# Credit Pricing by wda20026

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```									    Problems of Credit
Pricing and Portfolio Management

ISDA - PRMIA
October 2003
Con Keating
The Finance Development Centre   1

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

20

10
Returns %

0
0   2            4           6            8   10   12
-10

-20

-30

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

And correlations

0.6

0.4
Cross-correlations

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
From
To:           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%
One year above and Three year below
From
To:              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 portfolio
One Year Returns -Credit Migration Alone

The Set-Up
Px after
Initial Rating            Initial price                 Rating
Coupon   2            1           30 0.985982                   30        1 0.988659
Life     5            2           45 0.979064                   45        2 0.983051
3           70 0.967666                   70        3 0.973793
4          150 0.932274                 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                          2.25%
StDev                       0.015%
Skew                       -0.28155
Kurt                       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      0.023   0.023

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AA Returns Histograms

Histogram AA Returns

0.180

0.160

Mean      2.35%           0.140

StDev    0.083%           0.120
0.100
Skew     -4.0264
0.080
Kurt    20.18325
0.060

0.040

0.020

0.000
0.018   0.019   0.020   0.021   0.022   0.023   0.024

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A Returns Histograms

Histogram - A Returns
Mean     2.46%
0.100
StDev   0.139%              0.090

Skew     -1.365             0.080

0.070
Kurt      3.401             0.060

0.050

0.040

0.030

0.020

0.010

0.000

0.016   0.018   0.020     0.022     0.024   0.026

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Diversified AAA/AA/A/BBB Portfolio
Histogram "Diversified" Portfolio
Mean     2.43%
StDev   0.202%       0.100

Skew     -1.238      0.090

Kurt      2.308      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

60
No. Of
Days

50

40

30

20

10

0
-2

-1

0

1

2
-2.2

-1.8

-1.6

-1.4

-1.2

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2.2
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



 (1 F(x))dx
(r) :       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

2 .
The slope at the mean is 


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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 A                         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,

Risk (r)

  2.4
  2.2

  2.0

Risk (r) decreases as decreases and also

as we move away from the mean for fixed 

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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
100
Omega

10

1
0.0212         0.0216              0.022                 0.0224
0.1

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 - 30% Recovery                                              Histogram - 100% 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                                                          0.000
0.013    0.018         0.023         0.028                      0.017   0.019   0.021   0.023   0.025   0.027   0.029   0.031

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Security and Collateral - Omegas

10000
30% Recovery
1000
100% Recovery
100

10

1

03
3

5

6

8

9

1

2

4

5

7

8

1

3
0.1
01

01

01

01

01

02

02

02

02

02

02

03

03
0.
0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.
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

Mean 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%

• 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
25/10/02
(bps)
(3.90;144)

140

130

120                                                                        04/07/02
(4.49;114)

110

10/03/03
100          (2.98;104)

7/11/01
(3.67;99)
90

80
13/06/03
(2.64;75)
70
03/09/03                                       21/08/00
30/05/01
(3.63;65)                                      (5.30;69)
(4.76;66)
60
2.50              3.00        3.50                      4.00        4.50           5.00           5.50

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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
Con.Keating@FinanceDevelopmentCentre.com
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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
E[ max{x  r ,0}]
( r )  r                  
r                E[max{r  x,0}]
 F (r )dr
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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