FM 9528 Banking Analytics: PD and LGD Calibration for Capital Requirements CRISTIáN BRAVO [email protected] OFFICE 280 Defining Ratings A rating is a homogeneous pool of obligors that are similar in terms of default risk. Ratings are defined because scores are considered too fine granular and hence too volatile. Ratings provide an ordinal and stable measure of credit risk. Problem statement: Start from z-score (for example, log(odds) from logistic regression): Categorize z-score into ratings. zBBB BBB PDBB= Cutoff points …… Defining Ratings Discrimination should remain unaffected as much as possible. Approximate ROC curve piecewise linearly as closely as possible. Objective function then becomes: min(||AUCorig-AUCpiec.lin.||) Approaches Map onto rating agency scale Regression trees Exponential risk evolution Directly optimize objective function Map onto Agency Rating Scale Define internal ratings so as to mimic as closely as possible the one-year default rates reported by the rating agencies. Directly Optimize Objective Function Start from a given number of ratings (for example, between 7 and 15). Start from random cutoff values on the z-variable (or for example, uniformly distributed). Calculate ||AUCorig-AUCpiec.lin||. Shift the cutoff values so as to optimize ||AUCorig-AUCpiec.lin||. Can do it using trial and error, or you could use genetic algorithms/simulated annealing. Rating Philosophy Before validation starts, you need to answer the question “What is the rating system supposed to do ” “Rating philosophy is the term used to describe the assignment horizon of a borrower rating system.” (FSA, CP 05/03, par. 7.66) Rating philosophy should be clearly articulated in bank rating policy. Federal Register, wholesale exposures A bank’s rating policy must describe its ratings philosophy and how quickly obligors are expected to migrate from one rating to another in response to economic cycles. Analyze rating philosophy by means of migration analysis. Rating Philosophy Point-in-Time (PIT) Ratings Consider obligor specific, cyclical, and non-cyclical information. Rating changes rapidly with macroeconomic situation (lots of rating mobility). PD is the best estimate of the obligor’s default during next 12 months. IFRS 9 Through-the-Cycle (TTC) Ratings Consider only non-cyclical information. Obligors with same TTC rating share similar stressed PDs. Rating robust with respect to macro-economic situation (not much rating mobility). PD is best estimate of the obligor’s default during a credit cycle. Basel Many hybrids exist. PIT and TTC represent two ends of a continuum! Point-in-Time Through-the-Cycle When entering a downturn, a migration to a lower rating takes place. When entering a downturn, no migration takes place. The actual default rate in each rating remains unchanged, and the rating level PD remains unchanged. The actual default rate in each rating increases, and the rating level PD remains unchanged. Due to the downgrade, the capital requirements increase. There is no downgrade, so the capital requirements remain theoretically unchanged. Rating Philosophy (Section 12, UK’s PRA) Calibration Assign PD to each rating using long-term historical data. For model development, one needs high quality, multidimensional data, which is typically available for only a short-term period (for example, 2–3 years). Calibration tries to deal with three key sources of default rate variability: Sampling uncertainty Economic conditions Internal effects Basel says five years of data are needed to calibrate PD (longer data histories are advised). PD and LGD calibration is usually a regression of observed PDs against scorecard variables and economic conditions. Process to Calibrate a PD or LGD Model 1. Start from a dataset of the portfolios of loans at the end of each month, and the behavioural scorecard each customer obtained. Assume you have the last portfolios so that = , , with the customers available at that time = {1,… ,}. 2. Decide on the number of cuts to make to the scorecard curve. 3. Create the cuts by optimizing min(||AUCorig-AUCpiec.lin.||). 4. Collect a set of macroeconomic regressors at the beginning of each period , let’s call these regressors = (1,, 2,, … , ,). You can also lag these regressors for every month. Typical lags are 30, 60, 90 and 180 days. Process to Calibrate a PD or LGD Model 5. Run an ARIMAX model so that: = 1 + 1 + This model can be extended to include moving averages or any other time series analysis you consider appropriate. LASSO can help too with the selection of the vector 1 is there is no certainty what are the most useful economic factors. 6. Study the most adequate long-term parameters for the vector . These will usually come from your chief economist or from official projections. Use them to calculate the for every segment. If calculating LGD you have to choose the downturn estimates for , i.e., the parameters in stressed periods. 7. Profit! This is your long-term PD / downturn LGD. Recommended Reading – LGD Modelling Level 0: Definition of LGD LGD means the ratio of the loss on an exposure due to the default of a counterparty to the amount outstanding at default (=1-recovery rate) (Art. 4, EU). Economic loss versus Accounting loss Problems: Extra Costs Realizing collateral value (haircuts), administrative costs (letters, telephone calls), legal costs, time delays in what is recovered (economic loss!), … Extra Benefits Interest on arrears, penalties for delays, commissions, … Issue LGD versus issuer LGD If cash flows cannot be allocated to a specific issue, use issuer-weighted LGD. Level 0: Ways of Measuring LGD Workout LGD (corporate/consumer) Discount cash flows from workout/collections process. Direct and indirect costs (for example, operating costs of workout department). Default EAD=100 -5 -5+20 +70 time Recovered=70 RR=70% LGD=30% Discount factor i Workout period Level 0: LGD According to Basel “The definition of loss used in estimating LGD is economic loss.” (Basel II, par. 460) “The exposure weighted average LGD for all retail exposures secured by residential property and not benefiting from guarantees from central government shall not be lower than 10%” (Art. 164 EU, Basel II par. 266) Foundation IRB approach For Corporates/Sovereigns/Banks “Senior claims on corporates, sovereigns and banks not secured by recognised collateral will be assigned a 45% LGD.” (Art. 161 EU, Basel II par. 287) “All subordinated claims on corporates, sovereigns, and banks will be assigned a 75% LGD.” (Art. 161 EU, Basel II par. 288) Level 0: Constructing an LGD Data Set Data set should cover at least a complete business cycle. Default definition and cures LGD measurement Decide on workout period Incomplete workouts Discount factor Negative LGDs versus LGDs>100% Indirect costs Drivers of LGD Level 0: Complete Business Cycle For the retail portfolio Minimum five years For wholesale (corporates, sovereigns, banks) Minimum seven years Preferably, one or two downturn periods should be included. Do not need to attach equal importance to each year of data. See Art. 181 EU Level 0: Default Definition and Cures All defaults should be included. Same default definition as for PD! Differences in default definition impact PD and LGD, but not EL=PD.LGD. Cures LGD=0 (or close) Depends strongly on default definition (cf. supra)! Relaxing default definition typically increases cures. Multiple defaults Include only last default event. PD and EAD are also related to this. Level 0: Incomplete Workouts No regulatory prescription as to how incomplete workouts should be treated! Options: Look at current LGD of an incomplete workout and use this for estimation. Some banks systematically disregard recoveries after three or five years. Very conservative!
Use expert or predictive models estimating final LGD based on what has
been observed already (% collect
ed, time of collection, …). Ignore incomplete workouts when not relevant. Survival analysis models whereby recovery amount is considered as censored variable at, for example, 12 months. See, for example, Stoyanov S., Application LGD Model Development, Credit Scoring and Credit Control XI Conference, 2009. Level 0: Discount Rate Ongoing debate between supervisors and firms Common industry approaches Use contractual rate at time of default. Use risk free rate + a risk premium. Weighted average cost of capital (WACC). It might be useful to perform a sensitivity analysis (effect could be quite limited anyway)! Subject to discussion (for example, FSA Expert group on LGD) The EG agrees that the use of the contractual rate as the discount rate is conceptually inappropriate.
The group proposes that the discount rate for this asset class should
be close to the risk-free rate, so long as firms can evidence and justify
sufficient conservatism in their estimation of the downturn. One
potential approach to a discount rate for this asset class could be the risk-free rate plus an appropriate premium. “The PRA expects firms to ensure that no discount rate used to estimate LGD is less than 9%.” (Section 13, PRA) A Theoretical and Empirical Analysis of Alternative Discount Rate Concepts for Computing LGDs Using Historical Bank Workout Data, Global Credit Data, 2017. Level 0: Negative LGDs and LGDs > 100% Negative LGDs Recovery rate > 100% Reasons EAD measured at time of default whereas claim on the borrower increases afterward (fines, fees, …) and everything is recovered. Result of gain in collateral sales Cap negative LGDs to 0. “The PRA expects firms to ensure that no LGD estimate is less than zero” (section 13, PRA) LGDs > 100% Recovery rate < 0% Additional costs incurred but nothing recovered Also because of definition of EAD; additional drawings after time of default considered part of LGD. Cap LGD > 100% to 100% Level 0: Indirect Costs “The definition of loss used in estimating LGD is economic loss…This must include material discount effects and material direct and indirect costs associated with collecting on the exposure.” (Basel II, par 460) “Work-out and collection costs should include the costs of running the institution’s collection and work- out department, the costs of outsourced services, and an appropriate percentage of other ongoing costs, such as corporate overhead.” (par. 205, CEBS, CP10) “Cost data comprise the material direct and indirect costs associated with workouts and collections.” (Federal Register) Material indirect costs, costs of running the collection and workout department, costs of outsourced services, appropriate percentage of overhead, must be included (Federal Register). Level 0: Drivers for Predictive Modeling of LGD Borrower characteristics Creditworthiness (PD, rating, application/behavioral score, bureau score, trends in creditworthiness, rating changes, delinquency history, …) Marital status, gender ( ), salary, time at address, time at job, … Intensity of relationship (number of years client, number of products, …) Industry sector, sector indicators Size of the company Legal form of the company Age of the company Balance sheet information (revenue, total assets, solvency/profitability/liquidity ratios, …) continued… Level 0: Drivers for Predictive Modeling of LGD Macro-economic factors GDP (growth) Default rates Inflation, unemployment rate Interest rate (has impact on discount factor!) continued… Level 0: Drivers for Predictive Modeling of LGD Loan Characteristics Type and value of collateral (for example, real estate, cash, inventories, guarantees, …) Real estate: flat, apartment, villa, detached/semi-detached house, … Loan to Value (LTV) Ratio of the value of the loan (=exposure) to the value of the underlying asset; LTV at start versus LTV at or prior to default Estimate and apply haircuts! Haircut=(current market value-forced sales price)/current market value Current market value=current HPI/start HPI* start valuation Define LTV through time (compute trends, …) continued… Level 0: Drivers for Predictive Modeling of LGD EAD Debt seniority Absolute debt seniority: senior secured, subordinated, … Relative debt seniority: debt amount above/total debt; debt amount below/total debt Remaining maturity Country-related features Geographical region (for example, ZIP code) For example, how creditor-friendly is the bankruptcy regime Level 0: LGD Drivers: Seniority Loans Banks Shareholders Bondholders and Banks Se n io ri ty Senior secured Senior unsecured Senior subordinated Subordinated Junior subordinated Shares Level 1: LGD Modeling Approaches Observed in Industry One-stage Segmentation Expert based (for example, based on experience) Statistical: regression trees (for example, CART) Regression Linear regression Linear regression with beta transformation Logistic regression Random forests / XGBoosting Models Two-stage models See Leow, M., & Mues, C. (2012). Predicting loss given default (LGD) for residential mortgage loans: A two-stage model and empirical evidence for UK bank data. International Journal of Forecasting, 28(1), 183-195. Level 1: Example LGD Distributions Loterman G., Brown I., Martens D., Mues C., Baesens B, Benchmarking regression algorithms for loss given default modeling, International Journal of Forecasting, Vol. 28 (1), pp. 161-170, 2012. Level 1: Two-Stage Model for Mortgages Leow, Mues, Credit Risk Models for Mortgage Loans, 2011 Level 2: Mapping to LGD Rating Grades Map the output of the regression/segmentation model to LGD facility rating grades. Compute LGD per rating grade based on historical data. Additional standards for corporate, sovereign, and bank exposures
“Estimates of LGD must be based on a minimum data observation period
that should ideally cover at least one complete economic cycle but must in any case be no shorter than a period of seven years for at least one source.” (Basel II, par. 472)
“There is no specific number of facility grades for banks using the
advanced approach for estimating LGD.” (Basel II, par. 407) Additional standards for retail exposures The minimum data observation period for LGD estimates for retail exposures is five years. (Basel II, par 473). Level 2: Mapping to LGD Rating Grades A A B C D C B D LGD Rating PD Rating Check correlations between PD and LGD rating. Check migration between PD and LGD ratings over time. x xxx x xx x x x x x x x x x x x x x x x x x x Level 2: Mapping to LGD Rating Grades Map onto rating agency scale Regression trees Monotonicity constraint! Directly optimize objective function (Van Gestel, 2007) = + + + 1 1 2 1 2 1 K i ii ii LGDLGD Level 2: Paragraph 468 of the Basel II Accord (Art 181, EU) Standards for all asset classes (advanced approach) “A bank must estimate an LGD for each facility that aims to reflect economic downturn conditions where necessary to capture the relevant risks. This LGD cannot be less than the long-run default-weighted average loss rate given default calculated based on the average economic loss of all observed defaults within the data source for that type of facility… a bank must take into account the potential for the LGD of the facility to be higher than the default-weighted average during a period when credit losses are substantially higher than average…this cyclical variability in loss severities may be important and banks will need to incorporate it into their LGD estimates… banks may use averages of loss severities observed during periods of high credit losses, forecasts based on appropriately conservative assumptions, or other similar method…using either internal or external data.” Again, solution is to regress LGD over the economic conditions and stress these conditions! 1. Calculate LGDs per portfolio of last 60 months. 2. Regress these against economic conditions. 3. Estimate stressed conditions and use these.