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Jacobs Mdl Rsk Par Crdt Der Risk Nov2011 V17 11 7 11

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It is not difficult to find situations of marked change in variables and with unpredictable event risk implies estimation problems. E.g., Credit spreads in 2008 rise to levels that could never have been forecast based upon previous history. The subprime crisis of 2007/8: credit spreads & volatility rise to unseen levels & shift in debtor behavior (delinquency patterns) E.g., estimating the volatility from data in a calm (turbulent) period implies under (over) estimation of future realized volatility
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  • 1. Risk Parameter Modeling for Credit Derivatives Michael Jacobs, Ph.D., CFA Senior Financial Economist Credit Risk Analysis Division U.S. Office of the Comptroller of the Currency Risk / Incisive Media Training, November 2011 The views expressed herein are those of the author and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.
  • 2. Outline <ul><li>Introduction and Motivation </li></ul><ul><li>Basic Concepts in Credit Derivative Valuation & Credit Risk Model Parameters </li></ul><ul><ul><li>The Structural Modeling Approach </li></ul></ul><ul><ul><li>The Reduced–Form Modeling Approach </li></ul></ul><ul><li>The Credit Curve and Market Implied Default Probabilities </li></ul><ul><li>Estimating Credit Risk Parameters from Historical Data </li></ul><ul><ul><li>Probability of Default (PD) </li></ul></ul><ul><ul><li>Loss-Given-Default (LGD) </li></ul></ul><ul><ul><li>Exposure-at-Default (EAD) </li></ul></ul><ul><ul><li>Correlations </li></ul></ul><ul><li>Mapping Risk Neutral to Physical PDs </li></ul><ul><li>PD Estimation Based on CDS Quotes vs. Vendor Model </li></ul><ul><li>Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk </li></ul>
  • 3. Introduction and Motivation: Parameters & Historical Data <ul><li>It is not difficult to find situations of marked change in variables and with unpredictable event risk implies estimation problems </li></ul><ul><li>Credit spreads in 2008 rise to levels that could never have been forecast based upon previous history </li></ul><ul><li>Subprime crisis of 2007/8: credit spreads & volatility rise to unseen levels & shift in debtor behavior (delinquency patterns) </li></ul><ul><li>E.g., estimating the volatility from data in a calm (turbulent) period implies under (over) estimation of future realized volatility </li></ul><ul><li>Markit Indices: most active traded single-name CDS contracts </li></ul><ul><li>Europe: BP premium 5 yr CDS contracts for 125 investment grade 1OO/25 ind./financial </li></ul><ul><li>Crossover: same for 35 junk rated </li></ul>
  • 4. Introduction and Motivation: Historical Data (continued) <ul><li>Guha et al FT Feb 2008:changes in delinquency behavior of U.S. homeowners U.S. mean lenders face losses much faster & decreases market value all residential mortgage loans </li></ul><ul><li>The maxim that history is not always a good indication of the future applies in assigning values to unknown quantities in credit derivative pricing and portfolio models </li></ul><ul><li>Most true in correlation estimation (unstable, stress, etc.) for portfolio models but applies broadly to other parameters </li></ul><ul><li>We do not suggest that historical data is never useful, as it is usually the best starting point for estimates when available </li></ul><ul><li>Point to highlight is that historical data should be supplemented by fundamental analysis of the environment or expert judgment </li></ul><ul><li>This analysis can point to changes in patterns or behavior in the future, which in turn require adjustment of values of parameters </li></ul><ul><li>Such changes in behavior obviously also present a significant source of uncertainty for credit derivative models </li></ul>
  • 5. Introduction and Motivation: Credit Risk <ul><li>Credit Risk (CR) : the potential loss in value of claims on counterparties due to reduced likelihood of fulfilling payment obligations or reduced value of collateral securing the obligation </li></ul><ul><li>Claims can be loans made to obligors, bonds bought, derivative transactions with counterparties or guarantees to customers </li></ul><ul><li>Credit risk is the single most important factor in bank failure & CR contributes more to bank risk than any other risk type </li></ul><ul><li>As lending could be considered a bank’s core competency, may seem contradictory, but likely when one considers correlation </li></ul><ul><li>Even if has sound credit analysis & avoids moral risk, balancing prudence & growth, concentration of losses potentially remains </li></ul><ul><li>Important to be aware of choices & assumptions: has a material impact on the results & how an institution can fine-tune its models for derivatives pricing & risk to optimize profitability </li></ul>
  • 6. Basic Concepts in Credit Derivative Valuation: Structural Models <ul><li>Consider the Black-Scholes-Merton (BSM) model: a firm with asset value A and equity value E has a zero-coupon bond with face value K, maturity T & Z d is the value of a unit zero-coupon bond maturing at T: </li></ul><ul><li>In the basic framework default is defined as the event A(T)<K, the probability of default (PD) in this model is given by: </li></ul>(1) <ul><li>The value of the defaultable debt at T is: </li></ul>(2) (3) <ul><li>Where Q denotes risk-neutral measure </li></ul>
  • 7. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.) <ul><li>This implies that the recovery rate RR, the complement of the loss-given-default (LGD = 1 - RR) rate, is given by: </li></ul><ul><li>The value of a defaultable bond is then the value of a long risk-free bond and a shot put option p: </li></ul>(4) (6) <ul><li>The exposure-at default (EAD) is simply fixed at K: </li></ul>(5) <ul><li>The risk credit spread on the bond is given trivially as: </li></ul>(7) <ul><li>The higher the value of the put sold to shareholders, the wider the credit spread, and the higher the risk-neutral PD, and the lower is the firm’s credit quality </li></ul>
  • 8. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.) <ul><li>The BSM model can be solved by assuming that firm value follows a geometric Brownian motion (GBM) : </li></ul><ul><li>This implies that firm value is log-normally distributed, and under the assumption of constant interest rates R(t,T) =r yields: </li></ul>(8) (9) (10) <ul><li>The credit spread is increasing in the leverage ratio: </li></ul><ul><li>The PD is simply the delta on the put: </li></ul>(11)
  • 9. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.) <ul><li>We can value a credit default swap in a simple first passage mode l extension of the BSM framework, that allows default to occur at any point up to maturity when firm value breaches K </li></ul><ul><li>Define the risk-neutral survival probability Q of a firm not defaulting at T given that it has survived to t: </li></ul>(4) (12) <ul><li>Given the log-normal dynamics of A(t) in (8), Musiela and Rutkowski (1998) provide a closed-form solution for Q(t,T): </li></ul>(13)
  • 10. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.) <ul><li>We can price a plain vanilla credit default swap (CDS) written on this firm with this model. The time t value η of the premium leg for a protection buyer paying constant continuous spread S: </li></ul>(4) (14) <ul><li>Assuming unit notional, the value of the protection leg θ is a contingent claim that pays 1 – RR in the event of default: </li></ul>(15) <ul><li>As the CDS has zero market value at inception, we equate (14) and (15) to solve for the spread S: </li></ul>(16)
  • 11. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.) <ul><li>This can be simplified and the relation to the annualized PD can be shown as follows by assuming PD approximately constant: </li></ul>(4) (17) <ul><li>Then (16) becomes the more familiar expression: </li></ul>(18) <ul><li>This says that the credit spread is the expected loss (EL), under risk neutral measure, which is EL = LGDXPD = (1-RR)XPD, which depends upon LGD and PD parameters to be estimated </li></ul>
  • 12. Implementing Structural Models: KMV Portfolio Manager TM <ul><li>EAD: Default point is long-term debt & ½ of short term debt </li></ul><ul><li>Assets follow geometric Brownian motion and equity is a down-&-out call option with indefinite maturity </li></ul><ul><li>Asset volatility derived from historical vol & current value equity </li></ul><ul><li>PD ( expected default frequency- ”EDF”): distance-to-default in a statistical relationship to historical default rates </li></ul><ul><li>Value loans at 1-yr using term structure EDFs & assumed LGD </li></ul><ul><li>Model dependence in asset values with equity correlations & factor model (global, regional, industry & firm-specific) </li></ul><ul><li>For private firms cannot do this directly but use EDFs from comparable public firms (by industry, geography, etc.) </li></ul><ul><li>Model performance depends upon reasonableness of assumptions – not intuitive for small business or retail </li></ul>
  • 13. Implementing Structural Models: Basel II Asymptotic Single Risk Factor Framework (ASRF) <ul><li>Assumptions: an infinitely-grained, homogenous credit portfolio affected by a single factor </li></ul><ul><li>ν :firm i asset value at time t, ε (x): idiosyncratic (common time-specific), ρ : asset value correlation, T * = Φ -1 (PD): default point </li></ul><ul><li>Implies that asset value is conditionally Normal and then can solve this for the year t conditional default rate: </li></ul><ul><li>Evaluating this at a low quantile of X (e.g., -2.99) with constant LGD and EAD yields the formula for regulatory capital: </li></ul>
  • 14. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models <ul><li>Rather than model the fundamentals of an issuer, this approach take default to be exogenous and to occur at a random time τ : </li></ul>(19) <ul><li>The unconditional (since we do not specify what happens between t and T) forward PD for defaulting in interval (T ,U) is: </li></ul>(20) <ul><li>It follows from (19) that the forward probability of defaulting before time U>t PD: </li></ul>(21) <ul><li>By Bayes Rule the t conditional survival probability for (T,U) is: </li></ul>(22)
  • 15. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) <ul><li>It follows from (22) that the time t conditional PD for (T,U) is: </li></ul>(23) <ul><li>Now define the normalized measure H, (22) evaluated at t=T divided by time period T-t, a conditional forward average PD: </li></ul>(24) <ul><li>Similarly, the average forward PD for (T,U) is: </li></ul>(25)
  • 16. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) <ul><li>The instantaneous forward default rate follows from the limit: </li></ul>(26) <ul><li>Then an important result is the conditional survival probability (22) has the negative exponential representation: </li></ul>(27) <ul><li>h(t,T,U) is a sufficient statistic for calculating default probabilities over any time interval (t,U) conditioning on time t information & is related to the instantaneous PD by: </li></ul>(28)
  • 17. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) (29) <ul><li>Instantaneous forward PD h is related to a key quantity in RFMs, default intensity λ , which is the “spot” PD: </li></ul><ul><li>Else if λ is stochastic, due to changes in systematic or firm-specific factors, then under appropriate regularity conditions: </li></ul>(30) <ul><li>For t<s, λ (s)depends upon all information at s, but h(t,s) conditions only upon survival to s, and λ (s) = h(t,t) </li></ul><ul><li>They coincide if λ is deterministic: </li></ul>(31)
  • 18. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) (32) <ul><li>If we assume λ to be constant, we may estimate it from CDS quotes (assuming a fixed or known LGD), or fro the frequency of default events (e.g., λ is the mean of a Poisson process) </li></ul><ul><li>If we want to assume random λ , a simple & intuitive model is the square-root process of Cox, Ingersoll & Ross (1985; CIR): </li></ul><ul><li>We could estimate these parameters from historical loss rates or CDS spreads, or calibrate closed-form CIR solutions for risky bonds directly to prices quotes </li></ul><ul><li>The final piece that we need for pricing is the probability density of the random default time τ : </li></ul>(33)
  • 19. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) <ul><li>(29) and (30) imply that g is related to λ by: </li></ul>(35) (34) <ul><li>Which for deterministic λ reduces to: </li></ul><ul><li>Assuming zero recovery and interest rates independent of the default intensity, the value of a defaultable bond is: </li></ul>(36) <ul><li>The risky bond spread can be shown to be: </li></ul>(37)
  • 20. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) <ul><li>We can incorporate LGD as by valuing the defaultable bond as a portfolio of a zero-recovery bond and a contingent claim that pays RR = 1 – LGD at time t if there is default , otherwise zero: </li></ul>(40) (38) <ul><li>Assuming constant default intensity and interest rates yields: </li></ul>(39) <ul><li>In the simple model premium & protections legs of a CDS are given as follows and yield the equality of the spread and EL: </li></ul>(41)
  • 21. The Credit Curve and Market Implied Default Probabilities <ul><li>Now in discrete time, consider a CDS with $1 notional at time t with premium S n and payment dates [T 1 ,…,T n ]. In the event of default the protection seller pays 0 ≤ LGD = 1-RR ≤ 1. Then under risk neutrality and independence of default from risk-free interest rates, the present value of the premium leg can be written: </li></ul>(4) (41) <ul><li>Where Z(t,T i ) is the time t price of a unit zero-coupon bond that pays $1 at T j and Q(t,T i ) is the reference entity’s survival probability through T i conditional of surviving up to t. Similarly, the PV of protection leg may be written as: </li></ul>(42) <ul><li>Where τ is the random time of default and Pr t Q [T i-1 < τ ≤ T i ] is the time t conditional PD of the reference entity in this interval </li></ul>
  • 22. The Credit Curve and Market Implied Default Probabilities (cont’d.) <ul><li>We can rewrite (20) by noting that conditional PD is the minus the increment in Q (ie, the rate of decay in survival probability): </li></ul>(4) (44) <ul><li>We can solve for the survival probabilities Q recursively from the observed vector of market spreads by equating (19)&(20): </li></ul>(43) (45)
  • 23. The Credit Curve and Market Implied Default Probabilities (cont’d.) <ul><li>PB (JI) is highly (low) rated and has an increasing (decreasing) credit curve </li></ul><ul><li>Intuition? – PB has nowhere to go but down, but if JI with high short term PD survives more likely to be upgraded </li></ul><ul><li>If we make a flat CDS curve assumption of uniformity at the spread S = S 5 then the calibration is significantly simplified: </li></ul>(46)
  • 24. The Credit Curve and Market Implied Default Probabilities (cont’d.) <ul><li>The flat CDS curve assumption means that the 5-yr premium is a 1-yr PD measure under zero recovery </li></ul><ul><li>While PDs are increasing in LGD, for PB the sensitivity to horizon is slightly increasing for lower LGD, and the opposite for JI </li></ul>(4)
  • 25. Defining and Estimating Credit Risk Parameters: PD <ul><li>PD : estimate of the probability that a counterparty will default over a given horizon (should reflect obligor’s creditworthines) </li></ul><ul><li>Key to PD estimation is defining a default event: narrow (bankruptcy / loss) vs. broad (agencies / Basel II)->different magnitudes of estimate </li></ul><ul><li>Ideally PD is an obligor phenomenon (e.g., Basel 2), but in reality loan structure or 3 rd party support matters, which is a challenge </li></ul><ul><li>Point-in-time (e.g., SM, RF models) vs. through-the-cycle : (e.g., RMM models calibrated to agencies): EC estimate will inherit this </li></ul><ul><ul><li>In reality most banks rating systems are somewhere in between PIT & TTC </li></ul></ul><ul><li>Important to distinguish the system for rating customers from the method to assigning PD estimates </li></ul><ul><li>Common way to estimate PD is to take average default rates in ratings over a cycle -> TTC system (more common in C&I vs. retail) </li></ul><ul><li>Another popular way to rate is by a partly judgmental scorecard that may be backtested over time </li></ul><ul><li>Less common in wholesale: statistical/econometric models of PD </li></ul>
  • 26. PD Estimation for Credit Models: Rating Agency Data <ul><li>Credit rating agencies have a long history in providing estimates of firms’ creditworthiness </li></ul><ul><li>Information about firms’ creditworthiness has historically been difficult to obtain </li></ul><ul><li>In general, agency ratings rank order firms’ likelihood of default over the next five years </li></ul><ul><li>However, it is common to take average default rates by ratings as PD estimates </li></ul><ul><li&g
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