Download Omega:A Sharper Ratio

Problems of Credit
Pricing and Portfolio Management
ISDA - PRMIA
October 2003
Con Keating
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Spreads and Returns
The relation is well known
rt 1  y t 1  Dt ( yt  yt 1)
But this only applies to default free bonds
And the duration of a corporate is difficult to estimate,
the standard calculation does not apply.
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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
Returns %
20
10
0
0
2
4
6
8
10
12
-10
-20
-30
Yield Spread %
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Some Statistics
ExAnte Spread
4.76
1.98
1.77
3.04
Mean
StDev
Skew
Kurtosis
Return
1.88
11.42
-0.06
-0.25
And correlations
Cross-correlations ExAnte Spread / Return
0.6
Cross-correlations
0.4
0.2
0
-0.2
-0.4
-0.6
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
Lag
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Transition Matrices
To:
aaa
aa
a
bbb
bb
b
c
D
From
AAA
AA
A
92.06%
1.19%
7.20%
90.84%
0.74%
7.59%
0.00%
0.27%
0.00%
0.08%
0.00%
0.01%
0.00%
0.00%
0.00%
0.02%
0.05%
2.40%
91.89%
4.99%
0.51%
0.13%
0.01%
0.02%
BBB
0.05%
0.25%
5.33%
88.39%
4.87%
0.77%
0.16%
0.18%
One year above and Three year below
To:
aaa
aa
a
bbb
bb
b
c
D
From
AAA
AA
A
78.3%
3.0%
18.1%
75.9%
3.4%
19.2%
0.2%
1.7%
0.0%
0.2%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
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BBB
0.2%
6.2%
80.1%
12.4%
0.9%
0.2%
0.0%
0.0%
0.2%
1.1%
14.7%
78.7%
4.4%
0.7%
0.1%
0.2%
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Simulations
A 150 bond equal weight AAA portfolio
One Year Returns -Credit Migration Alone
The Set-Up
Coupon
Life
Initial RatingInitial spread
Initial price
2
1
30 0.985982
5
2
45 0.979064
3
70 0.967666
4
150 0.932274
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Px after
Trading spread
Rating
1 year
30
1 0.988659
45
2 0.983051
70
3 0.973793
150
4 0.944904
525
5 0.82317
650
6 0.787086
1000
7 0.696265
8
0.3
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The Results - AAA
Mean
StDev
Skew
Kurt
2.25%
0.015%
-0.28155
0.210952
Distribution
His togram AAA Re turns
0.300
0.250
0.200
0.150
0.100
0.050
0.000
0.022
0.022
0.022
0.022
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0.023
0.023
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AA Returns Histograms
Histogram AA Returns
0.180
0.160
Mean
StDev
Skew
Kurt
2.35%
0.083%
-4.0264
20.18325
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
0.018
0.019
0.020
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0.021
0.022
0.023
0.024
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A Returns Histograms
Mean
StDev
Skew
Kurt
2.46%
0.139%
-1.365
3.401
Histogram - A Returns
0.100
0.090
0.080
0.070
0.060
0.050
0.040
0.030
0.020
0.010
0.000
0.016
0.018
0.020
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0.022
0.024
0.026
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Diversified AAA/AA/A/BBB Portfolio
Mean
StDev
Skew
Kurt
2.43%
0.202%
-1.238
2.308
Histogram "Diversified" Portfolio
0.100
0.090
0.080
0.070
0.060
0.050
0.040
0.030
0.020
0.010
0.000
0.013
0.015
0.017
0.019
0.021
0.023
0.025
0.027
The skewness is not diversified away !
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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 managers
Remolona 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
His togram .02 De pe nde nce
0.180
0.160
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
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
Copulae are little better.
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.
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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
Long-Short tends to neutralise common factors
The Sharpe ratio for a long only strategy is bounded
above.
The Sharpe ratio for Long-Short is unbounded
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Higher Moment Approaches
A Hedge Fund trying to be Normal
Skew 0.06 Excess Kurtosis 0.36
Historical Daily Return Distribution
90
80
70
No. Of
Days
60
50
40
30
20
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
0
-2.2
10
Midpoint Of Range
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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 value
of 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
r
.
 F(x)dx

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Why is this important?
The Omega function of a distribution is mathematically equivalent
to 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… but
sometimes the approximate picture they give hides the
features of critical importance for terminal value.
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Graphically

The area outlined in black is:
I2 (r) :
 (1 F(x))dx
r

The area outlined in red is:
r
I1 (r) :
 F(x)dx

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Omega for a normal distribution
r
The slope at the mean is 
2 .

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
25
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
A
Asset A
A  7, A  3
B

Asset B
B  7, B  4

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:
1 d
Risk (r) :
(r) dr
For normally distributed returns, at the mean this
is simply determined by the standard deviation.

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Even for normally distributed returns,
Omega has more information
Risk(r)
  2.4
  2.2
  2.0

Risk(r) decreases as decreases and also
 for fixed 
as we move away from the mean
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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 Omega
Option Strategies
Omegas for two US mortgage-backed strategies
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Risk Profiles – Option Strategies
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Simulations show the potential impact on terminal value.
Losses were 250 times more
likely with BH than with CL
BH folded in September 2002 after a loss of 60% on a
gamble for redemption.
Loss ~ $500million. The SEC investigation continues…
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Returning to the earlier simulations
Omega AAA Simulations
1000000
100000
Iteration 1
10000
Iteration 2
1000
Iteration 3
Omega
100
10
1
0.0212
0.1
0.0216
0.022
0.0224
0.01
0.001
0.0001
0.00001
Return
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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
Covenants in public debt are good for shareholders
In a competitive investment market all of the gains
associated with lower funding cost accrue to the
company
Covenants serve to discipline management
Ratio test covenants of the income or asset coverage
genre may increase the likelihood of default and
distress
Ratings 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 - 100% Recovery
Histogram - 30% Recovery
0.080
0.070
0.070
0.060
0.060
0.050
0.050
0.040
0.040
0.030
0.030
0.020
0.020
0.010
0.010
0.000
0.013
0.000
0.018
0.023
0.028
0.017
0.019
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0.021
0.023
0.025
0.027
0.029
0.031
42
Security and Collateral - Omegas
10000
30% Recovery
1000
100% Recovery
100
10
0.
01
3
0.
01
5
0.
01
6
0.
01
8
0.
01
9
0.
02
1
0.
02
2
0.
02
4
0.
02
5
0.
02
7
0.
02
8
0.
03
0.
03
1
0.
03
3
1
0.1
0.01
0.001
0.0001
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.
100000
1000
Omega Price
10
-0.016
-0.012
-0.008
-0.004
0.1 0
0.004
0.008
0.001
0.00001
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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
Auto-correlation
• Auto-correlation - the degree to which today’s return forecasts
tomorrows.
• Skill?
• Or returns smoothing?
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Correcting for Auto-correlation
ConvertibleFRM
HFR
CSFB
Henn
Fixed Inc FRM
HFR
CSFB
Excess Returns
Mean Std Dev Info Ratio
0.682 1.065 0.640
0.524 1.033 0.507
0.494 1.371 0.361
0.357 1.235 0.289
0.470 1.370 0.343
0.045 1.320 0.034
0.166 1.176 0.141
Adjusted Returns
Mean Std Dev Info Ratio
0.670 1.624 0.413
0.503 1.594 0.315
0.485 2.618 0.185
0.349 1.865 0.187
0.439 2.574 0.171
0.037 1.931 0.019
0.162 1.882 0.086
Errors
Mean
1.76%
4.01%
1.82%
2.24%
6.60%
17.78%
2.41%
Std Dev
-52.49%
-54.31%
-90.96%
-51.01%
-87.88%
-46.29%
-60.03%
Info Ratio
35.47%
37.87%
48.75%
35.29%
50.15%
44.12%
39.01%
• The differences are meaningful
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Adding a security to a portfolio
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Autocorrellogram - Portfolio Ex
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But this isn’t enough
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Instantaneous Regression
Yields 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
Euro Corporate Spread vs Government Yield
150
(bps)
25/10/02
(3.90;144)
140
130
120
04/07/02
(4.49;114)
110
100
10/03/03
(2.98;104)
7/11/01
(3.67;99)
90
80
13/06/03
(2.64;75)
70
03/09/03
(3.63;65)
21/08/00
(5.30;69)
30/05/01
(4.76;66)
60
2.50
3.00
3.50
4.00
4.50
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5.00
5.50
54
Modigliani - Miller and Modern Finance
Few will not now know the M-M theorem, under which
corporate financial structure is irrelevant
Newer Theories exist - in many regards these look like
the pre-M-M world.
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
1000000
Equ 1
100000
Equ 2
10000
EL 1
1000
EL 2
100
10
1
-0.004
0.1
0.006
0.016
0.026
0.036
0.01
0.001
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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!
Notwithstanding that, some of the mono-lines look
over-exposed.
None of the models in use for pricing works with any
meaningful precision.
This will require full information pricing.
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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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In my experience linear factor models can “explain”
only 70% - 80% of what happens
And that isn’t enough for practical pricing
The work has really only just started
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
Further Papers: www.FinanceDevelopmentCentre.com
[email protected]
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Omega Interpretations
Omega may be interpreted as the ratio of a “virtual”
call to a “virtual” put.
b
 (1  F (r ))dr
( r )  r
r
 F (r )dr
E[ max{ x  r ,0}]

E[max{r  x,0}]
a
Omega may be viewed as the “fair game”
representation of the distribution.
And we might argue that this is the correct place from
which to measure Risk
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