Problems of Credit Pricing and Portfolio Management

The Finance Development Centre 1

Problems of Credit Pricing and Portfolio Management

ISDA - PRMIAOctober 2003Con Keating


Spreads and Returns

The relation is well known

And the duration of a corporate is difficult to estimate, the standard calculation does not apply.

But this only applies to default free bonds

)( 111 ttttt yyDyr


The Problem of Duration

Consider two five year zero coupon bonds, a AAA and a BBB yielding respectively 6% and 10% while the

equivalent government yields 5%

The AAA has a modified duration of 5/1.06 = 4.71 years

The BBB has a modified duration of 5/1.10 = 4.54 years

The govt. has a modified duration of 5/1.05 = 4.76 years

This suggests that lower credits are less risky and less volatile than governments of equivalent characteristics.


Is this a practical problem?

The relation between ex-ante spread and subsequent returns

A sub-investment grade Index 1979 -2002

Ex-Ante Spread / One Year Returns

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0 2 4 6 8 10 12

Yield Spread %

Retu

rns %


Some StatisticsExAnte Spread Return

Mean 4.76 1.88StDev 1.98 11.42Skew 1.77 -0.06Kurtosis 3.04 -0.25

And correlations

Cross-correlations ExAnte Spread / Return

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Lag

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ss-c

orre

lati

on

s


Transition Matrices

One year above and Three year below

FromTo: AAA AA A BBB

aaa 92.06% 1.19% 0.05% 0.05%aa 7.20% 90.84% 2.40% 0.25%a 0.74% 7.59% 91.89% 5.33%bbb 0.00% 0.27% 4.99% 88.39%bb 0.00% 0.08% 0.51% 4.87%b 0.00% 0.01% 0.13% 0.77%c 0.00% 0.00% 0.01% 0.16%D 0.00% 0.02% 0.02% 0.18%

FromTo: AAA AA A BBB

aaa 78.3% 3.0% 0.2% 0.2%aa 18.1% 75.9% 6.2% 1.1%a 3.4% 19.2% 80.1% 14.7%bbb 0.2% 1.7% 12.4% 78.7%bb 0.0% 0.2% 0.9% 4.4%b 0.0% 0.0% 0.2% 0.7%c 0.0% 0.0% 0.0% 0.1%D 0.0% 0.0% 0.0% 0.2%


Simulations

A 150 bond equal weight AAA portfolioOne Year Returns -Credit Migration Alone

The Set-Up

Initial RatingInitial spreadInitial price Trading spreadRatingPx after1 year

Coupon 2 1 30 0.985982 30 1 0.988659Life 5 2 45 0.979064 45 2 0.983051

3 70 0.967666 70 3 0.9737934 150 0.932274 150 4 0.944904

525 5 0.82317650 6 0.787086

1000 7 0.6962658 0.3


The Results - AAA

Distribution

Mean 2.25%StDev 0.015%Skew -0.28155Kurt 0.210952

Histogram AAA Returns

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0.022 0.022 0.022 0.022 0.023 0.023


AA Returns Histograms


Histogram AA Returns

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A Returns Histograms


Histogram - A Returns

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0.016 0.018 0.020 0.022 0.024 0.026


Diversified AAA/AA/A/BBB Portfolio

The skewness is not diversified away !


Histogram "Diversified" Portfolio

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0.013 0.015 0.017 0.019 0.021 0.023 0.025 0.027


Diversification of Corporates

Corporate spreads are largely a compensation for bearing credit risk, and one reason why they are so wide is that losses from default can easily differ substantially from expected losses.

Moreover, such risk of unexpected loss is evidently difficult to diversify away.

As corporate bond portfolios go, one with 1,000 names is unusually large, and yet our example shows it could still be poorly

diversified in that unexpected losses remain significant.

Reaching for yield: Selected issues for reserve managersRemolona and Schrijvers, BIS Quarterly Review, Sep 2003


Even small correlation can be harmful to your health

A distribution of defaults with .02 correlation

Histogram .02 Dependence

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0.000 20.000 40.000 60.000 80.000 100.000 120.000

98% independent 2% dependent


Correlation and Dependence

Higher moments are needed to capture dependence.

Correlation tells one little about the shape of the joint distribution

The presence of common factors tells much about dependence.

Common Factors diversify slowly if at all

The limits to (additive)diversification are well known

But in the presence of common factors diversification may be slow and inefficient.

Copulae are little better.


Common Factors

In the presence of common factors, tails can be arbitrarily thick.

In the previous example, 100 defaults occur 5 standard deviations from the mean.

This is the free lunch associated with CBO transactions

Diversification score construction cards are flawed in this regard.


One possible solution

In hedge funds, we have always countered high correlation by short selling.

Both are equally valid techniques for the reduction of variability.

Long-Short neutralises all odd moments

The Sharpe ratio for a long only strategy is bounded above.

The Sharpe ratio for Long-Short is unbounded

Long-Short tends to neutralise common factors


Higher Moment Approaches

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Midpoint Of Range

Historical Daily Return Distribution

A Hedge Fund trying to be NormalSkew 0.06 Excess Kurtosis 0.36


Log-Normal or Abnormal?

One of these is lognormal. The other 2 have infinite skew and kurtosis


Omega functions

The left bias is evident,even though skew can’t be used to measure it.


Omega HF and Normal

Red is analytic normal of same mean and variance

The (small) sample properties of the actual should make its Omega lie above on the downside and below on the upside.


Risk Profile HF

This Difference in Risk Profiles arises from Skew & Excess Kurtosis of just 0.06 and 0.36


The Omega function for a Distribution

This process may be carried out for any series. The valueof the Omega function at r is the ratio of probability weighted gains relative to r, to probability weighted losses relative to r. If F is the cumulative distribution then

(r) :(1 F(x))dx

r

F(x)dx

r

.


Why is this important?

The Omega function of a distribution is mathematically equivalentto the distribution itself.

(Note for the quantitatively inclined. There is a closed form expression for F given Omega, just as there is for Omega given F.)

None of the information is lost or left un-used.

Sometimes mean and variance are enough… butsometimes the approximate picture they give hides thefeatures of critical importance for terminal value.


Graphically

I2(r) : (1 F(x))dxr

I1(r) : F(x)dx

r

The area outlined in red is:

The area outlined in black is:


Omega for a normal distribution

The slope at the mean is

2

r

.


How can we reliably incorporate return levels and tail behaviour?

Omega – A Sharper Ratio – does precisely this.

•Assumes nothing about preference or utility•Works directly with the returns series•Is as statistically significant as the returns•Does not require estimation of moments•Captures all the risk-reward characteristics


Basic Properties of

• It is equivalent to the distribution itself• It is a decreasing function of r• It takes the value 1 at the mean • It encodes variance, skew, kurtosis and all higher

moments• Risk is encoded in the relative change in Omega

produced by a small change in the level of returns.• The shape of Omega makes risk profiles apparent

For two assets, the one with the higher Omega is, literally,A BETTER BET.


Returns for 2 normally distributed assets A and B with the same means

Asset B

Asset A

A 7,A 3

B 7,B 4

A

B

The Sharpe ratio says A is preferable to B.Omega says it depends on your loss threshold.Below the mean, A is preferable, above the mean, B is.


Returns for 2 normally distributed assets A and B with the same means

A

B

The superior portfolio is dependent upon the threshold level.If we measure performance based on a sample of mean 6.9, then we will see a preference reversal relative to 7.1.


Omega Risk Profiles

The risk is encoded in the way Omega responds to a small change in the level of returns:

Risk(r) :1

(r)

ddr

For normally distributed returns, at the mean thisis simply determined by the standard deviation.


Even for normally distributed returns, Omega has more information

2.4

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2.0

decreases as decreases and also as we move away from the mean for fixed

Risk(r)

Risk(r)


Omega Risk Profiles for a distribution with negative skew and a normal with the same mean and variance show dramatically different features.

Negative skew in green, Normal in Blue, mean is 8.5,Standard Deviation is 1.82


The Shape of OmegaOption Strategies

Omegas for two US mortgage-backed strategies


Risk Profiles – Option Strategies


BH folded in September 2002 after a loss of 60% on a gamble for redemption.

Simulations show the potential impact on terminal value.

Losses were 250 times more likely with BH than with CL

Loss ~ $500million. The SEC investigation continues…


Returning to the earlier simulations

Omega AAA Simulations

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0.0212 0.0216 0.022 0.0224

Return

Om

ega

Iteration 1

Iteration 2

Iteration 3


AA- Omega(s)


Rating Class - Omegas


Portfolio & Rating Class - Omegas


Covenants and Collateral

In a competitive investment market all of the gains associated with lower funding cost accrue to the

company

Covenants in public debt are good for shareholders

Covenants serve to discipline management

Ratio test covenants of the income or asset coverage genre may increase the likelihood of default and

distressRatings triggers are really death spirals.


Covenants and pricing

Covenants restrict the range of possible state prices of corporate bond.

Covenants increase the price of a bond

Covenants, ceteris paribus, lower the mobility of the transition matrix.


Security and Collateral

To the extent they reduce the loss in default, also help to reduce the diversification problem

Histogram - 30% Recovery

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Histogram - 100% Recovery

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0.017 0.019 0.021 0.023 0.025 0.027 0.029 0.031


Security and Collateral - Omegas

0.0001

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1

10

100

1000

1000030% Recovery

100% Recovery

This results in a higher mean return, and vastly better downside protection.


Omega - Bond pricing

The essence of pricing corporate bonds using Omega is to equate the Omegas over the range of support of

the function.

Omega Price

0.00001

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1000

100000

-0.016 -0.012 -0.008 -0.004 0 0.004 0.008


Dynamics of Corporate Bond Returns

We need to examine two distinct elements

The relation of returns to their prior returns - autocorrelation

We might also consider correlation to treasuries.


One Problem for the Statisticians

Partial autocorrelation - T Short

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• Auto-correlation - the degree to which today’s return forecasts tomorrows.

• Skill?• Or returns smoothing?

Auto-correlationPartial autocorrelation T

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Correcting for Auto-correlation

• The differences are meaningful

Excess Returns Adjusted Returns ErrorsMean Std Dev Info Ratio Mean Std Dev Info Ratio Mean Std Dev Info Ratio

ConvertibleFRM 0.682 1.065 0.640 0.670 1.624 0.413 1.76% -52.49% 35.47%HFR 0.524 1.033 0.507 0.503 1.594 0.315 4.01% -54.31% 37.87%CSFB 0.494 1.371 0.361 0.485 2.618 0.185 1.82% -90.96% 48.75%Henn 0.357 1.235 0.289 0.349 1.865 0.187 2.24% -51.01% 35.29%

Fixed Inc FRM 0.470 1.370 0.343 0.439 2.574 0.171 6.60% -87.88% 50.15%HFR 0.045 1.320 0.034 0.037 1.931 0.019 17.78% -46.29% 44.12%CSFB 0.166 1.176 0.141 0.162 1.882 0.086 2.41% -60.03% 39.01%


Adding a security to a portfolio

Partial autocorrelogram -Security

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Autocorrellogram - Portfolio Ex

Partial autocorrelogram - Portfolio Ex

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But this isn’t enough

Cross-correlations Security and Portfolio Ex

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Instantaneous RegressionYields and Rates


But the long run relation between spread and yield is more complex

And this is at odds with the earlier instantaneous result


The answer lies in the dynamics

And therein lies a trading strategy.


But before delivering too much optimism

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2.50 3.00 3.50 4.00 4.50 5.00 5.50

10/03/03(2.98;104)

03/09/03(3.63;65)

21/08/00(5.30;69)

25/10/02(3.90;144)

13/06/03(2.64;75)

7/11/01(3.67;99)

04/07/02(4.49;114)

30/05/01(4.76;66)

(bps)

Euro Corporate Spread vs Government Yield


Modigliani - Miller and Modern Finance

Newer Theories exist - in many regards these look like the pre-M-M world.

Few will not now know the M-M theorem, under which corporate financial structure is irrelevant

A simple test: If M-M applies the principal components of default variability would be constant across

countries - observed corporate financial structure differs markedly internationally.


Principal Components of Default

The data was pre-processed to remove cyclical (phase) effects which might otherwise bias the results.


An important warning

The principal components analysis suggests that the default process varies markedly among countries.

This suggests that different credit evaluation models are needed in each country.

If these are based upon financial statements, it would be as well to remember the different purposes for

which financial statements are produced.

This is rather more than differences in legal processes and systems.


An Afterthought

Portfolio Weighting by Different Schemes

A Comparison of Equal weighting and weighting by equal expected loss

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Equ 1

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EL 1

EL 2


Credit Derivatives

The Banks have bought a net $190 billion of protection.

The Insurance industry has written a net $300 billion of protection.

These are small sums - about a quarter of the UK mortgage market!

None of the models in use for pricing works with any meaningful precision.

This will require full information pricing.

Notwithstanding that, some of the mono-lines look over-exposed.


The justification for that last assertion

Lies in the non-normality of spread distributions


But we might try estimating econometric models

Quite a few have done precisely this.

Here’s our model results


The diagnostics for which are:


The Durbin-Watson suggests that something may be awry

Which is just as well as:

Grimmett is a set of earthquake data

Sparrow is a set of car number plates collected by my daughters

And that illustrates the econometric problem rather well

The data is sparse, noisy and not really suitable for mining exercises.

The out of sample performance usually abysmal.


The work has really only just started

Further Papers: www.FinanceDevelopmentCentre.com

[email protected]

In my experience linear factor models can “explain” only 70% - 80% of what happens

And that isn’t enough for practical pricing

By way of ending let me offer a final insight

Credit is an expectation of Liquidity

So maybe we should all be working on Liquidity


Omega Interpretations

Omega may be interpreted as the ratio of a “virtual” call to a “virtual” put.

Omega may be viewed as the “fair game” representation of the distribution.

}]0,[max{

}]0,max{[

)(

))(1(

)(xrE

rxE

drrF

drrF

rr

a

b

r

And we might argue that this is the correct place from which to measure Risk

Problems of Credit Pricing and Portfolio Management

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