e+08 1.75e+08

0e+0

02e

-08

4e-0

86e

-08

8e-0

81e

-07


VaR99.7%=4.79e+7, q2.5%=5.57e+7, q97.5%=5.00e+7, CV=8.96%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): PNC (1984-2008)

Dens

ity

4.4e+07 4.6e+07 4.8e+07 5.0e+07

0.0e

+00

5.0e

-08

1.0e

-07

1.5e

-07

2.0e

-07

2.5e

-07

3.0e

-07

3.5e

-07

Bootstrapping of Value-at-Risk (Correlations): Gaussian Copula (Top

200 Banks)Gaussian Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR99.7%=7.64e+8, q2.5%=7.37e+8, q97.5%=7.91e+8, CV=7.0899.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)

De

nsity

7.2e+08 7.4e+08 7.6e+08 7.8e+08 8.0e+08

0.0

e+

00

5.0

e-0

91

.0e

-08

1.5

e-0

82

.0e

-08

2.5

e-0

8

Bootstrapping of Diversification %: Gaussian Copula (Correlations)

Gaussian Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit

Div%-VaR99.7%=42.05%, q2.5%=37.01%, q97.5%=46.89%, CV=23.48%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)

Den

sity

0.35 0.40 0.45 0.50

05

1015


Div%-VaR99.7%=28.27%, q2.5%=23.48%, q97.5%=33.19%, CV=34.32%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JPMC (1984-2008)

Dens

ity

0.20 0.25 0.30 0.35

05

1015


Div%-VaR99.7%=38.31%, q2.5%=33.94%, q97.5%=43.30%, CV=24.44%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): BofA (1984-2008)

Dens

ity

0.30 0.35 0.40 0.45 0.50

05

1015


Div%-VaR99.7%=35.23%, q2.5%=30.01%, q97.5%=40.24%, CV=29.02%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Citibank (1984-2008)

Dens

ity

0.30 0.35 0.40 0.45

05

1015


Div%-VaR99.7%=33.97%, q2.5%=31.03%, q97.5%=36.91%, CV=17.30%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Wells Fargo (1984-2008)

Dens

ity

0.30 0.32 0.34 0.36 0.38

05

1015

2025


Div%-VaR99.7%=32.16%, q2.5%=28.20%, q97.5%=36.09%, CV=24.50%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): PNC (1984-2008)

Dens

ity

0.26 0.28 0.30 0.32 0.34 0.36 0.38

05

1015

20

Bootstrapping of Diversification % (Correlations): Gaussian Copula (Top

200 Banks)Gaussian Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit


Den

sity

0.35 0.40 0.45 0.50

05

1015

Bootstrapping of Value-at-Risk: Gaussian Copula (Margins)

Gaussian Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR

VaR99.7%=7.64e+8, q2.5%=6.26e+8, q97.5%=8.94e+8, CV=35.37%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

95e

-09

6e-0

9


VaR99.7%=2.30e+8, q2.5%=1.83e+8, q97.5%=2.78e+8, CV=41.38%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JPMC (1984-2008)

Dens

ity

1.6e+08 1.8e+08 2.0e+08 2.2e+08 2.4e+08 2.6e+08 2.8e+08 3.0e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08



Dens

ity

1.0e+08 1.5e+08 2.0e+08 2.5e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08



Dens

ity

1.2e+08 1.4e+08 1.6e+08 1.8e+08 2.0e+08 2.2e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

2.5e

-08



Dens

ity

1.2e+08 1.4e+08 1.6e+08 1.8e+08 2.0e+08 2.2e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

2.5e

-08



Dens

ity

3e+07 4e+07 5e+07 6e+07 7e+07

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08

6e-0

87e

-08

Bootstrapping of Value-at-Risk (Margins): Gaussian Copula

(Top 200 Banks)Gaussian Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR


De

nsity

5e+08 6e+08 7e+08 8e+08 9e+08 1e+09

0e

+00

1e

-09

2e

-09

3e

-09

4e

-09

5e

-09

6e

-09

Bootstrapping of Diversification %: Gaussian Copula (Margins)

Gaussian Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit


Den

sity

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

02

46



Dens

ity

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

02

46

8



Dens

ity

0.25 0.30 0.35 0.40 0.45 0.50 0.55

02

46

8



Dens

ity

0.25 0.30 0.35 0.40 0.45

02

46

810



Dens

ity

0.20 0.25 0.30 0.35 0.40 0.45

02

46

810



Dens

ity

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

02

46

8

Bootstrapping of Diversification % (Margins): Gaussian Copula

(Top 200 Banks)Gaussian Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit


Den

sity

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

02

46

Bootstrapping of Value-at-Risk: T- Copula (Correlations)

Student-T Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR99.7%=8.12+8, q2.5%=7.78e+8, q97.5%=8.47e+8, CV=8.49%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)

Dens

ity

7.6e+08 7.8e+08 8.0e+08 8.2e+08 8.4e+08 8.6e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08


VaR99.7%=2.38+8, q2.5%=2.27e+8, q97.5%=2.49e+8, CV=9.06%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JPMC (1984-2008)

Dens

ity

2.20e+08 2.25e+08 2.30e+08 2.35e+08 2.40e+08 2.45e+08 2.50e+08 2.55e+08

0e+0

02e

-08

4e-0

86e

-08


VaR99.7%=2.00+8, q2.5%=1.94e+8, q97.5%=2.16e+8, CV=11.09%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): BofA (1984-2008)

Dens

ity

1.9e+08 2.0e+08 2.1e+08 2.2e+08

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08

6e-0

8


VaR99.7%=1.72+8, q2.5%=1.83e+8, q97.5%=1.94e+8, CV=12.00%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Citibank (1984-2008)

Dens

ity

1.6e+08 1.7e+08 1.8e+08 1.9e+08 2.0e+08

0e+0

02e

-08

4e-0

86e

-08


VaR99.7%=1.72+8, q2.5%=1.57e+8, q97.5%=1.85e+8, CV=16.31%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Wells Fargo (1984-2008)

Dens

ity

1.5e+08 1.6e+08 1.7e+08 1.8e+08 1.9e+08

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08


VaR99.7%=5.01+7, q2.5%=4.74e+7, q97.5%=5.29e+7, CV=10.89%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): PNC (1984-2008)

Dens

ity

4.4e+07 4.6e+07 4.8e+07 5.0e+07 5.2e+07 5.4e+07 5.6e+07

0.0e

+00

5.0e

-08

1.0e

-07

1.5e

-07

2.0e

-07

2.5e

-07

Bootstrapping of Value-at-Risk (Correlations): Student-T Copula

(Top 200 Banks)Student-T Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR99.7%=8.12+8, q2.5%=7.78e+8, q97.5%=8.47e+8, CV=8.49%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Top 200 Banks (1984-2008)

Dens

ity

7.6e+08 7.8e+08 8.0e+08 8.2e+08 8.4e+08 8.6e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

Bootstrapping of Diversification %: T-Copula (Correlations)

Student-T Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit


Dens

ity

0.20 0.25 0.30 0.35 0.40

02

46

810

1214



Dens

ity

0.25 0.30 0.35 0.400

24

68

1012



Dens

ity

0.30 0.35 0.40 0.45

02

46

810

12



Dens

ity

0.20 0.25 0.30 0.35

02

46

810

12



Dens

ity

0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34

05

1015

20



Dens

ity

0.30 0.35 0.40

05

1015

Bootstrapping of Diversification % (Correlations): Student-T Copula

(Top 200 Banks)Student-T Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit


Dens

ity

0.20 0.25 0.30 0.35 0.40

02

46

810

1214

Bootstrapping of Value-at-Risk: T-Copula (Margins)

Student-T Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR


Dens

ity

5.0e+08 6.0e+08 7.0e+08 8.0e+08 9.0e+08 1.0e+09 1.1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

9


VaR99.7%=2.38e+8, q2.5%=1.86e+8, q97.5%=2.97e+8, CV=46.64%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JPMC (1984-2008)

Dens

ity

2.0e+08 2.5e+08 3.0e+080.

0e+0

02.

0e-0

94.

0e-0

96.

0e-0

98.

0e-0

91.

0e-0

81.

2e-0

8



Dens

ity

1.5e+08 2.0e+08 2.5e+08 3.0e+08

0.0e

+00

2.0e

-09

4.0e

-09

6.0e

-09

8.0e

-09

1.0e

-08

1.2e

-08



Dens

ity

1.0e+08 1.5e+08 2.0e+08 2.5e+08 3.0e+08

0.0e

+00

2.0e

-09

4.0e

-09

6.0e

-09

8.0e

-09

1.0e

-08

1.2e

-08



Dens

ity

3e+07 4e+07 5e+07 6e+07 7e+07

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08

6e-0

8



Dens

ity

1.2e+08 1.4e+08 1.6e+08 1.8e+08 2.0e+08 2.2e+08 2.4e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

Bootstrapping of Value-at-Risk (Margins) : Student-T Copula

(Top 200 Banks)Student-T Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR


Dens

ity

5.0e+08 6.0e+08 7.0e+08 8.0e+08 9.0e+08 1.0e+09 1.1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

9

Bootstrapping of Diversification %: T-Copula (Margins)



Dens

ity

0.20 0.25 0.30 0.35 0.40

02

46

810

1214

Student-T Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit


Dens

ity

0.1 0.2 0.3 0.4 0.5

02

46



Dens

ity

0.2 0.3 0.4 0.5 0.6

01

23

45

6



Dens

ity

0.1 0.2 0.3 0.4 0.5

01

23

45

6



Dens

ity

3e+07 4e+07 5e+07 6e+07 7e+07

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08

6e-0

8



Dens

ity

0.15 0.20 0.25 0.30 0.35 0.40

02

46

8

Bootstrapping of Diversification % (Margins): Student-T Copula

(Top 200 Banks)Student-T Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit


Dens

ity

0.20 0.25 0.30 0.35 0.40

02

46

810

1214

Bootstrapping of Value-at-Risk: Archimadean Gumbel (Correl’s)

Archimadean (Gumbel) Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR99.7%=9.30e+8, q2.5%=8.93e+8, q97.5%=9.92e+8, CV=10.56%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks 1984-2008)

Den

sity

9.0e+08 9.5e+08 1.0e+09

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08



Den

sity

9.0e+08 9.5e+08 1.0e+09

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08


VaR99.7%=2.47e+8, q2.5%=2.38e+8, q97.5%=2.57e+8, CV=7.53%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JP Morgan Chase 1984-2008)

Dens

ity

2.35e+08 2.40e+08 2.45e+08 2.50e+08 2.55e+08 2.60e+080e

+00

2e-0

84e

-08

6e-0

88e

-08

1e-0

7


VaR99.7%=2.07e+8, q2.5%=2.03e+8, q97.5%=2.25e+8, CV=10.80%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Bank of America 1984-2008)

Dens

ity

2.0e+08 2.1e+08 2.2e+08 2.3e+08 2.4e+08

0e+0

02e

-08

4e-0

86e

-08



Dens

ity

1.90e+08 1.95e+08 2.00e+08 2.05e+08 2.10e+08

0.0e

+00

4.0e

-08

8.0e

-08

1.2e

-07



Den

sity

1.90e+08 1.95e+08 2.00e+08 2.05e+08

0e+0

02e

-08

4e-0

86e

-08

8e-0

8

Bootstrapping of Value-at-Risk (Correlations) : Archimadean Gumbel

(Top 200 Banks)Archimadean (Gumbel) Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR


Den

sity

9.0e+08 9.5e+08 1.0e+09

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

Bootstrapping of Diversification %: Archimadean Gumbel (Correl’s)

Archimadean (Gumbel) Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit

Div%-VaR99.7%=14.51%, q2.5%=8.86%, q97.5%=19.81%, CV=75.46%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks 1984-2008)

Den

sity

0.10 0.15 0.20

02

46

810

12



Den

sity

0.10 0.15 0.20

02

46

810

12


Div%-VaR99.7%=25.93%, q2.5%=20.67%, q97.5%=29.67%, CV=8.91%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JP Morgan Chase 1984-2008)

Dens

ity

0.20 0.25 0.30

05

1015


Div%-VaR99.7%=25.22%, q2.5%=17.13%, q97.5%=28.92%, CV=46.76%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Bank of America 1984-2008)

Dens

ity

0.10 0.15 0.20 0.25 0.30

02

46

810

1214



Dens

ity

0.10 0.15 0.20

05

1015



Den

sity

0.08 0.10 0.12 0.14 0.16 0.18 0.20

05

1015

Bootstrapping of Diversification % (Correlations) : Archimadean Gumbel

(Top 200 Banks)Archimadean (Gumbel) Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit


Den

sity

0.10 0.15 0.20

02

46

810

12

Bootstrapping of Value-at-Risk: Archimadean Gumbel (Margins)

Archimadean (Gumbel) Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR


Den

sity

6.0e+08 7.0e+08 8.0e+08 9.0e+08 1.0e+09 1.1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

9


VaR99.7%=2.47e+8, q2.5%=2.05e+8, q97.5%=2.88e+8, CV=33.51%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JP Morgan Chase (1984-2008)

Den

sity

2.0e+08 2.2e+08 2.4e+08 2.6e+08 2.8e+08 3.0e+080.

0e+0

05.

0e-0

91.

0e-0

81.

5e-0

82.

0e-0

8


VaR99.7%=2.07e+8, q2.5%=1.68e+8, q97.5%=2.47e+8, CV=37.79%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Bank of America (1984-2008)

Den

sity

1.6e+08 1.8e+08 2.0e+08 2.2e+08 2.4e+08 2.6e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08



Den

sity

1.4e+08 1.6e+08 1.8e+08 2.0e+08 2.2e+08 2.4e+08 2.6e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08



Den

sity

1.4e+08 1.6e+08 1.8e+08 2.0e+08 2.2e+08 2.4e+08 2.6e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08



Den

sity

4.0e+07 4.5e+07 5.0e+07 5.5e+07 6.0e+07 6.5e+07

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08

6e-0

8

Bootstrapping of Value-at-Risk (Margins): Archimadean Gumbel

(Top 200 Banks)Archimadean (Gumbel) Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR


Den

sity

6.0e+08 7.0e+08 8.0e+08 9.0e+08 1.0e+09 1.1e+09

0e+0

01e

-09

2e-0

93e

-09

4e-0

9

Bootstrapping of Diversification %: Archimadean Gumbel (Margins)

Archimadean (Gumbel) Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit


Den

sity

0.18 0.20 0.22 0.24 0.26 0.28 0.30

05

1015

20



Den

sity

0.18 0.20 0.22 0.24 0.26 0.28 0.30

05

1015

20



Den

sity

0.18 0.20 0.22 0.24 0.26 0.28 0.30

05

1015

20



Den

sity

0.18 0.20 0.22 0.24 0.26 0.28 0.30

05

1015

20



Den

sity

0.18 0.20 0.22 0.24 0.26 0.28 0.30

05

1015

20


Div%-VaR99.7%=20.18%, q2.5%=17.68%, q97.5%=25.52%, CV=38.83%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): JP Morgan Chase (1984-2008)

Den

sity

0.16 0.18 0.20 0.22 0.24 0.26 0.280

510

1520


Div%-VaR99.7%=25.22%, q2.5%=21.88%, q97.5%=33.10%, CV=16.27%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.): Bank of America (1984-2008)

Den

sity

0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34

05

1015



Den

sity

0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14

010

2030



Den

sity

0.08 0.10 0.12 0.14 0.16

05

1015

20



Den

sity

0.16 0.18 0.20 0.22 0.24 0.26

05

1015

Bootstrapping of Diversification % (Margins): Archimadean Gumbel (Top

200 Banks)Archimadean (Gumbel) Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit


Den

sity

0.18 0.20 0.22 0.24 0.26 0.28 0.30

05

1015

20

Bootstrapping of Value-at-Risk: Archimadean Clayton (Correl.’s)

Archimadean (Clayton) Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR-99.7%=7.28e+8,q2.5%=7.02e+8,q97.5%=7.50e+899.97 Perc. VaR Div. Benefit for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks 1984-2008)

Den

sity

7.0e+08 7.2e+08 7.4e+08 7.6e+08 7.8e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

2.5e

-08



Den

sity

2.25e+08 2.30e+08 2.35e+08 2.40e+08 2.45e+08 2.50e+080e

+00

2e-0

84e

-08

6e-0

88e

-08

1e-0

7



Den

sity

2.0e+08 2.1e+08 2.2e+08 2.3e+08 2.4e+08

0e+0

02e

-08

4e-0

86e

-08



Den

sity

1.45e+08 1.50e+08 1.55e+08 1.60e+08

0.0e

+00

5.0e

-08

1.0e

-07

1.5e

-07



Den

sity

1.45e+08 1.50e+08 1.55e+08 1.60e+08

0.0e

+00

5.0e

-08

1.0e

-07

1.5e

-07



Den

sity

4.4e+07 4.5e+07 4.6e+07 4.7e+07 4.8e+07 4.9e+07

0e+0

01e

-07

2e-0

73e

-07

4e-0

75e

-07

Bootstrapping of Value-at-Risk (Correlations): Archimadean Clayton

(Top 200 Banks)Archimadean (Clayton) Copula Bootstrapped (Correlations) Distribution of 99.97 Percentile VaR

VaR-99.7%=7.28e+8,q2.5%=7.02e+8,q97.5%=7.50e+899.97 Perc. VaR Div. Benefit for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks 1984-2008)

Den

sity

7.0e+08 7.2e+08 7.4e+08 7.6e+08 7.8e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08

2.5e

-08

Bootstrapping of Diversification %: Archimadean Clayton (Correl.’s)

Archimadean (Clayton) Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit

Div%-99.7PercVaR=46.38%,q2.5%=43.19%,q97.5%=54.28%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks 1984-2008)

Den

sity

0.40 0.45 0.50 0.55

05

1015



Den

sity

0.20 0.25 0.30

05

1015



Den

sity

0.10 0.15 0.20 0.25 0.30

02

46

810

1214



Den

sity

0.40 0.45 0.50 0.55

02

46

810

12



Den

sity

0.35 0.40 0.45 0.50

05

1015



Den

sity

0.30 0.35 0.40

02

46

810

1214

Bootstrapping of Diversification % (Correlations): Archimadean Clayton

(Top 200 Banks)Archimadean (Clayton) Copula Bootstrapped (Correlations) 99.97 Perc. VaR Diversification Benefit

Div%-99.7PercVaR=46.38%,q2.5%=43.19%,q97.5%=54.28%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks 1984-2008)

Den

sity

0.40 0.45 0.50 0.55

05

1015

Bootstrapping of Value-at-Risk: Archimadean Clayton (Margins)

Archimadean (Clayton) Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR

VaR99.7%=7.28e+8, q2.5%=5.46e+8, q97.5%=8.68e+8, CV=44.29%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks(1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

9



Den

sity

2.0e+08 2.2e+08 2.4e+08 2.6e+08 2.8e+08 3.0e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08



Den

sity

8.0e+07 1.0e+08 1.2e+08 1.4e+08 1.6e+08 1.8e+08 2.0e+08 2.2e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08



Den

sity

1.6e+08 1.8e+08 2.0e+08 2.2e+08 2.4e+08 2.6e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08



Den

sity

1.0e+08 1.2e+08 1.4e+08 1.6e+08 1.8e+08 2.0e+08

0.0e

+00

5.0e

-09

1.0e

-08

1.5e

-08

2.0e

-08



Den

sity

3e+07 4e+07 5e+07 6e+07 7e+07

0e+0

01e

-08

2e-0

83e

-08

4e-0

85e

-08

Bootstrapping of Value-at-Risk (Margins): Archimadean Clayton (Top

200 Banks)Archimadean (Clayton) Copula Bootstrapped (Margins) Distribution of 99.97 Percentile VaR

VaR99.7%=7.28e+8, q2.5%=5.46e+8, q97.5%=8.68e+8, CV=44.29%99.97 Percentile Value-at-Risk for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks(1984-2008)

Den

sity

5e+08 6e+08 7e+08 8e+08 9e+08

0e+0

01e

-09

2e-0

93e

-09

4e-0

9

Bootstrapping of Diversification %: Archimadean Clayton (Margins)

Archimadean (Clayton) Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit

Div%-VaR99.7%=46.38%, q2.5%=39.84%, q97.5%=56.08%, CV=35.01%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks(1984-2008)

Den

sity

0.40 0.45 0.50 0.55 0.60

02

46

810



Den

sity

0.34 0.36 0.38 0.40 0.42 0.440

510

1520

2530



Den

sity

0.42 0.44 0.46 0.48 0.50

05

1015

20



Den

sity

0.3 0.4 0.5 0.6 0.7

01

23

45

67



Den

sity

0.30 0.35 0.40 0.45 0.50 0.55 0.60

02

46

8



Den

sity

0.45 0.50 0.55 0.60

02

46

810

1214

Bootstrapping of Diversification (Margins): Archimadean Clayton (Top

200 Banks)Archimadean (Clayton) Copula Bootstrapped (Margins) 99.97 Perc. VaR Diversification Benefit

Div%-VaR99.7%=46.38%, q2.5%=39.84%, q97.5%=56.08%, CV=35.01%99.97 Perc. VaR Div% for 5 Risk Types(Cr.,Mkt.,Ops.,Liqu.&IntRt.):200 Large Banks(1984-2008)

Den

sity

0.40 0.45 0.50 0.55 0.60

02

46

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Summary of Contributions and Major Findings

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Directions for Future Research• …

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Risk Aggregation Inanoglu Jacobs 6 09 V1

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