1 University of Chicago Masters in Financial Mathematics Week 3 Jon Frye Federal Reserve Bank of Chicago [email protected] The views expressed are the.

1

University of Chicago

Masters in Financial Mathematics

Week 3

Jon Frye

Federal Reserve Bank of Chicago

[email protected]

The views expressed are the author’s and do not necessarily represent the views of the management of the Federal Reserve Bank of Chicago or the Federal Reserve System.

Modeling Default Correlation

2

Modeling Default CorrelationOutline of presentation

A. Setting the stage

B. Correlation in variants of CreditManager

C. Summary and conclusion

3

Modeling Default CorrelationOutline of presentation

A. Setting the stage

1. Default correlation and portfolio loss

Two state (default-only) model

Default correlation

Multi-state (marked-to-market) model

2. Latent variable probability model

4

Default correlation and portfolio lossTwo state (default-only) model

The bank has exposure of $1 to each of two firms.

The credit capital analysis has a horizon (one year).

Either the maturity of the exposures equals the horizon, or the bank does not care about the marked-to-market value of its exposure portfolio.

Therefore, the bank experiences a loss only if a firm defaults. Loss given default is equal to 100%.

5

Default correlation and portfolio lossDefault correlation

Let D1 = 1 if firm 1 defaults, D1 = 0 otherwise

Let D2 = 1 if firm 2 defaults, D2 = 0 otherwise

Correlation is defined as follows:

)()(

),(),(

21

2121

DVarDVar

DDCovDDCorr

6


Var(D1) = E [ (D1 - E [ D1 ])2 ]

= E [ D12 - 2 D1 E [ D1 ] + (E [ D1 ])2 ]

= E [ D1 ] - 2 (E [ D1 ])2 + (E [ D1 ])2

= p1 - p12 = p1(1-p1)

where p1 = Prob [ D1 = 1 ]

7


Cov(D1, D2) = E [ (D1 - E [ D1 ]) (D2 - E [ D2 ]) ]

= E [ D1D2 ] - E [ D1 ] E [ D2 ]

= p12 - p1p2

where p12 = Prob [ D1 = 1 and D2 = 1 ]

8


For any two events (such as D1 and D2),

if we have their probabilities p1 and p2,

then p12 implies correlation and correlation implies p12

For example, if p12 = p1 p2, then Corr = 0

Next: Calculating default correlation

))(()()(

),(),(

2121

2112

21

2121

11 pppp

ppp

DVarDVar

DDCovDDCorr

9

Default correlation and portfolio lossMulti-state (marked-to-market) model

Firm 220% 30% 50% Correlations

Firm 1 D Junk Inv D Junk Inv70% Inv 14% 21% 35% Inv 0% 0% 0%20% Junk 4% 6% 10% Junk 0% 0% 0%10% D 2% 3% 5% D 0% 0% 0%

10


With 3 states, there are 4 free probabilities With 4 states, there are 9 free probabilities...

How can we estimate, for example, the joint probability that Firm 1 becomes junk and Firm 2 defaults?

We do not have the data we want. If we had a long historical series of a large number of firms similar to Firm 1 and Firm 2, we could hope to estimate the joint probabilities as observed frequencies.

11


A common way (and the way used by CreditManager) around the data problem is to assume the joint probabilities reflect a simpler underlying relationship between “latent variables” having fewer parameters.

Most commonly (and in CreditManager), the latent variables are modeled as bivariate normal.

12

2. Latent variable probability model

Properties of bivariate normal distribution

Latent variables, joint probabilities, and event correlations

13

BBB

BB A

CCCB

Default

AA

AAA

-3.19 -2.83 -2.54 -1.47 1.36 1.78 2.09

Latent variable probability modelNormal latent variable

For X ~ U[0, 1], default if X < p0 = 0.07%

For Z ~ N[0, 1], default if Z < -1(p0) = -3.19

X = .9411 <=> Z = 1.56 <=> transition to A rating

14

Latent variable probability modelNormal latent variable

The choice of latent variable distribution does not matter for modeling the marginal probabilities (probability that a single firm transitions to default or another state)

Modelers specify bivariate normal latent variables, in part, because the entire distribution (and therefore the probabilities of joint events, such as Firm 1 becomes junk and Firm 2 defaults) is controlled by the single parameter, .

15

Latent variable probability modelProperties of bivariate normal distribution

The fully elaborated bivariate normal density has five parameters, 1, 2, 1, 2, and .

When the variables are standardized, only is free.

2

221212221 2

1

1

2

1

12

1zzzzexp)z,z

(

16


Excel: the bivariate normal (2D)

17


Excel: two firm portfolio

18


Given , the density of the latent variables is known.

Given the density of the latent variables and both firms’ transition probabilities, the probabilities of joint events are known.

Given the probabilities of joint events (such as p12), the default (and other event) correlations are known.

Thus, determines all joint probabilities and all event correlations.

19


Generalizing to a portfolio of many firms, multivariate standard normal latent variables capture the probability distribution of all joint events, given:

– the transition probabilities for each firm– the correlation matrix of the latent variables

With a portfolio of thousands of firms, the correlation matrix itself becomes too big to estimate consistently. CreditMetrics assumes that the latent variables arise from a simpler structure of underlying factors.

20

Correlation in variants of CreditManager

21


B. Correlation in variants of CreditManager

3. Single factor model of latent variables

Model

Latent variable correlation matrix

Variance of default rate

Fitting the model to default data 4. A neglected correlation: recovery

5. Multi- factor model of latent variables

6. Correlation in multi-state CreditManager

22


So far, we have modeled joint probabilities by using latent variables. Now we model the latent variables as rising from underlying factors.

The factors are normal, so the latent variables are normal as before.

We start with the simplest case: one factor, in a two-state (default mode) model.

Then we move to the multi-factor, multi-state framework of conventional CreditManager.

23

Single factor model of latent variablesModel

Let Aj be the latent variable for firm j:

11

112

1

1

22

222

2

2

)(

)Var()()Cov()Var()Var(

0]E[]E[]E[

1] N[0, ~ and 1] N[0, ~ where,

jj

jjjjjjj

jjjj

jjjjj

ww

ZwZ,XwwXwA

ZwXwA

ZXZwXwA

Therefore, Aj is standard normal

24


jjjj ZwXwA 21

All firms are affected by X

X is called the systematic risk factor

wj is called firm j’s loading on the systematic risk factor

Each firm is affected by its own independent Zj

The Zj are called the idiosyncratic risk factors

25


) Var()()Var()Var( jjjj ZwXwA 22 1

The loading wj controls how much of the variance of Aj comes from X. The remainder comes from Zj.

A more “cyclical” firm will have a higher loading on the systematic risk factor than will a less cyclical firm.

Cyclical firms are affected by the same underlying factor that affects other (cyclical and non-cyclical ) firms

26

Single factor model of latent variablesLatent variable correlation matrix

kj

kkkjjj

kjkjk,j

ww

ZwXwZwXw

A,A)A,A(

),Cov(

)Cov(Corr

22 11

The correlations between latent variables are controlled by the loadings {wj} on the systematic risk factor.

If two firms have the same loading, w, the correlation between their latent variables is w2.

27


Suppose all firms have w = 0.5.

Aj = 0.5 X + 0.87 Zj

Corr(Aj, Ak) = 0.52 = 25%, for all j, k

The correlation matrix has 1’s along the diagonal and 25% in other elements.

28


Suppose there are two kinds of firms

Cyclical firms have w = 0.5.

Non-cyclical firms have w = 0.3.

Acyclical, j = 0.5 X + 0.87 Zcyclical, j

Anon-cyclical, k = 0.3 X + 0.95 Znon-cyclical, k

29


Cyclical Non-cyclical1 25% 25% 15% 15% 15%

Cyclical 25% 1 25% 15% 15% 15%25% 25% 1 15% 15% 15%15% 15% 15% 1 9% 9%

Non-cyclical 15% 15% 15% 9% 1 9%15% 15% 15% 9% 9% 1

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Single factor model of latent variablesVariance of default rate

We can simulate this by drawing the systematic risk factor and the idiosyncratic risk factors:

Acyclical, j = 0.5 X + 0.87 Zcyclical, j

Anon-cyclical, k = 0.3 X + 0.95 Znon-cyclical, k

We can use the simulated latent variables in either a two-state (default only) model or in a multistate (marked-to-market) model.

31


For simplicity, we will use a two-state model. A loss is experienced only if a firm defaults.

We will fit this single factor, two-state latent variable model to Moody’s default data.

The key feature of the Moody’s data is the variance (from year to year) of the default rate. First consider the variance of the default rate in general terms, and then perform the actual estimation.

32


Variance of the default rate depends on default correlations:

)p(p)p(p,DDN

D...DDDNN

D

kkjjj k

kj

Nj

111

1

2

12

Corr

)Var(Var 32

We know that higher loadings {wj, wk} produce higher latent variable correlations {j,k}, which produce higher default correlations {Corr(Dj, Dk)}, which produce a higher variance of the default rate.

Next we look at two examples of that chain of effect.

33


Example 1: Suppose wj = 0 for all firms.

Aj = 0 X + Zj

Each Aj depends only on its own independent Zj.

The {Zj} are independent, so the {Aj} are independent,

therefore the {Dj} are independent.

On each run, if there are many firms, the simulated default rate will equal the expected default rate (+/- minor sampling variation). There will be little variation in the default rate from year to year.

34


Example 2: Suppose wj = 0.9 for all firms.

Each Aj depends mostly on X, and hardly on Zj.

A high value of X causes most of the Aj to be high, so most of the firms do well. Only firms that get a very bad Zj will default.

A very low value of X causes most of the Aj to be low. Even firms with average values of Zj may default. The default rate varies significantly from year to year.

35


We will maintain for estimation purposes the (grossly simplified) assumption that all firms have the same loading on the systematic risk factor. Then we can use the year-to-year variation in the default rate to estimate this loading. To stay consistent with “Collateral Damage,” denote the common loading as p.

Since a higher p causes greater variability in the default rate, there is only one level of p consistent with the observed variation of default.

36

Single factor model of latent variablesFitting the model to default data

Default data universe:

US Domicile

Industry Group not Financial, Banking, Insurance, …

Moody’s Rated Aaa, Aa1, Aa2, Aa3, A1, A2, A3, Baa1, Baa2, Baa3, Ba1, Ba2, Ba3, B1, B2, B3, Caa, Ca, C

37

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00Aaa 61 72 48 37 37 42 46 49 48 43 42 30 27 25 24 27 24 16 17

A 312 1 1 1

A1 74 86 91 118 111 111 102 97 108 109 106 100 99 107 102 94 91 85

A2 153 148 158 155 141 132 139 137 114 120 105 104 111 130 136 142 161 174

A3 108 105 111 114 101 93 100 102 102 113 118 118 123 123 133 137 129 130

B 43

B1 66 62 61 85 131 173 200 201 128 96 116 148 186 191 224 283 289 297

B2 10 15 26 28 47 57 69 79 69 59 55 77 97 98 109 149 191 181

B3 39 41 50 65 81 91 83 95 104 96 79 89 102 116 126 124 154 151

Caa 22 18 15 23 28 42 46 55 54 82 97 71 56 61 73 66 29 8 6

Caa1 43 88 104

Caa2 23 47 55

Caa3 19 26 47

Ca 18 24 17 11 13 17 14 14 16 22 20 22 19 31 33 37 14 22 21

C 6 6 6 5 5 5 6 5 5 5 1 1 1 2 4 8 10 10 13

Number of Issuers By Rating and Year


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Determining The Sample Period

0

50

100

150

200

250

70 72 74 76 78 80 82 84 86 88 90 92 94 96 98

Co

un

t

Defaulting Issues

Defaulting Issueswith Default Price


39


A key assumption: the law of large numbers applies. There are enough firms that the observed default rate equals its expectation (conditioned on X):

2

1

2

1

12

1

11

1

p

pxPD

p

pxPDZ

PDZppx

xX|PDADR

jjj

jj

jjj

P

P

P

40


Adapting this key assumption to the data set, with T years: t = 1,…,T (1983-1997)and R rating grades: r = 1,…,R (Aaa, Aa1,…)

In year t, the default rate of a firm rated r is

Assume that PDr is the long-term average default rate in grade r

2

1

1 p

pxPDDR tr

t,r

41


Let hr,t be the fraction of firms rated r in year t, so:

Then the default rate for the portfolio in year t is:

thr

t,r yeareach for 1

)x(g

p

pxPDhDR tp

r

trt,rt

2

1

1

Note that the function g is monotonic in x.

42


The default rate is a monotonic function of the systematic risk factor, X, which has a known probability density:

1] N[0,~ ; X)X(gDR tpt

We can find the density of DRt by change of variable, and, assuming independence from year to year, we can find the density of {DRt}.

The density of {DRt} depends on the parameter p. Given the {DRt} actually observed, a certain value of p provides the maximum likelihood estimate.

43


2

2

21

1

t

t

t

DRgexp

dDR

DRdgThe density of DRt equals

By taking the derivative, and with independence year-to-year, we have the joint distribution of the set of observed default rates:

44


R

r

trt,r

t

T

t

p

DRpgPDhp

DRgexpp

12

11

212

1

12

21

The density of {DRt} is

p enters this expression both directly and through g-1. Maximizing the density with respect to p provides the maximum likelihood estimate.

45


The default rate of bonds has a certain variation from year to year. This variation stems from the randomness of X, which has a standard normal distribution.

The loading p connects the variation of X to the variation of the default rate: the greater is p, the more the default rate will vary from year to year.

There is only one level of p consistent with the default rate variation that is actually observed.

46


The maximum likelihood estimate of p is 0.23 for this data set. The implied level of correlation between the latent variables of two firms is (0.23)2 = 5.3%. (Note, this is below the average correlation between stocks, thought to be in the range of 20-30%.)

Given p, the monotonic function g is determined and we can “back out” the level of X consistent with each year’s default rate:

)DR(gX)X(gDR tpttpt1 ;

47

Default Rate and { }

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

3%

4%

5%

83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

De

fau

lt R

ate

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

Va

lue

of

X

Default Rate

X

X̂

Single factor model of latent variables

48


The loading of 0.23 provides the best fit for the variance of the default rate for this data set. A higher loading, such as suggested by stock price correlation, would produce too much default rate variance.

To fit a default-only model, we used only default data. A multi-state model could be fit using transition data, and would probably have a different average loading.

In practice, a risk manager wants to know the loading of each firm, not just an overall average, to detect cyclically sensitive credits.

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4. A neglected correlation: recovery

Evidence of systematic recovery risk

Model

Fitting the model to recovery data

50

A neglected correlation: recoveryEvidence of systematic recovery risk

Recovery universe:

US Domicile, Non-Financial

Bonds, Not Loans

Not backed by second entity

Where Moody’s has a Default Price

(observed one month after default,

measures recovery)

51

0%

5%

10%

15%

20%

25%

0-10

10-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

90-100

100+

Recovery, Percent of Par

Fre

qu

en

cy

1990-1991

Other Years


52

0%

10%

20%

30%

40%

50%

60%

0% 2% 4% 6% 8% 10%

Default Frequency

Av

era

ge

Re

co

ve

ry

1991

1990


53

A neglected correlation: recoveryModel

We model each recovery as depending on the systematic risk factor (with loading q), and on an independent factor idiosyncratic to that recovery.

We need two more parameters to model the mean and the variance of recovery. Mean recovery is assumed to depend on the seniority of the bond. The variance of recovery is assumed uniform for all bonds.

54

A neglected correlation: recoveryModel

There are T years: t = 1, …, T

There are J seniority classes: j = 1, …, J

Senior Secured (SS) Senior Subordinated (SR)

Senior Unsecured (SU) Subordinated (SB)

There are Nj,t bonds of class j in year t:

i = 1, …, Nj,t

Bond recovery is specified as follows:

i,t,jtji,t,j ZqqXR 21

55

A neglected correlation: recoveryFitting the model to recovery data

The {Xt} were estimated from the default data. (This is sequential MLE, rather than simultaneous MLE.)

Treating Xt as known, each recovery has a normal distribution that depends on μj, σ, and q. Therefore, average recovery in a year has a normal distribution that depends on {μj}, σ, and q.

Maximizing the joint density with respect to the parameters produces the estimates σ = 0.32, q = 0.17.

i,t,jtji,t,j ZqqXR 21

56


σ = 0.32: Recovery is risky. The difference between zero recovery and 96% recovery is only 3 SD.

q = 0.17: Recovery varies with the systematic risk factor. A low realization of X causes both a high level of default and a low level of recovery.

To obtain the relation between the default rate and the average recovery rate, we can substitute a range of X into the relevant equations. The following chart assumes the average portfolio distribution among rating grades and among seniority classes.

57


0%

10%

20%

30%

40%

50%

60%

0% 2% 4% 6% 8% 10%

Default Rate

Av

era

ge

Re

co

ve

ry

Historical

Fitted Model

58


Bond recovery might fall by 25% or more in a bad year, e.g., from 45% to 20%.

If in a bad year, loan recovery falls by 25%–say from an average of 75% to 50%–LGD doubles.

Conventional credit models, which ignore systematic recovery risk, might miss half the risk.

59


Existing portfolio credit models do not allow a role for systematic recovery risk. Therefore, they understate credit risk, perhaps by 50% in aggregate (see Depressing Recoveries, November 2000 Risk Magazine).

The understatement is greater for deals having high expected recovery. The performance of a default-only credit model might be improved by a work-around called the “EL-equivalent substitution” (see Collateral Damage, April 2000 Risk Magazine).

60

5. Multi-factor model of latent variables

Model

Structure of correlation matrix

61

Multi-factor model of latent variablesModel

)Corr( )(

,Corr

N[0,1]

211221

2211

22

212211 1

X,Xwwww

wwwwAA

~A

ZwwXwXwA

,k,j,k,j

,k,j,k,jkj

j

j,j,j,j,jj

This model has two systematic risk factors, X1 and X2. These risk factors are not usually uncorrelated. The correlation between two firms depends on the correlation between the risk factors.

Each firm also depends on an idiosyncratic factor.

62

Multi-factor model of latent variablesStructure of correlation matrix

Suppose there are two kinds of firms:

212111

222221

122122

112121

21211

122112

112111

20150

2504030

8704030

8704030

25050

870050

870050

X,X..A,A

...A,A

Z.X.X.A

Z.X.X.A

..A,A

Z.XX.A

Z.XX.A

Corr Corr

Corr

Corr

Suppose Corr(X1, X2) = 0

63


Industry 1 Industry 21 25% 25% 15% 15% 15%

Industry 1 25% 1 25% 15% 15% 15%25% 25% 1 15% 15% 15%15% 15% 15% 1 25% 25%

Industry 2 15% 15% 15% 25% 1 25%15% 15% 15% 25% 25% 1

64


A multi-factor model can:

– capture “pockets of correlation,” where the correlation is higher within a group than between groups.

– capture differences in the timing of the default (or other transition) cycle between industries or countries.

These features are important, if the credit portfolio can become concentrated in a particular industry or country or other pocket of correlation.

As before, the latent variables can drive a default-only model or a marked-to-market model.

65

6. Correlation in multi-state CreditManager

Orientation

A few modest assumptions

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Correlation in multi-state CreditManagerOrientation

Multi-factor model: Allows richer structure to latent variable correlation matrix, as seen before.

Multi-state model: The latent variables determine upgrades and downgrades as well as defaults. For example, there might be 8 states:

AAA, AA, A, BBB, BB, B, C, and D

67


There are many ways to calibrate a multi-state model.

The first generalizes the approach of Collateral Damage, which used variation of the annual default rate.

The generalization would fit loadings (or correlations) to data involving all rating transitions, not just transition to default. The model would fit the variation of transition frequencies from year to year.

This approach might still miss pockets of correlation that interest the practitioner.

68


A second approach would look directly to the objects of interest: the prices of credit-risky instruments such as bonds. The behavior of bond spreads might imply the parameters of the credit model.

This approach is limited by the quality of corporate bonds data. Bonds have embedded options of uncertain value, which confounds the value of the default option.

Further, when a credit becomes weak, the bond often becomes illiquid, and pricing becomes uncertain.

69


The third approach, which is the one used in CreditManager, has several advantages.

It uses data that is more available, consistent, and readily interpreted than corporate bond data.

It can identify pockets of correlation more readily than methods that rely on transitions data.

However, this approach requires a few modest assumptions.

70

Correlation in multi-state CreditManagerThe latent variables are asset returns!!!!

Merton model of default

If firm’s asset value < liability value, the stockholders give the firm to the debt holders and walk away.

The firm’s asset value has analogous thresholds for rating transitions to non-default rating grades. There are corresponding thresholds for asset returns.

71

Correlation in multi-state CreditManagerAsset returns can be inferred from stock returns!!!!

KMV applies option pricing theory to the stockholders’ option to default. The level and volatility of a firm’s asset value should imply the level and volatility of its stock price. Working backward, KMV observes the level and volatility of the stock price and implies the firm’s asset value each period. These in turn imply the asset return.

Another approach simply assumes a linear relationship between assets and stocks, so that the correlation between latent variables equals the correlation between stocks.

72

Correlation in multi-state CreditManagerStock returns can be proxied in a factor model!!!!

Stock returns are generally not available for all firms in the portfolio. If only some of the correlations are estimated from stock returns, and other correlations come from other sources, it is likely that the correlation matrix is not positive definite.

In practice, stock prices are not used; a factor representation of stock prices is used instead. The correlation matrix of a factor model is always positive definite.

73

Correlation in multi-state CreditManagerThe factors are country and industry indices!!!!

The factors used to explain stock prices are the set of available stock indices.

Any source of correlation beyond the movements of these indices is ignored.

74

Correlation in multi-state CreditManagerThe loadings can be estimated by intuition!!!!

For firms that have stock prices, the factor representation is the regression of the firm on the indices. This regression throws away information.

The rationale for the approach comes from extending it to firms that do not have (enough history of) a stock price. Then the loadings are estimated intuitively, based on a knowledge of the firm’s domicile, industry, and so forth.

75

Correlation in multi-state CreditManagerA few modest assumptions

The latent variables are asset returns!!!!

Asset returns can be inferred from stock returns!!!!

Stock returns can be proxied in a factor model!!!!

The factors are country and industry indices!!!!

The loadings can be estimated by intuition!!!!

What can be said about this chain of assumptions?

76

Correlation in multi-state CreditManagerA few modest assumptions

The assumptions made in CreditManager establish a chain of inference that begins with the multi-variate distribution of stock index returns and ends with the probability distribution of the ratings of all firms in the portfolio.

77

Correlation in multi-state CreditManagerA few modest assumptions-disadvantages

Though many of the assumptions made in CreditManager have intuitive plausibility, few of them have been subject to explicit tests, and none of them can be considered “true.”

In the aggregate, there is no assurance that the correlations that arise within the stock index factor framework are consistent with the severity observed in the default cycle or the rating transition cycle.

78

Correlation in multi-state CreditManagerA few modest assumptions-advantages

To make inferences about specific firms, one needs data on those firms. Of several sources of data on specific firms (bond prices, balance sheets, earnings forecasts), stock data probably is the most reliable.

79


80


What matters in variants of CreditManager are the probabilities of joint events, such as firm A defaults while firm B gets a double downgrade.

Many credit models assume the joint probabilities depend on correlated latent variables. The key question then becomes the level of correlation among the latent variables.

81

Summary and conclusion

One way to calibrate latent variable correlation is to assume it takes the level that matches historical variation in the default rate (e.g., Collateral Damage).

(By the way, when recovery is also allowed to correlate, an economically significant risk is revealed.)

This method could be extended to multi-factor models (which have a richer correlation structure) or to multi-state models (which match a richer set of transitions data than simply the transition to default).

82

Summary and conclusion

CreditManager finds latent variable correlation at the firm level, using stock index data.

This approach has the potential to discover pockets of correlation, but it depends on a long chain of strong assumptions.

The resulting correlation matrix generally will not reproduce the severity of the default cycle.

83

University of Chicago

Masters in Financial Mathematics

Week 3

Jon Frye

Federal Reserve Bank of Chicago

[email protected]

The views expressed are the author’s and do not necessarily represent the views of the management of the Federal Reserve Bank of Chicago or the Federal Reserve System.

Modeling Default Correlation

1 University of Chicago Masters in Financial Mathematics Week 3 Jon Frye Federal Reserve Bank of Chicago [email protected] The views expressed are the.

Documents

d slide

state default

prob d

firm portfolio slide

market model slide

correlation matrix

joint probabilities

probability distribution