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1、Stochastic Methods in Credit Risk Stochastic Methods in Credit Risk Modelling, Valuation Modelling, Valuation and Hedgingand HedgingIntroduction to Credit Risk and Credit Introduction to Credit Risk and Credit DerivativesDerivatives Tomasz R. Bielecki Northeastern Illinois University t-bieleckineiu.
2、eduIn collaboration with Marek RutkowskiPart 1: Portfolio Credit RiskPart 1: Portfolio Credit RiskMeasuring credit risk. Portfolio analysis.CVaR models.CreditMetrics.CreditGrades.Counterparty credit risk.Reference credit risk.2Part 2: Credit DerivativesPart 2: Credit DerivativesCounterparty credit r
3、isk.Reference credit risk.Classification of credit derivatives.Total return swaps.Credit default swaps.Spread linked swaps.Credit options. 3Part 3: Mathematical Part 3: Mathematical ModellingModellingMertons model of corporate debt.Black and Cox approach.Intensity-based approach to credit risk.Hybri
4、d models.Implied probabilities of default.Markov models of credit ratings.Market risk and term structure models.4Credit Risk: Modelling, Valuation Credit Risk: Modelling, Valuation and Hedgingand HedgingPart 1: Portfolio Credit RiskThe central point is the quantitative estimate of the amount of econ
5、omic capital needed to support a banks risk-taking activitiesMeasuring Credit Risk 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 ra
6、te), Exposure at default (loss given default), Portfolio diversification (correlation risk), Historical Probabilities vs Risk-Neutral Probabilities.6Portfolio Analysis I Portfolio Analysis I What is really important: Concentration risk, Basle Committee 25% rule; Herfindahl-Hirshman Index Diversifica
7、tion effect, Rating structure, CVaR, Credit Value-at-Risk Risk-adjusted performance measures, Capital optimisation, Sensitivity and stress test analysis. 7Portfolio Analysis II Portfolio Analysis II How should we define and measure credit risk of a portfolio of loans or bonds?What are the measures o
8、f capital profitability the bank should apply? What is the risk-return profile of the banks credit portfolio? What is the capital amount required for the assumed rating of the banks credit portfolio?Important questions to risk managers:8Portfolio Analysis III Portfolio Analysis III Which credit expo
9、sures represent the highest risk-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?9CVaR Models IC
10、VaR Models ITypes 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, - Uncondition
11、al. Default measurement: - Default mode, Two modes: default or no-default - Mark-to-market (model), Credit migrations accounted for. 10CVaR ModelsCVaR Models II II Currently proposed industry sponsored CVaR models: CreditMetrics (RiskMetrics), CreditGrades (RiskMetrics), Credit Monitor/EDF (KMV/Mood
12、ys), CreditRisk+ (Credit Suisse FB), CreditPortfolioView (McKinsey).11CVaR ModelsCVaR Models IIIIII12CreditMetrics ICreditMetrics I A tool for assessing portfolio risk due to changes in debt value caused by changes in obligor credit quality.Changes in value caused not only by possible default events
13、, but also by upgrades and downgrades in credit quality are included. The value-at-risk (VaR) - the volatility of value, not just the expected losses, is assessed.13CreditMetrics IICreditMetrics II Risk is assessed within the full context of a portfolio. The correlation of credit quality moves acros
14、s obligors is addressed. This allows to directly calculate the diversification benefits. Value changes are relatively small with minor up(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 ad
15、dress.14CreditMetrics IIICreditMetrics III 15CreditMetrics IV 16CreditGrades ICreditGrades I Is meant to provide a simple framework linking the 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
16、 to Merton (1973), Black and Cox (1976). Main deficiency are artificially low short-term credit spreads. CreditGrades corrects this by taking random default barrier and recovery rate.This is essentially a pricing model17CreditGrades IICreditGrades II Asset value V follows a lognormal proces with a c
17、onstant volatility (under real-world probability).Default occurs at the first crossing of the default barrier by V.Default barrier is the product of the expected global recovery of the firms liabilities and the current debt per share of the firm.The CreditGrade is the model-implied 5-year credit spr
18、ead.18CreditGrades III 19CreditGrades: Case Study 20CreditGrades: Spin Summary 21CreditGradesCreditGrades: No Spin Critique: No Spin Critique CG appears to mix statistical and risk neutral probabilities.CG assumes no-drift condition for asset value process, which appears to be unjustified.Transparen
19、t formulae for probabilities of default resulting in CG framework and, apparently, relying on market observables only, appear to be founded on questionable (in general) relationship between volatility of equity and volatility of the asset value process.22Credit Monitor ICredit Monitor ICredit Monito
20、r provides M-KMVs EDF credit measures on corporate and financial firms globally, updated on a monthly basis with up to five years of historical EDF information.EDF (expected default frequency) is a forward looking measure of actual probability of default. EDF is firm specific.Credit Monitor model fo
21、llows the structural approach to calculate EDFs. The credit risk is driven by the firms value process.23Credit Monitor IICredit Monitor IICredit Monitor deals with firms whose equities are publicly traded. The market information contained in the firms stock price and the balance sheet is mapped to t
22、he firms EDF.Credit Monitor used in M-KVMs Portfolio Manager24CreditRisk+ I CreditRisk+ I An approach focused only on default event; it ignores migration and market risk. For a large number of obligors, the number of defaults during a given period has a Poisson distribution. The loss distribution of
23、 a bond/loan portfolio is derived.Belongs to the class of intensity-based (or reduced-form) models. Default risk is not linked to the capital structure of the firm. 25CreditRisk+ II 26CreditPortfolioViewCreditPortfolioView A multifactor model focused on the simulation of the joint distribution of de
24、fault and migration probabilities for various rating groups.Default/migration probabilities are linked to the state of the economy through macroeconomic factors (an econometric model).Conditional probabilities of default are modelled as a logit function of the index:27Credit Risk: Modelling, Valuati
25、on Credit Risk: Modelling, Valuation and Hedgingand Hedging Part 2: Credit DerivativesThe central points are providing protection against credit risk and diversification of credit risk exposure CounterpartyCounterparty Credit Risk Credit RiskDerivatives trading generates exposure to the credit risk
26、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 netting agreements.29Reference Credit
27、 RiskReference Credit RiskCredit derivatives are privately held negotiable bilateral contracts that allow users to manage their exposure to credit risk, so-called reference credit risk. Credit derivatives are financial assets like forward contracts, swaps and options for which the price is driven by
28、 the credit risk of economic agents (private investors or governments). 30Why Credit Derivatives?Why Credit Derivatives?Credit derivatives connect the different fixed-income markets by being the “clearing-house” for credit risk transfer.Insurance against credit events to reduce borrowing costs.Diver
29、sification of exposure by means of synthetic loans.Assume positions in markets that might otherwise be inaccessible.Accounting and tax advantages.31Default Protection Default Protection Default protection: Suppose a bank concerned that one of its customers may not be able to repay a loan. The bank c
30、an protect itself against loss by transferring the credit risk to another party, while keeping the loan on its books. Useful links: 32Special FeaturesPay-out typically based on extremal event (for instance, the default event).Limited liquidity (currently).Insurance components may require actuarial
31、analysis (under statistical probability).Operational risk management important - cant buy perfect insurance, and tail events are extremal (Bankers Trust)33A Simplified TaxonomyCredit derivatives are usually rather involved. They can be divided into three basic classes:Swaps:- Total rate of return sw
32、ap, default swap, and spread-linked swap.Notes:- Default note, spread-linked note, and levered notes.Options:- Price, spread, and default options.34Spectrum35Total return (or asset) swap - TRS,Credit-linked note - CLN,Credit default swap (or option) - CDS,Securitized pool (of corporates) - CDO,Optio
33、n on a corporate bond,Credit spread swap (or option),Insured cash-flow stream (swap guarantee).Vanilla Credit DerivativesVanilla Credit Derivatives36Total Return Swap ITotal Return Swap IAsset Total ReturnFloating PaymentsUnderlying assets may be bonds, loans, or other credit instruments. Permits th
34、e separation of asset ownershipand economic exposure: balance sheet rental or out-sourcing, for example.37Total Return Swap IITotal Return Swap IITotal Rate of Return Swap is a derivative contract that simulates the purchase of an instrument (note, bond, share, etc.) with 100% financing, typically f
35、loating rate. The contract may be marked to market at each reset date, with the total return receiver receiving (paying) any increase in value of the underlying instrument, and the total return payer receiving (paying) any decrease in the value of the underlying instrument. 38Credit Default Swap ICr
36、edit Default Swap IDefault PremiumRecovery (after default)Recovery is paid only if there is a default, so this is a “pure” credit risk product. That is, price and spreadrisk is stripped away. Bs exposure is like that of an off-balance sheet loan.39Credit Default Swap IICredit Default Swap IICredit d
37、efault swap is a contract between a buyer and 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 a contingent exchange that occurs either at the maturity of the underlying instrument or at the swaps
38、 date of early termination. The trigger event for the contingent payoff is a defined credit event (a default on the underlying instrument or other related event).40Credit Default Swap IIICredit Default Swap III41Credit Default Swap IVCredit Default Swap IV42Credit Default Swap VCredit Default Swap V
39、43Spread-Linked SwapSpread-Linked SwapPeriodic paymentsPayments based on spreadBs 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.44Default Notes Default notes: For example,
40、 an issuer (credit card company, say) agrees to pay back $100 at maturity and 8% coupons semiannually, but if some default event occurs the coupons drop to 4%. The investor will pay less than he would for a similar note without credit-linkage in compensation for the option he has sold to the issuer.
41、Spread-linked notes: Like above, except that here the coupon paid by the investor depends on the credit spread for some reference security.45Levered Notes For example, corporate bonds might be pooled, and the cash-flows repackaged in the form of a note that pays a high (leveraged) coupon in return f
42、or accepting with this the risk that the payments will stop (or be significantly reduced) if there are one or more defaults in the pool. The cash-flows might also be packaged in the form of lower-yielding money market instruments, thus earning profits for the issuer (at the cost of accepting some of
43、 the credit risk). In this case, it is the issuer who assumes the levered position.46Credit 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 secu
44、rity tightens or widens, or there is a default event of the reference entity.47Exotic VariationsBasket credit derivatives (correlation-sensitive products).Event-contingent option (if a certain project is completed on time, say).Real options (sell real decision risk instead of market factor risk).Fix
45、ed-income products linked to earthquakes or other catastrophes.Notes linked to real earnings and inflation (less volatility in real rates).48Types of RisksCredit risk (obvious) and the price risk (since this affects profitability, and therefore credit quality).Operational risk (contingency planing f
46、or 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).49Benefits from Credit DerivativesBetter serve customer needs.Diversification of exposures.Efficient use of balance sheet.Profitin
47、g from market views.Traders receive information on order flow, customer interest, etc.50Credit Risk: Modelling, Valuation Credit Risk: Modelling, Valuation and Hedgingand HedgingPart 3: Mathematical ModellingThe central point is providing formal quantitative tools to properly serve the purposes list
48、ed in Parts 1 and 2Mertons Model of Corporate DebtLet 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 T if the total value of the firms assets at time T is lower than the face value L of the f
49、irms debt. 52 The process representing the total value of the firms assets is governed by the stochastic (random) equation:Dynamics of Firms AssetsDynamics of Firms Assetswhere is the standard Brownian motion (one-dimensional Wiener process). The interest rate and the dividend yield are constant.53
50、The time of default is given byMertons Default TimeMertons Default TimeThe recovery payoff at time equalsand thus the corporate bond satisfies 54 The price at time of a -maturity corporate bond equals:Mertons Valuation FormulaMertons Valuation Formula where is the time to maturity and55Black and Cox
51、 ModelBasic assumptions of Mertons model are preserved. Value of firms assets is lognormally distributed. The random instant of default is specified as the first moment the value of the firm crosses some barrier: premature default.The latter assumption is assumed to represent the so-called safety co
52、venants.Closed-form solution for the value of corporate debt is available (but it is rather involved).56Structural ApproachThe total value of the firms assets is not easily observed. The total value of shares can be taken as a proxy.The internal structure of the reference firm is an essential ingred
53、ient of the model. On the other hand, both the cross-default provision and the debts seniority structure are relatively easy to cover.57Intensity-Based ApproachValue of the firm is not explicitly modelled.The intensity of the random time of default plays the role of a models input.Valuation result f
54、or corporate bonds and credit derivatives are relatively simple, even in the case of basket credit derivatives.In practice, the intensity of default can be inferred from observed prices of bonds (the calibrated or implied default intensity).58Default TimeDefault TimeStructural approach: is a predict
55、able stopping time with 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 c
56、omes as a surprise.59Credit RatingsSome more recent methods take into account not only the default event, but also the current and futures rating of each firm.In most cases, the process that models the up/downgrades is a Markov process. Instead of a default intensity, the whole matrix of intensities
57、 of migrations is specified.Official ratings are given by specialized rating agencies; they do not necessarily reflect (risk-neutral) probabilities of credit migrations.60Intensities of Migrations The matrix of intensities of credit migrations has the following form where K is the number of credit r
58、atings and the K-th class represents default event. State K is an absorbing state. 61References 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 Risk Analysis. J. Wiley & Sons 2000. B. Schmid: Pricing Credit Linked Financial Instruments. Springer 2002. D. Duffie and K. J. Singleton: Credit Risk, Princeton University Press 2003.CreditGrades II 63
