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					Risk Analysis & Modelling

    Lecture 9: Credit Risk
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    What we will look at in this
             class
We will look at a new quantitative measure of risk:
Credit Risk
Credit Risk (also known as default risk and
counterparty risk) measures the loss that an
individual or institution might experience if a
counterparty fails to make a agreed payment (such
as the repayment on a loan)
Insurance Companies are primarily exposed to Credit
Risk through the bonds they hold in their investment
portfolios (Investment Credit Risk) and the
reinsurance treaties or contracts they have entered
into (Reinsurance Credit Risk)
   Reinsurance & Investment Credit
                Risk

Reinsurance Companies                       Governments & Companies
                                                  Issuing Bonds



   Reinsurance Credit Risk: Will the     Investment Credit Risk: Will the issuer of
Reinsurance Company become insolvent    the Bonds become bankrupt and be unable
 and be unable to pay the Reinsurance         to pay back the Principal Sum?
            Recoverables?




                            Insurance Company
Credit Risk Model (Binary Model)
             Probability
  Default/Insolvency/Bankruptcy: P

                                          Partial Repayment




                                          Full Repayment
             Probability Not
  Default/Insolvency/Bankruptcy : 1 - P
Assessing the Risk of Bankruptcy
         or Insolvency
The risk or probability of a company being unable to
meet its obligations depends on its “financial
strength”
Ratings agencies (such as S&P or AM Best – which
specialises in rating insurers and reinsurers) analyse
companies and give them a financial strength rating
such as AAA or D
Companies often pay the rating agency to give them
a credit rating and provide them with information that
is not publicly available
Rating Agencies look at trends in “financial strength”
as well as the current snap shot
The rating can be used to look up the probability of
the company becoming insolvent over a period of
time using tables issued by the ratings company
Probability of Insolvency (Default)
  Table for S&P Credit Ratings
      Rating      Default Probability Over
                         Next Year
       AAA                 0.02%
        AA                0.02%
        A                 0.03%
       BBB                0.07%
        BB                1.32%
        B                 5.58%
       CCC                18.6%
           Recovery Rate
In the event of a bankruptcy or insolvency
part of the amount owed is paid back or
recovered
The proportion of the amount owed that is
repaid or recovered is known as the
Recovery Rate
The Recovery Rate depends on the severity
of the bankruptcy or insolvency, the quantity
of the outstanding debts to be paid and the
relative priority of these debts (some debts
are paid first from the remaining capital)
 Example Credit Risk Model
An insurance company is owed £100,000
in one years time from a counter party with
an S&P Credit Rating of BB
From the table we see that a counterparty
with this credit rating has a 1.32%
probability of bankruptcy over the following
year
In the event of bankruptcy the insurer
might predict a 65% recovery rate (so
expects to be repaid £65,000 if the
counterparty defaults)
 Credit Risk Binary Model Diagram

   Probability Bankruptcy:     Partial Repayment:
           1.32%
                             £65,000 (65% Recovery)




                                Full Repayment:
Probability No Bankruptcy:         £100,000
 98.68% (100% - 1.32%)


  One Year Time Horizon
         Loss Given Default
Related to the Recovery Rate is the Loss
Given Default, which is the amount lost in
the event of a Default or Insolvency
The LGD (Loss Given Default) is simply:
 LGD = (1 – Recovery Rate) * Amount Owed
This is the amount lost if the counter party
defaults, the loss if no default occurs is
zero
   Binary Loss Given Default (LGD)
                Model
                                      LGD:
   Probability Bankruptcy:
           1.32%             (100% - 65%) * £100,000
                                    = £35,000




Probability No Bankruptcy:      £0 = NO LOSS
 98.68% (100% - 1.32%)


  One Year Time Horizon
          Statistical Model
The statistical is driven by a Bernoulli Random
(D) variable for which the probability of a 1
(default) is p:
                ~ ~
                C  D.LGD
The average of a Bernoulli Random Number is:
                           ~
                       E ( D)  p
And the variance is:

                     ~
               VaR ( D)  p.(1  p )
The average loss can be calculated as:
        ~         ~
     E (C )  E ( D.LGD )  E ( D). LGD  p.LGD

The variance of the loss can be calculated as:
     ~           ~               ~
Var (C )  Var ( D.LGD )  Var ( D). LGD 2  (1  p ). p.LGD 2

And the Standard Deviation

          ~
   StdDev(C )  LGD. p.(1  p)
        Review Question
You are owed £200,000 from a
counterparty with an S&P Credit rating of
CCC
In the event of default you expect the
Recovery Rate to be 50%
What is the Loss Given Default?
What is the Average Loss over the next
year?
What is the Variance of Loss over the next
year?
Aggregating Multiple Credit Risks
Financial Institutions are likely to have
multiple counterparties who expose them
to Credit Risk
We can try to combine these to estimate
the Aggregate Loss Distribution
An obvious way to do with would be via a
Monte Carlo Simulation in which we
simulate whether a default occurs for each
counter party the sum the losses
   Using Sorted Output From Credit
     Risk Monte Carlo Simulation
100.00%


90.00%


80.00%


70.00%


60.00%
                                  Average Loss: 114101
                                  Standard Deviation of Loss: 117861
50.00%


40.00%


30.00%


20.00%
          0      100000       200000      300000      400000      500000       600000      700000


  By Simulating the default risk for each counterparty and summing sorting the total loss we can
                                  obtain a credit loss distribution
Calculating the Mean and the Variance
    of the Aggregate Credit Loss
     When the default losses for each counterparty
     are independent we could calculate the average
     aggregate loss as:

              AGG  p1.LGD1  p2 .LGD 2  p3 .LGD 3  ......

     Where pN is the probability of the nth counterparty
     defaulting and LGDN is the loss given the default
     of the nth counterparty
     And the variance could be
  2 AGG  p1.(1  p1 ).LGD12  p2 .(1  p2 ).LGD2 2  p3.(1  p3 ).LGD32  ......
     Portfolio Credit Risk Variance in
                Matrix Form

                      2
                           AGG    L .C.L
                                        T



        LGD1                     p1.(1-p1)      0           0        …
        LGD2                        0        p2.(1-p2)      0        …
L=                     C=
        LGD3                        0           0        p3.(1-p3)   …
         …..                        …           …           …        …


     The diagonal elements are the variances of the Bernoulli
         Random Numbers underlying the credit model
   Default Risk in Solvency II
The Standard Model in Solvency II estimates the 99.5%
Probably Maximum Loss from counter parties defaulting
using the Binary Model
The Recovery Rate and the Amount Owed are taken as
best estimates or Averages and are not treated as
random quantities (for example the insurer takes the
average Recoverable from each Reinsurer)
Assuming the distribution of Losses from all
Counterparties is Log-Normal the 1 in 200 loss or 99.5%
Quantile is assumed to be located 3 Standard Deviations
above the mean
For further details:
https://eiopa.europa.eu/fileadmin/tx_dam/files/consultatio
ns/consultationpapers/CP28/CEIOPS-L2-Final-Advice-
SCR-SF-Counterparty-default-risk.pdf
SCR for Default Risk in Solvency II




                              SCR  3. LT .C.L


                                                  99.5% Quantile



                                                3. LT .C.L
Solvency II assumes the distribution of Aggregate Losses from counterparties defaulting
is Log-Normal and therefore the 1 in 200 loss is 3 standard deviations above the average
  Probability of Insolvency (Default)
Table for Solvency II by Credit Ratings
         Rating     Default Probability Over
                           Next Year
         AAA                0.002%
          AA                0.01%
           A                0.05%
         BBB                0.24%
          BB                 1.2%
           B                6.04%
         CCC                30.41%
 Solvency II Credit Risk Calculation
    An insurance company has 3 counterparty that
    exposes it to default risk:

1) A reinsurer with a credit rating of BB who on average
     will owe the insurer £3,000,000 and will have an
     average recovery rate of 50%
2) A portfolio of bonds with a credit rating of CCC with a
     value of £10,000,000 and an average recovery rate
     30%
3) A bank deposit of £5,000,000 with a credit rating of A
     and an average recovery rate of 70%

    Using the “Solvency II Credit Risk” sheet calculate the
    variance and standard deviation of the credit loss
    Calculate the SCR for Credit Risk given it is equal to 3
    standard deviations
Why Credit Risks are Correlated
 Assuming Defaults between counterparties are
 uncorrelated is likely to underestimate the Maximum
 Loss from a portfolio of counterparties defaulting
 The probabilities of different counterparties is likely to
 influence by an underlying common factor, such as the
 economic climate (recession/boom) or economic shocks
 (event leading to loss of net asset value of
 counterparties)
 This underlying factor will increase or decrease the
 chance of counterparties defaulting and will cause
 correlation between defaults
 In Solvency II this underlying factor is in itself treated as
 a random variable that follows a Pareto type distribution
This underlying random factor S can take on value
between 0 and 1, and its CDF or the probability of it
being less than a value X is

                   ~           a        for 0  x  1
               Pr( S  X )  X
The probability of default p is also assumed to be a
random variable the depends on this underlying random
factor:                              b
                                  ~
                  ~  b  (1  b).S b
                  p

Where b is a baseline or minimum default probability
when the random S factor is 0 the probability of default
will be b, we can also see that when S is at its maximum
value the default probability would be 1
The parameters a and b are selected by the regulator
and are currently set so the ratio of a/b is 4
Finding the Unknown Base Default
              Rate b
In the Solvency II s Default Model the default
probabilities quoted by rating agencies are
averages that relate to the average value for the
underlying shock factor, for example for a B rated
counterparty:
                                          ~ bb     
            E  ~B   bB  (1  bB ). E  S B
                p                                     6.04 %
                                                   
                                                   
Using Calculus we can obtain and estimate for b
interms of the expected value of p:
                                   E p
                        b
                             a
                               .(1  E  p )  1
                             b
Since the regulator sets the ratio of a/b to 4:
                         E p
                b
                   4.(1  E  p )  1

So the base default probability for a BBB rated
counterparty is equal to:
                  E  pBBB              0.0604
   bBBB                                               0.0127  1.27 %
            4.(1  E  pBBB )  1 4.(1  0.0604 )  1

We can apply this formula to calculate the base
default probabilities for all credit ratings…
          Base Default Rate
Rating     P                       B

AAA      0.002%   0.002%/(4*(100%-0.002%)+1) = 0.0004%

 AA      0.01%     0.01%/(4*(100%-0.01%)+1) = 0.002%

  A      0.05%      0.05%/(4*(100%-0.05%)+1) = 0.01%

BBB      0.24%     0.24%/(4*(100%- 0.24%)+1) = 0.048%

 BB      1.2%       1.2%/(4*(100%- 1.2%)+1) = 0.242%

  B      6.04%     6.04%/(4*(100%- 6.04%)+1) = 1.269%

CCC      30.41%   30.41%/(4*(100%- 30.41%)+1) = 8.037%
Solvency II Default Correlation between B
     and CCC rated Counterparties
                                                  ~
                                                  S

                                                                                               a
                                  a                     ~                                 ~
pB
                             ~
~  0.01269  (1  0.01269 ).S 0.01269                  pCCC    0.08037  (1  0.08037 ).S 0.08037




                 ~                                                             ~
                 D1                      Correlated Defaults                   D2


 Counterparty 1 has a credit rating of B and Counterparty 2 has a
credit rating of CCC, because the probabilities of defaults are both
 influenced by the underlying factor S the defaults are correlated
       Covariance Between Defaults
       Using calculus we can derive the covariance
       between two counterparties

                             4.1  b1 1  b2 .b1.b2
                                                          p1  b1  p2  b2 
                 ~ ~                    .
            Cov( D1 , D2 )                                          .
                                4.b1.b2  b2  b1


       So for example the covariance between the
       defaults of a B counterparty (b1 = 1.269% and
       p1 = 6.04%) and CCC rated counterparty (b2 =
       8.037% and p2 = 30.41%) would be
                 4.1  0.01269* 1  0.08037* 0.01269* 0.08037
                                                                   0.0604  0.01269* 0.3041 0.08037
     ~ ~
Cov( D1 , D2 ) 
                      4.0.01269* 0.08037  0.01269  0.08037
     Solvency II Calculation with
            Correlations

                          0.05675184    0.027467   0.000373
       1500000




L=     7000000       C=   0.027466674   0.211623   0.000278




       1500000            0.000372716   0.000278    0.0005




                          Made up Covariances!



     SCR  3. L .C.L  9987191
                 T
       Investment Credit Risk
41% of the total £1.1 trillion invested by insurance
companies in the UK is held in bonds
Half of this quantity is in government bonds, while
the other half is in corporate bonds
Bonds are tradable loans made to corporations
that can be sold on the market before the principal
sum is repaid
The issuer of the bond can become bankrupt or
default and be unable to repay the amount in full –
they expose the insurer to Credit Risk
On publicly traded financial instruments such as
bonds, Credit Risk can affect the value of the bond
even if the issuer does not become bankrupt…
Breakdown of Investments Held by
    UK Insurance Companies
 Credit Ratings & Bond Values
The value of a bond is in part determined by the
Credit Rating of the issuer
The market demands a higher interest rate on
bonds issued by companies or governments with
lower credit ratings (Credit Spread)
The value of a bond will decrease if the credit rating
of the issuer drops (Credit Spread increases)
This movement in the price of a bond through
changes in the credit rating is an important part of
the Investment Credit Risk
Because of this relationship between the value of a
bond and its credit rating we have to look at the
possibility that the credit rating will change in
addition to the possibility of default…
           Credit Transitions
The credit rating of a bond is not fixed across time
It can be upgraded or downgraded
These are known as Credit Events or Credit
Transitions
These upgrades and downgrades are ultimately
related to the solvency of the issuer
Bonds move up to a higher credit rating if the issuer
becomes more credit worthy
Conversely, bonds move down the credit ratings if
the issuer becomes less credit worthy or less solvent
These transitions affect the price of the bond
through the credit spread
            Transitions Matrix
Rating Agencies provide estimates of the probabilities
of Credit Transitions based upon the initial rating of the
bond
These estimates are based upon historical data and
analysts projections
The various credit transitions that can occur and their
probabilities are documented in the Transition Matrix
This publicly available information provides broad
estimates of the probabilities of transitions between
ratings over a period of time (often 1 year)
The Probability of a movement or transition in a
company’s credit rating is entirely dependent upon its
current credit rating, its past ratings do not matter
(Markov Property)
   S&P Rating Transition Matrix
Initial               Rating at year end (%)
Rating    AAA AA       A     BBB BB            B       CCC     Default
AAA       87.74 10.93 0.45 0.63       0.12     0.10    0.02    0.02
AA        0.84 88.23 7.47 2.16        1.11     0.13    0.05    0.02
A         0.27 1.59   89.05 7.40      1.48     0.13    0.06    0.03
BBB       1.84 1.89   5.00 84.21 6.51          0.32    0.16    0.07
BB        0.08 2.91   3.29 5.53       74.68    8.05    4.14    1.32
B         0.21 0.36   9.25 8.29       2.31     63.89   10.13   5.58
CCC       0.06 0.25   1.85 2.06       12.34    24.86   39.97   18.60
Default   0.0   0.0   0.0    0.0      0.0      0.0     0.0     100.0


• The probability of a bond with an initial rating of
  A decreasing to a bond with a rating of BB over
  the next year is 1.48%
       Markov Transition Matrix
The Transition Matrix provided by Credit Rating Agencies is an
example of a Markov Transition Matrix (T)
One important property of these matrices is that the probability
of transitions over N time periods (TN) can be calculated as:

                         TN  T     N

For example, if C describes the probabilities of credit
transitions over 1 year then the probability of credit transitions
over 2 years (C2) can be calculated as :


                   C2  C  C * C
                             2


It is possible to calculate the transition matrix for fractional
values of N using an Eigen Value Decomposition
            Recovery Rates
When a company goes into default some
percentage of the bond’s face value gets paid out of
the remaining asset wealth in the issuing company
Some bonds are more senior than others in the
company’s capital structure
The seniority of the bond determines if it gets paid
before other bonds
There are 5 categories of bond seniority
Senior Secured gets paid first out of the remaining
assets
Junior Subordinate gets paid last
In general the lower the seniority of a bond the
lower the recovery rate
           Seniority Vs Recovery
Remaining Asset Value           Recovered Capital

                                               Senior Secured Debt
                                                    Payment

                                             Senior Unsecured Debt
                                                    Payment

                                             Senior Subordinate Debt
                                                     Payment


                   No Payment               Subordinate Debt Payment

                                             Junior Subordinate Debt
                   No Payment
                                                     Payment
             Recovery Rates

                        Source Carty & Lieberman
 Seniority Class        Average        Standard Dev
 Senior Secured         53.80%         26.86%
 Senior Unsecured       51.13%         25.45%
 Senior Subordinated    38.52%         23.81%
 Subordinated           32.74%         20.18%
 Junior Subordinated    17.09%         10.90%

• As the seniority of the bond goes down the
  expected recovery rate goes down
• The recovery rate is random or uncertain
              Credit Metrics
Credit Metrics is a framework developed by JP Morgan to
assess the Credit Risk on portfolios of bonds
It models the risk of both the issuer defaulting and of its
credit rating changing over a period of time
Using a Monte Carlo Simulation based technique it tries
to measure the Credit VaR over a period of time on the
portfolio defined as:

            Credit VaR  P *  E ( P)
Where P* is the value of the portfolio of bonds such that
worse values will only be observed some percentage of
the time (1%, 5% etc) and E(P) is the expected value of
the portfolio
          Credit Metrics Example
Imagine we have a CCC bond
The face value of the bond is £100
The bonds seniority class is Senior Subordinate
We wish to measure the Credit Risk over a 1 year period so
firstly we need to assess the probabilities of Credit
Transitions for a CCC rated bond from the 1 year Rating
Transition Matrix
Secondly we need to assess the value of the bond for each
possible Credit Transition
This evaluation will either involve estimating the market
value of the bond for a given credit rating by discounting or
by estimating the average amount recovered in the event of
default (we can assume the recovery rate is itself a random
variable – see Appendix)
  S&P Rating Transition Matrix
Initial                   Rating at year end (%)
Rating    AAA     AA      A      BBB BB            B       CCC     Default
AAA       87.74   10.93   0.45   0.63     0.12     0.10    0.02    0.02
AA        0.84    88.23   7.47   2.16     1.11     0.13    0.05    0.02
A         0.27    1.59    89.05 7.40      1.48     0.13    0.06    0.03
BBB       1.84    1.89    5.00   84.21 6.51        0.32    0.16    0.07
BB        0.08    2.91    3.29   5.53     74.68    8.05    4.14    1.32
B         0.21    0.36    9.25   8.29     2.31     63.89   10.13   5.58
CCC       0.06    0.25    1.85   2.06     12.34    24.86   39.97   18.60


 The highlighted row shows all the possible credit
 transitions that can occur over the year for a bond
 with an initial rating of CCC
    Credit Risk On 1 Bond Diagram
                                         AAA: 0.06%
            Finite Number of Outcomes
                                         AA: 0.25%

                                         A: 1.85%

                                         BBB: 2.06%

Initial Rating                           BB: 12.34%
     CCC
                                         B: 24.86%

                                         CCC: 39.97%

                                         Default: 18.6%
                 One Year Time Horizon
                               Credit Transitions & Values
Increasing Bond Value Due to


                               Rating       Bond Value          Probability Cumulative
  Decreasing Credit Spread


                                                                            Probability
                                AAA            93.5               0.06%         100%
                                 AA            93.1               0.25%        99.93%
                                 A             92.8               1.85%        99.68%
                                BBB            92.3               2.06%        97.83%
                                 BB            91.7              12.34%        95.77%
                                 B             90.9              24.86%        83.43%
                                CCC             87               39.97%        58.57%
                               Default   100*0.3852 = 38.5        18.6%         18.6%

              The bond value in default is equal to the face value multiplied by the
              average recovery rate for senior subordinated debt. For other credit
              transitions, the value is estimated by discounting by different credit risk
              spreads*.
Probability Histogram of Values
         for CCC Bond
 0.45

  0.4
                  39.97%
 0.35

  0.3

 0.25
                           24.86%
  0.2
         18.6%
 0.15

  0.1                               12.34%


 0.05                                        2.06%   1.85%   0.25%   0.07%
   0
        Default   CCC       B        BB      BBB      A      AA      AAA
         Simulating Credit Transitions for
                CCC over 1 Year
            1
                                                               99.68%   99.93%   100%
                                             95.77%   97.83%


           0.8                      83.43%



           0.6
                                    Bond Migrates to B
Rand()




                           58.57%


           0.4

                   Bond Stays at CCC
           0.2
                  18.6%


            0
                 Default   CCC        B       BB      BBB        A      AA       AAA
Credit Metrics Simulation Approach
 JP Morgan’s Credit Metrics uses a Standard
 Normal Random Variable (mean 0 standard
 deviation 1) to simulate Credit Transitions
 It divides the values that this random variable
 can take into ranges (or buckets)
 These ranges are selected so that the
 probability of the Standard Normal falling into
 one of them reflects the probability of a Credit
 Transition
 This approach is more complex but has certain
 advantages, as we will see later..
 Transitions For A CCC rated Bond
Rating Default     CCC          B          BB              BBB               A       AA      AAA

Prob     18.6%     39.97%       24.86%     12.34%          2.06%             1.85%   0.25%   0.06%




                            -0.89               0.21   0.97 1.72 2.01 2.27   3.19
                                            0
       Randomly Selected Number Implying                   Randomly Selected Number Implying
                    Default                                        Upgrading to AA
      Cutting Up The Standard Normal
               using the CDF
                  NORMDIST(0.8,0,1,TRUE) = 0.788

                  1                       1

                 0.9

                 0.8                     0.8

                 0.7

                 0.6                     0.6

                 0.5

                 0.4                     0.4

                 0.3

                 0.2                     0.2
                 0.1

                  0                       0
-3    -2    -1         0   1   2    3           Default   CCC   B   BB   BBB   A   AA   AAA




     Normally Distributed Random = 0.8         Results in simulated B Credit Rating
    Adding a Second Bond
We will now add a second bond to make a
portfolio
This bond will have a face value of £200,
an initial rating of BB and a seniority class
of Senior Secured
Using the Transition Matrix and the
recovery rate table we can build a table of
the possible outcomes over the one year
period
                               Evaluating Each Outcome
Increasing Bond Value Due to



                                Rating      Bond Value       Probability Cumulative
  Decreasing Credit Spread



                                                                         Probability
                                AAA           189.5            0.08%        100%
                                 AA           188.8            2.91%       99.92%
                                  A           188.1            3.29%       97.01%
                                BBB           187.6            5.53%       93.72%
                                 BB           185.2            74.68%      88.19%
                                  B           182.6            8.05%       13.51%
                                CCC           176.9            4.14%        5.46%
                               Default   200*0.538 = 107.6     1.32%        1.32%

               Again we will take the estimated Bond Values as given although we
               could calculate them using the forward yield curve
   Stochastic Recovery Rates
Up until now we have been assuming that default rates are
fixed at their average value
The Credit Metrics simulations randomly samples default
rates from a standard beta-distribution
Random variables sampled from a standard beta-
distribution are always between 0 and 1 (just like the
recovery rate)
The standard beta distribution is entirely determined by the
mean and standard deviation of the random variable
This distribution is very useful for generating random
financial ratios
We can generate beta distributed random variables using
the inverse transform method combined with the inverse
CDF for the beta distribution given by the BETAINV function
in Excel
           Various Default Rate Beta
                 Distributions
4.5

 4
                       Junior Subordinate Debt Recovery Rate
3.5

 3

2.5
                               Subordinate Recovery Rate
 2
                                  Subordinate Unsecured Recovery Rate
1.5

 1

0.5

  0
 -0.01   0.09   0.19    0.29      0.39   0.49   0.59   0.69   0.79   0.89   0.99
Correlated Credit Transitions
The Credit Transitions between bonds issued by different
counterparties are often correlated
There are two problems that must be addressed if we are to
model correlated credit transitions
Firstly how can we measure the correlation between credit
transitions in a meaningful way
Credit Transitions do not occur frequently so the dataset is
highly limited
Credit Ratings are not numerical and to measure correlation
we require numbers, what number do you assign to AAA and
BBB?
The Second problem relates to how can we simulate
correlated Credit Transitions
Up until now we have learnt how to generate independent
random numbers using the CDF of different distributions but
these have had no correlation between them
We will see that we can use the correlation matrix to generate
random numbers
  Measuring the Correlation between
          Credit Transitions
One of the key determinants of the Credit Rating of a
company is its Net Worth
Companies with Net Worth close to zero have low credit
ratings and are close to bankruptcy and default
Movements in the Net Worth of a company are highly
correlated with movements in the company’s stock price
Using the correlations between the returns in the stock
price of companies we can estimate the correlations
between their Credit Transitions
Credit Metrics assumes this underlying asset value
distribution is normal
The inspiration for this measure of asset value
correlation is the Merton Model
Correlated Asset Values and Credit
            Transitions

                              If the Asset Value of two counterparties are
                           positively correlated then the Credit Transitions
                           will also be positively correlated (tend to upgrade
                                        and downgrade together)




Asset Value Distribution
     Cholesky Decomposition
To bring correlation into our simulation we will have
to learn about the Cholesky Decomposition
The Cholesky Decomposition is an important result
from linear algebra
The Cholesky Decomposition allows us to take
positive definite, symmetric matrices (like the
correlation matrix) and decompose them into a form
such that
                  CD.CD  P T


Where P is the Correlation Matrix and CD is the
Cholesky Decomposition of that matrix.
It can be thought of as the square root of a matrix
If we take the Cholesky Decomposition of the
Correlation Matrix CD and multiply it by a vector of
standard normal random variables N (with mean of
zero and std dev of 1) we produce a vector of
correlated standard normal variables V:

                 CD.N  V
This transformation can also be applied to normal
random variables with a non-zero mean and various
standard deviations (see appendix)
The use of the Cholesky Decomposition relies on the
addition of random variables which would distort non-
normal random variables (eg the sum of normally
distributed random variables is still normal, the sum of
Pareto distributed random variables is not Pareto)
           Cholesky Transformation
            =1                                              =1



     =0                                              =0


            =1            Cholesky                         =1
                         Decomposition
                      of Correlation Matrix
     =0                                              =0

            =1                                             =1



     =0                                             =0
Independent Normals                           Correlated Normals
Modelling Reinsurance Credit Risk
Reinsurance Credit Risk measures the risks that monies
owed to an insurance company by its reinsurers
(Reinsurance Recoverables) are not paid because of the
insolvency or impairment of the reinsurer
Unlike bonds or bank deposits the amount owed by the
Reinsurance Counterparty is highly volatile and depends
on the frequency and severity of losses over the time
period
The risk for an Insurer is that a Reinsurance Counterparty
who owes them a large sum (recoverables) and becomes
insolvent
To model Reinsurance Credit Risk we have to model the
amount owed as well as the risk of default, it is not
sufficient to just take the average amount owed….
       Modelling the Reinsurance
             Recoverables
The amount owed by the Reinsurance Company
(Recoverables) depends on the type of reinsurance
purchased and the level of claims
Since the level of claims is random the Reinsurance
Recoverable is also random
We will look at one of the simplest forms of Reinsurance, Stop
Loss or Aggregate XL Reinsurance
Stop Loss Reinsurance pays if the Aggregate Claim is above
a level known as the Retention (R) upto a maximum Limit (L)
sometimes written as L xs R
If the Aggregate Claim is below the Retention R then nothing
is paid, if the Aggregate Claim is above R+L then L is paid out
The Reinsurance Recoverable will depend on the random
Aggregate Claim Level which can be derived from the
frequency severity model
                                                                                           Simulating Reinsurance
                                                                                               Recoverables
                     Simulate Aggregate Claim Level
                                                                                                                               0.0025


              0.12


               0.1


              0.08
                                                                                                         Probability Denisty
                                                                                                                                0.002




                                                                                                                               0.0015
                                                                                                                                                                                             Aggregate Gross
Probability




                                                                                                                                                                                                Claim: X
              0.06
                                                                                                                                0.001


              0.04
                                                                                                                               0.0005

              0.02


                                                                                                                                   0
                0
                                                                                                                                        0   500   1000                1500   2000   2500
                     1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
                                                                                                                                                         Claim Size
                                                      Number Claims Per Week




                                                                                                                                                                                           Apply Stoploss L xs R


                                                                                                                                                                                                Recoverable:
                                                                                                                                                                                              IF X > R THEN
                                                                                                                                                                                                MIN(X-R,L)
                                                                        Simulating Losses With
                                                                        Reinsurance Credit Risk
                        Simulate Aggregate Claim Level
                                                                                                                                    0.0025




                                                                                                                                                                                                Gross Claim
                 0.12

                                                                                                                                     0.002
                  0.1




                                                                                                              Probability Denisty
                 0.08                                                                                                               0.0015
   Probability




                 0.06


                 0.04


                 0.02
                                                                                                                                     0.001




                                                                                                                                    0.0005
                                                                                                                                                                                                   Level
                                                                                                                                        0
                   0
                                                                                                                                             0   500   1000                1500   2000   2500
                        1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
                                                                                                                                                              Claim Size
                                                         Number Claims Per Week




                                                                                                            Calculate Reinsurance
                                                                                                                Recoverables

Simulate Insolvency
                                                                                                                                                                                                Net Claim
                                                                                                                                                                                                 Level
         Multiple Reinsurers
Insurance Companies often try to diversify their
Reinsurance Credit Risk by splitting their coverage
between multiple reinsurance
The defaults between reinsurers are likely to be
correlated
We can combine the Credit Metrics method of correlating
defaults from an underlying asset distribution with the
Frequency Severity Model to simulate the Recoverable
and Credit Loss
This will take into account the effect the frequency and
severity of loss will have on the amount owed by the
reinsurer given the type of reinsurance purchased and
the effect this will have on the loss given default, the
probability of a reinsurer becoming insolvent and the
correlation between reinsurers and the extent to which
Reinsurance Credit Risk is Diversified
For example, in our simulation on the “LGD
Reinsurance” Sheet the reinsurer purchased
50000 xs 270000
The insurer might split this into two layers with
different reinsurers to diversify the credit risk so:
25000 xs 270000 and 25000 xs 295000
In “LGD Reinsurance 2” we will see how the
Reinsurance Credit Risk is reduced by
diversifying between two reinsurers assuming
there is a correlation between the asset values
of the reinsurers of 50%
   Correlated Reinsurance Risks
                  Random Reinsurer 1 Insolvency
                                                                                                            Random LGD in
                                                                                                               Layer 1

                                                                                       Simulate Aggregate Claim Level
                                                                                                                                                                                 0.0025

                                                                0.12

                                                                                                                                                                                  0.002




50% Asset Value
                                                                 0.1




                                                                                                                                                           Probability Denisty
                                                                0.08                                                                                                             0.0015




                                                  Probability
                                                                0.06
                                                                                                                                                                                  0.001


                                                                0.04

                                                                                                                                                                                 0.0005




  Correlation                                                   0.02


                                                                  0
                                                                       1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
                                                                                                        Number Claims Per Week
                                                                                                                                                                                     0
                                                                                                                                                                                          0   500   1000
                                                                                                                                                                                                           Claim Size
                                                                                                                                                                                                                        1500   2000   2500




              Random Reinsurer 2 Insolvency

                                                                                                        Random LGD in
                                                                                                           Layer 2
Beginning of Risk Modelling Level 2:
              Copulas
 The Credit Metrics method of converting the underlying
 correlated normal distributions into correlated Credit
 Transitions is known as the Gaussian Copulas (Normal
 Distribution is also known as the Gaussian Distribution)
 What if we want to generate correlated Pareto, Gamma
 and Log-Normal random variables?
 Copulas are Correlated or Related Uniform Random
 variables
 These Correlated Uniform Random Variables are then
 used in the Inverse Transform to generate correlated
 random variables from other distributions
 In the case of the Gaussian Copula these correlated
 Uniform Random Variables are derived from correlated
 Normal Random Variables…..
                      Gaussian Copula
                                                          Inverse Transform
                                                            To LogNormal
                                                1
                                              0.9
                                              0.8
                                              0.7
                                              0.6


                 NORMDIST                     0.5
                                              0.4
                                              0.3
                                              0.2
                                              0.1
                                                0
                                                    0         2        4        6        8    10




                                                    Correlated Pareto and
Correlated Normally      Correlated Uniform              Lognormal
Distributed Random       Random Numbers –
      Numbers             Gaussian Copula           Inverse Transform to
                                                           Pareto
                                                100%
                                                    90%
                                                    80%
                                                    70%

                NORMDIST                            60%
                                                    50%
                                                    40%
                                                    30%
                                                    20%
                                                    10%
                                                    0%
                                                          0       10       20       30       40
Gaussian Copulas and Measuring
          Correlation
When correlating random variables from Non-Normal
distributions the correlation statistic does not provide a
complete description of the behaviour
One solution to this can be to transform the random
variables from the other distribution into Normally
distributed random variables by reversing the process
we used to generate the correlated random samples
So for example, we convert the Gamma and Pareto
distributed datasets into Normally Distributed datasets
and then measure the correlation between these
transformed values
The main alternative to the Correlation Matrix approach
used by the Standard Model in Solvency II is to
aggregate risks using Copulas
 Transforming Lognormal and Pareto
  to Normal to Measure Correlation
                                                          Log Normal Dataset
                                 LogNormalCDF        1
                                                   0.9
                                                   0.8
                                                   0.7
                                                   0.6


                 NORMINV                           0.5
                                                   0.4
                                                   0.3
                                                   0.2
                                                   0.1
                                                     0
                                                         0          2        4        6        8    10




                                                         Correlated Pareto and
    We measure             Correlated Uniform                 Lognormal
correlation between        Random Numbers –
  the transformed           Gaussian Copula                        Pareto Dataset
   normal dataset
                                       ParetoCDF     100%
                                                         90%
                                                         80%
                                                         70%

                 NORMINV                                 60%
                                                         50%
                                                         40%
                                                         30%
                                                         20%
                                                         10%
                                                         0%
                                                               0        10       20       30       40
  Appendix: Beta Distribution
The Beta distribution is useful for modelling
random variables such as ratios which are
between 0 and 1
Its density function is given by
                              a 1          b 1
                          x      (1  x)
           pdf ( x ) 
                                B (a , b )
Where B is the Beta function defined as
                          1
            B(a , b )   t a 1 (1  t ) b 1.dt
                          0
a and b are parameters which determine the
shape of the distribution
We can calculate a and b in terms of the mean
 and standard deviation  of the random
variable:

                  .(1   ) 
         a   .             1
                              
                        2



                      .(1   ) 
       b  (1   ).             1
                                  
                            2
      Appendix: The Cholesky
          Transformation
Imagine we have 2 random variables A and B
each of which are sampled from a standard
normal distribution (mean 0, standard deviation
1). We will put these in a vector S (S for
stochastic)
We would like to transform these variables into
random variables sampled from a distribution with
mean, variance and covariance described by the
following:
      A           ERA            VarA    CovA,B
S=           R=           C=
      B           ERB           CovA,B   VarB
Firstly we perform the Cholesky decomposition
on the covariance matrix (C) to obtain CD
Then we simply perform the following Cholesky
transformation

               S   CD.S  R

Where S’ is a vector of transformed random
variables sampled from a distribution described
by R and CV
                        A’
                 S’=
                        B’
Where A’ will have mean ERA and
variance VarA, B’ will have mean ERB and
variance VarB, and where A’ and B’ will
have Covariance CovA,B
            Appendix Cholesky
          Transformation: A Proof
      It is important to note that if we have a vector
      of stochastic variables S with mean zero ,
      then the E(S.ST) = CV:

      A                     A.A   A.B
E            A    B                           Var(A)    Cov(A,B)
      B               = E   B.A   B.B    =
                                             Cov(A,B)    Var(B)


    • If we perform the Cholesky transformation on a vector
      of standard normal variables to get a new set of
      random variables:

                            S   CD.S
  Now
      S .S T  CD.S .(CD.S )T  CD.S .S T .CDT
• Taking expectations:
           E ( S .S T )  CD.E ( S .S T ).CDT
• Since S is a vector of independent
  (uncorrelated) unit normal variables E(S.ST)
  will be the identity matrix (why?)
       E ( S .S  )  CD.I .CD  CD.CD  C
               T             T           T


• So the transformed random variables will
  have variance and covariance described by
  the covariance matrix (C) that the cholesky
  decomposition (CD) was derived from.

				
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