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Structured Credit Solutions
February 28, 2008
Robert Stamicar, Research - Methodology
Marianela Hoz de Vila, Account Management – Practical Application
Denny Yu, Account Management - Moderator
Measure Risk of Advanced Structured Credit Trading Strategies
www.riskmetrics.com 2Webcast Series
Agenda
Credit indices – CDX & iTraxxCredit Default Swaps – description and pricing framework
Credit Default Indices – description and pricing framework
Spread decomposition – theoretical fair spread plus index basis
Application to risk management
Analyzing synthetic CDO tranches using base correlationsStress tests using base correlationsCorrelation as a risk factorMapping bespoke tranches to standard index tranches
www.riskmetrics.com 3Webcast Series
Credit index modelWhat are we trying to achieve?
Pre-credit index approaches
Credit index instrument
CDS
(single-name model)
Index - No idiosyncratic risk
- No single name stresses
- Short histories for new series
S-CDO (0-100%)
(granular model)
Constituent CDS - Have history but index is ignored
- Calibration not handled
Key Risk Factors Comments
Single-name model Index -“Synthetic” series handles short
history
-Systemic risk
Basket model Constituent CDS +
index basis
- Calibration now possible
- Idiosyncratic risk captured
Synthetic historical series
www.riskmetrics.com 4Webcast Series
Agenda
Credit indices – CDX & iTraxxCredit Default Swaps – description and pricing framework
Credit Default Indices – description and pricing framework
Spread decomposition – theoretical fair spread plus index basis
Application to risk management
Analyzing synthetic CDO tranches using base correlationsStress tests using base correlationsCorrelation as a risk factorMapping bespoke tranches to standard index tranches
www.riskmetrics.com 5Webcast Series
Credit Default Swaps (CDS)
CDS - Contract referencing bond(s) issued by a reference entity betweenProtection buyer – pays a periodic premium (spread)
Protection seller – pays a loss amount if the reference entity defaults
Spread conventionsTypically set at outset such that entry MTM is zero (all running premium).
For distressed credit, running premium is fixed, and the buyer pays an upfront amount that varies according to the market.
MaturityFive years the most liquid, but most maturities from one to ten years trade.
Actual maturity is fixed according to quarterly ISDA dates (20th of Mar, Jun, Sep, Dec)
So maturity of “five-year” CDS ranges from 5 to 5.25 years.
Buyer Sellers
Reference
XYZ
1-Recovery
Buyer Sellers
Reference
XYZ
1-Recovery
www.riskmetrics.com 6Webcast Series
Pricing of CDS
CDS pricing: deterministic discount factors, hazard rates
Fair spread: the premium that makes the contract MTM zero
S – survival probability (=1-F) RC – recovery claim
f – default density (=d F/d t)
www.riskmetrics.com 7Webcast Series
Calibrating to observed prices
Default probabilities are related to piecewise constant hazard rates through
Common practice is to fit hazards to the spread curve for a given reference
entity using successively longer maturity CDS:Δtk are maturities of observed CDS premia
Find h1 to match fair spread at Δt1Given h1… hk, find hk+1 to match fair spread at Δtk+1
www.riskmetrics.com 8Webcast Series
Credit default indices (CDX)
Various mergers have produced standard index families: CDX (North
America) and iTraxx (Europe and Asia).
Credit index: Contract references a standard basket of n reference entitiesProtection buyer – pays a fixed running premium on the remaining notional
Protection seller – pays a loss amount when a reference entity defaults
Equally weighted
On a default eventNotional amount is reduced by 1/nProtection seller pays loss on a notional amount of 1/nContract continues on remaining basket until maturity
www.riskmetrics.com 9Webcast Series
Credit default index mechanics
Contracts are standardized: on a single contract, all participants tradeSame basket of names
Same maturity date
Same running premium
Every six months (ISDA dates of Mar and Sep), the contract “rolls”New basket defined (5-10% turnover is typical)
New maturity date, new (maybe) running premium
Investors may remain in previous contract (fixed basket), or roll into new on-the-run contract (fixed duration)
Roll schedule implies that maturity of “five-year” index ranges from 4.75 to
5.25 years.
Trade on a price basisRunning premium is fixed
Upfront amount paid by either the buyer or seller to enter the contract.
www.riskmetrics.com 10Webcast Series
Pricing of credit index (as a basket)
Credit Index Pricing: n reference credits
Observe…If all the default probabilities are equal, this reduces to the single name case.
This is not a correlation product! Only marginal default distributions enter.
Can define the theoretical fair spread (S*) as the ratio of these two terms –
the spread that would make the MTM equal zero.
www.riskmetrics.com 11Webcast Series
Risk measurementsWhat are the problems?
Limited history – how do we capture risk for newly issued credit indices?
20−Dec−2004 20−Mar−2005 20−Jun−2005 20−Sep−2005 20−Dec−200535
40
45
50
55
60
65
70
75
80
Date
Spr
ead
(bp)
Series 3Series 4Series 5
Previous series?
On-the-run spreads?
Theoretical spreads?
Theoretical spreads plus basis?
www.riskmetrics.com 12Webcast Series
What are the problems (cont’d)
Previous series / on-the-run spreadsEasy to use - single-name model with one time series – but …
Constituents are ignored (typical turnover is 5-10%)Difficult to stress names in hedged positions involving tranched CDOs
Risk numbers can be inaccurate – especially, when volatile names drop for a new issuance
Theoretical basket plus basisConstituent breakdown
Is the basis between the index spread and theoretical spread relevant?Basis volatility is significant
Correlation between basis and theoretical fair spread is low
www.riskmetrics.com 13Webcast Series
Spread decomposition – the index basis
Decomposition of (quoted) spread
This decomposition allows us to create synthetic indices for short historiesEnsures that constituents are “fixed” (S* derived strictly from constituent CDS data)
Use on-the-run basis
Is S* good enough? Do we require the basis as a risk factor?
Theoretical fair spread
Basis: defined as the difference between the observed and theoretical fair spreads
www.riskmetrics.com 14Webcast Series
Decomposition - Explaining the theoretical fair spread S*
Buying protection on the index is close to buying equal protection on each name, which would cost exactly SAVG , but S*≠SAVG
The biggest factor is heterogeneity. Consider the first default.Same loss payment, same notional reduction.Index buyer pays lower premium by index spread times notional reduction.Equal protection buyer pays lower premium by spread on defaulted name times notional reduction.
Larger spreads are those most likely to default.So equal protection buyer likely has a greater premium reduction than the index protection buyer.So index protection buyer should pay less at outset … S*<SAVG
The difference between the two can be interpreted as an indicator of heterogeneity
www.riskmetrics.com 15Webcast Series
Why the basis?
Different quoting convention
Documentation differences (definition of default),
Maturity mismatches due to roll schedule and ISDA dates.
Different supply and demand effects,
Liquidity differences,
Asynchronous observations,
Preference for upfront (sure) versus running (risky) premiumHigher spreads … greater upfront payment to protection seller
Seller prefers upfront, so is willing to discount versus theoretical spread
Pushes observed spread lower … pushes basis lower (or more negative)
www.riskmetrics.com 16Webcast Series
NAIG Series 3 spread decompositionBehavior of basis and nonlinearities in F/GM and Delphi events
20-Mar-2005 20-Jun-2005 20-Sep-2005 20-Dec-2005 20-Mar-2006 20-Jun-2006
30
40
50
60
70
80
90
Date
Spr
ead
(bp)
AverageNon LinearitiesBasisObserved
Ford and GMdowngrades
Delphidefault
Decrease ofdemand(old Series)
www.riskmetrics.com 17Webcast Series
CDX.NAHY Series 4Less influence from default events
www.riskmetrics.com 18Webcast Series
NAIG realized volatilitiesHow to estimate Series 5 risk on issuance date?
Series 3 includes F, GM, Delphi. Series 4 includes F, GM, not Delphi. Series 5 includes none.
20-Jun-2005 20-Sep-2005 20-Dec-2005 20-Mar-2006 20-Jun-2006
6
8
10
12
14
16
18
20
22
24
26
Date
Rea
lized
Vol
atilit
y ov
er 6
0 D
ays
(bp)
Series 3Series 4On-the-runSynthetic Series 5
Series 5
www.riskmetrics.com 19Webcast Series
Agenda
Credit indices – CDX & iTraxxCredit Default Swaps – description and pricing framework
Credit Default Indices – description and pricing framework
Spread decomposition – theoretical fair spread plus index basis
Application to risk management
Analyzing synthetic CDO tranches using base correlationsStress tests using base correlationsCorrelation as a risk factorMapping bespoke tranches to standard index tranches
www.riskmetrics.com 20Webcast Series
Synthetic CDO mechanics
Protection buyer pays spread for protection against a portion of portfolio
losses
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 CDOsCorrelation does not affect the portfolio expected loss,
But redistributes losses around the capital structure
www.riskmetrics.com 21Webcast Series
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
www.riskmetrics.com 22Webcast Series
Compound correlations are problematic
SCDO price is a function of:Asset correlation
CDS spreads
Maturity
Recovery rates
Compound correlation: Asset correlation inferred from SCDO priceMultiple solutions for mezzanine tranches
How do you price a tranche with non-standard attachment/detachment points?
www.riskmetrics.com 23Webcast Series
Base correlation framework
Each tranche is decomposed into two “virtual” equity tranchesIncorporate entire capital structure
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
),(),( ,0,0, aabbba sVsVV ρρ −=
www.riskmetrics.com 24Webcast Series
Stress tests
Base correlations can be stressed directly or indirectlyShift risk factors – user defined stress
Shift model parameters – generalized stress test
Indirect stresses involve market observables:Upfront fee of equity and junior tranches
Tranche fair spreads
Stresses should propagate up the capital structureEach stress shift involving market quotes is translated into a base correlation shift
Example:
0-3% Upfront fee increases bootstrap: calculate ρ3’
3-7% Tranche fair spread increases bootstrap: calculate ρ7’ (use ρ3’)
7-10% No stress is applied bootstrap: use ρ7’ (ρ10’ unaltered)
www.riskmetrics.com 25Webcast Series
Stress test mechanics for model parameters
We can derive (bootstrap) market quotes in various ways
0-3% p1 s1 P1, sd1 pc1
3-7% p2 s2 P2, sd2 pc2
Tranche 1
Tranche 2
A/D Base Corr Fair spreadUpfront / deal spread Comp corr
0-3% p1 s1 P1, sd1 pc1
3-7% p2 s2 P2, sd2 pc2
Stress Test
Bootstrap base correlations when market observables are shifted
Calibrated parameters
www.riskmetrics.com 26Webcast Series
Base correlation as a risk factorVolatility versus price level
iTraxx Europe (Jan-05 to Sept-07)
0.1 0.15 0.2 0.250
0.005
0.01
0.015
0.02
0.025
0.1 0.15 0.2 0.250.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Vol from absolute difference Vol from relative difference
www.riskmetrics.com 27Webcast Series
Pricing bespoke tranches under the base correlation framework
What’s a bespoke tranche?Non-standard custom CDO trancheThe investor chooses:
Reference portfolioAttachment/detachment pointsMaturityOther details
Term used to distinguish SCDO tranches from (liquid) index tranchesTypically, refers to a different reference portfolio
It’s illiquidCan 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%
www.riskmetrics.com 28Webcast Series
Mapping base correlations between bespoke and index portfolios
How do we price a bespoke tranche?Equivalently, how do we determine base correlations for a bespoke?
Given a base correlation surface ρΙ(X,T), can we determine ρΒ(X,T)?
Idea is to find an equivalent equity tranche on a standard index with strike XI
This mapping gives the bespoke base correlations:
www.riskmetrics.com 29Webcast Series
Mapping base correlations:Mechanics after “equivalent” strike is determined
Consider a risky bespoke portfolio:
Bespoke portfolio Index portfolio
a
b
a’
b’
Iρ
a’ b’
)'()()'()(
aabb
IB
IB
ρρρρ
==
Index base correlations
www.riskmetrics.com 30Webcast Series
Two mapping options are available
User-defined mappingUser explicitly enters the equivalent attachment/detachment points from the index portfolio
Expected tranche loss mappingEquivalent strikes between the bespoke and index portfolios are determined via an expected tranche loss calculation
Adjusting loss distribution implied by one-factor Gaussian copula
www.riskmetrics.com 31Webcast Series
Summary
Credit indices – Basis as a risk factorThe basis matters, especially if you are trading it.
The basket constituents matter, even for large, safe indices.
A proxy based on synthetic histories can pick up both effects.
Synthetic CDOs – base correlation frameworkBase correlation are fairly straightforward to implement
Entire capital structure is incorporated (for the same maturity)
Stress testsProvide flexibility by allowing shifts of market observables
Fair spreads, upfront fees
Expected tranche loss is useful:Pricing of bespoke tranches
Future work: Interpolation / extrapolation of base correlations
www.riskmetrics.com 33Webcast Series
Agenda
Analytic enhancementsCDS IndexSynthetic CDO
“Base Correlation Curve Spec”“Bespoke Tranche Mapping Method”“Zero Coupon Synthetic CDO”
Stress Test Base CorrelationsCredit Spread Market Observables
Fair Spread; Upfront Price
Reports
www.riskmetrics.com 34Webcast Series
CDS Index - Overview
One input : CDS Index Spread Time Series
www.riskmetrics.com 35Webcast Series
Synthetic CDO - Overview
Constituent SourceOne input : Index Spread Time Series
Issuer List: 125 names
Capital Structure
www.riskmetrics.com 36Webcast Series
Synthetic CDO - Overview
Base Correlation Time series
Bespoke Tranches
Zero Coupon Synth CDO
Market Tranche List
www.riskmetrics.com 37Webcast Series
Stress Test – Index Basis
CDS Spread Curve Shift
CDS Index Basis Shift
www.riskmetrics.com 38Webcast Series
Stress Test – Base Correlation
Tranche Base Correlation Shift (Explicit)
CDO Base Correlation Shift (Explicit)
Fair Spread Shift (Implicit)
Compound Correlation Shift (Implicit)
Upfront Payment Shift (Implicit)
www.riskmetrics.com 39Webcast Series
Reports – CDX Index VaR
..........
www.riskmetrics.com 40Webcast Series
Reports – CDX Index Sensitivities
www.riskmetrics.com 41Webcast Series
Reports – Synth CDO VaR
Base Correlation Risk
Bespoke Tranche
www.riskmetrics.com 42Webcast Series
Reports – Synth CDO Sensitivities
Worst one-day move to CDX.NA.IG S8: July 25-26, 2007
CDX spread: +23%
Upfront fee: +15%
0-3% Base Correlation: +17%
Implicit Base Correlation Shift
Upfront Fee + Spread
Index is linked to all constituents
www.riskmetrics.com 43Webcast Series
SummaryC
HA
LL
EN
GE
•Risk management in structured credit
EX
PO
SUR
ES
•Systemic risk (index spreads)•Idiosyncratic risk (single name defaults)•Correlation risk (often driven by market observables)
Measure and Manage Risk•Index spreads•Single name spreads•Correlation (across the entire capital structure)•Volatility (single name spreads AND correlations as risk factors!)
TO
OL
S
www.riskmetrics.com 44Webcast Series
Questions
Please send any questions or comments to your Account Manager…