Managing Synthetic CDO Tranches using Base Correlations

Managing synthetic CDO tranches
using base correlations
Robert Stamicar

Agenda
Synthetic CDO mechanics
Base correlations under Gaussian copula model
Stress tests
Mapping bespoke tranches to standard index tranches
Interpolation / extrapolation of base correlations
3

Synthetic CDO mechanics
losses
Protection buyer pays spread for protection against a portion of portfolio
Typically, premium is paid as a spread on remaining notional over deal’s life
Underlying portfolio is a portfolio of single-name CDSs
Correlation is a key factor for pricing CDOs
Correlation does not affect the portfolio expected loss,
But redistributes losses around the capital structure
4

Synthetic CDO Mechanics
Protection buyer
Protection Seller
Dealer
Tranche spread
Loss payments
CDO
Tranche
5

Protection buyer can hedge by selling individual
protection
Dealer sells
protection
Protection buyer
Protection Seller
Name 1
Name 2
.
.
.
Dealer
Tranche spread
Loss payments
CDO
Tranche
b
a
Name N
Credit
Collateral Pool
6

Tranche Pricing (standard version)
CDS price: deterministic discount factors, hazard rates
SCDO price: deterministic discount factors, hazard rates
What about loss distribution assumption
One-factor Gaussian copula is the market standard
7

Agenda
Synthetic CDO mechanics
Base correlations under Gaussian copula model
Stress tests
Mapping bespoke tranches to standard index tranches
Interpolation / extrapolation of base correlations
8

Gaussian Copula Model
Standard Model Assumptions: One-factor Gaussian copula model
F
i
( t)
= Pr[ τ
i
< t]
⇔ Φ(
Zi
) = F(
ti
)
Conceptual two name example
Z
i
= ρ Z + 1− ρε
i
2.5
Correlated normals
100
Correlated default times
2
1.5
1
0.5
t
t
1
2
=
=
F
F
−1
1
−1
2
( Φ(
z
( Φ(
z
1
2
))
))
90
80
70
60
0
50
-0.5
40
-1
30
-1.5
20
-2
10
-2.5
-3 -2 -1 0 1 2 3
0
0 5 10 15 20 25 30 35 40 45 50
9

Gaussian Copula Model (con’t)
Standard Model Assumptions: One-factor Gaussian copula model
F
i
( t)
= Pr[ τ
i
< t]
⇔ Φ(
Zi
) = F(
ti
)
Two name example: condition on the market factor
Z
i
= ρ Z + 1− ρε
i
X
1
⎧1
= ⎨
⎩0
prob p ( z)
1
prob (1 − p ( z))
1
X
2
⎧1
= ⎨
⎩0
prob p
2
( z)
prob (1 − p
2
( z))
Loss Distribution
⎧0
⎪
l1
L = ⎨
⎪l2
⎪
⎩l1
+ l
2
(1 − p ( z))(1
− p
p ( z)(1
− p
1
2
p ( z)
p
1
1
2
2
1
( z)
2
( z))
p ( z)(1
− p ( z))
( z))
Tranche Pricing
Derived from portfolio loss distribution
⎧0
L < a
⎪
L T
= ⎨L
− a a < L < b
⎪
⎩b
− a b < L
Many names …
Use FFT (fast Fourier transforms) or recursive method (Andersen, Sidenius, Basu (2003)) to
compute probabilities
10

Compound correlations are problematic
SCDO price is a function of:
Asset correlation
CDS spreads
Maturity
Recovery rates
Compound correlation: Asset correlation inferred from SCDO price
Multiple solutions for mezzanine tranches
How do you price a tranche with non-standard attachment/detachment points
11

Base correlation framework
Each tranche is decomposed into two “virtual” equity tranches
Incorporate entire capital structure
V
= V
( s,
ρb)
−V0,
( s,
ρ )
a, b 0, b
a a
Why
Some analogy with pricing equity options with multiple strikes
Consistency across a fixed maturity (inconsistent across different tenors)
More importantly, empirical evidence suggests that base correlations provide better
sensitivities than compound correlations.
From standard index tranches we can bootstrap base correlations
12

Agenda
Synthetic CDO mechanics
Base correlations under Gaussian copula model
Stress tests
Mapping bespoke tranches to standard index tranches
Interpolation / extrapolation of base correlations
13

Stress tests
Base correlations can be stressed directly or indirectly
Indirect stresses involve market observables:
Upfront fee of equity and junior tranches
Tranche fair spreads
Stresses should propagate up the capital structure
Each stress shift involving market quotes is translated into a base correlation shift
Example:
0-3% Upfront fee increases bootstrap: calculate ρ 3
’
3-7% Tranche fair spread increases bootstrap: calculate ρ 7
’(use ρ 3
’)
7-10% No stress is applied bootstrap: ρ 10
’(use ρ 7
’)
14

Recent market volatility: US subprime crisis
In late February, credit problems emerged in US subprime mortgages
6000
ABX.HE.07-1 BBB spread
BBB Spread
5000
4000
3000
Price deterioration:
June 1: 75%
July 31: 40%
2000
1000
0
Jan07 Mar07 May07 Jul07 Sep07
15

Subprime crisis has spread to …
Leveraged loans and credit markets
CDX.NA.IG S8 index, 0-3% upfront fee
90
0.7
Index (bp)
80
70
60
50
40
30
20
10
0.6
0.5
0.4
0.3
0.2
0.1
Upfront fee
Date Index Upfront fee
June 1 34 bp 23%
July 31 76 bp 48%
Aug 3 81 bp 47%
0
Nov06 Jan07 Mar07 May07 Jul07 Sep07
0
16

Implied values: Base correlation of equity tranche
CDX.NA.IG S8 index, 3% base correlation
90
0.3
Index (bp)
80
70
60
50
40
30
0.25
0.2
0.15
0.1
Base correlation
Date Index Correlation
June 1 34 bp 14%
July 31 76 bp 20%
Aug 3 81 bp 26%
20
10
0.05
0
Nov06 Jan07 Mar07 May07 Jul07 Sep07
0
17

Stress Test Example for CDX.NA.IG S8
July 25-26, 2007
Market observables (relative movements)
CDX spread: +23%
Upfront fee: +15%
Implied movement of base correlation
0-3% tranche: +17%
Recall a decrease in equity price can be caused by
Explained portion: Increase CDX spread (increase in default probabilities)
Unexplained portion: Decrease in correlation (“sporadic” defaults will increase)
18

Apply stress test
Assume stress applies to a “calm” market environment
Explicit base correlation shift
Base correlation
stress increases price
Positions
Tranche
notional ($M)
Upfront
price ($M)
DV01
($000)
Index
($000)
Base correl
($000)
Market
spreads ($000)
Sell 0-3% 3 0.73 -29 -217 64 -141
Sell 3-7% 4 0.00 -10 -82 -2 -78
Index 50 0.00 -22 -183 0 -183
Implicit base correlation shift
Positions
Tranche
notional ($M)
Upfront
price ($M)
DV01
($000)
Index
($000)
Base correl
$000)
Market
spreads ($000)
Sell 0-3% 3 0.73 -29 -217 -108 -349
Sell 3-7% 4 0.00 -10 -82 -63 -188
Index 50 0.00 -22 -183 0 -183
19

So which stress test makes sense
Is the base correlation moving in the wrong direction
Base correlation explains the unexpected movement in prices
In the actual market scenario (July 25-26), the Gaussian copula model
overestimates the losses
Adjust model by increasing correlation
In fact, during this period of systemic risk, investors sold equity protection and
bought index protection.
This pushed base correlations up
Hedging contributed to increased volatility (and accentuated spread widening)
Hedge ratios broke down in high volatility environment
Models used by dealers underestimated tranche deltas. Dealers rebalanced hedge
ratios by buying index protection (to hedge bought mezzanine protection)
CPDO (constant proportion debt obligations): as index spread widened, dealers
needed to buy protection
20

Base correlation as a risk factor
Volatility versus price level
iTraxx Europe (Jan-05 to Sept-07)
Vol from absolute difference
Vol from relative difference
0.025
0.1
0.09
0.02
0.08
0.07
0.015
0.06
0.01
0.005
0.05
0.04
0.03
0.02
0
0.1 0.15 0.2 0.25
0.01
0.1 0.15 0.2 0.25
21

Agenda
Synthetic CDO mechanics
Base correlations under Gaussian copula model
Stress tests
Mapping bespoke tranches to standard index tranches
Interpolation / extrapolation of base correlations
22

Pricing bespoke tranches under the base correlation
framework
What’s a bespoke tranche
Non-standard custom CDO tranche
The investor chooses:
Reference portfolio
Attachment/detachment points
Maturity
Other details
Term used to distinguish SCDO tranches from (liquid) index tranches
Typically, refers to a different reference portfolio
It’s illiquid
Can we use standard tranche indices to price and capture risk for bespokes
Standard index tranches:
CDX 0-3%, 3-7%, 7-10%, 10-15%, 15-30%
iTraxx 0-3%, 3-6%, 6-9%, 9-12%, 12-22%
23

Mapping base correlations between bespoke and
index portfolios
Given a base correlation surface ρ Ι
(X,T), can we determine ρ Β
(X,T)
Idea is to find an equivalent base tranche on a standard index with strike X I
X
a
B
X I
This mapping gives the bespoke base correlations:
24

Mapping base correlations:
Mechanics after “equivalent” strike is determined
Consider a risky bespoke portfolio:
Bespoke portfolio
Index portfolio
b
ρ I
a
b’
a’
a’ b’
ρ ( b)
= ρ ( b')
B
ρ ( a)
= ρ ( a')
B
I
I
25

Base correlation mapping methods
Finding an “equivalent” strike
Method 1: Degenerate Case
X = ρ ( X , T)
= ρ ( X , T)
I
X B
B
I
Example
20%
10% 10%
Bespoke
Standard
RR = 0% RR = 50%
26

Mappings
Method 2: ATM mapping
Adjusts for recovery rates
Two tranches are equivalent if their strikes lie in the same region of the capital structure with respect to E[PL]
But dispersion is not captured
Method 3: Probability Matching
Two tranches are priced with the same correlation if they have the same probability of being wiped out
Problem: Discrete losses need to be smooth cumulative loss distribution
27

Method 4: Expected Tranche Loss Proportion
Adjusting loss distribution implied by one-factor Guassian copula
Method 5: Spread Matching
Assume bespoke and index equivalent have the same spread
28

How do the mappings compare
Consider the following test case (from Baheti and Morgan(2007)):
Bespoke: iTraxx S6
Index: CDX IG7
29

Comparison of mappings for iTraxx S6:
January 31, 2007
Source: Lehman Brothers
30

Base correlation skew for iTraxx S6 5Y
Source: Lehman Brothers
31

Heterogeneous portfolios
Can we find a suitable index portfolio
Last example worked well since both the bespoke and index portfolios were
fairly similar (both contained investment grade names)
What about dispersion Suppose bespoke portfolio contains high yield and
investment credits
Map to multiple indices
Find greatest overlap in names across appropriate indices
Extend one-factor Gaussian copula model so that we can map a bespoke portfolio
to multiple index portfolios
32

Interpolation and extrapolation of base correlations
Price is extremely sensitive to the slope of the base correlation curve
Abrupt change in slope can lead to arbitrage violations
How should we interpolate to maintain no-arbitrage framework
How should we extrapolate
Base correlation curve is monotonically increasing
Equity tranches are riskier than constant correlation predicts
33

Consider linear interpolation
iTraxx 18-Sep-2007
1
0.9
0.8
0.7
1200
1000
800
Fair spreads of tranchlets
Base Correlation
0.6
0.5
0.4
Fair spread (bp)
600
400
Arbitrage
Negative spreads
0.3
0.2
200
0.1
0
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Strike
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Strike
Increase X, fair spread decreases
Low gradient, fair spread increases
34

Instead, interpolate expected tranche loss
Parcel and Wood (2007):
Easier to formulate no-arbitrage conditions with expected tranche loss
ETL(
X,
ρX
) = ∫ Pr( L > s)
ds
0
∂ETL(
X,
ρX
)
Pr( L > x)
=
> 0
∂X
2
∂ ETL(
X,
ρX
)
fL(
x)
= −
2
∂X
X
Eliminates negative spreads
Eliminates fair spreads
increasing
Turn extrapolation of base correlations into an interpolation exercise
Expected tranche loss of zero strike is zero
Expected tranche loss of 100% detachment point is equal to the expected portfolio
loss
35

Summary
Base correlation are useful
Fairly straightforward to implement
Entire capital structure is incorporated (for the same maturity)
Stress tests
Provide flexibility by allowing shifts of market observables
Fair spreads, upfront fees
Expected tranche loss is useful:
Pricing of bespoke tranches
Interpolation / extrapolation of base correlations
References
Baheti and Morgan (2007): Base correlation mapping, Lehman Brothers
Parcell and Wood (2007): Wiping the smile off your base (correlation curve),
Derivatives Fitch
36