Cime Ems 2003 Bielecki

7/27/2019 Cime Ems 2003 Bielecki

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Stochastic Methods in Credit Risk

Modelling, Valuationand HedgingIntroduction to Credit Risk and Credit

Derivatives

Tomasz R. Bielecki

Northeastern Illinois University

[email protected]

In collaboration with Marek Rutkowski
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Part 1: Portfolio Credit Risk

Measuring credit risk.

Portfolio analysis.

CVaR models.

CreditMetrics.

CreditGrades.

Counterparty credit risk.

Reference credit risk.


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Part 2: Credit Derivatives

Counterparty credit risk.

Reference credit risk.

Classification of credit derivatives.

Total return swaps.

Credit default swaps.

Spread linked swaps.

Credit options.


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Part 3: Mathematical Modelling

Mertons model of corporate debt.

Black and Cox approach.

Intensity-based approach to credit risk.

Hybrid models.

Implied probabilities of default.

Markov models of credit ratings.

Market risk and term structure models.


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Credit Risk: Modelling, Valuation

and Hedging

Part 1: Portfolio Credit Risk

The central point is the quantitative estimate ofthe amount of economic capital needed to

support a banks risk-taking activities


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Measuring Credit Risk

Credit risk models should capture:

Systematic vs Idiosyncratic Risk Sources

Credit spread risk,

Downgrade risk (credit rating), Default risk (default probability),

Recovery rate risk (recovery rate),

Exposure at default (loss given default),

Portfolio diversification (correlation risk), Historical Probabilities vs Risk-Neutral Probabilities.


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Portfolio Analysis I

What is really important:

Concentration risk, Basle Committee 25% rule; Herfindahl-Hirshman Index

Diversification effect,

Rating structure, CVaR, Credit Value-at-Risk

Risk-adjusted performance measures,

Capital optimisation,

Sensitivity and stress test analysis.


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Portfolio Analysis II

How should we define and measure credit riskof a portfolio of loans or bonds?

What are the measures of capital profitabilitythe bank should apply?

What is the risk-return profile of the bankscredit portfolio?

What is the capital amount required for theassumed rating of the banks credit portfolio?

Important questions to risk managers:


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Portfolio Analysis III

Which credit exposures represent the highestrisk-adjusted profitability?

What are the main factors affecting the banks

credit portfolio risk-adjusted profitability? What are the main sources of the banks

credit risk concentration and diversification?

How can the bank improve its portfolio

profitability?


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CVaR Models I

Types of Credit Risk Models: Risk aggregation:

- Top-down,Aggregate risk in consumer, credit card, etc., portfolios;default rates for entire portfolios

- Bottom-up, Individual asset level; default rates for individual obligors.

Systemic factors recognition:- Conditional,

- Unconditional.

Default measurement:

- Default mode, Two modes: default or no-default

- Mark-to-market (model), Credit migrations accounted for.


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CVaR Models II

Currently proposed industry sponsoredCVaR models:

CreditMetrics (RiskMetrics),

CreditGrades (RiskMetrics),

Credit Monitor/EDF (KMV/Moodys),

CreditRisk+ (Credit Suisse FB),

CreditPortfolioView (McKinsey).


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CVaR Models III


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CreditMetrics I

A tool for assessing portfolio risk due tochanges in debt value caused by changes inobligor credit quality.

Changes in value caused not only by possibledefault events, but also by upgrades anddowngrades in credit quality are included.

The value-at-risk (VaR) - the volatility of value,

not just the expected losses, is assessed.


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CreditMetrics II

Risk is assessed within the full context of aportfolio. The correlation of credit quality movesacross obligors is addressed. This allows todirectly calculate the diversification benefits.

Value changes are relatively small with minorup(down)grades, but could be substantial if

there is a default (rare event).

This is far from the more normally distributed

market risks that VaR models typically address.


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CreditMetrics III


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CreditMetrics IV


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CreditGrades I

Is meant to provide a simple framework linkingthe credit risk and equity markets (a first-passage-time model).

Tracks the risk-neutral default probabilities.

Based on the ideas of the structural approach,due to Merton (1973), Black and Cox (1976).

Main deficiency are artificially low short-termcredit spreads. CreditGrades corrects this by

taking random default barrier and recovery rate. This is essentially a pricing model


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CreditGrades II

Asset valueVfollows a lognormal proces with

a constant volatility (under real-world probability).

Default occurs at the first crossing of the defaultbarrier by V.

Default barrieris the product of the expectedglobal recovery of the firms liabilities and thecurrent debt per share of the firm.

The CreditGrade is the model-implied 5-yearcredit spread.


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CreditGrades III


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CreditGrades: Case Study


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CreditGrades: Spin Summary


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CreditGrades: No Spin Critique

CGappears to mix statistical and risk neutralprobabilities.

CG assumes no-drift condition for asset valueprocess, which appears to be unjustified.

Transparent formulae for probabilities of defaultresulting in CG framework and, apparently,relying on market observables only, appear to befounded on questionable (in general) relationship

between volatility of equity and volatility of theasset value process.


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Credit Monitor I

Credit Monitor provides M-KMVs EDF creditmeasures on corporate and financial firmsglobally, updated on a monthly basis with upto five years of historical EDF information.

EDF (expected defaul t frequency) is aforward looking measure of actualprobability of default. EDF is firm specific.

Credit Monitor model follows the structural

approach to calculate EDFs. [The credit riskis driven by the firms value process.]


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Credit Monitor II

Credit Monitor deals with firms whoseequities are publicly traded. The market

information contained in the firms stockprice and the balance sheet is mapped tothe firms EDF.

Credit Monitor used in M-KVMs Port fo l io

Manager


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CreditRisk+ I

An approach focused only on default event; itignores migration and market risk.

For a large number of obligors, the number of

defaults during a given period has a Poissondistribution. The loss distribution of a bond/loanportfolio is derived.

Belongs to the class of intensity-based (or

reduced-form) models. Default risk is not linkedto the capital structure of the firm.


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CreditRisk+ II


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CreditPortfolioView

A multifactor model focused on the simulationof the joint distribution of default and migrationprobabilities for various rating groups.

Default/migration probabilities are linked to thestate of the economy through macroeconomicfactors (an econometric model).

Conditional probabilities of default are modelled

as a logit function of the index:


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and Hedging

Part 2: Credit DerivativesThe central points are providing protection

against credit risk and diversification of

credit risk exposure


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Counterparty Credit Risk

Derivatives trading generates exposure to

the credit risk of the counterparty involved in

a given contract (typical examples: bonds,vulnerable options, defaultable swaps).

Counterparty credit risk is a function of:

Creditworthiness of the counterparty,

Size of profits accrued yet unrealised,

Ability to use legally binding nettingagreements.


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Reference Credit Risk

Credit derivatives are privately held negotiablebilateral contracts that allow users to manage theirexposure to credit risk, so-called reference creditrisk.

Credit derivatives are financial assets like forwardcontracts, swaps and options for which the price isdriven by the credit risk of economic agents (private

investors or governments).


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Why Credit Derivatives?

Credit derivatives connect the different fixed-incomemarkets by being the clearing-house for credit risktransfer.

Insurance against credit events to reduce borrowing

costs. Diversification of exposure by means of synthetic

loans.

Assume positions in markets that might otherwise beinaccessible.

Accounting and tax advantages.


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Default Protection

Default protection:

Suppose a bank concerned that one of itscustomers may not be able to repay a loan.

The bank can protect itself against loss bytransferring the credit risk to another party,while keeping the loan on its books.

Useful links: www.defaultrisk.com

www.margrabe.com


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Special Features

Pay-out typically based on extremal event (forinstance, the default event).

Limited liquidity (currently).

Insurance components may require actuarialanalysis (under statistical probability).

Operational risk management important - cantbuy perfect insurance, and tail events are

extremal (Bankers Trust)


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A Simplified Taxonomy

Credit derivatives are usually rather involved.

They can be divided into three basic classes:

Swaps:

- Total rate of return swap, default swap, and

spread-linked swap. Notes:

- Default note, spread-linked note, and leverednotes.

Options:

- Price, spread, and default options.


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Spectrum


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Total return (or asset) swap - TRS,

Credit-linked note - CLN,

Credit default swap (or option) - CDS,

Securitized pool (of corporates) - CDO, Option on a corporate bond,

Credit spread swap (or option),

Insured cash-flow stream (swap guarantee).

Vanilla Credit Derivatives


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Total Return Swap I

Party A Party BAsset Total Return

Floating Payments

Underlying assets may be bonds, loans, or other creditinstruments. Permits the separation of asset ownership

and economic exposure: balance sheet rental orout-sourcing, for example.


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Total Return Swap II

Total Rate of Return Swap is a derivative contractthat simulates the purchase of an instrument (note,bond, share, etc.) with 100% financing, typicallyfloating rate.

The contract may be marked to market at eachreset date, with the total return receiver receiving(paying) any increase in value of the underlyinginstrument, and the total return payer receiving(paying) any decrease in the value of the underlying

instrument.


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Credit Default Swap I

Party A Party B

Default Premium

Recovery (after default)

Recovery is paid only if there is a default, so this is apure credit risk product. That is, price and spreadrisk is stripped away. Bs exposure is like that of an

off-balance sheet loan.


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Credit Default Swap II

Credit default swap is a contract between a buyerand a seller of protection, in which:

(a) the buyer of protection pays the seller a fixed,regular fee,

(b) the seller of protection provides the buyer with acontingent exchange that occurs either at thematurity of the underlying instrument or at theswap's date of early termination. The trigger eventfor the contingent payoff is a defined credit event (adefault on the underlying instrument or other relatedevent).

C f S


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Credit Default Swap III

C dit D f lt S IV


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Credit Default Swap IV

C dit D f lt S V


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Credit Default Swap V

Spread Linked Swap


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Spread-Linked Swap

Party A Party B

Periodic payments

Payments based on spread

Bs payments are based on the credit spread of

a reference security. B may only make a final

payment at maturity based on the credit spread.A pays LIBOR plus a fixed spread, say.

Default Notes


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Default Notes

Default notes: For example, an issuer (credit cardcompany, say) agrees to pay back $100 at maturity and8% coupons semiannually, but if some default eventoccurs the coupons drop to 4%.

The investor will pay less than he would for a similar notewithout credit-linkage in compensation for the option hehas sold to the issuer.

Spread-linked notes: Like above, except that here the

coupon paid by the investor depends on the creditspread for some reference security.

Levered Notes


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Levered Notes

For example, corporate bonds might be pooled, andthe cash-flows repackaged in the form of a note thatpays a high (leveraged) coupon in return for acceptingwith this the risk that the payments will stop (or besignificantly reduced) if there are one or more defaults

in the pool.

The cash-flows might also be packaged in the form oflower-yielding money market instruments, thus earningprofits for the issuer (at the cost of accepting some ofthe credit risk). In this case, it is the issuer who assumesthe leveredposition.

Credit Options


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Credit Options

Security with the payoff contingent on the

following credit events:

the price of a reference security drops below

a strike price (determined by a strike spread), the credit spread for a reference security

tightens or widens, or

there is a default event of the referenceentity.

Exotic Variations


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Exotic Variations

Basket credit derivatives (correlation-sensitiveproducts).

Event-contingent option (if a certain project iscompleted on time, say).

Real options (sell real decision risk instead of market

factor risk). Fixed-income products linked to earthquakes or other

catastrophes.

Notes linked to real earnings and inflation (lessvolatility in real rates).

Types of Risks


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Types of Risks

Credit risk (obvious) and the price risk (sincethis affects profitability, and therefore credit

quality).

Operational risk (contingency planing for

worst-case scenario, for example). Liquidity risk (can be mitigated by doing

deals back-to-back, and including early

termination provisions).

Legal risk (Orange County).

Benefits from Credit Derivatives


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Benefits from Credit Derivatives

Better serve customer needs.Diversification of exposures.

Efficient use of balance sheet.

Profiting from market views.

Traders receive information on orderflow, customer interest, etc.


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and Hedging

Part 3: Mathematical ModellingThe central point is providing formal quantitative tools to

properly serve the purposes listed in Parts 1 and 2

Mertons Model of Corporate Debt


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Merton s Model of Corporate Debt

Let us denote: V- total value of the firms assets,

L - face value of the firms debt,

T- maturity of the debt, - (random) time of default.

Default occurs at time Tif the total value of the

firms assets at time Tis lower than the face

value Lof the firms debt.

Dynamics of Firms Assets


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The process representing the total value of thefirms assets is governed by the stochastic (random)equation:

Dynamics of Firm s Assets

where is the standard Brownian motion(one-dimensional Wiener process).The interest rate and the dividend yieldare constant.

Mertons Default Time


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The time of default is given by

Merton s Default Time

The recovery payoff at time equals

and thus the corporate bond satisfies

Mertons Valuation Formula


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The price at time of a -maturity corporatebond equals:

Merton s Valuation Formula

where is the time to maturityand

Black and Cox Model


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Black and Cox Model

Basic assumptions of Mertons model are preserved.Value of firms assets is lognormally distributed.

The random instant of default is specified as the firstmoment the value of the firm crosses some barrier:premature default.

The latter assumption is assumed to represent the so-called safety covenants.

Closed-form solution for the value of corporate debt isavailable (but it is rather involved).

Structural Approach


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Structural Approach

The total value of the firms assets is noteasily observed. The total value of sharescan be taken as a proxy.

The internal structure of the reference firm

is an essential ingredient of the model.

On the other hand, both the cross-defaultprovision and the debts seniority structure

are relatively easy to cover.

Intensity-Based Approach


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Intensity Based Approach

Value of the firm is not explicitly modelled. The intensity of the random time of default

plays the role of a models input.

Valuation result for corporate bonds and

credit derivatives are relatively simple, evenin the case ofbasket credit derivatives.

In practice, the intensity of default can be

inferred from observed prices of bonds

(the calibrated orimplied default intensity).

Default Time


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Default Time

Structural approach: is a predictable stopping timewith respect to the filtration generated by the value

process. Default is announced by a sequence of

stopping times.

Intensity-based approach: is a totally inaccessible

stopping time with respect to the reference filtration

(including the observations of the default time. Default

comes as a surprise.

Credit Ratings


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Credit Ratings

Some more recent methods take into accountnot only the default event, but also the currentand futures rating of each firm.

In most cases, the process that models theup/downgrades is a Markov process.

Instead of a default intensity, the whole matrixof intensities of migrations is specified.

Official ratings are given by specialized ratingagencies; they do not necessarily reflect (risk-neutral) probabilities of credit migrations.

Intensities of Migrations


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g

The matrix of intensities of credit migrations has thefollowing form

where Kis the number of credit ratings andthe K-th class represents default event.State Kis an absorbing state.

References


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M. Ammann: Credit Risk Valuation: Methods,Models, and Applications. Springer 2001.

A. Arvanitis and J. Gregory: Credit Risk:The Complete Guide. Risk Books 2001.

T. R. Bielecki and M. Rutkowski: Credit Risk:Modelling, Valuation and Hedging. Springer 2002.

D. Cossin and H. Pirotte:Advanced Credit RiskAnalysis. J. Wiley & Sons 2000.

B. Schmid: Pricing Credit Linked FinancialInstruments. Springer 2002.

D. Duffie and K. J. Singleton: Credit Risk, PrincetonUniversity Press 2003.

CreditGrades II


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Documents

Cime Ems 2003 Bielecki