Credit Derivatives and the Default Risk of Large Complex Financial Institutions

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Credit Derivatives and the Default Risk of Large Complex Financial Institutions

Giovanni Calice¹ Christos Ioannidis² Julian Williams³

1School of Management, University of Southampton, SO17 1BJ [email protected]

2Department of Economics, University of Bath, BA2 7AY [email protected]

3Business School, University of Aberdeen [email protected]

June 3, 2010

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Introduction and Literature Summary and Theoretical basis

Executive Summary

We develop a dynamic model of bank default risk utilizing the Merton approach as our underlying theoretical framework.

For our empirical work we utilize a vector-autoregressive multivariate ARCH model to forecast direction and volatility of banks equity, co-evolving with two broad credit default swap indices, to proxy for factors affecting assets.

Combining these two models we create a measure of bank default risk and test it over 16 systemically important large complex financial institutions (LCFIs).

Finally we reverse engineer the model and back out a capitalization stress test based on future volatility scenarios generated from forward looking simulations.

We conclude that bank re-capitalizations could be far larger than expected given even fairly benign future asset volatility.

CIW 2010 (Soton, Bath & Aberdeen) CDs and the Default Risk of LCFIs June 3, 2010 2 / 48

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Literature

In the literature, there is no unambiguous answer to the question of whether these derivatives have a positive or negative effect on financial stability.

Bystrom (2005), Stock price volatility is also found to be significantly correlated with CDS spreads and the spreads are found to increase (decrease) with increasing (decreasing) stock price volatilities. Furthermore, the other interesting finding in this paper is the significant positive autocorrelation present in all the studied iTraxx indices.

Arping (2004), shows how CDs can facilitate banks’ quest for more effective lending relationships. He argues that CDs can have ambiguous effects on financial stability, and that disclosure requirements can strengthen the efficacy of the CDs market.

Verdier (2004), argues that, from a broader stability perspective, CDs in their current form increase the likelihood of future sovereign defaults.

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Literature

Instefjord (2005)analyzes risk taking by a bank that has access to CDs for risk management purposes. He finds that innovations in CDs markets lead to increased risk taking because of enhanced risk management opportunities.

Wagner and Marsh (2004), These authors demonstrate that CRT mechanism, under certain conditions, is generally welfare enhancing.

Wagner (2005), shows that the increased portfolio diversification possibilities introduced by CRT can increase the probability of liquidity-based crises.

Rajan (2005), has suggested that the hedging opportunities afforded by CDs and other risk management techniques are transforming the banking industry.

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Literature

Partnoy and Skeel (2006), suggest that CDOs are too complex. The transaction costs are high, the benefits questionable. They conclude that CDOs are being used to transform existing debt instruments that are accurately priced into new ones that areovervalued.

Wagner (2007), argues that new credit derivative instruments would improve the banks’ ability to sell their loans making them less vulnerable to liquidity shocks. However, this again might encourage banks to take on new risks because a higher liquidity of loans enables them to liquidate them more easily in a crisis. This effect would offset the initial positive impact on financial stability.

Wagner and Marsh (2006), on the other hand, argue that especially the transfer of credit risk from banks to non-banks would be beneficial for financial stability,

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Literature

A recent study byHu and Black (2008), concludes that, thanks to the explosive growth in CDs, debt-holders such as banks and hedge funds have often more to gain if companies fail than if they survive.

Allen and Gale (2006), develop a model of banking and insurance and show that, with complete markets and contracts, inter-sectoral transfers are desirable. However, with incomplete markets and contracts, CRT can occur as the result of regulatory arbitrage and this can increase systemic risk.

Heyde and Neyer (2007), analyze the consequences of CDs contracts in which both the protection buyer and the protection seller is a bank for the stability of the banking sector. Overall, they show that in a macroeconomic downturn, CDs reduce the stability of the banking sector.

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Credit Derivatives and Financial Stability

Table: Size of Global and UK CDs Market, Source: Fitch Global

Market 2002 2004 2006 2008est.

Global Market $1,952B $5,021B $20,207B $33,120B London Market $1,036B $2,450B $8,083B

Table: Global Participants in the Credit Derivatives Market

Institution 2002 2006

Banks 39% 36%

Investment Companies 16% 17%

Hedge Funds 12% 13%

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Methodology Merton Model of Default Risk

Methodology

Merton (1974a) proposed the classic distance to default (D2D) approach to the pricing of corporate debt.

The model treats equity,V_E∈R+, as a European call option on the value of the bank’s total assets,

which is assumed to follow a geometric Brownian motion

dV_A=V_A(µ_Adt+σ_AdW_t) (1) The model imputes the value,V_A∈R+, and volatility,σ_A, of assets from the equity market value and liabilities.

Liabilities,VL∈R+, of the firm are measured from the firm’s balance sheet.

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Solving for the Value and Volatility of Firm Assets I

the value of the firm’s equity is given as:

V_E=V_AN(d₁)−exp (−rt,T)V_LN(d₂) (2) where,

d₁ = log

VA

VL

+ rt,T +¹₂σ²_A (T−t) σA

√T−t (3)

d2 = log

V_A V_L

+ r_t,T −¹₂σ²_A (T−t) σ_A√

T−t (4)

= d1−σA

√

T−t (5)

here,N(z)is the evaluation of the standard cumulative normal distribution atz.

N µ, σ²

is the univariate normal probability density function, with meanµ

2

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Solving for the Value and Volatility of Firm Assets II

The volatility of equity,σ_E using the following expression, σ_E=

V_A VE

∂V_E

∂VA

σ_A (6)

this is effectively the delta of equity,

∂V_E

∂VA

≡ N(d₁) (7)

and the volatility of equity can be computed as:

σE= VA

VE

N(d1)σA. (8)

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Solving for the Value and Volatility of Firm Assets III

Rearranging and combining into a system of non-linear equations, f =

"

VAN(d1)−exp (−rt,T)VLN(d2)−VE

V_A V_E

N(d1)σA−σE

#

(9) and settingf =0, the estimated valueVˆ_A and volatilityσˆ_A of equity is computed using quadratic optimization or a similar technique,

f VˆA,σˆA

= min

VA,σA

[e⁰f f⁰e|VA, σA] (10) whereeis a 2-element unit vector. The expected number of standard deviations,ηt,T, from insolvency over the period,t→T, is thed2 of the call option of the value of assets over the value liabilities, i.e. the ”distance” that this option is away from beingout-of-the-money.

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Finding the Distance to Default D2D

The number of standard deviations from insolvency is therefore,

ηt,T = log_ˆ

VA

VL

+ µ−¹₂σˆ²_A (T −t) ˆ

σA

√T−t (11)

The market clearing probability of default is therefore,

P(ηt,T) =N(−ηt,T) (12)

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Methodology VAR-MV-GARCH Modelling

Forecasting Equity Volatility via Multivariate ARCH modelling

Consider the co-evolution of the value of equity and the credit derivatives indices,

yt= [∆ log (VE,t),∆ log (CDXt),∆ log (IT RAXXt)] (13) The volatility of equity, may then be modeled as a multivariate ARCH type model,

Σ_i,t=





σ_i² σ_i,CDX σ_{i,iT raxx} σ_i,CDX σ_CDX² σiT raxx,CDX

σi,iT raxx σiT raxx,CDX σ_{iT raxx}²





t

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Methodology Forecasting Equity Volatility via Multivariate ARCH modelling

Multivariate-ARCH Specification I

The system equation is,

yt = µt+Σ

1 2

tεt (15)

ε_t ∼ N(0,I) (16)

The evolution of the conditional covariance matrix is described by a BEKK type matrix autoregressive process,

Σi,t = KK⁰+A⁰Σi,t−1A+B⁰ Σ

1 2 i,t−1εt

Σ

1 2 i,t−1εt

⁰! B

θ = [ivechK⁰, vecA⁰, vecB⁰]⁰ (17) Maximum likelihood estimation is used to find the optimal parameter vector θ, the log likelihood function,ˆ F(.), is therefore,

F(θ) =−¹₂ nτlog (2π) +

τ

X

t=1

log|Σ_t|+ (y_t−µ_t)⁰Σ_t(y_t−µ_t)

! (18)

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Methodology Forecasting Equity Volatility via Multivariate ARCH modelling

Multivariate-ARCH Specification II

The volatility of assets,σA, is restated as a dynamic process, conditional on the covariance process underlying the equity volatility,

σ_A,t = ψ(σ_E,t|Σi,t) (19) σA,iT raxx,t = λ(ψ(σE,t)|Σi,t) (20) Once the implied asset volatilityσA,t has been computed, the implied covariationσA,iT raxx/CDX/Ratio,t between the asset volatility and any one of the components of the MV-ARCH spec is found by simply nesting, 6, into the MV-ARCH structure.

The model ascribes the contemporaneous and lagged dynamics of the credit indices endogenously within the volatility structure, making this model more flexible than a simple univariate GARCH-X type model.

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Methodology Data and Analysis

LCFIs Marsh and Stevens, BoE FSR (2003)

Q1−050 Q2−05 Q3−05 Q4−05 Q1−06 Q2−06 Q3−06 Q4−06 Q1−07 Q2−07 Q3−07 Q4−07 Q1−08 Q2−08

1 2 3 4 5 6 7 8 9

Time Index

Data Index

Time Series Plot

MORGAN STANLEY

GOLDMAN SACHS GROUP INC.

LEHMAN BROTHERS HOLDINGS INC.

DEUTSCHE BANK AG BNP PARIBAS

SOCIETE GENERALE BEAR STEARNS

HSBC HOLDINGS PLC ORD $0.50 (UK REG) BANK OF AMERICA CORPORATION

JP MORGAN CHASE & CO.

UBS AG

CREDIT SUISSE GROUP BARCLAYS PLC ORD

THE ROYAL BANK OF SCOTLAND GROUP CITIGROUP INC.

MERRILL LYNCH & CO., INC.

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Banks’ Financial Data

The Dataset consists of daily closing observations running from Jan 4th 2005 to 31st April 2009.

The dataset consists of individual financial institutions data drawn from the group of LCFIs as defined by the Bank of England (2001).

To derive the value of liabilitiesVL, the total liabilities excluding equity is collected for each quarter.

The data is then interpolated to a daily frequency and matched to the market capitalization dates, using a cubic spline.

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Credit Derivatives Data

The most widely traded of the indices is the iTraxx Europe index composed of the most liquid 125 CDSs referencing European investment grade credits, subject to certain sector rules as determined by the IIC(International Investment Company).

CDX is the brand-name for the family of Credit Default Swap Index products of a portfolio of 125 5-year default swaps, covering equal principal amounts of debt of each of 125 named North American investment-grade issuers.

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Results

Total Liabilities

5 10 15 20 25 30 35 40 45 50

$USD100Billions

Total Liabilities

GOLDMAN SACHS GP MORGAN STANLEY MERRILL LYNCH LEHMAN BROSHDG CITIGROUP BANK OF AMERICA JP MORGAN CHASE & CO BEAR STEARNS DEAD DEUTSCHE BANK UBS ’R’

CREDIT SUISSE GROUP N HSBC HDG (ORD $050) ROYAL BANK OF SCTLGP BARCLAYS BNP PARIBAS SOCIETE GENERALE

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Results

Equity, Market Capitalisation

20030 2004 2005 2006 2007 2008 2009

0.5 1 1.5 2 2.5 3x 10⁵

Time Index

Data Index

Time Series Plot

MORGAN STANLEY GOLDMAN SACHS GROUP INC.

LEHMAN BROTHERS HOLDINGS INC.

DEUTSCHE BANK AG BNP PARIBAS SOCIETE GENERALE BEAR STEARNS HSBC HOLDINGS PLC ORD $0.50 (UK R BANK OF AMERICA CORPORATION JP MORGAN CHASE & CO.

UBS AG CREDIT SUISSE GROUP BARCLAYS PLC ORD THE ROYAL BANK OF SCOTLAND GRO CITIGROUP INC.

MERRILL LYNCH & CO., INC.

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Results

CDX Index Source: Thomson-Reuters

Q1−0520 Q2−05 Q3−05 Q4−05 Q1−06 Q2−06 Q3−06 Q4−06 Q1−07 Q2−07 Q3−07 Q4−07 Q1−08 Q2−08

40 60 80 100 120 140 160 180 200

Time Index

Data Index

Time Series Plot

5Y IG CDX

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Results

ITRAXX Index Source: Thomson-Reuters

Q1−0520 Q2−05 Q3−05 Q4−05 Q1−06 Q2−06 Q3−06 Q4−06 Q1−07 Q2−07 Q3−07 Q4−07 Q1−08 Q2−08

40 60 80 100 120 140 160 180

Time Index

Data Index

Time Series Plot

ITRAXX IG 5Y

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Results Volatility of Assets

Volatility of Assets for US Banks

20030 2004 2005 2006 2007 2008 2009 2010

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Time Index

Data Index

Time Series Plot

GOLDMAN SACHS GP MORGAN STANLEY MERRILL LYNCH LEHMAN BROSHDG CITIGROUP BANK OF AMERICA JP MORGAN CHASE & CO BEAR STEARNS DEAD

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Volatility of Assets for UK Banks

20030 2004 2005 2006 2007 2008 2009 2010

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Time Index

Data Index

Time Series Plot

HSBC HDG (ORD $050) ROYAL BANK OF SCTLGP BARCLAYS

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Volatility of Assets for Swiss/German Banks

2003 2004 2005 2006 2007 2008 2009 2010

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time Index

Data Index

Time Series Plot

DEUTSCHE BANK UBS ’R’

CREDIT SUISSE GROUP N

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Volatility of Assets for French Banks

2003 2004 2005 2006 2007 2008 2009 2010

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Time Index

Data Index

Time Series Plot

BNP PARIBAS SOCIETE GENERALE

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Results Implied Distance and Probability to Default

Distance to default for US Banks

20030 2004 2005 2006 2007 2008 2009 2010

1 2 3 4 5 6 7 8 9 10

Time Index

Data Index

Time Series Plot

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Historical Probability of default for US Banks

20030 2004 2005 2006 2007 2008 2009 2010

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Time Index

Data Index

Time Series Plot

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Distance to default for UK Banks

20030 2004 2005 2006 2007 2008 2009 2010

2 4 6 8 10 12 14

Time Index

Data Index

Time Series Plot

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Historical Probability of default for UK Banks

20030 2004 2005 2006 2007 2008 2009 2010

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time Index

Data Index

Time Series Plot

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Distance to default for Swiss/German Banks

20030 2004 2005 2006 2007 2008 2009 2010

1 2 3 4 5 6 7 8

Time Index

Data Index

Time Series Plot

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Historical Probability of default for Swiss/German Banks

20030 2004 2005 2006 2007 2008 2009 2010

0.05 0.1 0.15 0.2 0.25 0.3 0.35

Time Index

Data Index

Time Series Plot

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Distance to default for French Banks

20030 2004 2005 2006 2007 2008 2009 2010

1 2 3 4 5 6 7

Time Index

Data Index

Time Series Plot

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Historical Probability of default for French Banks

20030 2004 2005 2006 2007 2008 2009 2010

0.05 0.1 0.15 0.2 0.25

Time Index

Data Index

Time Series Plot

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Results Dynamic Covariances

Example Covariance Analysis of GS

2003 2004 2005 2006 2007 2008 2009 2010

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Conditional Volatility

Time Index σi,t

GOLDMAN SACHS GP. − TOT RETURN IND (~U$) CDX

iTraxx

2003 2004 2005 2006 2007 2008 2009 2010

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8

Conditional Correlation

ρi,j,t

Equity,CDX Equity,iTraxx CDX,iTraxx

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Example Impulse Response Analysis of GS

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.3

−0.2

−0.1 0 0.1

Response in Mean, to a unit shock in CDX5YIG

GOLDMAN SACHS GP CDX5YIG iTraxx5YIG

0 0.5 1 1.5 2 2.5 3 3.5 4

−2 0 2 4

Response in Standard Deviation, to a unit shock in CDX5YIG

0 0.5 1 1.5 2 2.5 3 3.5 4

−1 0 1 2 3

Response in Correlation, to a unit shock in CDX5YIG

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Example Covariance Analysis of UBS

2003 2004 2005 2006 2007 2008 2009 2010

0.02 0.03 0.04 0.05 0.06 0.07

Time Index σi,t

UBS ’R’ − TOT RETURN IND (~U$) CDX

iTraxx

2003 2004 2005 2006 2007 2008 2009 2010

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6

ρi,j,t

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Example Impulse Response Analysis of UBS

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.4

−0.2 0 0.2 0.4

iTraxx5YIG CDX5YIG UBS ’R’

0 0.5 1 1.5 2 2.5 3 3.5 4

−1 0 1 2 3

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.5 0 0.5 1 1.5

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Example Covariance Analysis of BNP Paribas

2003 2004 2005 2006 2007 2008 2009 2010

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time Index σi,t

BNP PARIBAS − TOT RETURN IND (~U$) CDX

iTraxx

2003 2004 2005 2006 2007 2008 2009 2010

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

ρi,j,t

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Example Impulse Response Analysis of BNP Paribas

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.4

−0.2 0 0.2 0.4

BNP PARIBAS CDX5YIG iTraxx5YIG

0 0.5 1 1.5 2 2.5 3 3.5 4

−2 0 2 4

0 0.5 1 1.5 2 2.5 3 3.5 4

−2 0 2 4

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KLIC Test of GS Model Stability

2004 2005 2006 2007 2008 2009

−12

−10

−8

−6

−4

−2 0 2 4

KLIC Test, Recursive versus whole sample (mu=0.2) for GOLDMAN SACHS GP.

Test stat at time, t 5% Error Bound 10% Error Bound

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Capital Injections for Goldman Sachs

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 2 3 4 5 6 7 8 9

Capital Injection, $USD 100 Billions

Distance to Default

Distance to Default and Additional Capital Requirements for: GOLDMAN SACHS GP

Volatility Assets: 0.015604 Volatility Assets: 0.022666 Volatility Assets: 0.029727 Volatility Assets: 0.036789 Volatility Assets: 0.043851 Volatility Assets: 0.050912 Volatility Assets: 0.057974 Volatility Assets: 0.065036 Volatility Assets: 0.072097 Volatility Assets: 0.079159

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Capital Injections for Lehman Brothers

1 2 3 4 5 6 7 8 9

Distance to Default

Distance to Default and Additional Capital Requirements for: LEHMAN BROSHDG

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Capital Injections for UBS

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

2 4 6 8 10 12 14 16

Distance to Default

Distance to Default and Additional Capital Requirements for: UBS ’R’

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Capital Injections for Royal Bank of Scotland

2 4 6 8 10 12 14 16 18 20 22

Distance to Default

Distance to Default and Additional Capital Requirements for: ROYAL BANK OF SCTLGP

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Capital Injections for BNP Paribas

1.5 2 2.5 3 3.5 4 4.5

2 4 6 8 10 12 14 16 18 20

Distance to Default

Distance to Default and Additional Capital Requirements for: BNP PARIBAS

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Capital Injections for Soci´ et´ e G´ en´ erale

2 4 6 8 10 12 14 16 18

Distance to Default

Distance to Default and Additional Capital Requirements for: SOCIETE GENERALE

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Conclusions

Concluding Remarks

At present this paper demonstrates a distance to default model with some striking results regarding the potential default risk of several international LCFIs.

Our preliminary results suggest that default risk is highly correlated across international boundaries and that information contained in the market prices of Credit Derivatives impacts of the of equity and asset volatilities of LCFIs.

Given this transmission mechanism we suggest that there is significant evidence that large default events can propagate across institutions and that this shock propagation to the collective asset volatility is enough to create significant default events in those institutions heavily exposed in these markets.

We believe that credit default swap index factor augmented models of default risk will be a major area of risk analysis for future research.