FM 9528
Banking Analytics:
PD and LGD Calibration for
Capital Requirements
CRISTIÁN BRAVO
CBRAVORO@UWO.CA
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 (% collected, 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.
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Level 2: Mapping to LGD Rating Grades
◦ Map onto rating agency scale
◦ Regression trees
◦ Monotonicity constraint!
◦ Directly optimize objective function (Van Gestel, 2007)

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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.