Estimating Credit Exposure and Economic Capital Using Monte Carlo Simulation Ronald Lagnado Vice Pre

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Estimating Credit Exposure and Economic Capital
Using Monte Carlo SimulationRonald LagnadoVice
President, MKIRiskIPAM Conference on Financial
MathematicsJanuary 11, 2001

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Monte Carlo Simulation for Integrated
Market/Credit Risk

• Random sampling generates potential future paths
  of market/credit risk sources
• Provides time profile of credit exposure and
  distribution of losses
• Facilitates effective management of credit limits
  and optimal allocation of capital

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Benefits of Monte Carlo Simulation for Credit
Risk Analysis

• Efficient Capital Allocation
• Avoid overstating credit exposure by correctly
  aggregating across master agreements, time, and
  market scenarios
• Account for netting, collateral,
  less-than-perfect correlation, mean reversion,
  etc.
• Prudent Capital Allocation
• Account for default correlation, risky
  collateral, margin call lags, correlation
  instability, etc.

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MKI Integrated Risk Management Solution
Source Systems
Optional Middleware
Limit Management RV Limits
Source Systems
Consolidation Database - RV Data
Source systems
A P I 's
Source systems
Manual Entry
Price Feed Sources
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Monte Carlo Simulation
Value
Begin With Current Mark-to-Market
Base Mark- to- Market
Time Nodes
1 2 3 4
5 6 7 8 9
Time (Nodes)
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Monte Carlo Simulation
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Monte Carlo Simulation
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Monte Carlo Simulation
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Monte Carlo Simulation
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Monte Carlo Simulation
Value
NEW MARKET DATA
VALUE EVERY DEAL
Base Mark- to- Market
ASSIGN TO PORTFOLIOS
APPLY NETTING, COLLATERAL, ETC.
Time Nodes
1 2 3 4
5 6 7 8 9
Time (Nodes)
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Monte Carlo Simulation
Value
Base Mark- to- Market
Repeat for Successive Time Nodes
Time Nodes
1 2 3 4
5 6 7 8 9
Time (Nodes)
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Monte Carlo Simulation
Distribution of Portfolio Values, Exposures,
etc.
Value
Base Mark- to- Market
Runs
Time Nodes
1 2 3 4
5 6 7 8 9
Time (Nodes)
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Credit Exposure Profiles
Portfolio Exposure Dynamics
Exposure
Max Exposure
Future Potential Exposure
1 Std Dev
Y Std Dev
Mean
Current Exposure
0 1
T
Future Simulation Dates
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Credit Relationships
Counterparty C - Guaranteed or not
Counterparty B - Guaranteed or not
Counterparty A - Guaranteed or not
Master Agreement A2
Master Agreement A1
CSA A12
CSA A11
Trade 10003
Trade 10002
Collateral
Trade 10001
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Counterparty Exposure (Netting)

• Net credit exposure to Counterparty i

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Market Risk Drivers

• Interest Rates
• Base Term Structures
• Spread Term Structures
• Exchange Rates
• Equities
• Indexes
• Individual Stocks
• Commodities
• Spot Prices
• Forward Prices
• Implied Volatility Surfaces

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Example Interest Rate Process

• r vector of interest rates drivers
• ? vector of mean reversion levels
• A matrix of mean reversion speeds
• ? instantaneous covariance matrix
• Z vector of independent Brownian motions

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Example Interest Rate Process

• Integrate over time step discrete VAR(1)
  process

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Parameter Estimates USD Libor

• rates 1m 3m 6m 1y 2y
  3y 5y 7y 10y
• speed 0.51 0.37 0.42 0.51
  0.50 0.64 0.78 0.80 0.78
• volatility 0.23 0.19 0.20 0.20
  0.16 0.16 0.15 0.14 0.13
• correlation 1.
• 0.39 1.
• 0.34 0.48 1.
• 0.24 0.35 0.53
  1.
• 0.23 0.35 0.40 0.51 1.
• 0.22 0.33 0.38 0.49 0.97
  1.
• 0.20 0.31 0.36 0.46 0.93
  0.95 1.
• 0.19 0.29 0.34 0.44 0.88
  0.91 0.96 1.
• 0.17 0.27 0.31 0.42 0.83
  0.87 0.93 0.96 1.

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Option Exposure Comparison of Exact Results
with Monte Carlo

• Equity Index Call Option
• expiration 2 years
• implied volatility 20
• initially at-the-money
• Underlying Stochastic Parameters
• drift 15
• volatility 20
• Monte Carlo Simulation Weekly Time-Steps
• Exact Results Obtained with Gauss-Hermite
  Quadrature

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Simulation of Dynamic Collateral and Margin
Call Lags

• Example
• Single Counterparty
• Single Transaction 2-year equity call option
• Margin Call Parameters
• Threshold 30 Million
• Margin Call Lag 4 weeks
• Delivery Lag 1 week
• Excess Collateral Returned Immediately
• Monte Carlo Simulation 10000 paths

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Losses and Capital Calculation

• Model Requirements
• Exposure Profiles
• Credit Quality Migration and Default (Correlated)
• Stochastic Recovery
• Benefits
• Loss Reserves and Economic Capital
• Capital Allocation across Business Units
• Performance Measures (RAROC)
• Incremental Capital and Capital-Based Pricing

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The Losses Distribution
Distribution of Losses (Integrated Market/Credit
Risk Simulation)
Losses PDF
0
PV(Losses))
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The Losses Distribution
Distribution of Losses (Integrated Market/Credit
Risk Simulation)
Losses PDF
Expected Losses
0
PV(Losses))
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The Losses Distribution
Distribution of Losses (Integrated Market/Credit
Risk Simulation)
Losses PDF
Expected Losses
Unexpected Losses
0
PV(Losses))
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The Losses Distribution
Distribution of Losses (Integrated Market/Credit
Risk Simulation)
Losses PDF
Expected Losses (Reserves)
Unexpected Losses (Economic Capital)
0
PV(Losses))
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Credit Migration Model

• Markov chain with transition probability
  matrix
• probability of
  migrating from rating to rating during the
  time interval

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Credit Migration Model

• Time Inhomogeneous
• Time Homogeneous

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Typical Transition Matrix (1-Year)
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Credit Quality Migration and Default
Correlation

• Factor Model for Asset Value Return
• For each counterparty
  

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Credit Migration Quantiles
BBB
BB
A
B
AA
CCC
AAA
D
0
Change in Firm Value (Normalized)
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Relating Asset Returns to Default Correlation

• Asset-Return Correlation
• Default Correlation

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Losses

• discrete time nodes
• market risk driver path
• idiosyncratic credit driver path
• default stopping time

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Loss Statistics (Simplified Case)

• Single-period Independent exposure and default

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Loss Statistics (Simplified Case)

• Single-period
• Constant and identical exposures
• Identical default probabilities and correlations

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Loss distributions 500 counterparties, constant
exposures, p 0.05
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Tolerance Intervals

• Ordered sample of losses from Monte Carlo
  simulation
• Estimated quantile
• Distribution of order statistics

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Tolerance Intervals

• Construct non-parametric
  confidence interval for estimated quantile

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Convergence of Unexpected Losses

• 500 counterparties, 550 deals, 1 year horizon

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Summary

• Monte Carlo simulation is preferred approach for
  integrated market/credit risk analysis
• Reveals distributions of future credit exposure
  and losses to default
• Facilitates efficient capital allocation by
  correctly aggregating exposure across time and
  market scenarios
• Leads to prudent capital allocation by accounting
  for market complexities