Securitization and Copula Functions Advanced Methods of Risk Management Umberto Cherubini

1. Securitization and Copula Functions Advanced Methods of Risk Management Umberto Cherubini

2. In this lecture you will learn To evaluate basket credit derivatives using Marshall-Olkin distributions and copula functions. To analyze and evaluate securitization deals and tranches To evaluate the risk of tranches and design hedges Learning Objectives

3. Assume we have a portfolio of exposures (for simplicity with the same LGD). We can distinguish between a very large number of exposures and a limited number of them. In a retail setting we are obviously interested in the former case, even though to set up the model we can focus on the latter one (around 50-100). We want define the probability of loss on the portfolio. We define Q(k) the probability of observing k defaults (Q(0) being survival probability of the portfolio). Expected loss is Portfolios of exposures

4. Consider a credit derivative, that is a contract providing “protection” the first time that an element in the basket of obligsations defaults. Assume the protection is extended up to time T. The value of the derivative is FTD = LGD v(t,T)(1 – Q(0)) Q(0) is the survival probability of all the names in the basket: Q(0) Q(1 > T, 2 > T…) “First-to-default” derivatives

5. As an extension, consider a derivative providing protection on the first x defaults of the obligations in the basket. The value of the derivative will be “First-x-to-default” derivatives

6. Securitization deals Senior Tranche Originator Junior 1 Tranche Special Purpose Vehicle SPV Sale of Assets Junior 2 Tranche … Tranche Equity Tranche

7. Arbitrage (no more available): by partitioning the basket of exposures in a set of tranches the originator used to increase the overall value. Regulatory Arbitrage: free capital from low-risk/low-return to high return/high risk investments. Funding: diversification with respect to deposits Balance sheet cleaning: writing down non performing loans and other assets from the balance sheet. Providing diversification: allowing mutual funds to diversify investment The economic rationale

8. Securitization deal structures are based on three decisions Choice of assets (well diversified) Choice of number and structure of tranches (tranching) Definition of the rules by which losses on assets are translated into losses for each tranches (waterfall scheme) Structuring securitization deals

9. The choice of the pool of assets to be securitized determines the overall scenarios of losses. Actually, a CDO tranche is a set of derivatives written on an underlying asset which is the overall loss on a portfolio L = L1 + L2 +…Ln Obviously the choice of the kinds of assets, and their dependence structure, would have a deep impact on the probability distribution of losses. Choice of assets

10. A tranche is a bond issued by a SPV, absorbing losses higher than a level La (attachment) and exausting principal when losses reach level Lb (detachment). The nominal value of a tranche (size) is the difference between Lb and La . Size = Lb – La Tranche

11. Equity tranche is defined as La = 0. Its value is a put option on tranches. v(t,T)EQ[max(Lb – L,0)] A seniortranche with attachmentLa absorbs losses beyond La up to the value of the entire pool, 100. Its value is then v(t,T)(100 – La) – v(t,T)EQ[max(L – La,0)] Kinds of tranches

12. If tranches are traded and quoted in a liquid market, the following no-arbitrage relationships must hold. Every intermediate tranche must be worth as the difference of two equity tranches EL(La, Lb) = EL(0, Lb) – EL(0,La) Buyng an equity tranche with detachmentLa and buyng the corresponding senior tranche (attachmentLa) amounts to buy exposure to the overall pool of losses. v(t,T)EQ[max(La – L,0)] + v(t,T)(100 – La) – v(t,T)EQ[max(L – La,0)] = v(t,T)[100 – EQ (L)] Arbitrage relationships

13. Different “tranches” have different risk features. “Equity” tranches are more sensitive to idiosincratic risk, while “senior” tranches are more sensitive to systematic risk factors. “Equity” tranches used to be held by the “originator” both because it was difficult to place it in the market and to signal a good credit standing of the pool. In the recent past, this job has been done by private equity and hedge funds. Risk of different “tranches”

14. Cash CDO vs Synthetic CDO: pools of CDS on the asset side, issuance of bonds on the liability side Funded CDO vs unfunded CDO: CDS both on the asset and the liability side of the SPV Bespoke CDO vs standard CDO: CDO on a customized pool of assets or exchange traded CDO on standardized terms CDO2: securitization of pools of assets including tranches Large CDO (ABS): very large pools of exposures, arising from leasing or mortgage deals (CMO) Managed vs unmanaged CDO: the asset of the SPV is held with an asset manager who can substitute some of the assets in the pool. Securitization zoology

15. Synthetic CDOs Senior Tranche Originator Junior 1 Tranche Protection Sale Special Purpose Vehicle SPV Junior 2 Tranche CDS Premia Interest Payments Investment … Tranche Collateral AAA Equity Tranche

16. CDO2 Originator Senior Tranche Tranche 1,j Junior 1 Tranche Tranche 2,j Special Purpose Vehicle SPV Junior 2 Tranche Tranche i,j … Tranche Tranche … Equity Tranche

17. Since June 2003 standardized securitization deals were introduced in the market. They are unfunded CDOs referred to standard set of “names”, considered representative of particular markets. The terms of thess contracts are also standardized, which makes them particularly liquid. They are used both to hedged bespoke contracts and to acquire exposure to credit. 125 American names (CDX) o European, Asian or Australian (iTraxx), pool changed every 6 months Standardized maturities (5, 7 e 10 anni) Standardized detachment Standardized notional (250 millions) Standardized CDOs

18. i-Traxx CDX Tranche Bid Ask Tranche Bid Ask 0-3% 23.5* 24.5* 0-3% 44.5* 45* 3-6% 71 73 3-7% 113 117 6-9% 19 22 7-10% 25 30 9-12% 8.5 10.5 10-15% 13 16 12-22% 4.5 5.5 15-30% 4.5 5.5 (*) Amount to be paid “up-front” plus 500 bp on a running basis Source: Lehman Brothers, Correlation Monitor, September 28th 2005. i-Traxx and CDX quotes, 5 year, September 27th 2005

19. The standard technique used in the market is based on Gaussian copula C(u1, u2,…, uN) = N(N – 1 (u1 ), N – 1 (u2 ), …, N – 1 (uN ); ) where ui is the probability of event i T and iis the default time of the i-th name. The correlation used is the same across all the correlation matrix.The value of a tranche can either be quoted in terms of credit spread or in term of the correlation figure corresponding to such spread. This concept is known as implied correlation. Notice that the Gaussian copula plays the same role as the Black and Scholes formula in option prices. Since equity tranches are options, the concept of implied correlation is only well defined for them. In this case, it is called base correlation. The market also use the term compound correlation for intermediate tranches, even though it does not have mathematical meaning (the function linking the price of the intermediate tranche to correlation is NOT invertible!!!) Gaussian copula and implied correlation

20. Cholesky decomposition A of the correlation matrix R Simulate a set of n independent random variables z = (z1,..., zn)’ from N(0,1), with N standard normal Set x = Az Determine ui = N(xi) with i = 1,2,...,n (y1,...,yn)’ =[F1-1(u1),...,Fn-1(un)] where Fidenotes the i-th marginal distribution. Monte Carlo simulationGaussian Copula

21. Cholesky decomposition A of the correlation matrix R Simulate a set of n independent random variables z = (z1,..., zn)’ from N(0,1), with N standard normal Simulate a random variable s from 2 indipendent from z Set x = Az Set x = (/s)1/2y Determine ui =Tv(xi) with Tv the Student t distribution (y1,...,yn)’ =[F1-1(u1),...,Fn-1(un)] where Fidenotes the i-th marginal distribution. Monte Carlo simulationStudent t Copula

23. Base correlation

24. Default Probability Correlation 0% Correlation 20% Correlation 95% MC simulation pn a basket of 100 names

25. Example of iTraxx quote

26. Tranches can be hedged, by: Taking offsetting positions in the underlying CDS Taking offsetting positions in other tranches (i.e. mezz-equity hedge) These hedging strategies may fail if correlation changes. This happened in May 2005 when correlation dropped to a historical low by causing equity and mezz to move in opposite directions. Tranche hedging

27. Large CDO refer to securitization structures which are done on a large set of securities, which are mainly mortgages or retail credit. The subprime CDOs that originated the crisis in 2007 are examples of this kind of product. For these products it is not possible to model each and every obligor and to link them by a copula function. What can be done is instead to approximate the portfolio by assuming it to be homogeneous . Large CDO

28. Assume a model in which there is a single factor driving all losses. The dependence structure is gaussian. In terms of conditional probabilility where M is the common factor and m is a particular scenario of it. Gaussian factor model (Basel II)

29. Vasicek proposed a model in which a large number of obligors has similar probability of default and same gaussian dependence with the common factor M (homogeneous portfolio. Probability of a percentage of losses Ld: Vasicek model

30. Vasicek density function

31. The mean value of the distribution is p, the value of default probability of each individual Value of equity tranche with detachment Ld is Equity(Ld) = (Ld – N(N-1(p); N-1 (Ld);sqr(1 – 2)) Value of the senior tranche with attachment equal to Ld is Senior(Ld) = (p – N(N-1(p); N-1 (Ld);sqr(1 – 2)) where N(N-1(u); N-1 (v); 2) is the gaussian copula. Vasicek model