Securitization and Copula Functions

Advanced Methods of Risk Management Umberto Cherubini

Learning Objectives

• In this lecture you will learn 1. To evaluate basket credit derivatives using Marshall-Olkin distributions and copula functions. 2. To analyze and evaluate securitization deals and tranches 3. To evaluate the risk of tranches and design hedges

Portfolios of exposures

• 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

EL



LGD k n

  1

kQ

 

“First-to-default” derivatives

• • 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-x-to-default” derivatives

• 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

FTD

  

LGD k x

  1

kQ

  

xLGD k n

 

x

 1

Q

Originator Sale of Assets

Securitization deals

Special Purpose Vehicle SPV Senior Tranche Junior 1 Tranche Junior 2 Tranche … Tranche Equity Tranche

The economic rationale

• 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

Structuring securitization deals

• 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)

Choice of assets

• 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

=

L

1 +

L

2 +…

L n

• Obviously the choice of the kinds of assets, and their dependence structure, would have a deep impact on the probability distribution of losses.

Tranche

• A

tranche

is a bond issued by a SPV, absorbing losses higher than a level

L a

(

attachment

) and exausting principal when losses reach level

L b

(

detachment

).

• The nominal value of a tranche (size) is the difference between

L b

and

L a

.

Size = L b

–

L a

Kinds of

tranches

•

Equity tranche

is defined as

L a

is a put option on tranches.

= 0. Its value

v

(

t

,

T

)

E Q

[max(

L b

–

L

,0)] • A

senior tranche

losses beyond

L a

with

attachment L a

pool, 100. Its value is then absorbs up to the value of the entire

v

(

t

,

T

)(100 –

L a

) –

v

(

t

,

T

)

E Q

[max(

L

–

L a

,0)]

Arbitrage relationships

• 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

(

L a

,

L b

) =

EL

(0,

L b

) –

EL

(0,

L a

) • Buyng an equity tranche with

detachment L a

and buyng the corresponding senior tranche (

attachment L a

) amounts to buy exposure to the overall pool of losses.

v

(

t

,

T

)

E Q

[max(

L a v

(

t

,

T

)(100 – –

L

,0)] +

L a

) –

v

(

t

,

T

)

E Q

[max(

L v

(

t

,

T

)[100 –

E Q

(

L

)] –

L a

,0)] =

Risk of different “

tranches

”

• 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.

Securitization zoology

• 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

assets or exchange traded CDO on standardized terms • CDO 2 CDO vs standard CDO: CDO on a customized pool of : 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.

Synthetic CDOs

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

CDO

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

Standardized CDOs

• 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)

i-Traxx and CDX quotes, 5 year, September 27 th 2005 Tranche 0-3% 3-6% 6-9% 9-12% i-Traxx Bid

23.5* 71 19 8.5

Ask

24.5* 73 22 10.5

Tranche 0-3% 3-7% 7-10% 10-15% 12-22%

4.5

5.5

15-30%

(*) Amount to be paid “up-front” plus 500 bp on a running basis Source: Lehman Brothers,

Correlation Monitor

, September 28 th 2005.

CDX Bid

44.5* 113 25 13 4.5

Ask

45* 117 30 16 5.5

Gaussian copula and implied correlation

• • • The standard technique used in the market is based on Gaussian copula C(u where u i i-th name.

1 , u 2 ,…, u N ) = N(N – 1 (u 1 is the probability of event ), N  i  – 1 (u 2 ), …, N – 1 T and  i (u N );  ) is the default time of the 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!!!)

Monte Carlo simulation Gaussian Copula

1. Cholesky decomposition

A

of the correlation matrix

R

2. Simulate a set of

n z =

(

z 1 ,..., z n

)

’

normal independent random variables from

N

(

0,1

), with

N

standard 3. Set

x = Az

4. Determine

u i = N

(

x i

) with

i = 1,2,...,n

5. (

y 1 ,...,y n

)

’ =

[

F 1 -1 (u 1 ),...,F n -1 (u n )

] the

i

-th marginal distribution.

where

F i

denotes

Monte Carlo simulation Student t Copula

1. Cholesky decomposition

A

matrix

R

of the correlation 2. Simulate a set of

n z =

(

z 1 ,..., z n

)

’

from independent random variables

N

(

0,1

), with

N

standard normal 3. Simulate a random variable

s

from 

2

 indipendent from

z

4. Set

x = Az

5. Set

x = (



/s) 1/2 y

6. Determine

u i

distribution

= T v

(

x i

) with

T v

the Student t 7. (

y 1 ,...,y n

)

’ =

[

F 1 -1 (u 1 ),...,F n -1 (u n )

] where

F i

the

i

-th marginal distribution.

denotes

Base correlation

Correlation 0%

Default Probability

Correlation 20% Correlation 95% MC simulation pn a basket of 100 names

Example of iTraxx quote

Tranche hedging

• 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.

Large CDO

• 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 .

Gaussian factor model (Basel II)

• Assume a model in which there is a single factor driving all losses. The dependence structure is gaussian. In terms of conditional probabilility Pr 

Default M



m

 

N

 

N

 1 1      2 

m

  where M is the common factor and m is a particular scenario of it.

Vasicek model

• 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 L d : Pr 

L



L d

 

N

  1   2

N

 1    2

d



N

 1   

16 14 4 2 0 0 12 10 8 6

Vasicek density function

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Rho = 0.2

Rho = 0.6

Rho = 0.8

Vasicek model

• The mean value of the distribution is

p

, the value of default probability of each individual • Value of equity tranche with detachment L d is Equity(L d ) = (L d – N(N -1 (p); N -1 (L d );sqr(1 –  2 )) • Value of the senior tranche with attachment equal to L d is Senior(L d ) = (p – N(N -1 (p); N -1 (L d );sqr(1 –  2 )) where N(N -1 (u); N -1 (v);  2 ) is the gaussian copula.