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
                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.




              The Finance Development Centre                2
             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.


             The Finance Development Centre            3
                          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

                                       Yield Spread %


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


                                                      And correlations
                                         Cross-correlations ExAnte Spread / Return

                     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




                                               The Finance Development Centre                          5
                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%
            The Finance Development Centre                      6
                      Simulations
           A 150 bond equal weight AAA portfolio
          One Year Returns -Credit Migration Alone

                                      The Set-Up
                                                                            Px after
                        Initial spread
           Initial Rating            Initial price                 Rating
                                                        Trading spread      1 year
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


        The Finance Development Centre                      8
                    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




                   The Finance Development Centre                                      9
                    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




                  The Finance Development Centre                                    10
         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 !
                  The Finance Development Centre                                          11
              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


                The Finance Development Centre                      12
 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
                          The Finance Development Centre                           13
            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.

              The Finance Development Centre             15
             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
           The Finance Development Centre           16
                                        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



                                              The Finance Development Centre                                                                             17
Log-Normal or Abnormal?



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




 The Finance Development Centre                    18
     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.

                      The Finance Development Centre                20
                         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.

                       The Finance Development Centre         23
                                  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
             




                        The Finance Development Centre                24
           Omega for a normal distribution




                                      r



                                2 .
The slope at the mean is 
                                


                The Finance Development Centre   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




                  The Finance Development Centre             26
                        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.
                      The Finance Development Centre               27
                                          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.

                      The Finance Development Centre                  28
                                        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.

                    The Finance Development Centre                  29
                  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.
   


               The Finance Development Centre        30
            Even for normally distributed returns,
               Omega has more information


         Risk (r)

                                                        2.4
                                                        2.2

                                                        2.0



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

                     The Finance Development Centre             31
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 Finance Development Centre                32
             The Shape of Omega
               Option Strategies




Omegas for two US mortgage-backed strategies


           The Finance Development Centre      33
Risk Profiles – Option Strategies




   The Finance Development Centre   34
    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…
                   The Finance Development Centre                 35
                  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




                      The Finance Development Centre                           36
        AA- Omega(s)




The Finance Development Centre   37
 Rating Class - Omegas




The Finance Development Centre   38
Portfolio & Rating Class - Omegas




    The Finance Development Centre   39
           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.

           The Finance Development Centre             40
                 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.


               The Finance Development Centre              41
                                                 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




                                           The Finance Development Centre                                                           42
           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.

               The Finance Development Centre                   43
                     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



                   The Finance Development Centre                     44
    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.




          The Finance Development Centre            45
                  One Problem for the Statisticians
                              Auto-correlation




• Auto-correlation - the degree to which today’s return forecasts
  tomorrows.
• Skill?
• Or returns smoothing?
                       The Finance Development Centre               46
                              Correcting for Auto-correlation

                  Excess Returns              Adjusted Returns            Errors
                    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




                                  The Finance Development Centre                                     47
Adding a security to a portfolio




  The Finance Development Centre   48
Autocorrellogram - Portfolio Ex




  The Finance Development Centre   49
   But this isn’t enough




The Finance Development Centre   50
Instantaneous Regression
     Yields and Rates




The Finance Development Centre   51
 But the long run relation between spread and yield is
                     more complex




And this is at odds with the earlier instantaneous result

                The Finance Development Centre              52
  The answer lies in the dynamics




And therein lies a trading strategy.

      The Finance Development Centre   53
        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




                          The Finance Development Centre                                                     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.



              The Finance Development Centre         55
              Principal Components of Default




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

                  The Finance Development Centre                      56
                  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.

                The Finance Development Centre            57
                      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


                The Finance Development Centre                      58
                   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.
              The Finance Development Centre           59
    The justification for that last assertion
Lies in the non-normality of spread distributions




           The Finance Development Centre           60
But we might try estimating econometric models
     Quite a few have done precisely this.

           Here’s our model results




          The Finance Development Centre         61
The diagnostics for which are:




  The Finance Development Centre   62
 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 Finance Development Centre            63
  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
                The Finance Development Centre          64
                 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


               The Finance Development Centre              65

								
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