1. Dinosaurs in a portfolio Marc Freydefont Moody’s Investors Service Ltd. London, 13 March 2002 ISDA - PRMIA Seminar

2. Contents Section 1Introduction Section 2Rating CDOs Section 3The BET approach Section 4The Monte Carlo approach and the log-normal method Section 5Portfolio analysis Section 6Characteristic functions / Fourier transforms Section 7Conclusions

3. Introduction Section 1

4. CDOs: Another Record Year in Europe

5. Collateralised Risk Obligations ? CLO:“Collateralised Loan Obligations” - Securitisation of a portfolio of corporate loans CBO:“Collateralised Bond Obligations” - Securitisation of a portfolio of corporate bonds CDO:“Collateralised Debt Obligations” - Can include CLOs, CBOs, or a combination thereof CSO:“Collateralised Synthetic Obligations” - Securitisation of a portfolio of synthetic exposures via Credit Default Swaps CFO:“Collateralised Fund Obligations” - Securitisation of a portfolio of exposures to hedge funds CEO:“Collateralised Equity Obligations” - Securitisation of a portfolio of equity or private equity exposures COO: “Collateralised Options Obligations” - Securitisation of a portfolio of Options CRO: “Collateralised Risk Obligations” - Securitisation of a portfolio of Risks Section 4Portfolio Analysis Section 5Characteristic Functions applied to Portfolio Analysis Conclusions

6. Rating CDOs Section 2

7. Rating CDOs Issuing SPV... Asset Pool B2 Ba3 B1 Principal & Interest Senior Investors Junior Investors Swap (Aa2)

8. Rating CDOs: Expected Loss  Expected Loss is the probability-weighted mean of losses arising from credit events  Expected Loss “=” Probability x Severity  Expected Loss =,  Where  L :Severity of credit loss  p(L) :Probability density of credit loss

9. Rating CDOs: Various Inputs  Scheduled cash flow from assets  Cash flow allocation within the transaction  Probability of default of the assets  Diversification within the portfolio  Recovery rate in case of default

10. Rating CDOs: EL Calculation... Default Scenarios Scenario 1 Scenario 2 Scenario 3 Senior Loss 1 Sub Loss 1 Senior Loss 2 Sub Loss 2 Senior Loss 3 Sub Loss 3 PV Losses to Notes... Cash Flows from Assets Cash Flows to Notes Senior Expected Loss Junior Expected Loss Mean of Losses (Weighted by Scenario Probability) Cash Flow Model Priorities Reserves OC & IC Triggers Swaps Leverage

11. Analysing CROs  Measuring the credit performance of a CRO means the following:  determining the likelihood of occurrences of defaults and losses on the underlying portfolio of assets  determining the likelihood of occurrences of defaults and losses on each class of notes issued by the CRO vehicle  Three classical approaches:  Binomial / Multinomial Expansion Trees (“BET”) - Homogeneous pools  Monte Carlo simulations - Heterogeneous pools  Log-normal method  Innovative approach:  Characteristic functions and Fourier transforms

12. The BET approach Section 3

13. The BET approach Description The total expected loss of a pool of N assets having the same default probability p and the same recovery rate RR is calculated using the binomial formula Key variables and assumptions Homogeneous pool of identical independent assets Main disadvantages Does not account for heterogeneity in size, in risk (default probability), in recovery rate, in correlations

14. The BET approach : Binomial Expansion Tree  Real portfolio represented as a lesser number D of independent and identical assets  Each scenario, with defaults ranging from 0 to D, is considered  The loss for each tranche is recorded and then multiplied by the probability of the scenario  Probability of n defaults =

15. The BET approach : Binomial or multinomial

16. The BET approach : Cash flow allocation

17. The BET approach : Loss calculation

18. The Monte Carlo approach and the log-normal method Section 4

19. The Monte Carlo approach Description The total expected loss of a portfolio can be calculated as the average of the losses generated by running a high number of default simulations on the pool of assets and applying to each defaulted asset the relevant recovery rate Key variables and assumptions Default probability, recovery rates, default correlations Main disadvantages Difficult to implement, convergence problems, calculation time

20. The log-normal method

21. Portfolio analysis Section 5

22. Portfolio analysis In the following, consider an ever increasing number of independent assets in the pool: N = 1 bond A2 cases : no default or A defaults N = 2 bonds A and B4 cases : no default, A or B defaults, A and B default N = 3 bonds A, B and C8 cases : no default, A, B or C defaults, (A and B) (B and C) or (A and C) default and (A, B and C) default... N bonds2 N cases

23. Portfolio analysis N=1 N=4N=3 N=2 Losses in % Prob.(%) Prob.(%) Prob.(%) Prob.(%)

24. Portfolio analysis N=5 N=30N=20 N=10 Losses in % Prob.(%) Prob.(%) Prob.(%) Prob.(%)

25. Portfolio analysis N=40 N=200N=100 N=50 Losses in % Prob.(%) Prob.(%) Prob.(%) Prob.(%)

26. Portfolio analysis N=300 N=600N=500 N=400 Losses in % Prob.(%) Prob.(%) Prob.(%) Prob.(%)

27. Portfolio analysis N=700 N=1000N=900 N=800 Losses in % Prob.(%) Prob.(%) Prob.(%) Prob.(%)

28. The log-normal method

29. Characteristic functions Section 6

30. Characteristic functions In probability theory, a characteristic function is defined by:

31. Characteristic functions The characteristic function of simple random variables can be computed easily For instance, when X is a Bernouilli variable : X = 1 with probability p X = 0 with probability q = 1 - p Therefore for a bond with a default probability p and a size S:

32. Characteristic functions  Consider now a portfolio of N independent bonds / assets : Sizes: S 1, S 2, … S N Default probabilities: p 1, p 2, …, p N  Define X (random variable) as being the defaulted amount of the portfolio: X = S 1. X 1 + S 2. X 2 + S 3. X 3 + … S N. X N

33.  In a Nutshell: Fourier transform theory Functions of a space variable (x) Real Space Functions of a time/ frequency variable (t) Fourier Dual Space Fourier Tranform Inverse Fourier Tranform   If you know the Fourier Transform of a function, it is easy (at least theoretically) to get the original function by applying the inverse Fourier Transform (hence the name of the inverse Fourier Transform)

34. Portfolio analysis : why did dinosaurs come into the picture ?

35. Conclusion  In most cases, it is impossible to derive tractable formulas for default distributions in the space domain.  However, under certain sets of modelling assumptions, the formulas simplify if we translate ourselves in the Fourier domain.  In order to get back to the “real” space, apply the Inverse Fourier Transform.  Computing an Inverse Fourier Transform basically costs nothing in terms of computation time. Fast Fourier Transform algorithms were discovered some 50 years, (it is a powerful technique that made possible technical revolutions in many industries - electronics, CD, DVD, radio, telecommunications, medical systems,…  This is precisely because computing a Fourier Transform costs practically nothing in terms of computation times that this new numerical method is of interest for getting default distributions.

36. Conclusion  Theoretical framework that permits to justify the log-normal approach  Framework that permits much more: analysis of tricky portfolio risk profile that results from aggregation of heterogeneous assets and analysis of dependence between recovery rates and default rates (not really addressed so far)  Avoids Monte Carlo simulations and calibration/convergence issues  Could potentially be applied to any kind of portfolio indicators (default, losses, but also PV, etc,)

37.  Surprisingly, almost nothing was written by academics on portfolio loss/default distributions. However, very recently, Vasicek (1997- KMV) and Finger (1999 - Credit Metrics) obtained very promising results  Using a simple factor model (i.e. a model that assumes that default correlation between the loans is created by exposure of all the loans to a common market index), they find that the portfolio defaults have a normal inverse probability distribution Tackling dependencies between assets ?

38.  Default correlation is primarily the result of individual companies being linked to one another through the general economy  Beyond that induced by the general economy, default correlation exists between firms in the same industry because of industry- specific economic conditions  Default correlations also exist between companies in different industries that rely on the same production inputs and among companies that rely on the same geographical market

39. Tackling dependencies between assets ?  Given the probability of default for each asset, we calculate a threshold  such that given a variable Z normally distributed (i.e. having a density of probability function N(0,1)):   is the threshold against which we will need to compare the random gaussian variable to determine if a default on that specific obligor has occurred

40. Tackling dependencies between assets ?  The model’s assumptions:  Let’s define Z j as a (normalized) credit risk measure of the j th debtor at the end of the horizon. The lower Z j, the higher the credit risk of the j th debtor.  Debtor j defaults during the period if Z j <  j, i.e.  j is determined by Pr(Z j <  j ) = p j.   j is the default threshold. If Z j is normalized,  j =  -1 (p j ), where  (x) is the cumulative standard normal distribution function.  The credit risk indicator of the j th debtor is split between a systemic risk (exposure to a common normalised market index Z - for instance, economic growth, … ) and an idiosyncratic risk (normalised risk that can be only attributed to the j th debtor): Systemic risk Idiosyncraticrisk Idiosyncratic risk Exposure to Common Index (correlation parameter)

41. Tackling dependencies between assets ?  The j th debtor will default if Z j <  j  i.e. if  Using the standard normality of  j, the default probability of the j th debtor (conditional to a fixed Z) will simply be  Conditional Fourier transform of asset j (given Z) is:  Conditional Fourier transform of the portfolio (given Z) is:  Fourier transform of the portfolio is: