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Risk Analysis & Modelling Lecture 9: Credit Risk www.angelfire.com/linux/riskanalysis RiskCourseHQ@Hotmail.com 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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