Transcript Document

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
n
EL  LGD kQk 
k 1
“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
x
n
k 1
k  x 1
FTDx   LGD kQk   xLGD  Qk 
Securitization deals
Senior Tranche
Originator
Junior 1 Tranche
Sale of
Assets
Special
Purpose
Vehicle
SPV
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 = 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.
Tranche
• 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
Kinds of tranches
• Equity tranche is defined as La = 0. Its value
is a put option on tranches.
v(t,T)EQ[max(Lb – L,0)]
• A senior tranche with attachment La 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)]
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(La, Lb) = EL(0, Lb) – EL(0,La)
• Buyng an equity tranche with detachment La and
buyng the corresponding senior tranche (attachment
La) 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)]
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 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.
Synthetic CDOs
Senior Tranche
Originator
Junior 1 Tranche
Protection
Sale
CDS Premia
Interest
Payments
Collateral
AAA
Special
Purpose
Vehicle
SPV
Investment
Junior 2 Tranche
… Tranche
Equity Tranche
CDO2
Originator
Senior Tranche
Tranche 1,j
Junior 1 Tranche
Tranche 2,j
Tranche i,j
Tranche …
Special
Purpose
Vehicle
SPV
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 27th 2005
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.
Gaussian copula and
implied correlation
•
•
•
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 i is 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!!!)
Monte Carlo simulation
Gaussian Copula
1. Cholesky decomposition A of the correlation
matrix R
2. Simulate a set of n independent random variables
z = (z1,..., zn)’ from N(0,1), with N standard
normal
3. Set x = Az
4. Determine ui = N(xi) with i = 1,2,...,n
5. (y1,...,yn)’ =[F1-1(u1),...,Fn-1(un)] where Fi denotes
the i-th marginal distribution.
Monte Carlo simulation
Student t Copula
1. Cholesky decomposition A of the correlation
matrix R
2. Simulate a set of n independent random variables
z = (z1,..., zn)’ from 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)1/2y
6. Determine ui = Tv(xi) with Tv the Student t
distribution
7. (y1,...,yn)’ =[F1-1(u1),...,Fn-1(un)] where Fi denotes
the i-th marginal distribution.
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
 N 1 u   m 

PrDefaultM  m  N 
2


1




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 Ld:
 1   2 N 1 L   N 1  p  
d

PrL  Ld   N 
2





Vasicek density function
16
14
12
10
Rho = 0.2
Rho = 0.6
Rho = 0.8
8
6
4
2
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Vasicek model
• 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.