# Risk Analysis & Modelling by HVoNSIT

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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:
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

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
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
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
These are known as Credit Events or Credit
Transitions
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
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

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
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
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
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

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

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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