Nominal (E)Nominal (E)Probability (%)

(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

(2)

●Outlines

Part I: Problem Definition

Part II: Hybrid optimization method

Part III: Application to portfolio optimization

Conclusion and perspectives

Outlines

• Problem definition:

-Credit derivative product description

-Loss modeling + performance indicators

-Optimization problems

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●Outlines

Outlines

• Problem definition:

-Credit derivative product description

-Loss modeling + performance indicators -Optimization problems

• Optimization algorithm:

-Semi-deterministic algorithm -Genetic algorithm

-Hybdridation

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●Outlines

Outlines

• 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

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●Outlines

●Historical context

●Considered derivative credit product

●Objective of the work

●

CDO2

model: Data

●

CDO2

model: Loss evaluation

●Merton’s model

●KMV’s model

●Default times model

●Density function

βL

●Performance Indicators

●Risk measures

●Optimization problems

Part I: Problem Definition

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Historical context

• Object: derivative credit products:

-Development of those products (In 2004: 2300 M ¯ $)

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Historical context

• 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 method (Merton 1974...), Risk measure

(Rockafellar, Artzner 1998...)

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Historical context

• 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 method (Merton 1974...), Risk measure (Rockafellar, Artzner 1998...)

2-Profitability improvement

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Considered derivative credit product

Collateralized Debt Obligations (CDO)

Facility N

Senior

Tranche A

Tranche B Equity 3%

30%

37%

Facility 1 Facility 2

-Objective: buy securitization in order to protect from eventual

defaults.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Considered derivative credit product

CDO ² / Master CDO

JUNIOR

AA AA AA

6%

4%

90%

CDO1 CDO2 CDO3 MASTER CDO

TRANCHE A

TRANCHE C TRANCHE B

SENIOR Facility 1 Facility 2 Facility5 Facility3

Facility4 ABS 1

ABS 2

AA A

AAA BB

AAA AAA AAA

BBB BBB BBB

MEZZANINE

JUNIOR

SENIOR

S&P RATING

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Objective of the work

Find the nominal coefficients, using an optimization algorithm, that improve desired portfolio characteristics:

■ Reduction of Risk measure.

■ Augmentation of the income.

■ Augmentation of the profitability.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

CDO ² model: Data

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.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

CDO ² model: Data

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 = P N

i=1 Sp _i × N ⁱ

■ P N = P N

i=1 N _i

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

CDO ² model: Loss evaluation

In order to compute more complex indicator. We consider:

L the CLO ² ’s amount of losses: Random variable.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

CDO ² model: Loss evaluation

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

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Merton’s model

1974: Model of the firm:

The liabilities are divided :

■ Debt D

■ Equities E

■ Assets A

Linked by: A = D + E

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Merton’s model

( 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+σ ² /2)t σ √

t and d 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

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

KMV’s model

It’s factor model based on Merton’s model: A = D + E.

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, r ⁱ _t = ^A

i

t −A ⁱ _t−1

∆t , is given by:

r _t ⁱ = R _i

" ₁₄ X

k=1

α ⁱ _k F _t ^k + β ^C ⁱ C _t ⁱ + β ^I ⁱ I _t ⁱ

# +

q

1 − R ² i ε ⁱ _t

where ε ⁱ _t is the specific rate of return of firm i.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

KMV’s model

Monte-Carlo Algorithm:

■ Simulate realizations of the latent variables ¡

F _t ^k ¢

1≤k≤14 ,

¡ I _t ⁱ ¢

1≤i≤61 , ¡ C _t ⁱ ¢

1≤i≤45 , ¡ ε ⁱ _t ¢

1≤i≤N until horizon H

■ Deduce the rates of return of the portfolio’s assets

■ 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:

X

i def ault

LGD _i × N ⁱ

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

KMV’s model

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

KMV’s model

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.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Default times model

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 + q

1 − ρ ² _i .ǫ _i

where for i = 1, · · · , n , Y, ǫ _i , i = 1, · · · , n is a standard gaussian

random variable.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Default times model

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 )

=

Z +∞

−∞

Ã _n Y

i=1

Φ

Ã Φ ⁻ ¹ (F i (t i )) − ρ ⁱ .Y p 1 − ρ ² i

!!

e ⁻ ^t2 ²

√ 2π dt

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.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Default times model

Monte-Carlo approach to compute β _L :

■ Simulate KMV coefficients.

■ Compute default times.

■ Compute loss amount: P

i def ault LGD _i × N ⁱ . Case 1:

SENIOR

AA

SUPER SENIOR SUPER SENIOR

AA AA

EQUITY EQUITY EQUITY

SUPER SENIOR

CDO1 CDO2 CDO3 MASTER CDO

EQUITY TRANCHE A

TRANCHE C TRANCHE B

Case 2:

SENIOR

AA

SUPER SENIOR SUPER SENIOR

AA AA

SUPER SENIOR

TRANCHE A

TRANCHE C TRANCHE B

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Density function β _L

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Performance Indicators

◦ Expected Loss (EL):

The expected value, over some specified horizon, of portfolio losses due to default. Given by:

EL(β _L ) =

Z portf olio ^′ s nominal

0 β _L (x)

portf olio ^′ s nominal dx

◦ Risk Adjusted Return On Capital (RAROC):

Profitability. RAROC is given by:

RAROC = ( P

portf olio (Sp _i ) × x ⁱ ) − EL(β ^L )

EC

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Risk measures

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).

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Risk measures

VaR: The smallest nominal loss of the worst α % of losses:

V aR _α (L) = inf [L ^′ |

Z 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

X α

p=0

inf {x|P [L ≤ x] ≥ p}dp

CVaR is often preferred: In some case it’s coherent and convex

risk measure.

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●Outlines

●

CDO2

model: Data

●

CDO2

●Merton’s model

●KMV’s model

●Density function

βL

●Risk measures

Optimization problems

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.

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●Outlines

●Global Optimization and Dynamical system

●Recursive semi-deterministic methods

●Genetic algorithm

●Hybrid algorithm

Part II: Hybrid optimization method

(31)

●Outlines

●Hybrid algorithm

Global Optimization and Dynamical system

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.

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●Outlines

●Hybrid algorithm

Global Optimization and Dynamical system

Most deterministic minimization algorithms can be seen as discretizations of Dynamical Systems.

A numerical global optimization 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.

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●Outlines

●Hybrid algorithm

Recursive semi-deterministic methods

Remove the over determination:

 



 

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.

(34)

●Outlines

●Hybrid algorithm

Genetic algorithm

General overview:

(35)

●Outlines

●Hybrid algorithm

Genetic 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 )



 



(36)

●Outlines

●Hybrid algorithm

Genetic algorithm

■ Selection:

X ^i+1/3 = S ⁱ X ⁱ

■ Crossover:

X ^i+2/3 = C ⁱ X ^i+1/3

■ Mutation:

X ⁱ⁺¹ = X ^i+2/3 + E ⁱ

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●Outlines

●Hybrid algorithm

Genetic algorithm

The new population can be written as:

X ⁱ⁺¹ = C ⁱ S ⁱ X ⁱ + E ⁱ ⁽¹⁾

Thus genetics algorithms can be associated to:

X ˙ (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 }

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●Outlines

●Hybrid algorithm

Genetic algorithm

 



 

X(t) = Λ ˙ ₂ (t, X(t), p)X(t)Λ ₃ (t, X(t), p) − X(t) X(0) = X ⁰

J(X(Z)) ≈ J b ^m

Where:

- J(X) = min(J(x b _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

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●Outlines

●Hybrid 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 ) _J(o ^o i ⁱ )−J(v ^−v ⁱ ⁱ )

EndFor

• Output: A ₁ (X ⁰ , N, ǫ) = argmin {J(o ^k ), k = 1, ..., 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 optimization problem:

-Microfluidic device

-Optical fiber synthesis

-Academic test cases

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●Outlines

●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

Part III: Application to portfolio

optimization

(41)

●Outlines

●Parameterization

●Cost function

Considered Portfolio

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 %

(42)

●Outlines

●Parameterization

●Cost function

Parameterization

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.

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●Outlines

●Parameterization

●Cost function

Cost function

Optimization problem is of the form:

min _x∈Ω P 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 e

ncons X

j=1

(max(uc _j −C ^j (x), 0)+max(C _j (x)−lc ^j , 0))

(44)

●Outlines

●Parameterization

●Cost function

Computational time reduction

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 optimization: 2000 functional evaluations → 6H

computation.

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●Outlines

●Parameterization

●Cost function

Var min/ IC constraint

Nominal Nom. ICLO Nom. SN IC VaR _0,1% RAROC Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90%

Sen. - - - 3% 7% 2%

P 1 2e9 5.5e8 1.45e9 2.1e7 1.3e8 87%

Evo. 0% -26% 11% 0% -31% -3%

Sen. - - - 1% 1% 1%

(46)

●Outlines

●Parameterization

●Cost function

Var min/ IC constraint

10 20 30

0 2 4 6 8 10

12 x 10

⁷

Inner CDOs

Facility

Nominal (E)

20 40 60

0 2 4 6 8 10

12 x 10

⁷

Single−Names

Facility

Nominal (E)

.2 .3 .3

Loss density

EL VaR

6 8 10

12 x 10

⁷

Inner CDOs

6 8 10

12 x 10

⁷

Single−names

(47)

●Outlines

●Parameterization

●Cost function

IC max/ VaR constraint

Nominal Nom. ICLO Nom. SN IC VaR _0,1% RAROC Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90%

Sen. - - - 3% 7% 2%

P 2 3.2e9 7.5e8 2.7e9 3.e8 1.9e8 92%

Evo. 57% 6% 88% 28% 0% 2%

Sen. - - - 2% 1% 1%

(48)

●Outlines

●Parameterization

●Cost function

IC max/ VaR constraint

10 20 30

0 2 4 6 8 10

12 x 10

⁷

Inner CDOs

Facility

Nominal (E)

20 40 60

0 2 4 6 8 10

12 x 10

⁷

Single−Names

Facility

Nominal (E)

.2 .3 .3

Loss density

EL VaR

6 8 10

12 x 10

⁷

Inner CDOs

6 8 10

12 x 10

⁷

Single−names

(49)

●Outlines

●Parameterization

●Cost function

RAROC max/ IC constraint

Nominal Nom. ICLO Nom. SN IC VaR _0,1% RAROC Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90%

Sen. - - - 3% 7% 2%

P 3 2.6e9 9.e8 1.7e9 2.7e7 1.75e8 113%

Evo. 27% 23% 29% 24% -8% 26%

Sen. - - - 3% 3% 1%

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●Outlines

●Parameterization

●Cost function

RAROC max/ IC constraint

10 20 30

0 2 4 6 8 10

12 x 10

⁷

Inner CDOs

Facility

Nominal (E)

20 40 60

0 2 4 6 8 10

12 x 10

⁷

Single−Names

Facility

Nominal (E)

.2 .3 .3

Loss density

EL VaR

6 8 10

12 x 10

⁷

Inner CDOs

6 8 10

12 x 10

⁷

Single−names

(51)

●Outlines

●Parameterization

●Cost function

VaR min, IC max, RAROC max

Nominal Nom. ICLO Nom. SN IC VaR _0,1% RAROC Initial 2.e9 7.5e8 1.25e9 2.1e7 1.9e8 90%

Sen. - - - 3% 7% 2%

P 4 2.7e9 73e8 2.e9 2.5e7 1.4e8 96%

Evo. 31% -16% 57% 17% -26% 7%

Sen. - - - 5% 1% 2%

(52)

●Outlines

●Parameterization

●Cost function

VaR min, IC max, RAROC max

10 20 30

0 2 4 6 8 10

12 x 10

⁷

Inner CDOs

Facility

Nominal (E)

20 40 60

0 2 4 6 8 10

12 x 10

⁷

Single−Names

Facility

Nominal (E)

.2 .3 .3

Loss density

EL VaR

6 8 10

12 x 10

⁷

Inner CDOs

6 8 10

12 x 10

⁷

Single−Names

(53)

●Outlines

●Conclusion and perspectives

Conclusion and perspectives

(54)

●Outlines

Conclusion and perspectives

• Results are in adequacy with financial intuition.

(55)

●Outlines

Conclusion and perspectives

• Results are in adequacy with financial intuition.

• Hybrid method + simplification techniques: New efficient and

fast tool (To be compared with linear search methods).

(56)

●Outlines

Conclusion and perspectives

• Results are in adequacy with financial intuition.

• Hybrid method + simplification techniques: New efficient and fast tool (To be compared with linear search methods).

• Perspectives: Try with More complete models, extension to

other kind of facilities (hedge)...

(57)

●Outlines