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Problems of Credit Pricing and Portfolio Management
ISDA - PRMIAOctober 2003Con Keating
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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
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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.
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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
-30
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-10
0
10
20
30
0 2 4 6 8 10 12
Yield Spread %
Retu
rns %
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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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-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Lag
Cro
ss-c
orre
lati
on
s
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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%
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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
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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
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AA Returns Histograms
Mean 2.35%StDev 0.083%Skew -4.0264Kurt 20.18325
Histogram AA Returns
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0.018 0.019 0.020 0.021 0.022 0.023 0.024
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A Returns Histograms
Mean 2.46%StDev 0.139%Skew -1.365Kurt 3.401
Histogram - A Returns
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0.016 0.018 0.020 0.022 0.024 0.026
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Diversified AAA/AA/A/BBB Portfolio
The skewness is not diversified away !
Mean 2.43%StDev 0.202%Skew -1.238Kurt 2.308
Histogram "Diversified" Portfolio
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0.013 0.015 0.017 0.019 0.021 0.023 0.025 0.027
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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
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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
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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.
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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.
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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
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Higher Moment Approaches
0
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90
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No
. O
f
Da
ys
Midpoint Of Range
Historical Daily Return Distribution
A Hedge Fund trying to be NormalSkew 0.06 Excess Kurtosis 0.36
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Log-Normal or Abnormal?
One of these is lognormal. The other 2 have infinite skew and kurtosis
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Omega functions
The left bias is evident,even though skew can’t be used to measure it.
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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.
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Risk Profile HF
This Difference in Risk Profiles arises from Skew & Excess Kurtosis of just 0.06 and 0.36
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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
.
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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.
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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:
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Omega for a normal distribution
The slope at the mean is
2
r
.
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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
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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.
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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.
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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.
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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.
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Even for normally distributed returns, Omega has more information
2.4
2.2
2.0
decreases as decreases and also as we move away from the mean for fixed
Risk(r)
Risk(r)
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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
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The Shape of OmegaOption Strategies
Omegas for two US mortgage-backed strategies
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Risk Profiles – Option Strategies
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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…
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Returning to the earlier simulations
Omega AAA Simulations
0.00001
0.0001
0.001
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1000000
0.0212 0.0216 0.022 0.0224
Return
Om
ega
Iteration 1
Iteration 2
Iteration 3
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AA- Omega(s)
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Rating Class - Omegas
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Portfolio & Rating Class - Omegas
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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.
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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.
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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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0.013 0.018 0.023 0.028
Histogram - 100% Recovery
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0.017 0.019 0.021 0.023 0.025 0.027 0.029 0.031
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Security and Collateral - Omegas
0.0001
0.001
0.01
0.1
1
10
100
1000
1000030% Recovery
100% Recovery
This results in a higher mean return, and vastly better downside protection.
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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
0.001
0.1
10
1000
100000
-0.016 -0.012 -0.008 -0.004 0 0.004 0.008
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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.
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One Problem for the Statisticians
Partial autocorrelation - T Short
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Lag
• Auto-correlation - the degree to which today’s return forecasts tomorrows.
• Skill?• Or returns smoothing?
Auto-correlationPartial autocorrelation T
-1
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Lag
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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%
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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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Lag
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Instantaneous RegressionYields and Rates
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But the long run relation between spread and yield is more complex
And this is at odds with the earlier instantaneous result
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The answer lies in the dynamics
And therein lies a trading strategy.
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But before delivering too much optimism
60
70
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100
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150
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
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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.
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Principal Components of Default
The data was pre-processed to remove cyclical (phase) effects which might otherwise bias the results.
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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.
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An Afterthought
Portfolio Weighting by Different Schemes
A Comparison of Equal weighting and weighting by equal expected loss
0.001
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-0.004 0.006 0.016 0.026 0.036
Equ 1
Equ 2
EL 1
EL 2
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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.
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The justification for that last assertion
Lies in the non-normality of spread distributions
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But we might try estimating econometric models
Quite a few have done precisely this.
Here’s our model results
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The diagnostics for which are:
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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.
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The work has really only just started
Further Papers: www.FinanceDevelopmentCentre.com
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
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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
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