credit risk, economic capital, and supercomputing · zkb motivation model implementation results...

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Motivation Model Implementation Results

Credit Risk, Economic Capital, andSupercomputing

FINRISK Conference on Risk and Portfolio Management

Dr. D. Egloff1

1Manager Financial ComputingZürcher Kantonalbank

Switzerland

January 26, 2006

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Outline

MotivationCredit RiskEconomic Captial for Credit Risk

ModelModel ComponentsModelling CreditworthinessTransaction ModellingData Requirements and Calibration

ImplementationSimulation FrameworkHigh Performance Cluster Implementation

ResultsEC Reduction as Function of Model Complexity

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Credit Risk

• Credit risk is major risk for all commerical and retail banks.

• Commerical banks are exposed to a multitude ofcounterparties.

• Credit events are low probability events with severe impact.

• Effective measurement and management of overall creditrisk is emerging as a core business in the financial industry.

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Economic Captial

DefinitionBank’s own assessment of the capital it requires to cover itsrisky business activities.

Principal usage:

• Risk-based capital management to improve strategic andtactical planning.

• Capital attribution to risky business activities.

• Pricing economic capital consumption at transaction level.

• Regulatory and economic capital might converge also forcredit risk.

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Economic Capital and Profit & Loss Distribution

Realistic economic capital is derived from a P&L distribution,e.g. mark-to-market as opposed to default mode paradigm.

• Time horizon [0, T ], usually over multiple years.

• Mark-to-market paradigm

P&L = VT − B(0, T )V0 + I[0,T ] − C[0,T ] ,

where• Vt value of portfolio at time t ,• I[0,T ] compounded repayment, amortization, interest

income,• C[0,T ] compounded refinancing, capital, operation costs,• B(t , T ) zero bond prices for compounding and discounting.

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How to model P&L realistically and to calculate it efficiently?

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Key Model Components

Modelling P&L based on mark-to-market paradigm requires

• modelling creditworthiness of counterparties,

• modelling credit transactions,

• valuation of credit transactions,

• definition of P&L as functional of all risk factors.

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Modelling Creditworthiness

• Rating classes as a discrete measures of credit quality.

• Rating dynamics Ri(t) a discrete time Markov chain.

• Latent variables A = (Ai)i=1,...,N

Ai ∼ N(0, 1).

• Transition probabilities

P (Ri(t + 1) = q | Ri(t) = r) = P(

Ai ∈[

θi(r , q + 1), θi(r , q)))

.

(1)

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Dependence Structure

• Dependence between creditworthiness of obligers iscrucial in credit risk modelling.

• Joint defaults is main risk in large loan portfolios.• Sources of dependence

• Economy: common factors affecting all obligers.• Example: interest rates, economic growth of industry

sectors.• Microstructure: direct business and legal relations.• Example: Swissair & suppliers, Enron & Arthur Andersen.

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• Represent latent variables as

A = dY D(√

1 − v2)

Y + dεD (v) ε(Y , ε) ∈ RN , (2)

Y ∼ N(0,Σ), ε ∼ N(0, 1N), and dY , dε scaling matrices .

• Macroeconomic dependence through common factor Y .

• Microstructural dependence through

ε(Y , ε) = ΞA + D(η)ε (3)

as function of other obligers’ latent variables.

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• Solve for dY , dε in (2) such that Ai ∼ N(0, 1).

• Variance condition Var(Ai) = 1 does not change marginallaw (1).

• Resulting structure

A = CY Y + Cεε (4)

multivariate Gaussian, Y ∼ N(0,Σ), ε ∼ N(0, 1N).

• CY , Cε fixed points of a nonlinear map.

• Convergence: Banach fixed point theorem.

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Modelling Credit Transactions

• Translate transaction contract details into transactioncashflows.

• Detailed transaction cashflows, including• variable, fixed or administered interest rates,• decomposition of interest payments according to pricing

and costing regime,• amortization,• prepayments and early repayment,• utilization of credit lines.

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Valuating Credit Transactions

• Modelling credit spreads for different rating classes.

• Static spreads.

• Affine or quadratic term structure models.

• What is the market price of credit risk?

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Data Requirements for Modelling Creditworthiness

• Transition thresholds θi in (1) calibrated from empiricaltransition matrices.

• Sector weights and correlation matrix v , Σ in (2) calibratedfrom historical sector default frequencies.

• Microstructure weights Ξ in (3) from expert judgment, η

residual.• Proxies for Ξ:

• Business volume (e.g. rental income), turnover.• Investments in affiliates, intercompany participations.

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Data Requirements for Credit Transactions

• Transaction contract details.

• Pricing and costing details.

• Market data to calibrate risk free term structure.

• Market data to calibrate credit spreads and market price ofrisk.

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Simulation Framework

Simulating P&L distribution.

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Calculation Procedure

• Select portfolio aggregation hierarchies.

• Select risk measures for each sub-portfolio.

• Choose a Monte Carlo simulation method (standard, staticor adaptive importance sampling).

1. Simulate risk factors (ratings, credit spreads, ...).

2. Evaluate transaction cashflows given new risk factors.

3. Aggregate transaction cashflows at all sub-portfolio levels.

4. Update risk measures of sub-portfolios and their samplingerrors.

5. Update Monte Carlo simulation parameters.

6. Continue with 1. until desired precision is reached.

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Calculation Procedure

• Relevant risk measures:• Tail probabilities, quantiles, conditional tail means.

• Difficulties:• High quantile levels α ∈ [0.95, 0.9995].• Massive data in order of 1 to 10 TB.• Sequential update of risk measures.• No independent identically distributed sampling for adaptive

importance sampling.

• High performance Cluster implementation:• Massivly parallel simulatioin application.• Distributed memory infrastructure.• Message passing.

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Mare Nostrum – High Performance Cluster

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Linux High Performance Cluster

Our setup “slightly” smaller.

• Intel Xeon based dual CPU Linux cluster.

• Operating system Debian Gnu/Linux.

• Open source whenever possible (Boost C++ libraries,MPICH2, MySQL, ...).

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EC Reduction as Function of Model Complexity

EC

1 Factor Default Mode Fixed 1 Year Horizon Aggregated Transactions

1 Factor Default Mode Transaction maturity structure Aggregated Transactions

N Factor Default Mode Transaction maturity structure Aggregated Transactions

N Factor Mark-to-Market Mode Transaction maturity structure Cashflow based transactions Microstructure dependence

1

0.43

0.35

0.24

credit risk, economic capital, and supercomputing · zkb motivation model implementation results...

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