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Monte Carlo Simulation for Integrated Market/Credit Risk

Estimating Credit Exposure and Economic Capital Using Monte Carlo Simulation Ronald Lagnado Vice President, MKIRisk IPAM Conference on Financial Mathematics January 11, 2001. Monte Carlo Simulation for Integrated Market/Credit Risk.

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Monte Carlo Simulation for Integrated Market/Credit Risk

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

  2. 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

  3. 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.

  4. MKI Integrated Risk Management Solution Manage Data consistent, complete, timely, accurate • Collect Data • Trades/deals • Static Data • Prices, Curves, ... Evaluate & Monitor Risk Distribute Information Enquiries ! Irregularity notifications Portfolio Analytics RV CARMA Reports 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

  5. 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)

  6. Monte Carlo Simulation Value Advance to a Future Date Base Mark- to- Market Time Nodes 1 2 3 4 5 6 7 8 9

  7. Monte Carlo Simulation Value EVOLVE RISK DRIVERS Base Mark- to- Market Time Nodes 1 2 3 4 5 6 7 8 9

  8. Monte Carlo Simulation Value EVOLVE RISK DRIVERS VALUE EVERY DEAL Base Mark- to- Market Time Nodes 1 2 3 4 5 6 7 8 9

  9. Monte Carlo Simulation Value EVOLVE RISK DRIVERS VALUE EVERY DEAL Base Mark- to- Market ASSIGN TO PORTFOLIOS Time Nodes 1 2 3 4 5 6 7 8 9

  10. 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)

  11. 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)

  12. 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)

  13. 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

  14. 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 Collateral Trade 10002 Trade 10001

  15. Counterparty Exposure (Netting) • Net credit exposure to Counterparty i:

  16. 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

  17. 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

  18. Example: Interest Rate Process • Integrate over time step: discrete VAR(1) process

  19. 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.

  20. 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

  21. 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

  22. 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

  23. The Losses Distribution Distribution of Losses (Integrated Market/Credit Risk Simulation) Losses PDF 0 PV(Losses))

  24. The Losses Distribution Distribution of Losses (Integrated Market/Credit Risk Simulation) Losses PDF Expected Losses 0 PV(Losses))

  25. The Losses Distribution Distribution of Losses (Integrated Market/Credit Risk Simulation) Losses PDF Expected Losses Unexpected Losses 0 PV(Losses))

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

  27. Credit Migration Model • Markov chain with transition probability matrix: • probability of migrating from rating to rating during the time interval

  28. Credit Migration Model • Time Inhomogeneous: • Time Homogeneous:

  29. Typical Transition Matrix (1-Year)

  30. Credit Quality Migration and Default Correlation • Factor Model for Asset Value Return • For each counterparty

  31. Credit Migration Quantiles BBB BB A B AA CCC AAA D 0 % Change in Firm Value (Normalized)

  32. Relating Asset Returns to Default Correlation • Asset-Return Correlation: • Default Correlation:

  33. Losses • discrete time nodes: • market risk driver path: • idiosyncratic credit driver path: • default stopping time:

  34. Loss Statistics (Simplified Case) • Single-period; Independent exposure and default

  35. Loss Statistics (Simplified Case) • Single-period • Constant and identical exposures • Identical default probabilities and correlations

  36. Loss distributions: 500 counterparties, constant exposures, p = 0.05

  37. Tolerance Intervals • Ordered sample of losses from Monte Carlo simulation: • Estimated quantile: • Distribution of order statistics:

  38. Tolerance Intervals • Construct non-parametric confidence interval for estimated quantile:

  39. Convergence of Unexpected Losses • 500 counterparties, 550 deals, 1 year horizon

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