Presentation Title Page

1
Dr. Donald R. van Deventer Chairman and Chief
Executive Officer 2222 Kalakaua Avenue, 14th
Floor Honolulu, Hawaii USA 96815 dvandeventer_at_kama
kuraco.com 1-808-791-9888, extension
8888 www.kamakuraco.com
Implications of Alternative CDO and Credit
Portfolio Modeling Techniques, October 19, 2007
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Lessons from the Shanghai Night Market
Most of the vendors in the Shanghai night market
failed to graduate from the sixth grade. Those
vendors, though, know the value of everything
they buy and sell, every single day and for every
single trade. How many investors in CDOs can say
the same thing?
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Background for Todays Presentation

• Findings presented by Mich Araten of
  JPMorganChase in Geneva, December 2006
• Typical credit portfolio modeling approaches were
  single period in nature
• Default simulation therefore means there are only
  two possible times at which default can be
  captured
• All defaults happen at time zero (the beginning
  of the single period)
• All defaults happen at time T (the end of the
  single period)
• We gain a lot of realism by taking a multiple
  models approach and a multiperiod look at the
  credit portfolio modeling problem.
• Mich is not responsible for any of the opinions
  that follow, however!

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Consequences of Being Wrong are Substantial

• On October 5 it was announced that Merrill Lynch
  would take 5.5 billion in write-downs on
  subprime mortgages and highly leveraged loans.

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Todays Example

• Hypothetical synthetic CDO with 500 reference
  names
• Multiple correlation modeling assumptions
• No correlation base case
• Historical simulation
• Copula simulation
• correlations from 0 to 1.0 in 0.05 increments
• Contrary to single period convention, modeled as
  60 1-month periods
• Macro factor driven default probabilities
• Impact of Sampling Error on Accuracy

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Reference Portfolio 491 BBBs and 9 BBs
The 491 BBB rated companies were selected by
using the ranking screen in KRIS by rating to
identify all BBB-rated companies on that day..
The BB companies were randomly selected from the
BBs identified by the ranking screen.
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We analyze a hypothetical 7 tranche synthetic CDO
We have 500 counterparties in the reference
portfolio and exposure of 1 million in notional
principal to each of them. The most subordinated
of the 7 tranches is tranche 1, which takes the
first losses up to 5 million.
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Default Probability Alternatives

• The choice of models is arbitrary. For todays
  presentation we use the Jarrow-Chava version 4.1
  reduced form model, but we could have chosen any
  of these alternatives.
• KDP-jc3 Jarrow-Chava Reduced Form Model Version
  3.0
• KDP-jc4 Jarrow-Chava Reduced Form Model Version
  4.0
• KDP-ms4 Merton Structural Model Version 4.0
• KDP-jm4 Jarrow Merton Hybrid Model Version 4.0
• Modeling can be based on default probabilities of
  any of these maturities
• 1 Month
• 3 Month
• 6 Month
• 1 Year
• 2 Year
• 3 Year
• 5 Year

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Options for Modeling Correlated Defaults

• Constant default probabilities with random
  occurrence of defaults
• No default correlation
• Copula driven with arbitrary correlation
• Note that this assumption implicitly ignores
  measurement error in the default probabilities
• Random default probabilities
• Historical random sampling
• Macro factor driven, in which the simulation of
  PD movements explicitly includes a simulation of
  the measurement error as well (i.e. movements in
  the individual company PDs that are unexplained
  by macro factors alone)

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Consider the Error in Assuming CCC Default Rate
is Constant at Its Long Run Average

• Credits at almost all quality levels show
  variation in default probabilities over the
  business cycle. They differ in the degree to
  which default rates rise as business conditions
  deteriorate.

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Cumulative Loss Distribution All Collateral,
Assuming No Correlation in Defaults
We used the current 1 month default probabilities
in KRIS, which reflect the April 2007 very good
credit conditions. The worst case of the 10,000
scenarios showed 3 million in losses (5
defaults) but the 95th percentile in losses was
only 1.2 million in losses.
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All losses are born by Tranche 1.
This is because the losses dont exceed the 5
million notional amount of tranche 1 in any
scenario.
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We solve for coupons which produce fair value
using these assumptions as base case.
To get fair value coupons on the assumption
that our analysis is correct, we solve for the
coupons that produce values of approximately zero
(plus or minus 10,000). Of course they show
very low coupons for the most senior tranches.
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Even tranche 1 shows no losses more than 50 of
the time
Over the 10,000 scenarios, even tranche 1 bears
no losses more than 50 of the time.
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Even if we use 5 year KDPs, losses are very small.
Instead of the 1 month KDPs, we could have
selected 5 year default probabilities since the
total length of the modeling period is 5 years.
In this case the 100th percentile of losses
begins almost immediately and peaks at 3.6
million in losses.
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The value of Tranche 1 is much lower.
This seemingly minor change in assumptions lowers
the value of Tranche 1 considerablyit goes from
a value of 3,509 to a value of -248,710.
That shows this assumption is worth a quarter
million dollars in thought!
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Losses remain modest when we sample from
historical KDPs.
When we select historical sampling, KRIS-CDO
selects KDPs for each reference name from N
periods in history. Because each names KDP
comes from the same period in time, correlation
is implicit in the PDs. KRM gives users the
additional option of sampling from history in
sequence. The historical sampling again produces
cumulative losses of 1.8 million at the 95th
percentile level. The median loss is the green
line, the 50th percentile level, at a loss of
600,000. Even the worst case scenario is only
4.8 million in losses.
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Tranche 1 value shows a 139,212 loss.
Again, Tranche 1 bears all the losses since
losses never exceed the 5 million in notional
principal of Tranche 1. Under this assumption,
Tranche 1 has a negative value of 139,212.
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Copula Simulation with Correlation0

• If the pair wise correlation for all 500 x 499/2
  pairs of companies is set to zero, the simulation
  produces losses identical to the zero correlation
  case. The only difference is that run times are
  slower because the normal distribution used for
  copulas is not as fast as the uniform
  distribution used in the no correlation
  simulation.

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Impact of Correlation on Tranche Values

• On a multiperiod basis, both the equity tranche
  and other risky tranches can and do decline in
  value.

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The equity tranche rises and falls.

• Viewed from close up, the multiperiod simulation
  makes it clear that even the equity tranche is
  not long correlation as many market
  participants believe. The tranches 2-4 also show
  value declines as correlation rises.

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We now turn to macro factor simulation, which has
a number of advantages over the copula method.

• The copula method produces loss distributions
  that vary with correlations but
• Positions are not hedgeable because common
  drivers of risk are not identified
• Pair-wise correlations are assumed equal for all
  pairs of companies, which is clearly not true
• It is implicitly assumed that there is one common
  risk factor for all companies when in fact there
  are many risk factors and they affect each
  company differently.
• The copula method is computationally intensive in
  spite of these disadvantages from an accuracy and
  hedging point of view. Its slower than the macro
  factor approach but has no advantages over the
  macro factor approach.

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Business Cycles Have a Huge Impact on Correlated
Losses
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Reduced Form PD Models Driven by Macro Factors
Capture This Movement

• The constant term in the logistic regression
  function and the coefficients on the macro
  factors allow the variation in PDs over the
  business cycle to be modeled in a realistic way.

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Systematic and Idiosyncratic Movements in PDs

• Macro factor movements typically explain 50-80
  of historical movements in an individual
  companys default probability
• The remaining movement in the default probability
  is idiosyncratic credit risk
• KRIS-CDOs macro factor correlation modeling
  captures the impacts of both types of risk.
• All monte carlo simulations involve sampling
  error because the scenarios selected are a subset
  of everything that could possibly happen
• We can measure the importance of sampling error.

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Example Creating Macro Driven PDs in Logistic
Form

• Transform true default probabilities to a time
  series of the variable Zi such that
  Zi-ln(1-PDi)/PDi
• Run the ordinary least squares regression
• Zia b1(SP 500 2 year changei)
• b2 (UST 10 year yieldi)
• b3 (oil pricesi)
• ei

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We embed the linear regression in our model

• We now have a PD equation for GM that is a
  function of only macro-economic factors and
  idiosyncratic risk embedded in ei.
• We know by OLS assumptions that ei is normally
  distributed, and the standard deviation of this
  distribution is 0.353 from the linear regression
  results.

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Systematic versus Idiosyncratic PD Movement

• Macro factors explain 53 of the variation in GM
  PDs. If we ignore the remaining 47 of
  variation, that due to idiosyncratic risk, we
  dramatically understate the risk in an analysis
  of CDO tranches.

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Random Default Probability Modeling Using Macro
Factor Driven Method

• 27 candidate macro factors used in the
  simulation, with a different macro factor
  relationship for all 500 names

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Random Default Probability Modeling Using Macro
Factor Driven Method

• Historical volatilities of the 27 macro factors
  are derived based on month-end data from January
  1999 to July 2006

Volatility numbers are annualized
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Random Default Probability Modeling Using Macro
Factor Driven Method

• The chart at the right shows the macro factors
  that were most often statistically significant on
  18,000 global companies for the 1 month
  Jarrow-Chava version 4.1 reduced form default
  probabilities. The results for other maturities
  of the model are similar.

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50th percentile losses are about 15 million
As you would expect, macro-factor driven
simulation produces a higher loss rate. The
median loss rate of 15 million over 5 years
represents a default rate of 5 out of 500
reference names a year, or a 1.00 default rate.
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Tranche 1 suffers total losses at all
percentiles, the only difference is timing.
From the 5th percentile of losses to the 95th
percentile of losses, the total notional
principal of Tranche 1 is wiped out. This
happens within a year at the 95th percentile of
losses. It takes about 56 months to happen at the
5th percentile of losses.
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Tranche 2 is also wiped out with certainty.
For Tranche 2, losses for the 95th percentile
begin to be incurred in month 1. Losses kick in
at month 4 in the 5th percentile, consistent with
the prior slide. Even in the best case scenario,
Tranche 2 is wiped out before maturity.
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Tranche 3 suffers losses from the 25th percentile
scenario
In Tranche 3, all principal is wiped out at the
50th percentile scenario and above. No losses
are incurred at the 5th and 10th percentile
scenarios.
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Tranche 4 takes losses from the 50th percentile
on up.
In Tranche 4, losses are incurred from the 50th
percentile on up. All principal is lost from the
75th percentile on up.
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Tranche 1 value is a loss of 4.7 million
Tranche 1s net present value is a loss of 4.7
million since all notional principal is lost in
all scenarios. The break-even coupon of 1.53
is nowhere near enough to compensate for these
losses.
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For Fair Value, Coupons Need to be MUCH Higher
than the No Correlation Case Indicates if Macro
Factors Drive Defaults

• If one assumes no correlation, the break-even
  coupon tranche one is 1.53, but if one assumes
  macro-factor driven default, the coupon has to be
  more than 100 because the tranches value is
  wiped out by losses very quickly.

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Implications for CDO Risk Analysis

• Analytical assumptions make a dramatic difference
  in the perceived risk and return of a CDO tranche
• VALUATION differences represent the only
  difference between buyer, seller or structurer in
  a CDO trade because the underlying reference
  collateral (credit default swaps) is a commodity
  product
• Very serious attention should be given to the
  implications of changes in valuation technology
  or the firm will be the arbitragee, not the
  arbitrageur!

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How important is sampling error in CDO
analysis?We study this using the macro-factor
driven case and KRIS-CDO to simulate 10 million
scenarios. We use the higher coupon levels shown
in the previous slide.
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We start with 100 runs of 10,000 scenarios each

• The valuations for tranche 4 vary widely by run,
  giving dramatically different images of perceived
  value.

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Sampling Error for Tranche 4

• The graph at the right shows the distribution of
  values for tranche 4 based on 100 different runs
  with 10,000 scenarios each.
• The graph makes it obvious that an accurate
  valuation for a CDO tranche requires an intensive
  calculation with a high number of scenarios.

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We increase scenarios to 10 million

• As the number of scenarios increases (in this
  case to 10,000,000), the resulting valuations
  approach the mean of the distribution shown on
  the previous page.
• KRIS-CDO allows users to measure how high a
  scenario count is needed for a highly accurate
  valuation given the modeling assumptions selected
  by the user.
• We explore this in later slides

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We fit sampling error as a function of number of
scenarios

• Fishman (Monte Carlo Concepts, Algorithms, and
  Applications) shows sampling error should be
  proportional to the square root of 1/N, where N
  is sample size.

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Fitting Sampling Error with 96.8 Accuracy

• For tranche 4, sampling error as a percentage of
  notional principal is
• Error-.0014 4.79 (square root 1/N)

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How many scenarios do we need?

• A rational criterion for the number of scenarios
  is that the estimated value of the tranche be
  within the bid-offered spread with a high
  percentage (90, 95, or 99)
• If this is not the case, the valuation will
  indicate that a trade makes money when in fact
  the perceived profit is merely sampling error
• We use the derived function to solve for how many
  scenarios are needed to value tranche 4 if
  bid-offered spreads as a percent of notional
  principal are consistent with other markets.

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Bid-Offered Spreads as Percent of Value

• In other securities markets, bid-offered spreads
  are 0.015 to 0.08 of the value of the security
• If we assume the synthetic CDO tranche market has
  bid-offered spreads in this range, how many
  scenarios do we need?

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We Need a High Scenario Count

• The results show that one would need a high 6
  million to 11 million scenarios to be within
  bid-offered spreads typical of other markets when
  value tranche 4.

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Do I really need to worry about these obscure
modeling issues?
On second thought, even though I hate swimming, I
should probably take it up.
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Conclusions

• Most market participants cannot meet the
  Shanghai night market standard of valuation in
  the synthetic CDO market
• In order to meet that standard, one needs a high
  scenario count and a rich monte carlo simulation
  capability
• Conventional copula calculations lead to
  substantially different valuations than a
  macro-factor driven reduced form approach because
  of (a) a single period modeling effort, (b)
  constant PDs, and (c) an assumption that there is
  only a single driver of movements in PDs

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Synthetic CDOs tend to be all or nothing
securities despite their ratings.

• Modern credit modeling makes it more apparent
  that either a total loss or no loss at all
  becomes more and more likely as the width of the
  CDO tranche narrows, even for very highly rated
  synthetic CDO tranches.

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