Frontiers In Portfolio Credit Risk Computation

1
Frontiers In Portfolio Credit Risk Computation

• Andrew Abrahams
• Derivatives Research
• JPMorganChase
• Rice University CoEFS Conference
• November 8, 2002

2
Outline

• Business model
• Portfolio credit risk overview
• Active Management of Counter-party Exposure
• Counter-party exposure for derivatives
• Exposure definitions
• Mathematical problem
• Simulation Framework
• Relatedness
• The computational problem
• Unified measurement and management of portfolio
  credit risks

3
Overview of risks in a credit portfolio

• Products
• Traditional credit Loans, commitments etc
• OTC Derivatives with counter-party risk
• Measures
• Capital potential loss under extreme events
• Price of credit exposure, hedging costs
• Methodology
• Credit Portfolio simulation, generation of loss
  distributions
  
• Market simulation for derivative exposure
• Credit derivative pricing
• Convergence
• Relatedness credit quality related to exposure
• Realistic modeling of portfolio hedging

4
Derivatives credit business model

• Various markets business take part in derivatives
  transactions
  
• In most cases, these contracts have counter-party
  risk
  
• Business pays up-front credit charge and
  transfers risk to central unit
  
• Counter-party exposures are marked-to-market
• Central unit compensates businesses for losses
  when default occurs
  

Client A
Client B
Client C
Trading desk
Trading desk
Credit Mgmt
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Central credit management for derivatives

• Structuring
• Helps arrange credit friendly transactions
• Requires marginal analysis tools
• Manages collateral agreements as credit risk
  mitigants
  
• Risk management
• Hedges market risk on aggregate Portfolio
• Manages credit risk on single name and Portfolio
  basis
  
• Market hedges fund rebalance of credit hedges and
  visa-versa
  
• For some books, more active market/credit
  hedging
  

6
Definitions of exposure quantities

• Exposure
• What we could lose if counter-party defaulted
• A function of time, state of world
• Expected Exposure
• A time profile obtained by averaging exposure
  over states of the world
  
• Used in hedging, reserving, capital measurement
• Credit Valuation Adjustment (CVA)
• Default probability weighted expected exposure
• Market price of credit default swap protection
  with payout defined by Expected Exposure profile
  
• Peak Exposure
• Worst case (defined by particular confidence
  interval) loss
  
• Used for credit approval of particular
  transactions
  
• Diversified peak measures for concentration
  analysis
  

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Simple counter-party exposure example

• 11-year Interest rate swap, receive floating, pay
  fixed
  
• Exposure increases over time as rates are
  volatile but decreases as there are fewer
  cashflows remaining
  
• If Counter-party defaults at some date, bank
  could lose current (positive) MTM value
  
• Single trade can be treated fairly easily with
  closed-form or numerical integration methods
  

Fixed regular payment
Bank
Client
Regular payment tied to current libor
8
General strategy for single trade counter-party
exposure

• For more complex derivatives -- involving
  complicated payoffs and multiple underlying
  market factors, the simple IR swap example is
  extended in a straightforward fashion
• Given processes for underlying markets (including
  volatility), generate joint-distribution of
  market factors at the time horizon of interest
  
• Use this to compute distribution of forward MTM
  values and from that exposure statistics and
  moments
  

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Problem for a portfolio of trades

• What is the is the distribution of potential
  portfolio values (MTM) at a forward date?
  
• Potential exposure can be strongly dependent on
  whether we can net assets and liabilities (ITM
  and OTM positions) when the default occurs
  
• Distribution should be conditioned on default of
  counter-party, i.e. what does the information
  that the counter-party is defaulting at that date
  tell us about the distribution
• Exposure can also be mitigated by collateral
  usually called for on the basis of a MTM
  agreement
  
• To account for collateral we need to know
  portfolio MTM at time T conditional on default at
  time Tdt in state-of-world S

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Why is this difficult?

• Equivalent to pricing a credit-contingent
  multi-asset option
  
• Very high dimensionality with a particular
  counter-party we might deal in tens or even
  hundreds of underlying markets (IR, FX, equity,
  credit)
• With optionality in the portfolio, collateral
  agreements and path-dependent payoffs, the
  effective dimensionality is greatly increased.
  
• Relatedness calculations amount to a jump process
  (possibly in many underliers)
  
• Direct evaluation of integrals via simulation is
  preferable to uncontrolled approximations
  

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Netting impact on exposure

• Portfolio several hundred swaps, FRAs, swaptions
  USD,JPY,AUD
  
• Reduces average exposure from 60MM to 4.6MM
• Reduces maximum peak exposure (97.5 confidence)
  from 330MM to 60MM

12
Collateral impact on exposure

• Portfolio several hundred swaps, swaptions,USD,
  JPY, EUR, GBP
  
• Collateral agreement has zero threshold and daily
  MTM
  
• Reduces average exposure from 40MM to 0.6MM
• Reduces maximum peak exposure from 250MM to
  20MM
  

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Structure of simulation engine
Inputs
Outputs
Perturb market inputs
Transactions
Generate forward market scenarios (paths)
Exposure measures by counter-party
Price transactions (MTM) under simulated market
environments
Market data
Aggregate and apply portfolio effects
Market sensitivities of credit risk
Generate statistics of exposures
Portfolio, legal reference data
Determine market price of credit exposures
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Simulation overview I

• Evolve relevant market variables
• Multi-factor Heath-Jarrow-Morton model for
  interest rates and credit spreads
  
• Lognormal diffusion for equity and FX with
  volatility term structure
  
• Implied volatilities where there are liquid
  markets
  
• Risk neutral drifts
• Generate portfolio (possibly path-dependent)
• Option exercise, termination agreements
• Condition rates on default of counter-party
  (relatedness)
  
• Calculate MTMs for every trade on every path and
  exposure date
  
• Uses exact pricing models where feasible in
  other cases approximations are built
  

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Simulation overview II

• Counter-party level portfolio effects
• Netting applied where legally enforceable
• trades of different types with different
  underlying market dependencies can be netted in
  the simulation framework
  
• Collateral
• Simulates unilateral and bilateral agreements
• collateral thresholds (including rating dependent
  ones)
  
• call frequency, time to recognize default
  accounted for
  
• Legal confidence in netting and collateral
  agreements factored into exposure
  
• Compute statistics of exposure distributions and
  prices

MTM
Threshold
Call
Default
Time
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Simulation overview III

• Pricing the credit exposure CVA
• Expected exposure profile treated as (irregularly
  amortizing) contingent leg on Credit Default
  Swap
  
• Market credit spreads and recovery rates used to
  generate curve of (risk neutral) default
  probabilities
  
• CVA is computed on a counter-party basis and
  aggregated across Portfolio

Exposure X(t)
Time t
?CVA -X(t) (1-R) ?(Cum_Prob_def)
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Relatedness

• Situation is complicated when the credit of the
  counter-party tied to various underlying rates
  (and volatilities). Thus we compute exposure
  quantities conditional on counter-party default
• Example of wrong way exposure cross currency
  swap with sovereign like counter-party where we
  pay foreign currency and receive dollars
  
• FX sensitive derivatives
• Jump treatment applied for wrong way exposure
• If sovereign or counter-party highly related to
  sovereign defaults, one would expect this to be
  connected to currency devaluation
  
• Credit derivatives
• model correlation of credit spreads and rates
• sample multiple default scenarios
• Collateral simulation requires careful treatment
  of time-scales

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Scale of calculation

• 400,000 derivative trades
• Simulations extend beyond 50 years
• Large number of market factors
• 70 FX rates
• 100 IR curves
• 1200 Equity spot
• 1500 Credit curves
• Each simulation requires billions of MTM
  evaluations
  
• Modes of operation
• CVA MTM, PL calculations
• sensitivities for hedging (market and credit) and
  predict
  
• marginal pricing (real time credit charge
  calculation with portfolio effects used to
  structure credit sensitive transactions)
  

19
The computational problemMethods for high
dimensional integrals

• Algorithmic approaches
• Mixed mode calculation using standard Monte
  Carlo, supercube, Brownian bridge, and path
  weighting techniques
  
• Exposure sampling and simulation time-step
  adapted for stability of results
  
• Number of simulation paths adapted to complexity
  of counter-party
  
• Need flexible, scalable parallel computing
• Path parallelism is embarrassing but useful
• For better scaling, portfolio can be subdivided
• Challenges
• Feed collection from a variety of systems
• Significant amounts of input data
• Business processes require reliable/timely
  results
  
• Efficient usage of cached results
• Evolving from shared memory multi-processing to
  grid style distributed computing

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Full portfolio view

• Portfolio is a mixture of exposures with
  different trading characteristics
  
• Liquid Can be hedged with name-specific
  instruments
  
• Semi-liquid Can be hedged on a portfolio basis
  (leaving basis risk)
  
• Illiquid
• A completely unified treatment of these is
  necessary for
  
• coherent measures of portfolio concentration and
  tail risk
  
• better understanding of economic capital
  requirements
  
• ability to evaluate complex hedges and ascertain
  associated costs
  
• Need to generate full distribution of potential
  gains and losses
  
• Portfolio value changes due to MTM and default
• Mesh exposure distributions for counter-party
  exposures with portfolio simulation of defaults
  and changes in credit spreads
  
• Availability of large scale distributed computing
  environments will make practical the use of
  Portfolio level marginal calculations to support
  active decision processes