An Hybrid Optimization Method For Risk Measure Reduction Of A Credit Portfolio Under Constraints

Transcription

1 An Hybrid Optimization Method For Risk Measure Reduction Of A Credit Portfolio Under Constraints Ivorra Benjamin & Mohammadi Bijan Montpellier 2 University Quibel Guillaume, Tehraoui Rim & Delcourt Sebastien BNP-Paribas - Asset Management Paris April 4, 2006 Seminaire Montpellier 04/04/06 - p. 1/31

2 Outlines Part II: Hybrid Problem definition: -Credit derivative product description -Loss modeling + performance indicators -Optimization problems April 4, 2006 Seminaire Montpellier 04/04/06 - p. 2/31

3 Outlines Part II: Hybrid Problem definition: -Credit derivative product description -Loss modeling + performance indicators -Optimization problems Optimization algorithm: -Semi-deterministic algorithm -Genetic algorithm -Hybdridation April 4, 2006 Seminaire Montpellier 04/04/06 - p. 2/31

4 Outlines Part II: Hybrid Problem definition: -Credit derivative product description -Loss modeling + performance indicators -Optimization problems Optimization algorithm: -Semi-deterministic algorithm -Genetic algorithm -Hybdridation Problem resolution: -Results -Financial analysis April 4, 2006 Seminaire Montpellier 04/04/06 - p. 2/31

5 Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 3/31

6 Historical context Historical context Considered derivative credit product Objective of the work Object: derivative credit products: -Development of those products (In 2004: 2300 M $) CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 4/31

7 Historical context Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Object: derivative credit products: -Development of those products (In 2004: 2300 M $) Main problems: 1-Risk management: - Risk of a trading partner not fulfilling his obligations on the due date Loss. - Example: Bankruptcy (Enron/Worldcom in 2001/2002) Variation of rating (Telecom. in 2000) Loss evaluation (Merton ), Risk measure (Rockafellar, Artzner ) April 4, 2006 Seminaire Montpellier 04/04/06 - p. 4/31

8 Historical context Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Object: derivative credit products: -Development of those products (In 2004: 2300 M $) Main problems: 1-Risk management: - Risk of a trading partner not fulfilling his obligations on the due date Loss. - Example: Bankruptcy (Enron/Worldcom in 2001/2002) Variation of rating (Telecom. in 2000) Loss evaluation (Merton ), Risk measure (Rockafellar, Artzner ) 2-Profitability improvement April 4, 2006 Seminaire Montpellier 04/04/06 - p. 4/31

9 Considered derivative credit product Collateralized Debt Obligations (CDO) Historical context Considered derivative credit product Objective of the work CDO 2 model: Data Facility 1 Facility 2 37% Senior CDO 2 model: Loss evaluation Merton s model KMV s model Default times model 30% Tranche A Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Facility N 30% 3% Tranche B Equity -Objective: buy securitization in order to protect from eventual defaults. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 5/31

10 Considered derivative credit product CDO 2 / Master CDO S&P RATING AAA AA A BB Historical context Considered derivative credit product Objective of the work CDO 2 model: Data Facility 1 Facility 2 Facility5 Facility3 ABS 1 Facility4 CDO 2 model: Loss evaluation Merton s model ABS 2 KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid SENIOR MEZZANINE 90% 4% AAA AAA AAA AA AA AA SENIOR TRANCHE A TRANCHE B JUNIOR 6% BBB BBB BBB CDO1 CDO2 CDO3 -Objective: Resilient to low losses. -Main default: When Losses: Fast and highly severe. TRANCHE C JUNIOR MASTER CDO April 4, 2006 Seminaire Montpellier 04/04/06 - p. 5/31

11 Objective of the work Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Find the nominal coefficients, using an algorithm, that improve desired portfolio characteristics: Reduction of Risk measure. Augmentation of the income. Augmentation of the profitability. Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 6/31

12 CDO 2 model: Data Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems For each facility i, market gives: A nominal (N i ). A maturity (T i ) date. A spread (Sp i ). A loss given default (LGD i ). geographical/zone/stability Coefficients. Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 7/31

13 CDO 2 model: Data Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid For each facility i, market gives: A nominal (N i ). A maturity (T i ) date. A spread (Sp i ). A loss given default (LGD i ). geographical/zone/stability Coefficients. We can directly compute: Income: IC = N i=1 Sp i N i PN = N i=1 N i April 4, 2006 Seminaire Montpellier 04/04/06 - p. 7/31

14 CDO 2 model: Loss evaluation Historical context Considered derivative credit product Objective of the work In order to compute more complex indicator. We consider: L the CLO 2 s amount of losses: Random variable. CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 8/31

15 CDO 2 model: Loss evaluation Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid In order to compute more complex indicator. We consider: L the CLO 2 s amount of losses: Random variable. We need to determine is Loss density function β L : Merton s Model Kealhofer, McQuown and Vasicek s Model (Factor Model) Default Time s Model April 4, 2006 Seminaire Montpellier 04/04/06 - p. 8/31

16 Merton s model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems 1974: Model of the firm: The liabilities are divided : Debt D Equities E Assets A Linked by: A = D + E Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 9/31

17 Merton s model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid { A = sφ(d 1 ) xe rt φ(d 2 ) E = xe rt φ( d 2 ) sφ( d 1 ) where: - A originally corresponds to call price and E to put price. - φ is the normal probability density function. - d 1 = log(s/x)+(r+σ2 /2)t σ and d t 2 = d 1 σ t. - s the price of the underlying stock. - x is the strike price. - r is the continuously compounded risk free interest rate. - t is the time in years until the expiration of the option. If at debt maturity: E >= D no default. E < D default. TIME CONSUMING April 4, 2006 Seminaire Montpellier 04/04/06 - p. 9/31

18 KMV s model It s factor model based on Merton s model: A = D + E. Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Introducing random variables normally distributed and independent of each other: 14 global factors (F k ) 1 k 14 that model the global economic trends. Local factors: Facility activity sectors C (61 sectors) and Facility geographical areas I (45 areas). The rate of return of firm, rt i = Ai t Ai t 1 t, is given by: [ 14 ] rt i = R i αkf i t k + β C i Ct i + β I i It i + 1 Ri 2εi t k=1 where ε i t is the specific rate of return of firm i. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 10/31

19 KMV s model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Monte-Carlo Algorithm: Simulate realizations of the latent variables ( ) Ft k ( ) I i t 1 i 61, ( ) Ct i 1 i 45, ( ) ε i t until horizon H 1 i N Deduce the rates of return of the portfolio s assets 1 k 14, For each facility and for each scenario, compare the asset value (A) with the default point (D). For each scenario: evaluate the total loss at given horizon: LGD i N i i def ault April 4, 2006 Seminaire Montpellier 04/04/06 - p. 10/31

20 KMV s model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 10/31

21 KMV s model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Inconveniences: The grid country/industry used to classify firms is not always satisfying. Time of computation. The determination of default events is achieved by comparing asset values and default points at horizon. A default should occur as soon as the asset value trajectory meets the default point trajectory. Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 10/31

22 Default times model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid We introduce default times τ = (τ 1,..., τ i,..., τ n ). The default times are computed from Gaussian vector: A = (A 1,..., A i,..., A n ) with zero mean and unit variances and covariance matrix Σ given by KMV model: Considering that A i is given by KMV formulation, of the form: A i = ρ i.y + 1 ρ 2 i.ǫ i where for i = 1,, n, Y, ǫ i, i = 1,, n is a standard gaussian random variable. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 11/31

23 Default times model Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid We define the joint default probability function given by (Copula theory): = F(t 1,, t n ) = IP(τ 1 < t 1,, τ n < t n ) ( n ( )) Φ 1 (F i (t i )) ρ i.y e t2 2 Φ dt 1 ρ 2 i 2π + i=1 with F i (t) = IP(τ i t) the default time s marginal repartition function. Default times are given by: τ i = F 1 (Θ(A i )) where Θ(.) denotes the standard normal gaussian density function. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 11/31

24 Default times model Historical context Considered derivative credit product Objective of the work Monte-Carlo approach to compute β L : Simulate KMV coefficients. Compute default times. Compute loss amount: i default LGD i N i. Case 1: CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Case 2: SUPER SENIOR AA EQUITY CDO1 SUPER SENIOR AA EQUITY CDO2 SUPER SENIOR AA EQUITY CDO3 SENIOR TRANCHE A TRANCHE B TRANCHE C EQUITY MASTER CDO SUPER SENIOR AA SUPER SENIOR AA SUPER SENIOR AA SENIOR TRANCHE A TRANCHE B EQUITY EQUITY EQUITY TRANCHE C EQUITY CDO1 CDO2 CDO3 MASTER CDO April 4, 2006 Seminaire Montpellier 04/04/06 - p. 11/31

25 Density function β L Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 12/31

26 Performance Indicators Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Expected Loss (EL): The expected value, over some specified horizon, of portfolio losses due to default. Given by: EL(β L ) = portfolio s nominal 0 Risk Adjusted Return On Capital (RAROC): Profitability. RAROC is given by: β L (x) portfolio s nominal dx RAROC = ( portfolio (Sp i) x i ) EL(β L ) EC April 4, 2006 Seminaire Montpellier 04/04/06 - p. 13/31

27 Risk measures Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid We consider: -Ω a finite set of states of nature. -A random variable by X. -The probability space L (Ω, I, IP) (view as set of all risks: changes in values between two dates). Any mapping ρ : L (Ω, I, IP) IR is called risk measure. Important in finance: A mapping ρ : L (Ω, I, IP) IR satisfying the Sub-additivity propriety: ρ(x 1 + X 2 ) ρ(x 1 ) + ρ(x 2 ) (more generally coherent). April 4, 2006 Seminaire Montpellier 04/04/06 - p. 14/31

28 Risk measures Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid VaR: The smallest nominal loss of the worst α % of losses: V ar α (L) = inf[l L 0 β L (x)dx < (1 α)] C-VaR (Expected Shortfall): The average of the worst α % of losses. In the case of the discrete distribution β L, the ES α is given by: ES α (L) = 1 α α inf{x P[L x] p}dp p=0 CVaR is often preferred: In some case it s coherent and convex risk measure. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 14/31

29 Optimization problems Historical context Considered derivative credit product Objective of the work CDO 2 model: Data CDO 2 model: Loss evaluation Merton s model KMV s model Default times model Density function β L Performance Indicators Risk measures Optimization problems Part II: Hybrid Optimize Facilities nominal of BNP-Paribas Portfolio-Management (PM) portfolio in order to: Reduce risk measure keeping the income higher than the initial value: α = 0.1 V ar. Both measures have led to the same kind of results. Maximize Income keeping risk measure lower than the initial value. Maximize profitability keeping the income higher than the initial value: We use the RAROC as profitability measure to be maximized. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 15/31

30 Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 16/31

31 Global Optimization and Dynamical system Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm We consider a function J : Ω ad IR to be minimized. We assume: -J C 1 (Ω ad, IR) - Ω ad IR N compact The minimum of J in Ω ad is denoted by J m. In practice, when J m is unknown, we set J m to a lower value and look for the best solution for a given complexity and computational effort. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 17/31

32 Global Optimization and Dynamical system Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm Most deterministic minimization algorithms can be seen as discretizations of Dynamical Systems. A numerical global of J with one of those algorithms is possible if the following BVP has a solution: First or second order dynamical system Initial conditions J(x(Z)) J m < ǫ where: Z IR a given time and ǫ the approximation precision. This BVP is over-determined. Idea: Remove over determination. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 17/31

33 Recursive semi-deterministic s Remove the over determination: Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm First or second order dynamical system Initial conditions J(x(Z)) J m < ǫ One of the initial condition is considered as a variable v Then apply BVP techniques: Example: SHOOTING METHOD. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 18/31

34 Genetic algorithm General overview: Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm April 4, 2006 Seminaire Montpellier 04/04/06 - p. 19/31

35 Genetic algorithm Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm One matrical representation: Initial population: X 0 = {x 0 l Ω ad, l = 1,..., N p } i th Population: X i = x i 1(1)... x i 1(N) x i N p (1)... x i N p (N) April 4, 2006 Seminaire Montpellier 04/04/06 - p. 19/31

36 Genetic algorithm Selection: Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm Crossover: X i+1/3 = S i X i Mutation: X i+2/3 = C i X i+1/3 X i+1 = X i+2/3 + E i April 4, 2006 Seminaire Montpellier 04/04/06 - p. 19/31

37 Genetic algorithm The new population can be written as: Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm X i+1 = C i S i X i + E i (1) Thus genetics algorithms can be associated to: Ẋ(t) = Λ 2 (t,x(t),p)x(t)λ 3 (t,x(t),p) X(t) Where -P are a set of fixed parameters -X of the form: X = {x i / i = 1,..., N p x i Ω ad } April 4, 2006 Seminaire Montpellier 04/04/06 - p. 19/31

38 Genetic algorithm Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm Ẋ(t) = Λ 2 (t, X(t), p)x(t)λ 3 (t, X(t), p) X(t) X(0) = X 0 Ĵ(X(Z)) J m Where: -Ĵ(X) = min(j(x i)/x i X) -P are a set of fixed parameters - X of the form: X = {x i / i = 1,..., N p x i Ω ad } Inconveniences of GA: -Slow convergence/ computational complexity -Lack of precision Idea: Combine our SD with GA: X 0 Considered as a new variable. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 19/31

39 Hybrid algorithm Part II: Hybrid Global Optimization and Dynamical system Recursive semi-deterministic s Genetic algorithm Hybrid algorithm Input: X 0, N, ǫ For i going from O to N o i = GA(X i, I, ǫ) If min{j(o k ), k = 1,..., i} < J m + ǫ EndFor We construct X i+1 : x i X i x i+1 = x i J(o i ) EndFor Output: A 1 (X 0, N, ǫ) = argmin{j(o k ), k = 1,..., i} o i v i J(o i ) J(v i ) We can build recursively algorithm A 2, A 3,... replacing GA by A 1, A 2... GA is taken with small population/generation values. Have been tested and compared with classical GA on various problem: -Microfluidic device -Optical fiber synthesis -Academic test cases April 4, 2006 Seminaire Montpellier 04/04/06 - p. 20/31

40 Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max April 4, 2006 Seminaire Montpellier 04/04/06 - p. 21/31

41 Considered Portfolio Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max PM portfolio: CLO 2 structure Compound by 500 facilities dispatched into 40 ICLOs tranches and 54 Single-Names ( SN ). Portfolio nominal: 2.e9 Euros (E) ICDO nominal: 7.5e8 E. SN nominal: 1.25 E. Income: 2.1e7 E. VaR: 1.9e8 E. RAROC: 90 % April 4, 2006 Seminaire Montpellier 04/04/06 - p. 22/31

42 Parameterization Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Parameters : nominal of each ICDO s tranches and SN included in the entire PM universe (1500 facilities) For versatility and respecting the BNP investment guideline, we consider constraints on facility: Avoid too much concentration in one facility: maximum nominal 1.e8 E. Minimum facility investment: if nominal <5e6 E nominal set to 0. Facility quality: Depending on quality coefficients: The nominal can be raised, decreased or unmodified. Thus the Parameter number: 65. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 23/31

43 Cost function Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Optimization problem is of the form: min x Ω nchar n=1 ξ i J n (x) lc 1 C 1 (x) uc 1. lc ncons C ncons (x) uc ncons We rewrite using wall functions: J(x) = J(x)+ϑ ncons j=1 (max(uc j C j (x), 0)+max(C j (x) lc j, 0)) April 4, 2006 Seminaire Montpellier 04/04/06 - p. 24/31

44 Computational time reduction Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Sensitivity analysis is difficult: - At least 1e6 Monte-Carlo iterations to capt variations 20 minutes (real time). Thus we use HSGA + computational reduction techniques: - Default times are computed once (Need lot of memory 1e5=512 Mb) - When constraint is not satisfied: intend to project portfolio on admissible space border using proportional coefficient. One : 2000 functional evaluations 6H computation. April 4, 2006 Seminaire Montpellier 04/04/06 - p. 25/31

45 Var min/ IC constraint Nominal Nom. ICLO Nom. SN IC VaR 0,1% RAROC Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90% Sen % 7% 2% P 1 2e9 5.5e8 1.45e9 2.1e7 1.3e8 87% Evo. 0% -26% 11% 0% -31% -3% Sen % 1% 1% April 4, 2006 Seminaire Montpellier 04/04/06 - p. 26/31

46 Var min/ IC constraint Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Nominal (E) Nominal(E) Probability (%) 12 x 107 Inner CDOs EL Facility 12 x 107 Inner CDOs Nominal (E) 12 x Single Names Facility Loss density 12 x Single names 107 VaR 10 Nominal (E) April 4, 2006 Loss Amount 0 (E) Seminaire 60 Montpellier 04/04/06 - p. 26/31 Facility Facility

47 IC max/ VaR constraint Nominal Nom. ICLO Nom. SN IC VaR 0,1% RAROC Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90% Sen % 7% 2% P 2 3.2e9 7.5e8 2.7e9 3.e8 1.9e8 92% Evo. 57% 6% 88% 28% 0% 2% Sen % 1% 1% April 4, 2006 Seminaire Montpellier 04/04/06 - p. 27/31

48 IC max/ VaR constraint Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Nominal (E) Nominal Probability (E) (%) 12 x 107 Inner CDOs Facility.3 Inner CDOs x EL Nominal (E) 12 x Single Names Facility Loss density 12 x Single names 107 VaR 10 Nominal (E) April 4, 2006 Loss Amount 0 (E) Seminaire 60 Montpellier 04/04/06 - p. 27/31 Facility Facility

49 RAROC max/ IC constraint Nominal Nom. ICLO Nom. SN IC VaR 0,1% RAROC Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90% Sen % 7% 2% P 3 2.6e9 9.e8 1.7e9 2.7e7 1.75e8 113% Evo. 27% 23% 29% 24% -8% 26% Sen % 3% 1% April 4, 2006 Seminaire Montpellier 04/04/06 - p. 28/31

50 RAROC max/ IC constraint Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Nominal (E) Nomina Probability l(e) (%) 12 x 107 Inner CDOs EL Facility 12 x 107 Inner CDOs Nominal (E) 12 x Single Names Facility Loss density 12 x Single names 107 VaR 10 Nomina l(e) April 4, 2006 Loss Amount 0 (E) Seminaire 60 Montpellier 04/04/06 - p. 28/31 Facility Facility

51 VaR min, IC max, RAROC max Nominal Nom. ICLO Nom. SN IC VaR 0,1% RAROC Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90% Sen % 7% 2% P 4 2.7e9 73e8 2.e9 2.5e7 1.4e8 96% Evo. 31% -16% 57% 17% -26% 7% Sen % 1% 2% April 4, 2006 Seminaire Montpellier 04/04/06 - p. 29/31

52 VaR min, IC max, RAROC max Part II: Hybrid Considered Portfolio Parameterization Cost function Computational time reduction Var min/ IC constraint IC max/ VaR constraint RAROC max/ IC constraint VaR min, IC max, RAROC max Nominal (E) Nominal Probability (E) (%) 12 x 107 Inner CDOs EL Facility 12 x 107 Inner CDOs Nominal (E) 12 x Single Names Facility Loss density 12 x Single Names 107 VaR 10 Nominal (E) April 4, 2006 Loss Amount 0 (E) Seminaire 60 Montpellier 04/04/06 - p. 29/31 Facility Facility

53 Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 30/31

54 Results are in adequacy with financial intuition. Part II: Hybrid April 4, 2006 Seminaire Montpellier 04/04/06 - p. 31/31

55 Part II: Hybrid Results are in adequacy with financial intuition. Hybrid + simplification techniques: New efficient and fast tool (To be compared with linear search s). April 4, 2006 Seminaire Montpellier 04/04/06 - p. 31/31

56 Part II: Hybrid Results are in adequacy with financial intuition. Hybrid + simplification techniques: New efficient and fast tool (To be compared with linear search s). Perspectives: Try with More complete models, extension to other kind of facilities (hedge)... April 4, 2006 Seminaire Montpellier 04/04/06 - p. 31/31

57 Part II: Hybrid Results are in adequacy with financial intuition. Hybrid + simplification techniques: New efficient and fast tool (To be compared with linear search s). Perspectives: Try with More complete models, extension to other kind of facilities (hedge)...!!! Thank You!!! April 4, 2006 Seminaire Montpellier 04/04/06 - p. 31/31