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Comptroller of the Currency

Comptroller of the Currency. Administrator of National Banks.

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Comptroller of the Currency

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  1. Comptroller of the Currency Administrator of National Banks Theoretical Modeling of Ultimate Loss-Give-Default: Undiversifiable Recovery Risk and Downturn EffectsMichael Jacobs, Ph.D., CFASenior Financial Economist – Credit Risk ModelingRisk Analysis DivisionWashington, DC 20219Presentation to the CRM Discussion Group 2/6/08michael.jacobs@occ.treas.govThe views expressed herein are solely those of the author and do not reflect necessarily the policies or procedures of the Office of the Comptroller of the Currency or of the US Department of the Treasury.

  2. CRM, 2-08 Outline • Introduction and Summary • Motivation and Background Issues • Review of the Literature • Theoretical Credit Risk Models • Empirical Evidence & Recent Literature • Models for downturn LGD (Barco, 2007) and possible extensions • Theoretical Framework • Estimation Methodology • Downtuen LGD • Directions for Future Research

  3. CRM, 2-08 Motivation • Loss Given Default (LGD) – ultimate economic loss per dollar of outstanding balance at default (or one minus the Recovery Rate) • LGD is a critical parameter in various facets of credit risk modeling – expected loss /allowance, pricing, capital • Basel II Internal Ratings Based (IRB) advanced approach to regulatory credit capital requires banks to estimate LGD • May be measured either on a nominal (undiscounted) or economic (discounted) basis – we care about the latter • Here we are concerned with a theoretical framework for quantifying downturn economic LGD

  4. CRM, 2-08 Background and Measurement Issues • Choice of discount rate – risk free vs. risk adjusted? • The definition of default - bankruptcy vs. broader concept? • Unit of observations - obligor (firm or estate) vs. instrument (facility) • “Actuarial” approach (discount cash flows) vs. market for distressed debt (trading or settlement prices) • Consistency with other credit risk parameters - Exposure at Default (EAD) and Probability of Default (PD)

  5. CRM, 2-08 Background and Measurement Issues (continued) • Many extant credit risk models assume LGD to be fixed despite evidence it is stochastic and predictable with respect to other variables • The boundedness of LGD gives rise to unique statistical issues – boundary bias (not adressed directly here) • Determinants of ultimate LGD considered: • Contractual features – collateral, seniority, debt cushion, facility type • Capital structure: % secured / bank debt, number creditor classes • Borrower – profitability, industry, liquidity, size, leverage • Systematic factors – macroeconomic state, debt market • LGD at default • Cumulative abnormal equity returns

  6. CRM, 2-08 Literature Review: Theoretical Models • Structural models: Merton(1974), Black and Cox (1976), Geske(1977), Vasicek(1984), Kim at al (1993), Hull & White (1995), Longstaff & Schwartz (1995) • Endogenous PD but LGD not independently modeled • Reduced form models: Litterman & Iben (1991), Madan & Unal (1995), Jarrow & Turnbull (1995), Jarrow et al (1997), Lando (1998), Duffie & Singleton (1999), Duffie (1998) • LGD is exogenous but may be correlated with PD process • Credit VaR models: Creditmetrics™ (Gupton et al, 1997), KMV™ • Typically models LGD as exogenous but stochastic • Hybrid approaches: Frye (2000), Jarrow (2001), Jokivuolle et al (2003), Carey & Gordy (2003), Bakshi et al (2001) • Realistic LGD assumptions: correlation with PD & systematic factors

  7. CRM, 2-08 Literature Review: Dependence between PD and LGD Various recent academic studies have appeared on this topic: • Frye (2000) • Pykhtin (2003) – “collateral damage detected” • Dullman and Trapp (2004) – ML estimation • Giese (2005) • Roche & Scheule (2005) • Hildebrand (2006) • Miu and Ozdemir (2006) – systematic vs. contagion effects

  8. CRM, 2-08 Downturn LGD • Vasicek (2002) model does not account for systematic correlation between PD & LGD -> requirement to consider “downturn LGD” • Basel Committee (2004) suggested establishing a functional relationship between LGD & PD -> “mark-up” factor to long-run LGD • Miu and Ozdemir (2006): tabulates a relationship through simulation • Barco (2007): derives an analytical relationship between long-run and downturn LGD • Merton (1974) framework & 3-paramter log-normal recovery distribution • Builds upon work of Frye (2000) and Pykhtin (2003) • Solves for LGD in single factor model that gives the same capital as his model (requires LGD estimate for EL vs. loss quantile)

  9. CRM, 2-08 Theoretical Model • Start with the “work-horse” Merton (1974) single-factor framework: • Ai: asset or firm value process (for ith obligor or segment), Xm: standard normal systematic factor, Xis: idiosyncratic factor (also N(0,1) & independent of Xm), ρiPD: loading on systematic factor (or square-root of “asset value correlation” in single factor model) • Introduce a separate process for normalized return on loan or collateral depends upon a separate systematic factor: (1) (2) • Ri: recovery or return on collateral process, Xm: standard normal systematic factor (same as in (1)), Ym: independent systematic factor (also N(0,1) & independent of Xm), ρiR: collateral value correlation • Note that by construction Ai and Ri are both standard normal and have correlation:

  10. CRM, 2-08 Theoretical Model (continued) • The loss variable here (the “loss-rate”), not the LGD, is the truncated loss on the collateral value (assumes no gain to the bank upon default): • Where µiR and σiR are the mean and volatility of the non-standardized return on collateral (will be loan or segment specific). • Note the strong (but very convenient) parametric assumptions – losses are modeled as log-normal random variables • In this framework, the expected loss-given-default µLGD is the expected value of (1) conditional upon the obligor’s default: (3) (3.1) • Where µiPD is the exogenously specified expected PD and Φ-1 is the inverse of the standard normal distribution. • It can be shown that expected loss (EL) is the product of expected PD and expected LGD in (3.1) whether or not there is of PD-LGD correlation: • The trick here is condition upon event of default inside E[] & pull out the PD (3.2)

  11. CRM, 2-08 Theoretical Model (continued) • The key to this framework is that we can derive expressions for long-run PD, LGD and EL in terms of the underlying parameters of the model in (1)-(3) • First, “spot” PD and LGD are given by the following: (4) (5) • Spot LGD can be solved (note the similarity to an option pricing equation – just the expectation of a truncated log-normal variable): (6)

  12. CRM, 2-08 Theoretical Model (continued) • Important point here: while taking the expectation of spot PD in (4) gives long-run PD, the same is not true for LGD (5): (6.1) (6.2) • i.e., the unconditional expectation of loss ratio LiLR, defined as µiLR, is not equal to long-run LGD, µiLGD. Rather, the latter is given by the following: (6.3) • (6.3) is just mean default weighted LGD.

  13. CRM, 2-08 Theoretical Model (concluded) • To close out the model, we can find an analytic expression for EL in terms of all of the underlying parameters, as derived by Pykhtin (2003): (9) • We get an expression involving bivariate normal distributions in the default point Φ-1(µiPD), minus the “normalized recovery” -µiR/σiR and correlation between systematic factors ρiR ρiPD (similar structure to a compound option) • We recover an expression for long-run LGD in terms of the model parameters according to the relationship (6.3), where we substitute in (9) for the numerator: (9.1)

  14. CRM, 2-08 Estimation Methodology • Barco (2007) follows a 2-stage procedure, relying upon annual time series averages of default and recovery rates at the pooled level. • First, the parameters of the asset value process (ρiPD &µiPD) and latent factor Xm, according to large-sample approximations. Note that while subscripted these by I in the paper, they are constant across segments. • Second, the parameters of the recovery process (ρiR, µiPD & σiPD for ith loan type) by maximum likelihood. • Given observed default rate for the ith segment in year t, pit, Dullman and Trapp (2004) show that the following are asymptotically consistent ML estimates: • Given these estimates, the latent factor is estimated by inverting the expression for spot-PD in each year t:

  15. CRM, 2-08 Estimation Methodology (continued) • Conditional on the latent factor Xm, the distribution of the recovery rate (or log 1 minus the loss-ration) is a log-normal truncated to the unit interval: • The density function for liR,t is given by:

  16. CRM, 2-08 Estimation Methodology (concluded) • Under the usual (not-discussed) technical conditions, this gives rise to the log-likelihood function: • Note that estimates for XtR,t and Xm,t are substituted in to form the empirical version of this, leading to additional uncertainty • Barco claims that this can be maximized using a 3-dimensional Newton-Ralphson algorithm coupled with Guassian quadrature for the integral

  17. CRM, 2-08 Directions for Future Research • A separate process for the default point? • Contagion vs. systematic correlation of PD/LGD? • Different distributions for recovery? • Capital structure? • Upgrade the estimation methodology (e.g., finer segmentation or loan level)?

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