Michael Jacobs, Ph.D., CFA Senior Financial Economist Credit Risk Analysis Division

1. The LGD Discount Rate for Basel II IRB Quantification: Requirements, Theory, Evidence and Bank Practice Michael Jacobs, Ph.D., CFA Senior Financial Economist Credit Risk Analysis Division Office of the Comptroller of the Currency 2011 Risk Quantification Forum Washington DC, June 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 • Background and Motivation • Rule Requirements • Alternative Approaches • Theoretical Framework • Empirical Evidence: Jacobs (2010) • Regulatory Capital Impact • Benchmark Analysis of LGD Discount Rates • Survey of Bank Practices

3. Background and Motivation The discount factor for recoveries on defaulted debt is a required component of economic LGD under Basel 2 IRB Final Rule in the U.S. (OCC et al, 2007, Page 450): “Where positive or negative cash flows on a wholesale exposure to a defaulted obligor…occur after the date of default, the economic loss must reflect the net present value of cash flows … using a discount rate appropriate to the risk of the defaulted exposure.” Rationale? - lengthy resolution durations, non-marketability & low liquidity of defaulted loans, uncertainty of cash flow timing & magnitude While theoretical arguments for risk adjustment exist, designing an internally consistent model is very complicated Maintained hypothesis: the proper objective for capital measurement is the estimation of loss distributions under physical measure Implication for IRB institutions: potential of not assigning enough regulatory capital to instruments with high recovery risk Apart from Basel II IRB, relevance for Pillar II, credit risk modeling, defaulted asset investors or risk managers, or finance academicians

4. Alternative Approaches • Carey & Gordy (2007) suggest the risk-free term structure (pricing under risk neutral measure): depends upon things like hedgibility / diversifiability of recovery risk • The “opportunity cost of funds” approaches (WACC, cost of debt, hurdle rate, etc.) assume the defaulted loan is replaced with one of typical risk in the portfolio & financed • A comparable risky rate (punitive or contract rate at default; see Araten et al (2001)-the “JPMC LGD” study) may be more appropriate to defaulted exposures as a separate asset class • A few studies have investigated the influence of varying the discount rate by segment on economic LGD or regulatory capital: Machlachlan (2004), Brady et al (2006) and Jacobs (2010) • Option adjusted spread (OAS) methodology of Kupiec (2007) argues that the above approaches leads to bias due to timing & magnitude of recovery cash-flow uncertainty • Basel 2 IRB practitioner suggestions:Cost of funds measures (debt/equity, WACC), contract rate in reference data or on some portion of current non-defaulted portfolio, arbitrary punitive rate, distressed index rate

5. Theoretical Framework • Determination of rsD : start with Merton (1974) structural model of credit risk and asymptotic single risk factor model of Gordy (2000) • Stochastic process describing the instantaneous evolution of the ith representative firm’s (or PD segment) asset return at time t as: • Vi,t :asset value, μi: drift, and Wi,t: standard Weiner process,Xt & Zi,t :systematic & idiosyncratic risk factors,& factor loadingρi,X • We consider an extension with systematic and idiosyncratic variation in the recovery process: • XtR and Zi,tR: systematic & idiosyncratic risk factors, factor loading ρi,X is for loans in “recovery class”, implies asset-value correlation (AVC) amongst LGD segments s and u is given by:

6. Theoretical Framework • As in the Basel framework, assuming the factor loading to be constant amongst exposures in an LGD segments implies ρs,W2=RS • If we identify this with the correlation to a market portfolio then it follows from the standard CAPM that the beta relating the market’s to the return on the defaulted asset is: • In this setting the proper discount rate for LGD in the sth segment, riD, is equal to the expected return on the defaulted exposure, which is given by the risk-free rate rrf and the debt-specific risk-premium δs: • Where rM is return on a market index and σs (σM) is volatility of the defaulted asset’s (market) return

7. Benchmarking Alternative LGD Discount Rate Frameworks (contd.) • Calibrate a 2-factor version of the Basel capital model with systematic recovery risk to Moody’s annual default and loss rate data 1987-2010 • Recovery process correlation 15% (8%) loans (bonds) similar (lower) higher than Frye (2000) single factor model 17% • 8% AVC estimate in ballpark with previous literature that calibrates to loss data • High 64% correlation between systematic risk factor estimates support single factor • rf=5%,MRP=7%,σM=10%->LGD discount of about 12% for loans

8. Summary of Jacobs (2010)* • Overall mean RDD 28.6% with standard error of mean 3.11% • We decide to include the 76 out-of-court settlements despite higher return / risk (mean / std err 37.2 / 15.3%) • Maximum 893.8% even after eliminating 35 clear outliers (all > 30K% RDD!) • Loans have slightly mean & std err RDD 32.2% & 5.5% • Revolvers slightly lower mean (higher std err mean) RDD of 25.9% (7.3%) * Jacobs, Jr., M., 2010, An empirical study of the returns on defaulted debt and the discount rate for loss-given default, Working paper, U.S. Office of the Comptroller of the currency.

9. The Regulatory Capital Impact of the Discount Rate for LGD • Examining the distributions of portfolio regulatory capital in Figures 13.5 and 13.6, where we see that the density mass is shifted right-ward under the RDD model relative to either the contract rate or the 25% punitive discount rate. • But there is less peakedness and skewness under the RDD model as compared to the other discounting, so most of the difference is from the body & not the tails of the distribution (although the standard deviation is higher)

10. Benchmarking Alternative LGD Discount Rate Frameworks

11. Wholesale Qualification Team Survey of Bank Practices