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., …

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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  • Eg,: credit (PD, LGD, EAD), market (expected returns), insurance (frequency/severity claims, life expectancy)
  • CR incl loss to dflt Note that bonds/loans may either be held to mat or for trading purp (bnk vs trd bk); loans may be under accr or fair value acc. Imp. of der. (or CP) CR illustrated by the Lehman failure (Moodys 2008): CDS trades ref LB=72B, LBsold CDS 2-3T & CP in many other der contr (FX, IR) See Warren Buiffet’s sharholder letter2001 in the context of insurance Eg, Cont Illinois 1984, Nippon Credit Bank 1998, Danish Roskilde Bank 2008. Contin: comp EC to avail cap, prof.: RAROC
  • While theor EDF can be solved for, this treatment compensate for unreal mdl ass (& still calibr isues remian -
  • Eg., CDS premia reflect PD & rec rate (EL=PD*LGD) – if make assumpt on LGD (constant) can back out PD There is a diff flavor of this that econometrically est PD from obs dflt – “hazzard rate mdls” Probl w/rtg trans from ag – loose link to issuer or ref ent ( Li 2000: if copula (just a mult cum dsn unif rvs –can link diff & arb marg dsns ) Guassian -> same as CreditMetr E.g., last appr Hull & White 2008 JD
  • Note that disc factors Z comp from term str LIBOR & swap rates cons with interp of CDS prem as a floater yld spr on ref entity rel to libor
  • Note that disc factors Z comp from term str LIBOR & swap rates cons with interp of CDS prem as a floater yld spr on ref entity rel to libor
  • Note that disc factors Z comp from term str LIBOR & swap rates cons with interp of CDS prem as a floater yld spr on ref entity rel to libor
  • May be direct inp (RMM) or der obl inf (SM) We want our est not to refl things out of contr obl –e.g., trans&conv event for country freezes outflows Eg coll matters: AIG tripped received govt loan to post coll & avoided dflt – is this real or not? Dflt def: ag (bankrupt,ren debt, missed payment-they claim basically B2), B2 (bank det unl to pay or obl 90dpd any mat obs-now typically same as nonaccr, but some diff do exist) PIT: PD refl curr sit->obl quickly upgr/downgr & DRs by rating same acr cycle, TTC: stable ratings but DRs fluctuate Scrcrd sys:popular because don’t rely on extensive internal default data (esp. for low dflt portf.) Stat mdls: more prev in rtl due to much dflt data
  • May be direct inp (RMM) or der obl inf (SM) We want our est not to refl things out of contr obl –e.g., trans&conv event for country freezes outflows Eg coll matters: AIG tripped received govt loan to post coll & avoided dflt – is this real or not? Dflt def: ag (bankrupt,ren debt, missed payment-they claim basically B2), B2 (bank det unl to pay or obl 90dpd any mat obs-now typically same as nonaccr, but some diff do exist) PIT: PD refl curr sit->obl quickly upgr/downgr & DRs by rating same acr cycle, TTC: stable ratings but DRs fluctuate Scrcrd sys:popular because don’t rely on extensive internal default data (esp. for low dflt portf.) Stat mdls: more prev in rtl due to much dflt data
  • May be direct inp (RMM) or der obl inf (SM) We want our est not to refl things out of contr obl –e.g., trans&conv event for country freezes outflows Eg coll matters: AIG tripped received govt loan to post coll & avoided dflt – is this real or not? Dflt def: ag (bankrupt,ren debt, missed payment-they claim basically B2), B2 (bank det unl to pay or obl 90dpd any mat obs-now typically same as nonaccr, but some diff do exist) PIT: PD refl curr sit->obl quickly upgr/downgr & DRs by rating same acr cycle, TTC: stable ratings but DRs fluctuate Scrcrd sys:popular because don’t rely on extensive internal default data (esp. for low dflt portf.) Stat mdls: more prev in rtl due to much dflt data
  • May be direct inp (RMM) or der obl inf (SM) We want our est not to refl things out of contr obl –e.g., trans&conv event for country freezes outflows Eg coll matters: AIG tripped received govt loan to post coll & avoided dflt – is this real or not? Dflt def: ag (bankrupt,ren debt, missed payment-they claim basically B2), B2 (bank det unl to pay or obl 90dpd any mat obs-now typically same as nonaccr, but some diff do exist) PIT: PD refl curr sit->obl quickly upgr/downgr & DRs by rating same acr cycle, TTC: stable ratings but DRs fluctuate Scrcrd sys:popular because don’t rely on extensive internal default data (esp. for low dflt portf.) Stat mdls: more prev in rtl due to much dflt data
  • May be direct inp (RMM) or der obl inf (SM) We want our est not to refl things out of contr obl –e.g., trans&conv event for country freezes outflows Eg coll matters: AIG tripped received govt loan to post coll & avoided dflt – is this real or not? Dflt def: ag (bankrupt,ren debt, missed payment-they claim basically B2), B2 (bank det unl to pay or obl 90dpd any mat obs-now typically same as nonaccr, but some diff do exist) PIT: PD refl curr sit->obl quickly upgr/downgr & DRs by rating same acr cycle, TTC: stable ratings but DRs fluctuate Scrcrd sys:popular because don’t rely on extensive internal default data (esp. for low dflt portf.) Stat mdls: more prev in rtl due to much dflt data
  • A competitor to the well-known KMV model – the structural EDF based on Merton (1973) Refs: van Deventer & Imai book (2003), academic paper Chava & Jarrow RF 2004, Hosmer & Lemeshow (2000) bk log regr Just as diff classes of EC mdl, same for the drivers (and as PD is driver of EC, PD has its own drivers) Allows different expl var’s/mdls for diff hor
  • Mlt LGD: avail only for mark debt, subj to ill/swings inv sent; W.O. / ult LGD: takes many years to get data, the B II std for many banks (esp middle mkt or priv debt portfolios), probl in meas (need all mat costs-coll costs, dir + indir) Diff WO prac -> banks see diff d-LGD behavior in diff portf (also
  • Mlt LGD: avail only for mark debt, subj to ill/swings inv sent; W.O. / ult LGD: takes many years to get data, the B II std for many banks (esp middle mkt or priv debt portfolios), probl in meas (need all mat costs-coll costs, dir + indir) Diff WO prac -> banks see diff d-LGD behavior in diff portf (also
  • Mlt LGD: avail only for mark debt, subj to ill/swings inv sent; W.O. / ult LGD: takes many years to get data, the B II std for many banks (esp middle mkt or priv debt portfolios), probl in meas (need all mat costs-coll costs, dir + indir) Diff WO prac -> banks see diff d-LGD behavior in diff portf (also
  • Dflt Rate Serv d.b. – mkt LGD , MURD: ult LGD
  • Dflt Rate Serv d.b. – mkt LGD , MURD: ult LGD
  • Dflt Rate Serv d.b. – mkt LGD , MURD: ult LGD
  • Facility ultimate LGD de(in)creasing in creditor rank, collateral quality, tranche thickness (time-to-maturity,EAD,ultimate obligor LGD, market LGD) Firm ultimate LGD de(in)creasing in leverage, liquidity, cash flow, size, profitability,industry utility/profit,time-between defaults,% secured or bank debt,CARs, prepack,S&P return, investment grade at origination (intangibility,Tobin’s Q, industry tech, # creditor classes, obligor market LGD, bankruptcy filing,recession period,Moody’s default rate)
  • Typically borr going into dflt will try to draw down on credit lines as liqu or alt funding dries up Der. WWE ex.: 1. cross-FX swap with weaker curr CP: more likely to dflt just when curr weakens & bank is in the $ 2. CDS purch prot & insurer is deter same time as the ref entity As either borr deteriorates or in downturn, EAD risk may become lower as banks cut lines
  • Looked at dflt rev in Moody’s MURD database & traced exposure back in fin filings (10Q &10K reports) Similar to JPMC (2001) study, added a few variables, and tried alt meas EAD risk to LEQ factor Caveat: onlt defaults up to early 2009, somewhat sens to the part meas, r^2 still low given # var’s ,judg calls in reading fin statements
  • Contag.: phen that it is not only gen ec that makes firms default, but 2 nd order feedback eff (eg, real est./subpr crsis-dflt->suply overhang & neg wealth eff->depr ec cond further->more defaults) E.g., high frequ equ price (daily, weekly) corr can show small corr betw cycl & oncycl ind, but longer term (quart, ann) loss data can show high dep->need to analyze sens of estm to this Eg, incr lev & PD->decr value equ, which is consis with decr asset vol (equ is call opt); emp evid Gordy and HeitfeldL (2002) Eg, data sources: losses, equities, CDS
  • Jacobs, Michael. (2010) “ Modeling the Time Varying Dynamics of Correlations: Applications for Forecasting and Risk Management ,” (with Ahmet Karagozoglu). Working Paper. Estimates over longer moving windows are smoother overall, but shorter window estimates can look to be zero over shorter time periods Corr can go from very negative to very pos from one time period to another – structural breaks Different sectors can have very diff avg corr to the broader market-implic for div
  • Case of strured prod (tranche of RMBS) this is an order of magn more sens
  • Starting pnt usually an annual migr matr See JPM 1997 tech doc CQV sim to AV process in str mdls & dflt thrhld like dflt buckets
  • Ques. re equ corr as proxy dflt corr: De Servigny and Renault fin doverall equ only slightly hiugher, but large dev @ ind lvl -> reas for well div acr ind portf Kiesel et al: incl spr vol -> sign incr EC (esp high cr qual); called specific risk SM have av of mdl firm spec PDs & equ pr very resp chngs in cr qual (but many bank’s portf priv – but can use int ratings) CPV not widely used: diff in est rel spec DR & macro var’s acr ind & geog’s Bangia et al have a similarappr that est migr matr cond on the econom stste gd vs. bad & prob of state (switching mdl)
  • Starting pnt usually an annual migr matr See JPM 1997 tech doc CQV sim to AV process in str mdls & dflt thrhld like dflt buckets
  • 22 facilities, 8 names (comb fac for same name – same as 100% corr)

Transcript

  • 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
    • Introduction and Motivation
    • Basic Concepts in Credit Derivative Valuation & Credit Risk Model Parameters
      • The Structural Modeling Approach
      • The Reduced–Form Modeling Approach
    • The Credit Curve and Market Implied Default Probabilities
    • Estimating Credit Risk Parameters from Historical Data
      • Probability of Default (PD)
      • Loss-Given-Default (LGD)
      • Exposure-at-Default (EAD)
      • Correlations
    • Mapping Risk Neutral to Physical PDs
    • PD Estimation Based on CDS Quotes vs. Vendor Model
    • Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk
  • 3. Introduction and Motivation: Parameters & Historical Data
    • It is not difficult to find situations of marked change in variables and with unpredictable event risk implies estimation problems
    • Credit spreads in 2008 rise to levels that could never have been forecast based upon previous history
    • 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
    • Markit Indices: most active traded single-name CDS contracts
    • Europe: BP premium 5 yr CDS contracts for 125 investment grade 1OO/25 ind./financial
    • Crossover: same for 35 junk rated
  • 4. Introduction and Motivation: Historical Data (continued)
    • 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
    • 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
    • Most true in correlation estimation (unstable, stress, etc.) for portfolio models but applies broadly to other parameters
    • We do not suggest that historical data is never useful, as it is usually the best starting point for estimates when available
    • Point to highlight is that historical data should be supplemented by fundamental analysis of the environment or expert judgment
    • This analysis can point to changes in patterns or behavior in the future, which in turn require adjustment of values of parameters
    • Such changes in behavior obviously also present a significant source of uncertainty for credit derivative models
  • 5. Introduction and Motivation: Credit Risk
    • 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
    • Claims can be loans made to obligors, bonds bought, derivative transactions with counterparties or guarantees to customers
    • Credit risk is the single most important factor in bank failure & CR contributes more to bank risk than any other risk type
    • As lending could be considered a bank’s core competency, may seem contradictory, but likely when one considers correlation
    • Even if has sound credit analysis & avoids moral risk, balancing prudence & growth, concentration of losses potentially remains
    • 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
  • 6. Basic Concepts in Credit Derivative Valuation: Structural Models
    • 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:
    • In the basic framework default is defined as the event A(T)<K, the probability of default (PD) in this model is given by:
    (1)
    • The value of the defaultable debt at T is:
    (2) (3)
    • Where Q denotes risk-neutral measure
  • 7. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.)
    • This implies that the recovery rate RR, the complement of the loss-given-default (LGD = 1 - RR) rate, is given by:
    • The value of a defaultable bond is then the value of a long risk-free bond and a shot put option p:
    (4) (6)
    • The exposure-at default (EAD) is simply fixed at K:
    (5)
    • The risk credit spread on the bond is given trivially as:
    (7)
    • 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
  • 8. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.)
    • The BSM model can be solved by assuming that firm value follows a geometric Brownian motion (GBM) :
    • This implies that firm value is log-normally distributed, and under the assumption of constant interest rates R(t,T) =r yields:
    (8) (9) (10)
    • The credit spread is increasing in the leverage ratio:
    • The PD is simply the delta on the put:
    (11)
  • 9. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.)
    • 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
    • Define the risk-neutral survival probability Q of a firm not defaulting at T given that it has survived to t:
    (4) (12)
    • Given the log-normal dynamics of A(t) in (8), Musiela and Rutkowski (1998) provide a closed-form solution for Q(t,T):
    (13)
  • 10. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.)
    • 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:
    (4) (14)
    • Assuming unit notional, the value of the protection leg θ is a contingent claim that pays 1 – RR in the event of default:
    (15)
    • As the CDS has zero market value at inception, we equate (14) and (15) to solve for the spread S:
    (16)
  • 11. Basic Concepts in Credit Derivative Valuation: Structural Models (cont.)
    • This can be simplified and the relation to the annualized PD can be shown as follows by assuming PD approximately constant:
    (4) (17)
    • Then (16) becomes the more familiar expression:
    (18)
    • 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
  • 12. Implementing Structural Models: KMV Portfolio Manager TM
    • EAD: Default point is long-term debt & ½ of short term debt
    • Assets follow geometric Brownian motion and equity is a down-&-out call option with indefinite maturity
    • Asset volatility derived from historical vol & current value equity
    • PD ( expected default frequency- ”EDF”): distance-to-default in a statistical relationship to historical default rates
    • Value loans at 1-yr using term structure EDFs & assumed LGD
    • Model dependence in asset values with equity correlations & factor model (global, regional, industry & firm-specific)
    • For private firms cannot do this directly but use EDFs from comparable public firms (by industry, geography, etc.)
    • Model performance depends upon reasonableness of assumptions – not intuitive for small business or retail
  • 13. Implementing Structural Models: Basel II Asymptotic Single Risk Factor Framework (ASRF)
    • Assumptions: an infinitely-grained, homogenous credit portfolio affected by a single factor
    • ν :firm i asset value at time t, ε (x): idiosyncratic (common time-specific), ρ : asset value correlation, T * = Φ -1 (PD): default point
    • Implies that asset value is conditionally Normal and then can solve this for the year t conditional default rate:
    • Evaluating this at a low quantile of X (e.g., -2.99) with constant LGD and EAD yields the formula for regulatory capital:
  • 14. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models
    • Rather than model the fundamentals of an issuer, this approach take default to be exogenous and to occur at a random time τ :
    (19)
    • The unconditional (since we do not specify what happens between t and T) forward PD for defaulting in interval (T ,U) is:
    (20)
    • It follows from (19) that the forward probability of defaulting before time U>t PD:
    (21)
    • By Bayes Rule the t conditional survival probability for (T,U) is:
    (22)
  • 15. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.)
    • It follows from (22) that the time t conditional PD for (T,U) is:
    (23)
    • Now define the normalized measure H, (22) evaluated at t=T divided by time period T-t, a conditional forward average PD:
    (24)
    • Similarly, the average forward PD for (T,U) is:
    (25)
  • 16. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.)
    • The instantaneous forward default rate follows from the limit:
    (26)
    • Then an important result is the conditional survival probability (22) has the negative exponential representation:
    (27)
    • 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:
    (28)
  • 17. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) (29)
    • Instantaneous forward PD h is related to a key quantity in RFMs, default intensity λ , which is the “spot” PD:
    • Else if λ is stochastic, due to changes in systematic or firm-specific factors, then under appropriate regularity conditions:
    (30)
    • 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)
    • They coincide if λ is deterministic:
    (31)
  • 18. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.) (32)
    • 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)
    • If we want to assume random λ , a simple & intuitive model is the square-root process of Cox, Ingersoll & Ross (1985; CIR):
    • 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
    • The final piece that we need for pricing is the probability density of the random default time τ :
    (33)
  • 19. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.)
    • (29) and (30) imply that g is related to λ by:
    (35) (34)
    • Which for deterministic λ reduces to:
    • Assuming zero recovery and interest rates independent of the default intensity, the value of a defaultable bond is:
    (36)
    • The risky bond spread can be shown to be:
    (37)
  • 20. Basic Concepts in Derivative Valuation and Parameter Estimation: Reduced-Form Models (cont’d.)
    • 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:
    (40) (38)
    • Assuming constant default intensity and interest rates yields:
    (39)
    • In the simple model premium & protections legs of a CDS are given as follows and yield the equality of the spread and EL:
    (41)
  • 21. The Credit Curve and Market Implied Default Probabilities
    • 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:
    (4) (41)
    • 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:
    (42)
    • 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
  • 22. The Credit Curve and Market Implied Default Probabilities (cont’d.)
    • We can rewrite (20) by noting that conditional PD is the minus the increment in Q (ie, the rate of decay in survival probability):
    (4) (44)
    • We can solve for the survival probabilities Q recursively from the observed vector of market spreads by equating (19)&(20):
    (43) (45)
  • 23. The Credit Curve and Market Implied Default Probabilities (cont’d.)
    • PB (JI) is highly (low) rated and has an increasing (decreasing) credit curve
    • Intuition? – PB has nowhere to go but down, but if JI with high short term PD survives more likely to be upgraded
    • If we make a flat CDS curve assumption of uniformity at the spread S = S 5 then the calibration is significantly simplified:
    (46)
  • 24. The Credit Curve and Market Implied Default Probabilities (cont’d.)
    • The flat CDS curve assumption means that the 5-yr premium is a 1-yr PD measure under zero recovery
    • While PDs are increasing in LGD, for PB the sensitivity to horizon is slightly increasing for lower LGD, and the opposite for JI
    (4)
  • 25. Defining and Estimating Credit Risk Parameters: PD
    • PD : estimate of the probability that a counterparty will default over a given horizon (should reflect obligor’s creditworthines)
    • Key to PD estimation is defining a default event: narrow (bankruptcy / loss) vs. broad (agencies / Basel II)->different magnitudes of estimate
    • 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
    • Point-in-time (e.g., SM, RF models) vs. through-the-cycle : (e.g., RMM models calibrated to agencies): EC estimate will inherit this
      • In reality most banks rating systems are somewhere in between PIT & TTC
    • Important to distinguish the system for rating customers from the method to assigning PD estimates
    • 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)
    • Another popular way to rate is by a partly judgmental scorecard that may be backtested over time
    • Less common in wholesale: statistical/econometric models of PD
  • 26. PD Estimation for Credit Models: Rating Agency Data
    • Credit rating agencies have a long history in providing estimates of firms’ creditworthiness
    • Information about firms’ creditworthiness has historically been difficult to obtain
    • In general, agency ratings rank order firms’ likelihood of default over the next five years
    • However, it is common to take average default rates by ratings as PD estimates
    • The figure shows that agency ratings reflect market segmentations
  • 27. PD Estimation: Rating Agency Data – Migration & Default Rates
    • ,….
    • Migration matrices summarize the average rates of transition between rating categories
    • The default rates in the final column are often taken as PD estimates for obligor rated similarly to the agency ratings
    • Default rates are increasing for worse ratings & as the time horizons increase
  • 28. PD Estimation: Rating Agency Data – Default Rates*
    • ,….
    • Default rates tend to rise in downturns and are higher for speculative than investment grade ratings in most years
    • Investment grade default rates are very volatile and zero in many years, with an extremely skewed distribution
    *Reproduced with permission from: Moody’s Investor Services / Credit Policy, Special Comment: Corporate Default an and Recovery Rates 1970-2010, 2 -28-11.
  • 29. PD Estimation: Rating Agency Data – Performance of Ratings
    • ,….
    • Issuers downgraded to the B1 level as early as five years prior to default, B3 among issuers that defaulted in 2010
    • Cumulative accuracy profile (CAP) curve for 2010 bows towards the northwest corner more than the one for the 1983-2010 period, which suggests recent rating performance better than the historical average
    • 1-year accuracy ratio (AR) is positively correlated with the credit cycle, less so at 5 years
  • 30. PD Estimation for Credit Models: Kamakura Public Firm Model*
    • This vendor provides a suite of PD models (structural, reduced-form & hybrid) all based upon logistic regression techniques
    • Similar to credit scoring models in retail: directly estimate PD using historical data on defaults and observable explanatory variables
    • Kamakura Default Probability (KDP) estimate of PD:
      • X: explanatory variables
      • α , β : coefficient estimates
      • Y: default indicator (=1,0 if default,survive)
      • i,j,t, τ : indexes firm, variable, calendar time, time horizon
    • “ Leading” Jarrow-Chava model: based on 1990-2010 actual defaults all listed companies N. America (1,764,230 obs. & 2,064 defaults)
    • Variables included in the final model:
      • Accounting: net income, cash, total assets & liabilities, number of shares
      • Macro: 1 mo. LIBOR, VIX, MIT CRE, 10 govt. bond yld, GDP, unemployment rate, oil price
      • 3 stock price-related: firm & market indices, firm percentile rank
      • 2 other variables: industry sector & month of the year
    *Reproduced with permission from: Kamakura Corporation (Donald van Deventer), Kamakura Pubic Firm Model: Technical Document, September, 2011.
  • 31. PD Estimation for Credit Models: Kamakura Public Firm Model (cont.)
    • Area Under the Receiver Operating Curve (AUROC) : measure rank ordering power of models to distinguish default risk at different horizon & models decent but reduced form dominates structural model
    • Comparison of predicted PD vs. actual default rate measures accuracy of models: broadly consistent with history & RFM performs better than SFM
    • Issues & supervisory concerns with this: overfitting (“kitchen sink” modeling) and concerns about out-sample-performance
    *Reproduced with permission from: Kamakura Corporation (Donald van Deventer), Kamakura Pubic Firm Model: Technical Document, September, 2011. *
  • 32. PD Estimation for Credit Models: Bayesian Model*
    • Jacobs & Kiefer (2010): Bayesian 1 (Binomial – rating agencies), 2 (Basel II ASRF) & 3-parameter extension (Generalized Linear Mixed Models) models
    • Combines default rates for Moody’s Ba rated credits 1999-2009 in conjunction with an expert elicited prior distribution for PD
    • Coherent incorporation of expert information (formal elicitation & fitting of a prior) with limited data & in line with supervisory validation expectations
    • A secondary advantage is access to efficient computational methods such as Markov Chain Monte Carlo (MCMC)
    • Evidence that expert information can result in a reasonable posterior distribution of the PD given limited data information
    • Findings: Basel 2 asset value correlations may be mispecified (too high) & systematic factor mildly (positively) autocorrelated
    * Jacobs Jr., M., and N. M. Kiefer (2010) “The Bayesian Approach to Default Risk: A Guide,” (with.) in Ed.: Klaus Boecker, Rethinking Risk Measurement and Reporting (Risk Books, London)..
  • 33. PD Estimation for Credit Models: Bayesian Model (cont.)
    • Ba default rate 0.9%, both prior & posterior centered at 1%, 95% credible interval = (0.7%, 1.4%)
    • Prior on rho a diffuse beta distribution centered at typical Basel 2 value 20%, posterior mean 8.2%, 95%CI = (4%,13%),
    • Prior on tau uniform centered at 0%, posterior mean 16.2%, 95% CI (-.01%, 29.2%)
  • 34. Loss Given Default Estimation for Credit Models
    • LGD : estimate of the amount a bank loses if a counterparty defaults (expected PV of economic loss / EAD or 1 minus the recovery rate)
    • Depends on claim seniority, collateral, legal jurisdiction, condition of defaulted firm or capital structure, bank practice, type of exposure
    • Measured LGDs depend on default definition: broader (distressed exchange/reneg.) vs. narrow (bankruptcy,liquidation)->lower/higher
    • Market vs. workout LGD: prices of defaulted debt shortly after default vs. realized discounted ultimate recoveries up to resolution
    • LGDs on individual instruments tends to be either very high (sub or unsecured debt) or very low (secured bonds or loans) - “bimodal”
    • Downturn LGD : intuition & evidence that should be elevated in economic downturns – but mixed evidence & role of bank practice
    • Note differences across different types of lending (e.g., enterprise value & debt markets is particular large corporate)
  • 35. LGD Estimation for Credit Models: Capital Structure
    • Contractual features: more senior and secured instruments do better.
    • Absolute Priority Rule: some violations (but usually small)
    • More senior instruments tend to be better secured.
    • Debt cushion as distinct from position in the capital structure.
    • High LGD for senior debt with little sub-debt?
    • Proportion of bank debt
    • The “Grim Reaper” story
    • Enterprise value
    SENIORITY Bank Loans Senior Secured Senior Unsecured Senior Subordinated Junior Subordinated Preferred Shares Common Shares Employees, Trade Creditors, Lawyers Banks Bondholders Shareholders
  • 36. LGD Estimation for Credit Models: Default Process*
    • Bankruptcies (65.2%) have higher LGDs than out-of-court settlements (55.8%)
    • Firms reorganized (emerged or acquired) have lower LGDs (43.9%) than firms liquidated (68.9%)
    *Diagram reproduced from: Jacobs, M., et al., 2011, Understanding and predicting the resolution of financial distress, Forthcoming Journal of Portfolio Management (March,2012), page 31. 518 defaulted S&P/Moody’s rated firms 1985-2004.
  • 37. LGD Estimation for Credit Models: Collateral and Seniority
    • Distributions of Moody’s Defaulted Bonds & Loan LGD (DRS Database 1970-2010)
    • Lower the quality of collateral, the higher the LGD
    • Lower ranking of the creditor class, the higher the LGD
    • And higher seniority debt tends to have better collateral
    * Reproduced with permision: Moody’s Analytics.Default Rate Service Database, 10-15-10. * Reproduced with permission: Moody’s, URD, Release 10-15-10.
  • 38. LGD Estimation for Credit Models: The Business Cycle*
    • Downturns: 1973-74, 1981-82, 1990-91, 2001-02, 2008-09
    • As noted previously, commonly accepted that LGD is higher during economic downturns when default rates are elevated
    • Lower collateral values
    • Greater supply of distressed debt
    • The cycle is evident in time series, but note all the noise
    * Reproduced with permission: Moody’s Analytics. Default Rate Service Database, Release Date 10-15-10.
  • 39. LGD Estimation for Credit Models: Judgmental Decision Tree for Corporate Unsecured
  • 40. LGD Estimation for Credit Models: Statistical Model
    • Jacobs & Karagozoglu (2011)* study ultimate LGD in Moody’s URD at the loan & firm level simultaneously
    • Empirically models notion that recovery on a loan is akin to a collar option on the firm/enterprise level recovery
    • Firm (loan) LGD depends on fin ratios, capital structure, industry state, macroeconomy, equity market / CARs (seniority, collateral quality, debt cushion)
    • Feedback from ultimate obligor LGD to the facility level & at both level ultimate LGD depends upon market
    * Jacobs, Jr., M., and Karagozoglu, A, 2011, Modeling ultimate loss given default on corporate debt, The Journal of Fixed Income, 21:1 (Summer), 6-20.
  • 41. Exposure at Default Estimation for Credit Models
    • EAD : an estimate of the dollar amount of exposure on an instrument if there is a counterparty / obligor default over some horizon
    • Typically, a borrower going into default will try to draw down on credit lines as liquidity or alternative funding dries up
    • Correlation between EAD & PD for derivatives exposure: wrong way exposure (WWE) problem: higher exposure & more default risk
    • Derivative WWE examples
      • A cross-FX swap with weaker a currency counterparty: more likely to default just when currency weakens & banks are in the money
      • A bank purchases credit protection through a CDS & the insurer is deteriorating at the same time as the reference entity
    • Although Basel II stipulates “margin of conservatism” for EAD, in the case of loans greater monitoring->negative correlation with PD
    • As either borrower deteriorates or in downturn conditions, EAD risk may actually become lower as banks cut lines
  • 42. EAD Estimation for Credit Models: Defaultable Loans
    • Typically banks estimate EAD by a loan equivalency quotient (LEQ): fraction of unused drawn down in default over total current availability:
    • Where O: outstanding, L: limit, t: current time, τ : time of default, T: horizon, X : vector of risk factors , E t (.) mathematical expectation
    • For traditional credit products depends on loan size, redemption schedule, covenants, bank monitoring, borrower distress, pricing
    • Case of unfunded commitments (e.g., revolvers): EAD anywhere from 0% to 100% of line limit (term loans typically just face value)
  • 43. EAD Estimation for Credit Models: Defaultable Loans - Example
  • 44. Exposure at Default Estimation for Credit Models - Derivatives
    • Many institutions to employ internal expected potential exposure (EPE) estimates of defined netting sets of counterparty credit risk (CCR) exposures in computing EAD
    • Models commonly used for CCR estimate a time profile of expected exposure (EE) over each point in the future, which equals the average exposure, over possible future values of relevant market risk factors (e.g., interest rates, FX rates, etc.)
    • Short-dated securities financing transactions (SFT): problem measuring EPE since EE time considers current transactions
    • Therefore Effective EE t = max (Effective EE t-1 , EE t )
    • Also applied to short-term OTC transactions
  • 45. EAD Example for Credit Models: Jacobs (2010) Study
    • EAD risk increasing in time-to-default; loan undrawn or limit amount; firm size or intangibility; % bank or secured debt
    • EAD risk decreasing in PD ( worse obligor rating or aggregate default rate); firm leverage or profitability; loan collateral quality or debt cushion
    *Jacobs Jr., M., 2010, An empirical study of exposure at default, The Journal of Advanced Studies in Finance, Volume 1, Number 1
  • 46. Correlation Estimation for Credit Risk Models
    • Correlations of creditworthiness between counterparties critical to credit of model, but hard to estimate & models results is sensitive to it
    • The 1 st source is the state of the economy, but extent & timing of the rise in default rates varies by industry & geography
    • Also depends upon degree to which firms are diversified across activities (often proxied for by size: larger->less correlation)
    • Contagion : apart from the broader economy, default itself implies more defaults (interdependencies), which can worsen the economy
    • Time horizon over which correlations are measured matters – shorter (longer) can imply see little (much) dependence between sectors
    • Some credit models have asset correlation decrease in PD (Basel II), but weak evidence for this & not intuitive->need economic source
    • May use various types of data having sufficient history, but beware of structural change & time variation (cyclicality-increases in downturn)
  • 47. Correlation Estimation for Credit Risk Models – Empirical Example
    • Jacobs et al (2010)*: while not directly related to credit or default, these show important facts about correlations
    • The plot shows that correlations are time-varying and can differ according to time horizon
    • The table shows how correlations amongst different sectors’ indices can vary widely
    * Jacobs, Jr., M., and Karagozoglu, A, 2011 (June), Performance of time varying correlation estimation methods, Forthcoming,. Quantitative Finance (December, 2011).
  • 48. Correlation Estimation for Credit Risk Models – Sensitivity Analysis
  • 49. Mapping Risk Neutral to Physical Probabilities of Default
    • Given an LGD assumption or model, a term structure of CDS spreads can be related to a term structure of risk-neutral PDs by equating PVs of the default & fee legs of CDS contracts under a no-arbitrage argument
    • Since a term structure of CDS is not sufficient to fully specify the full term structure of risk-neutral PDs, make a parametric assumption on the risk-neutral survival function:
    • Where Ψ is some distribution characterizing the risk-neutral survival function, Z is the risk-free discount curve, θ PD & θ LGD are parameter vectors describing PD and LGD, respectively
    (47) * *Reproduced with permission from: Moody’s Analytics, Special Comment, CDS Implied EDF Credit Measures and Fair Value Spreads, 10-11-03. *
  • 50. Mapping Risk Neutral to Physical Probabilities of Default (cont’d.)
    • This formulation is much richer S=PDXLGD approximation as it captures the full term structure of risk-neutral PD and all contingent future cash flows
    • The difference between risk-neutral & physical PD is driven by the risk premium determined by the market price of risk, the level of systematic risk, as well as the tenor of the contract
    • LGD can be assumed to be fixed by broad segment or from a regression model, while for Ψ we may make a convenient Weibul assumption, which implies risk neutral PD has the form:
    (48) * * *Reproduced with permission from: Moody’s Analytics, Special Comment, CDS Implied EDF Credit Measures and Fair Value Spreads, 10-11-03.
  • 51. Mapping Risk Neutral to Physical Probabilities of Default (cont’d.)
    • Motivated by the BSM structural credit risk framework, we translate between these PDs probability using the formula:
    • Where λ is the market price of risk (MRP; or Sharpe Ratio) and ρ is the correlation of the issuer’s assets to the market
    (49) (50)
    • This model is implemented by estimating MRPs and LGDs by region, sector & rating class
    • The figure shows MRPs for investment grade firms
  • 52. Mapping Risk Neutral to Physical Probabilities of Default (cont’d.)
    • During the beginning of 2009, the high sector LGD for NA Utilities reflects elevated spreads relative to other sectors and similar PD credit measures
    • A rapid increase in LGD typically reflects spreads increasing in the sector without a comparable increase in the PD
    • Risk premiums increased significantly during the “great recession” as retail investors hoarded cash & capital markets around experienced severe credit crunch
    • The figure shows LGD for the North American sector
  • 53. Mapping Risk Neutral to Physical Probabilities of Default (cont’d.)
    • A typical bank portfolio does not have PD-CDS measure as most exposures do not trade in the CDS market, but we can still make a conservative PD measure utilizing both, the maximum of PD-CDS and another PD when both exist
    • This figure reports power curves comparing out CDS implied PDs (PD-CDS) to PDs estimated from a vendor model (PD-KDP)
    • Predictive power as measured by the Accuracy Ratio the maximum of PD-CS & PD-KDP is 84.6%, much higher than 77.3% & 79.4% of either alone, respectively
  • 54. Probability of Default Estimation Based on CDS Quotes
    • Jacobs, Karagozoglu & Peluso (2010)* analyzes daily 333 CDS contracts from Bloomberg with S&P ratings 2003-08
    • CDS implied ratings (JKP’s) are formed by ranking daily CDS quotes
    • Build a regression model to explain CDS spreads, were LGD JK is the Jacobs & Karagozoglu (2010) regression model for LGD discussed previously
          • We compare our CDS and LGD model based PDs to the Kamakura vendor model PDs discussed previously
    *Jacobs, Jr., M., and Karagozoglu, A., 2010 (July), Measuring credit risk: CDS spreads vs. credit ratings, Working paper. Under review for The Journal of Credit Risk .
  • 55. PD Estimation Based on CDS Quotes vs. Vendor Model: Distributions of Output by Rating (Investment Grade)
  • 56. PD Estimation Based on CDS Quotes vs. Vendor Model: Distributions of Output by Rating (Speculative Grade)
  • 57. PD Estimation Based on CDS Quotes vs. Vendor Model: Output Over Time by Rating
  • 58. PD Estimation Based on CDS Quotes vs. Vendor Model: Output Over Time by Rating (continued)
    • The 2 models generally track each other, except that JKP is systematically higher
    • The models do not track very well during the downturn and the estimates become volatile
  • 59. PD Estimation Based on CDS Quotes vs. Vendor Model: Performance Comparison
    • The models rank order default risk about equally
    • Bur, KDP is built on much limited data, and not on actual default as is KAM
    • However, this does not mean that KDP accurately predicts levels of PD as well as KAM
  • 60. Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk
    • Rating migration models (RMM) link potential changes in the value of credit exposures to changes in credit ratings of obligor
    • A downgrade decreases exposure’s value & increases the probability of default -> greater potential unexpected losses
    • CreditMetrics TM developed by JP Morgan in the 90’s is a well-known migration model adopted internally by many banks
    • These can be used to value complex credit derivatives, but that is beyond scope (but can accommodate portfolio CDS hedges)
    • Assumes an unobserved Gaussian credit quality variable for each firm, realization of which determines the rating
    • Difference to structural models: instead of asset value based on firm’s equity/debt, use rating transitions
  • 61. Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk (cont’d.)
    • Correlations model similar to KMV: equities & factor model
    • Similar to structural models issue of if equity correlations are reasonable proxies for default or asset value correlations
    • Empirical evidence suggests a slight overstatement overall but large deviations at industry level (de Servigny et al 2003)
    • Revaluation of loans at horizon at rating specific credit curve: does not model spread risk (but extensions: Kiesel et al 2003)
    • Fundamental drawback of RMMs: assumes all firms in a rating have same migration probabilities & credit spread curve
    • Depending upon level of simulated default rates can adjust migration matrices -> capture migration & default correlation
  • 62.
    • The TTC matrix is the conventional average 1-year non-overlapping transition rates published by the agencies
    • The PIT matrix is constructed from transitions amongst PD bands in the year prior to & model 1-year PDs at observation point
    • PIT matrix has higher (lower) transitions to default (rates on the diagonal) than TTC
    Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk (cont’d.)
  • 63.
    • Sample of 9 Moody’s speculative rated vanilla loans as of 4Q08 (in fact all in DRS meeting exclusions)
    • Due to higher PDs, EL is about $100m (0.8% of FV) higher under PIT than TTC
    • Across confidence levels PIT capital is 200-220MM (1.67% of FV) higher due to both higher PDs & more volatile transitions
    • But if we looked at capital year over year, PIT would be more volatile than TTC
    Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk (cont’d.)
  • 64.
    • The TTC $ capital add-on is roughly constant across CI’s, but increasing slightly in % of FV terms from 99.9 th to 99.97 th percentile
    Rating Transitions Based on Agency Data vs. PD Model & Portfolio Credit Value-at-Risk (cont’d.)
  • 65. References
    • Araten, M. and M. Jacobs Jr., 2001, Loan equivalents for defaulted revolving credits and advised lines, The Journal of the Risk Management Association, May, 34-39.
    • Araten, M., Jacobs Jr., M., and P. Varshney, 2004, Measuring LGD on commercial loans: An 18-year internal study, The Journal of the Risk Management Association, May, 28-35.
    • Bangia, A., Diebold, F., and A. Kronimus,2002, Ratings migration and the business cycle, with application to credit portfolio stress testing, Journal of Banking and Finance 26, 445-474.
    • Basel Committee on Banking Supervision (2006), &quot;International convergence of capital measurement and capital standards: A revised framework”.
    • Chava, S.,and R. Jarrow, 2004, Bankruptcy prediction with industry effects , Review of Finance, 8(4), 537-569.
    • Crouhy, M., Galai, D., and R. Mark, 2006, “The Essentials of Risk Management”, Forlag: McGraw Hill.
    • De Servigny, A., and O. Renault, 2003, Correlations: evidence, Risk, July, 90-94.
    • Gordy, M., and E. Heitfield, 2002, Estimating default correlations from short-panels of credit rating.
    • performance data, Working Paper, US Federal Reserve Board,Working paper.
    • Guha, K., and G. Tett, 2008, ”Last Year’s Model: Stricken U.S. Homeowners Confound Predictions”, Financial Times, Financial Times , February:11.
    • Hosmer, D,W., and S. Lemeshow (2000). &quot;Applied Logistic Regression, 2nd Edition.&quot; Wiley.
    • Hull, J., and A. White, 2008, Dynamic models of portfolio credit risk: A simplified approach, Journal of Derivatives, Summer, 9-28.
    • Jacobs Jr., M., 2010, An empirical study of exposure at default, The Journal of Advanced Studies in Finance, Volume 1, Number 1 (Summer.)
    • Jacobs, Jr., M., Karagozoglu, A., and Layish, D., 2012, Resolution of corporate financial distress: an empirical analysis of processes and outcomes, The Journal of Portfolio Management, Spring, Forthcoming.
    • Jacobs, Jr., M., and Karagozoglu, A, 2011, Modeling ultimate loss given default on corporate debt, The Journal of Fixed Income , 21:1 (Summer), 6-20.
    • Jacobs Jr., M., and A. Karagozoglu, 2010, Modeling the time varying dynamics of correlations: applications for forecasting and risk management, Working paper.
  • 66. References (continued)
    • Jacobs, Jr., M., and Karagozoglu, A, 2011 (June), Performance of time varying correlation estimation methods, Forthcoming Quantitative Finance (December, 2011).
    • Jacobs Jr., M., Karagozoglu, A., and C. Pelusso, 2010, Measuring Credit Risk: CDS Spreads vs. Credit Ratings. Hofstra University & Goldman Sachs, Working paper.
    • Jacobs Jr., M., and N. M. Kiefer (2010) “The Bayesian Approach to Default Risk: A Guide,” (with.) in Ed.: Klaus Boecker, Rethinking Risk Measurement and Reporting (Risk Books, London).
    • J.P. Morgan, 1997, “CreditMetrics - Technical Document”.
    • Kamakura Corporation (Donald van Deventer), Kamakura Pubic Firm Model: Technical Document, September, 2011.
    • Kiesel, R. Perraudin, W., and A.P. Taylor, 2003, The structure of credit risk: Spread volatility and ratings transitions Journal of Risk 6, 1-36.
    • Koedij, K.C.G. , Campbell, R.A.J, and P. Kofman, 2002, Increased correlation in bear markets, Financial Analysts Journal 58, 287-94.
    • Koyluoglu, H., and A. Hickman, 1998, Reconcilable differences, October, 56-62.
    • Li, D., 2000, On default correlation: A copula approach, Journal of Fixed Income 9, 43-54.
    • Merton, R., 1974, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 29, 4449-470.
    • Moody’s Analytics / Credit Policy, Special Comment: Corporate Default an and Recovery Rates 1970-2010, 2 -28-11.
    • Moody’s Analytics, Special Comment, CDS Implied EDF Credit Measures and Fair Value Spreads, 10-11-03.
    • Moody’s Analytics, Default Rate Service Database, Release Date 10-15-10.
    • Moody’s Analytics, Ultimate Recovery Database, Release Date 9-31-10.