Risk Analysis & Modelling

Risk Analysis & Modelling Lecture 9: Credit Risk and Copulas

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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) is the risk that an individual or institution might not be able to pay back their Creditors in full (such as a company failing to repay 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 they have entered into (Reinsurance Credit Risk) • In learning about the Credit Risk Models we will learn about a new statistical technique call the (Gaussian) Copula, and how it came to be known as the “Formula that Killed Wall Street”

Reinsurance & Investment Credit Risk Reinsurance Companies Governments & Companies Issuing Bonds Reinsurance Credit Risk: Will the Reinsurance Company become insolvent and be unable to pay the Reinsurance Recoverables? Investment Credit Risk: Will the issuer of the Bonds become bankrupt and be unable to pay back the Principal Sum? 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 Default • The risk or probability of a company being unable to pay its creditors depends on its “financial strength” • Ratings Agencies analyse companies and give them a financial strength rating such as AAA or CCC • We will focus on the ratings from the largest Rating Agency S&P • Companies often pay the Rating Agency to give them a Credit Rating and provide them with information that is not publicly available • Although there is some controversy surrounding the accuracy of Credit Ratings (such as AIG being issued with a AAA rating days before its collapse) they are often the best tool to assess a counterparties credit worthiness • Ratings Agencies provide tables which can be used to look up the probability of a counterparty defaulting (PD) over the next year

Probability of Insolvency (Default) Table for S&P Credit Ratings (Source: S&P Global Corporate Transition Matrix)

Recovery Rate • In the event of a default or insolvency part of the amount owed is paid back or recovered • The proportion of the amount owed that is recovered is known as the Recovery Rate (RR) • The amount Recovered (R) after the default can be written as: • Where E is the Exposure or the amount owed by the counterparty defaulting • The Recovery Rate depends on the remaining worth of the counter party after default and the quantity and priority of other outstanding debts

Example Credit Risk Model • An insurance company is owed £100,000 (Exposure) 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 an average recovery rate of 65% (so expects to be repaid £65,000 if the counterparty defaults)

Binary Credit Risk Model Diagram Probability Default: 1.32% Partial Repayment: £65,000 (£100000 * 65%) Full Repayment: £100,000 Probability No Default: 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) * Exposure • This is the amount lost if the counter party defaults • If no default occurs the loss is zero

Binary Loss Given Default (LGD) Model LGD: (100% - 65%) * £100,000 = £35,000 Probability Default: 1.32% £0 = NO LOSS Probability No Default: 98.68% (100% - 1.32%) One Year Time Horizon

Average and Variance of Loss • This statistical model is driven by a Bernoulli Random (D) variable for which the probability of a 1 (default) is p: • Where L is the random loss due to the counterparty defaulting • The Average of a Bernoulli Random Number is: • And the Variance is:

The average loss can be calculated as: • The variance of the loss can be calculated as: • And the standard deviation:

Review Question • You are owed £1,000,000 from a counterparty with an S&P Credit rating of CCC • In the event of default you expect the Recovery Rate to be 75% • What is the Loss Given Default? • What is the Average Loss over the next year? • What is the Variance and Standard Deviation 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 combine the Binary Credit Models for each counterparty to estimate the Aggregate Loss Distribution due to counterparties defaulting • An obvious way to do with would be via a Credit Metrics style Monte Carlo Simulation in which we simulate whether a default occurs for each counter party and then sum the losses • Using this distribution we could calculate the 5% and 1% Credit VaR (Absolute) by taking the loss such that we would only observe losses greater than that 5% or 1% of the time

Distribution of Total Losses due to Credit Defaults across the Portfolio Only 5% of the time the Aggregate Loss will be greater than 387500 this is the Credit VaR at 5% confidence By Simulating the default risk for each counterparty, summing the losses and then sorting we can obtain a Empirical Distribution of losses Across the Portfolio

Calculating the Mean and the Variance of the Aggregate Loss • The average of the Loss Given Default mAGG across a portfolio is equal to: • 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 of the total loss (sAGG ) would be:

Portfolio Credit Risk Variance in Matrix Form L= C = 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 this 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/consultations/consultationpapers/CP28/CEIOPS-L2-Final-Advice-SCR-SF-Counterparty-default-risk.pdf

SCR for Default Risk in Solvency II 99.5% Quantile 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

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 AA who on average will owe the insurer £20,000,000 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 80% 3) A deposit of £50,000,000 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

Credit Risk on Bonds • 45% of the total £1.8 trillion is invested by UK Insurance Companies is held in bonds • 20% of this total is held in Government Bonds, while the remaining 25% is held in Corporate Bonds • Bonds are tradable loans made issued by Governments and Corporations that can be sold or traded on the market • 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 • Bonds also expose the Insurance Company to Credit Risk through movements in the Credit Rating of the issuer effecting the market value…

Breakdown of Investments Held by UK Insurance Companies (Total Value of all Investments £1.8 Trillion; Source ABI 2012)

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 Credit Risk on bond portfolios • Because of this relationship between the value of a bond of Credit Transitions…

Transitions Matrix • Rating Agencies provide estimates of the probabilities of Credit Transitions based upon the initial rating of the bond • 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 (Source S&P: Global Corporate Transition Matrix) • 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: • For example, if T describes the probabilities of credit transitions over 1 year then the probability of credit transitions over 2 years (T2) can be calculated as : • It is possible to calculate the transition matrix for fractional values of N using an Eigen Value Decomposition

Recovery Rates for Bonds • 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 Subordinate Debt Payment No Payment Junior Subordinate Debt Payment No Payment

Recovery Rates Source Carty & Lieberman Seniority ClassAverageStandard 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 Example • Imagine we have a CCC bond • The face value of the bond is £100 • The bonds seniority is Senior Subordinate • We wish to model 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

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 S&P Rating Transition Matrix • 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 Credit Transitions are Discrete Random Variables AAA: 0.06% AA: 0.25% A: 1.85% BBB: 2.06% Initial Rating CCC BB: 12.34% B: 24.86% CCC: 39.97% Default: 18.6% One Year Time Horizon

Credit Transitions & Values Increasing Bond Value Due to Decreasing Credit Spread 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 39.97% 24.86% 18.6% 12.34% 2.06% 1.85% 0.25% 0.07%

Simulating Credit Transitions for CCC over 1 Year 100% 99.68% 99.93% 97.83% 95.77% 83.43% Bond Migrates to B 58.57% Rand() Bond Stays at CCC 18.6%

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

Credit Metrics VaR Measure • Although it has been used in a number of context JP Morgan’s Credit Metrics was originally developed to measure the Credit Risk on a portfolio of bonds • Using a Monte Carlo Simulation based technique it measure the Credit VaR over a period of time on the portfolio defined as: • Where P* is the value of the portfolio of bonds such that lower values will only be observed some percentage of the time (1%, 5% etc) and E(P) is the expected value of the portfolio • This is a Relative VaR measure

Stochastic Recovery Rates • Up until now we have been assuming that Recovery Rates are fixed at their average value • The Credit Metrics 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 ratios (we also saw its use in the RMS model) • 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 Junior Subordinate Debt Recovery Rate Subordinate Recovery Rate Subordinate Unsecured Recovery Rate

Generating Correlated Random Samples • Our Credit Risk Simulations have also assumed that there is no correlation between Credit Transitions and Defaults • The original JP Morgan’s Credit Metrics model used a technique called the Gaussian Copula to capture the correlation between credit transitions • The Gaussian Copula is sometimes known as “The Formula that Killed Wall Street” and its misuse is widely blamed as a major contributing factor to the 2008 Financial Crisis • Before we learn about the Gaussian Copula we will have to learn a mathematical trick that will let us generate correlated random numbers…

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 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:

Multivarite Normal Sample s=1 s=1 m=0 m=0 Cholesky Decomposition of Correlation Matrix s=1 s=1 m=0 m=0 s=1 s=1 m=0 m=0 Correlated Normals (Multi-Variate Normal) Independent Normals

The Gaussian Copula • Copula’s are sets of Uniformly Distributed Random numbers with some statistical interdependence or correlation • The Gaussian Copula is basically a set of Correlated Normally Distributed random numbers that have been converted into a set of Correlated Uniformly Distributed random numbers • Just as we can use the CDF to transform Uniformly Distributed random numbers into Normally Distributed random numbers we can also use it to transform Normally Distributed random numbers into Uniform!

The Inverse Transform Uncorrelated Uniform Random Numbers NORMINV Uncorrelated Normal or Gaussian Random Numbers NORMINV Using the Inverse of the Normal CDF (NormInv) we convert Uncorrelated Uniformly Distributed Random Numbers into Uncorrelated Normally Distributed Random Numbers.

The Gaussian Copula Gaussian Copula (Correlated Uniform Random Numbers) NORMDIST NORMDIST Correlated Normal (Gaussian) Random Numbers Using the Normal CDF (NORMDIST) we convert Correlated Normally Distributed Random Numbers into the Gaussian Copula

Correlated Defaults with the Gaussian Copula Gaussian Copula (Correlated rand!) Does B Default? Does A Default? Because the uniform random numbers used to simulate the defaults are correlated the defaults are also correlated

Risk Analysis & Modelling