Probability of default (PD) 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
Basel II is the second of the Basel Accords, which are recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision. It is now extended and partially superseded by Basel III.
The Basel II Accord was publis ...
, it is a key parameter used in the calculation of
economic capital
In finance, mainly for financial services firms, economic capital (ecap) is the amount of risk capital, assessed on a realistic basis, which a firm requires to cover the risks that it is running or collecting as a going concern, such as market r ...
or
regulatory capital
A capital requirement (also known as regulatory capital, capital adequacy or capital base) is the amount of capital a bank or other financial institution has to have as required by its financial regulator. This is usually expressed as a capital ad ...
for a banking institution.
PD is closely linked to the
expected loss Expected loss is the sum of the values of all possible losses, each multiplied by the probability of that loss occurring.
In bank lending (homes, autos, credit cards, commercial lending, etc.) the expected loss on a loan varies over time for a num ...
, which is defined as the product of the PD, the
loss given default
Loss given default or LGD is the share of an asset that is lost if a borrower defaults.
It is a common parameter in risk models and also a parameter used in the calculation of economic capital, expected loss or regulatory capital under Basel II f ...
(LGD) and the
exposure at default
Exposure at default or (EAD) is a parameter used in the calculation of economic capital or regulatory capital under Basel II for a banking institution. It can be defined as the gross exposure under a facility upon default of an obligor.
Outside o ...
(EAD).
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 score
A credit score is a numerical expression based on a level analysis of a person's credit files, to represent the creditworthiness of an individual. A credit score is primarily based on a credit report, information typically sourced from credit bu ...
s, 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 (average) obligor of the group. In comparison, a PD for a bond or commercial loan, are typically determined for a single entity.
Under
Basel II
Basel II is the second of the Basel Accords, which are recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision. It is now extended and partially superseded by Basel III.
The Basel II Accord was publis ...
, 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
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
not only depends on the risk characteristics of that particular
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
but also the economic environment and the degree to which it affects the
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
. 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
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
s.
* Obligor specific information like revenue growth (wholesale), number of times delinquent in the past six months (retail), etc. - this information is specific to a single
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
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
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
will default over a particular time horizon considering the current macroeconomic as well as
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
specific information. This implies that if the macroeconomic conditions deteriorate, the PD of an
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
will tend to increase while it will tend to decrease if economic conditions improve.
A stressed PD is an estimate that the
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
will default over a particular time horizon considering the current
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
specific information, but considering "stressed" macroeconomic factors irrespective of the current state of the economy. The stressed PD of an
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
changes over time depending on the risk characteristics of the
obligor
A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to tr ...
, 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.
[BIS:Studies on the Validation of Internal Rating Systems](_blank)
/ref>
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 (TTC) or point-in-time (PIT) 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 (CDS-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 probability of default can be inferred by the price.
CDS provide risk-neutral probabilities of default, which may overestimate the real world probability of default unless risk premiums are somehow taken into account. One option is to use CDS implied PD's in conjunction with EDF (Expected Default Frequency) credit measures.
Deriving point-in-time and through-the-cycle 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 (or their proxies) 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 (ASRF).
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 (DRs), 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 (KDPs), 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” (DD). 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 (enumerated as default distance) 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.
Most PD models output PDs that are of a hybrid nature: they are neither perfectly Point-In-Time (PIT) nor through-the-cycle (TTC). The long-run average of Observed Default Frequency ODF is often regarded as a TTC PD. It is argued that when considered over a long period, the systematic effects averages close to zero. However, defining the appropriate period of reference for calculating such an average is often challenging, e.g. multiple business cycles in the historical data can over or underestimate the average PD which is considered a biased estimate. Furthermore, the assumption of a constant TTC PD for a pool of obligors is not realistic in practice. In fact, idiosyncratic risk of a portfolio can vary over time. A classic example is changes in the distribution of the portfolio due to in- and out-flows of the obligors but also due to decisions taken by the bank, such as modifications of lending conditions or policies.
Estimation
There are many alternatives for estimating the probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of default. Default probabilities may be estimated from a historical data base of actual defaults using modern techniques like logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear function (calculus), linear combination of one or more independent var ...
. Default probabilities may also be estimated from the observable prices of credit default swaps
A credit default swap (CDS) is a financial swap agreement that the seller of the CDS will compensate the buyer in the event of a debt default (by the debtor) or other credit event. That is, the seller of the CDS insures the buyer against som ...
, 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
Moody's Investors Service, often referred to as Moody's, is the bond credit rating business of Moody's Corporation, representing the company's traditional line of business and its historical name. Moody's Investors Service provides international ...
for estimating PDs from historical default experience. For small business default probability estimation, logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear function (calculus), linear combination of one or more independent var ...
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
A credit score is a numerical expression based on a level analysis of a person's credit files, to represent the creditworthiness of an individual. A credit score is primarily based on a credit report, information typically sourced from credit bu ...
" as a euphemism
A euphemism () is an innocuous word or expression used in place of one that is deemed offensive or suggests something unpleasant. Some euphemisms are intended to amuse, while others use bland, inoffensive terms for concepts that the user wishes ...
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.The Basel II Risk Parameters
/ref>
* Linear regression
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...
* Discriminant analysis
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features ...
* Logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.
Mathematically, the logit is the ...
and probit
In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and s ...
Models
* Panel models
* Cox proportional hazards model
Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional hazar ...
* Neural networks
A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
* Classification tree
Classification chart or classification tree is a synopsis of the classification scheme, designed to illustrate the structure of any particular field.
Overview
Classification is the process in which ideas and objects are recognized, differentia ...
s
See also
* Jarrow–Turnbull model
The Jarrow–Turnbull model is a widely used "reduced-form" credit risk model.
It was published in 1995 by Robert A. Jarrow and Stuart Turnbull.
Under the model, which returns the corporate's probability of default, bankruptcy is modeled as a ...
* Merton model The Merton model,
developed by Robert C. Merton in 1974, is a widely used credit risk model.
Analysts and investors utilize the Merton model to understand how capable a company is at meeting financial obligations, servicing its debt, and weighing ...
References
Reading
*
* {{cite book , author = Duffie, Darrell and Kenneth J. Singleton , title = Credit Risk: Pricing, Measurement, and Management , year = 2003 , publisher = Princeton University Press, isbn= 978-0-691-09046-7
External links
* https://ssrn.com/abstract=1921419 Through-the-Cycle EDF Credit Measures methodology paper
* http://www.bis.org/publ/bcbsca.htm Basel II: Revised international capital framework (BCBS)
* http://www.bis.org/publ/bcbs107.htm Basel II: International Convergence of Capital Measurement and Capital Standards: a Revised Framework (BCBS)
* http://www.bis.org/publ/bcbs118.htm Basel II: International Convergence of Capital Measurement and Capital Standards: a Revised Framework (BCBS) (November 2005 Revision)
* http://www.bis.org/publ/bcbs128.pdf Basel II: International Convergence of Capital Measurement and Capital Standards: a Revised Framework, Comprehensive Version (BCBS) (June 2006 Revision)
Loans
Basel II
Banking terms
Credit risk