# Probability of default

Probability of default is a financial

term describing the likelihood of a default over a particular time horizon.

It provides an estimate of the likelihood that a borrower will be

unable to meet its debt obligations. PD is used in a variety of credit

analyses and risk management frameworks. Under Basel II, it is a key parameter

used in the calculation of economic capital or regulatory capital for a

banking institution. Overview

The probability of default is an estimate of the likelihood that the

default event will occur. It applies to a particular assessment horizon, usually

one year. Credit scores, such as FICO for

consumers or bond ratings from S&P, Fitch or Moodys for corporations or

governments, typically imply a certain probability of default.

For group of obligors sharing similar credit risk characteristics such as a

RMBS or pool of loans, a PD may be derived for a group of assets that is

representative of the typical obligor of the group. In comparison, a PD for a

bond or commercial loan, are typically determined for a single entity.

Under Basel II, a default event on a debt obligation is said to have occurred

if it is unlikely that the obligor will be

able to repay its debt to the bank without giving up any pledged collateral

the obligor is more than 90 days past due on a material credit obligation

Stressed and Unstressed PD The PD of an obligor not only depends on

the risk characteristics of that particular obligor but also the economic

environment and the degree to which it affects the obligor. Thus, the

information available to estimate PD can be divided into two broad categories –

Macroeconomic information like house price indices, unemployment, GDP growth

rates, etc. – this information remains the same for multiple obligors.

Obligor specific information like revenue growth, number of times

delinquent in the past six months, etc. – this information is specific to a

single obligor and can be either static or dynamic in nature. Examples of static

characteristics are industry for wholesale loans and origination “loan to

value ratio” for retail loans. An unstressed PD is an estimate that the

obligor will default over a particular time horizon considering the current

macroeconomic as well as obligor specific information. This implies that

if the macroeconomic conditions deteriorate, the PD of an obligor will

tend to increase while it will tend to decrease if economic conditions improve.

A stressed PD is an estimate that the obligor will default over a particular

time horizon considering the current obligor specific information, but

considering “stressed” macroeconomic factors irrespective of the current

state of the economy. The stressed PD of an obligor changes over time depending

on the risk characteristics of the obligor, but is not heavily affected by

changes in the economic cycle as adverse economic conditions are already factored

into the estimate. For a more detailed conceptual

explanation of stressed and unstressed PD, refer.

Through-the-cycle(TTC) and Point-in-Time(PIT)

Closely related to the concept of stressed and unstressed PD’s, the terms

through-the-cycle or point-in-time can be used both in the context of PD as

well as rating system. In the context of PD, the stressed PD defined above

usually denotes the TTC PD of an obligor whereas the unstressed PD denotes the

PIT PD. In the context of rating systems, a PIT rating system assigns

each obligor to a bucket such that all obligors in a bucket share similar

unstressed PDs while all obligors in a risk bucket assigned by a TTC rating

system share similar stressed PDs. Credit default swap-implied

probabilities of default are based upon the market prices of credit default

swaps. Like equity prices, their prices contain all information available to the

market as a whole. As such, the probabliity of default can be inferred

by the price. CDS implied PD’s can be used with EDF credit measures to improve

accuracy. Deriving Point-in-Time(PIT) and

Through-the-cycle(TTC) PDs There are alternative approaches for

deriving and estimating PIT and TTC PDs. One such framework involves

distinguishing PIT and TTC PDs by means of systematic predictable fluctuations

in credit conditions, i.e. by means of a “credit cycle”. This framework,

involving the selective use of either PIT or TTC PDs for different purposes,

has been successfully implemented in large UK banks with BASEL II AIRB

status. As a first step this framework makes use

of Merton approach in which leverage and volatility are used to create a PD

model. As a second step, this framework assumes

existence of systematic factor(s) similar to Asymptotic Risk Factor Model.

As a third step, this framework makes use of predictability of credit cycles.

This means that if the default rate in a sector is near historic high then one

would assume it to fall and if the default rate in a sector is near

historic low then one would assume it to rise. In contrast to other approaches

which assumes the systematic factor to be completely random, this framework

quantifies the predictable component of the systematic factor which results in

more accurate prediction of default rates.

As per this framework, the term PIT applies to PDs that move over time in

tandem with realized, default rates, increasing as general credit conditions

deteriorate and decreasing as conditions improve. The term TTC applies to PDs

that exhibit no such fluctuations, remaining fixed overall even as general

credit conditions wax and wane. The TTC PDs of different entities will change,

but the overall average across all entities won’t. The greater accuracy of

PIT PDs makes them the preferred choice in such current, risk applications as

pricing or portfolio management. The overall stability of TTC PDs makes them

attractive in such applications as determining Basel II/II RWA.

The above framework provides a method to quantify credit cycles, their systematic

and random components and resulting PIT and TTC PDs. This is accomplished for

wholesale credit by summarizing, for each of several industries or regions,

MKMV EDFs, Kamakura Default Probabilities, or some other,

comprehensive set of PIT PDs or DRs. After that, one transforms these factors

into convenient units and expressed them as deviations from their respective,

long-run-average values. The unit transformation typically involves the

application of the inverse-normal distribution function, thereby

converting measures of median or average PDs into measures of median or average

“default distance”. At this point, one has a set of indices measuring the

distance between current and long-run-average DD in each of a

selected set of sectors. Depending on data availability and portfolio

requirements, such indices can be created for various industries and

regions with 20+ years covering multiple recessions.

After developing these indices, one can calculate both PIT and TTC PDs for

counterparties within each of the covered sectors. To obtain PIT PDs, one

introduces the relevant indices into the relevant default models, re-calibrate

the models to defaults, and apply the models with current and projected

changes in indices as inputs. If a PD model weren’t otherwise PIT, the

introduction of the indices will make it PIT. The specific model formulation

depends on the features important to each, distinguished class of

counterparties and data constraints. Some common approaches include:

Factor Ratio Model: Calibration of financial/non-financial factors and

credit-cycle indices to defaults. This approach works well with large number of

defaults, e.g. SME portfolios or large-corporate portfolios calibrated to

external default samples. Scorecard model: Calibration of score

and credit-cycle indices calibrated to observed internal or external defaults.

This approach works with smaller number of defaults where there is not enough

data to develop a ratio model. E.g. Funds portfolio

Agency Direct model: Calibration of ECAI grades and credit indices to ECAI

defaults and applying it to Agency and internal co-rated entities. This

approach works well where there is a large co-rated dataset but not enough

internal defaults e.g. Insurance portfolio

Agency Replication model: Calibrate financial/non-financial

factors/scorecard score to PDs estimated from the Agency Direct model. This

approach works well where there is a large, co-rated dataset but a small

sample of internal defaults—e.g. Insurance portfolio

External vendor model: Use of models such as MKMV EDF model with credit cycle

indices. At this point, to determine a TTC PD,

one follows three steps: Converting the PIT PD to PIT DD

Subtracting the credit cycle index from the PIT DD, thereby obtaining the TTC

DD; and Converting the TTC DD to TTC PD.

In addition to PD models, this framework can also be used to develop PIT and TTC

variants of LGD, EAD and Stress Testing models.

PD Estimation There are many alternatives for

estimating the probability of default. Default probabilities may be estimated

from a historical data base of actual defaults using modern techniques like

logistic regression. Default probabilities may also be estimated from

the observable prices of credit default swaps, bonds, and options on common

stock. The simplest approach, taken by many banks, is to use external ratings

agencies such as Standard and Poors, Fitch or Moody’s Investors Service for

estimating PDs from historical default experience. For small business default

probability estimation, logistic regression is again the most common

technique for estimating the drivers of default for a small business based on a

historical data base of defaults. These models are both developed internally and

supplied by third parties. A similar approach is taken to retail default,

using the term “credit score” as a euphemism for the default probability

which is the true focus of the lender. Some of the popular statistical methods

which have been used to model probability of default are listed below.

Linear Regression Discriminant analysis

Logit and Probit Models Panel models

Cox proportional hazards model Neural Networks

Classification Trees See also

Expected loss and its three factors Loss given default magnitude of likely

loss on the exposure, expressed as a percentage of the exposure

Probability of default probability of default of a borrower

Exposure at default amount to which the bank was exposed to the borrower at the

time of default, measured in currency For the effects of correlation between

PD and LGD see Expected loss References

^ Bankopedia:PD Definition ^ FT Lexicon:Probability of default

^ Introduction:Issues in the credit risk modelling of retail markets

^ Basel II Comprehensive Version, Pg 100 ^ a b BIS:Studies on the Validation of

Internal Rating Systems ^ Slides 5 and 6:The Distinction between

PIT and TTC Credit Measures it-Measures-and-Fair-Value-Spreads.ashx>

1/aguais_et_al_basel_handbook2_jan07.pdf ^ Aguais, S. D., et al, 2004,

“Point-in-Time versus Through-the-Cycle Ratings”, in M. Ong, The Basel Handbook:

A Guide for Financial Practitioners ^ Merton, Robert C., “On the Pricing of

Corporate Debt: The Risk Structure of Interest Rates”, Journal of Finance,

Vol. 29, No. 2,, pp. 449-470 ^ Gordy, M. B. A risk-factor model

foundation for ratings-based bank capital rules. Journal of Financial

Intermediation 12, 199 – 232. ^ http:www.bis.orgirbriskweight.pdf

^ The Basel II Risk Parameters Reading

de Servigny, Arnaud and Olivier Renault. The Standard & Poor’s Guide to Measuring

and Managing Credit Risk. McGraw-Hill. ISBN 978-0-07-141755-6.

Duffie, Darrell and Kenneth J. Singleton. Credit Risk: Pricing,

Measurement, and Management. Princeton University Press. ISBN

978-0-691-09046-7. External links

Implied Default Probability from CDS – QuantCalc, Online Financial Math

Calculator ode282731.pdf?abstractid=1921419&mirid=1

Through-the-Cycle EDF Credit Measures methodology paper

http:www.bis.orgbcbsca.htm Basel II: Revised international capital framework

http:www.bis.orgbcbs107.htm Basel II: International Convergence of Capital

Measurement and Capital Standards: a Revised Framework

http:www.bis.orgbcbs118.htm Basel II: International Convergence of Capital

Measurement and Capital Standards: a Revised Framework

http:www.bis.orgbcbs128.pdf Basel II: International Convergence of Capital

Measurement and Capital Standards: a Revised Framework, Comprehensive Version

just reciting everything from wikipedia

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