K.P.P. Munniksma
BMI Paper
Vrije Universiteit
Faculty of Sciences
Business Mathematics and
Informatics
De Boelelaan 1081a
1081 HV Amsterdam
Preface
This paper is a compulsory element
of the study Business Mathematics and Informatics. During an internship at a
financial institution I got interested in the subject of this paper, Credit
Risk Measurement under Basel II. The main goal of this paper is that the
student learns to do literary research and that the student learns to write a
paper in a scientific way. This paper has been the last subject I have done and
can be seen as one of the main achievements of 5 years of study.
I hereby want to thank my
supervisor Menno Dobber for his time and advice. I enjoyed writing this paper
and his way of guiding me through this process has certainly attributed to
that! Menno, good luck with writing your PhD thesis.
Koen Munniksma
August 2006
Abstract
In this paper the credit risk measurement under the
Basel II Capital Accord is discussed. The Basel II Capital Accord is the
successor of the Basel I Capital Accord (1988). The Basel I Capital Accord
represented the first step toward risk-based capital adequacy requirements.
However, since 1988 the financial world has changed dramatically and there
existed a growing need within the financial world for a new capital accord. In 2007 a new capital accord
will become operative, and it will improve the Basel I Capital Accord on
several aspects.
The Basel II Capital Accord seeks to improve on the existing rules by aligning regulatory capital
requirements more closely to the underlying risks that banks face. One of the
risk types described in the Capital Accord is credit risk. Banks
need to hold capital to cover the credit risk on their credit portfolio. The focus of this paper lies on the analysis of the
Internal Ratings Based (IRB) Capital Requirements function for credit risk. On
the basis of the work of Merton and Vasicek, the Basel Committee on Banking
Supervision (BCBS) decided to adopt the assumptions of a normal distribution
for the systematic and idiosyncratic risk factors of a credit portfolio. The
model behind the capital requirements function is called an Asymptotic Single
Risk Factor (ASRF) model. The Basel Committee had as important requirement that
the capital requirements function should be portfolio invariant. Gordy has
shown that essentially only ASRF models are portfolio invariant. and therefore
the Basel Committee has chosen for an ASRF model. Although the Basel II Capital
Accord is a clear improvement of the Basel I Capital Accord, it still has some
clear weaknesses. For example, the assumption of portfolio invariance made by
the Basel Committee ignores the existence of concentration risk.
Table of Contents
1. Introduction................................................................................................................................ 4
2. Bank for
International Settlements.............................................................................................. 5
3. Basel I Capital
accord................................................................................................................. 8
4. Basel II Capital
Accord.............................................................................................................. 10
.. 4.1 Introduction......................................................................................................................... 11
.. 4.2 The three pillars of the new
Accord...................................................................................... 12
5. Value at Risk and
other financial risk measures........................................................................ 13
.. 5.1 Introduction......................................................................................................................... 13
.. 5.2 VaR Approaches.................................................................................................................. 13
.. 5.3 Financial Risk Measures...................................................................................................... 15
6. Basel II credit
risk apporaches.................................................................................................. 16
.. 6.1 Introduction......................................................................................................................... 16
.. 6.2 The Standardized Approach................................................................................................. 16
.. 6.3 Internal Ratings Based Approach......................................................................................... 17
.. 6.3.1 Minimal Regulatory Capital
Requirements (K)............................................................... 17
.. 6.3.2 Economic Concepts behind the IRB
Capital Requirements Function............................. 18
.. 6.4 Interpreting the IRB
Capital Requirements function.............................................................. 20
.. 6.5 Differences between the Foundation
and Advanced IRB approaches.................................... 27
Conclusion................................................................................................................................... 28
Abbreviations............................................................................................................................... 29
References.................................................................................................................................... 30
1. Introduction
In recent years the needs for
professional skills in the modelling and management of credit risk have rapidly
increased and credit risk modelling has become an important topic in the field
of finance and banking. While in the past most interests were in the
assessments of the individual creditworthiness of an obligor, more recently
there is a focus on modelling the risk inherent in the entire banking portfolio.
This shift in focus is caused in greater part by the change in the regulatory
environment of the banking industry. Banks need to retain capital as a
buffer for unexpected losses on their credit portfolio. The level of capital
that needs to be retained is determined by the central banks. In 2007 a new capital accord,
the Basel II Capital Accord, will become operative. This accord will be the
successor of the Basel Basel ,
Switzerland
Due to these changes in the banking and finance
industry the use of risk measures in credit portfolio management has increased
dramatically. One of the most used risk measures is Value at Risk (VaR). VaR is used in finance as a technique that uses
statistical analysis of historical data to estimate the likelihood that a given
portfolio’s losses will exceed a certain amount. There are various methods to
calculate VaR and these methods have a large impact on the way a bank manages
its credit portfolio. Furthermore VaR is incorporated in the Basel II Capital
Accord.
In
this paper the reader will be introduced to the economic concepts and
mathematical models underlying credit risk measurement under Basel II. The
development of credit risk measurement will intensify because of the New
Capital Accord and new models and methods will arise to improve risk
management. A step is made in the right direction, but the end destination has certainly
not been reached!
In
Chapters 2 till 4 comprise out of a general discussion of the Bank for
International Settlements and the Basel Capital Accords. In Chapter 5 VaR and
other financial risk measures are discussed. Thereby a short description is
given of the concept of coherent risk measure. In chapter 6 the Basel II
approaches for credit risk measurement are broadly discussed. In the final
chapter some conclusions are drawn on the limitations of the Basel II Capital
accord for credit risk measurement.
2. Bank for International Settlements
The Bank for International
Settlements (BIS) was established in 1930. It is the world’s oldest
international financial institution and remains the centre for international
central bank cooperation. The BIS was established in the context of the Young
Plan (1930). This plan dealt with the issue of the reparation payments imposed
on Germany Berlin
The BIS was also created to act
as a trustee for the Dawes and Young Loans, international loans issued to
finance reparations, and to promote central bank cooperation in general. The
functions involving the issue of reparation payments quickly became obsolete.
The Bank focused thereby its activities entirely on the cooperation between central
banks and other agencies in pursuit of monetary and financial stability.
In the immediate aftermath of the
Second World War and until the early 1970s the BIS focus was shifted primarily
on implementing and defending the Bretton Woods system. On July 1st 1944 , politicians
from all 44 allied nations came together in Bretton Woods New Hampshire to
construct a new economical system. This system is called the Bretton Woods
system. The chief features of the Bretton Woods system were an
obligation for each country to adopt a monetary policy that maintained the exchange
rate of its currency within a fixed value (plus or minus one percent) in terms
of gold. In the face of increasing strain, the system collapsed in 1971,
following the United States
The central bank cooperation at
the BIS has taken place through the regular meetings in Basel
In 1988, the BCBS has proposed
guiding principles for the banking industry in the Basel Capital Accord. The
Basel Capital Accord was gradually adopted by the central banks. Although the
BIS does not occupy itself with the implementation of regulation, its
reputation and influence is of such nature that its recommendations are
considered as best practise. More recently the issue of financial stability in
the wake of economic integration and globalisation has received a lot of
attention.
The Basel 2007 in many of the over
100 countries currently using the Basel
Table 2.1
Evolution of the Regulatory Environment
|
|
|
|
1996
|
Market Risk
Capital Amendment
|
|
1996 – 1998
|
Ad hoc rules
for credit derivatives
|
|
Jun. 1999
|
Consultative
Document from
|
|
Jan. 2001
|
Basel
Committee proposes New Capital Accord
|
|
Nov. 2005
|
|
Apart from fostering monetary
policy cooperation, the BIS performs and has performed numerous other
functions:
-
The BIS has always performed "traditional"
banking functions for the central bank community. It has done gold and foreign
exchange transactions and employed trustee and agency functions. The BIS was
the agent for the European Payments Union (EPU, 1950-58) and has acted as the
agent for various European exchange rate arrangements. One of these exchange
rate arrangements was the European Monetary System (EMS, 1979-94) which
preceded the move to a single currency.
-
The BIS has provided or organised emergency financing
to support the international monetary system when needed. During the 1931-33
financial crisis, the BIS organised support credits for both the Austrian and
German central banks. In the 1960s, the BIS arranged special support credits
for the French franc (1968). More recently, the BIS has provided finance in the
context of IMF-led stabilisation programmes (Mexico in 1982 and Brazil in 1998).
-
The BIS has developed its own research in financial
and monetary economics and makes an important contribution to the collection,
compilation and dissemination of economic and financial statistics.
3. Basel
The Basel I Capital Accord
represented the first step toward risk-based capital adequacy requirements. The
accord was an agreement by the members of the BCBS with respect to minimum
regulatory capital for credit risk. Credit risk is the possibility of a loss as
a result of a situation that those who owe money to the bank may not fulfil
their obligation. Regulatory capital refers to the risk-based capital
requirements under the Capital Accord. The purpose of regulatory capital is to
ensure adequate resources are available to absorb bank-wide unexpected losses.
Under the rules of the Basel
Regulatory
Capital = Risk Weight ´
Exposure ´ 8%
= Risk-Weighted-Assets x 8%
In the
above formula the constant 8% is the minimum ratio of regulatory capital to
total risk-weighted assets (RWA). This value was determined by the Basel
Committee. In the following paragraphs the formula for the
calculation of regulatory capital is discussed. At the end of this Chapter the
shortcomings of the Basel I Capital Accord are discussed.
Risk Weight
The risk weights reflect the relative
credit riskiness across different types of exposures. The Risk
Weight for a transaction is determined by characteristics of the obligor. In
Table 3.1 the risk weights from the Basel I Capital Accord are given. Roughly
speaking, three kinds of obligors can be distinguished: Sovereigns, Banks and
Corporates with corresponding risk weights of 0%, 20% and 100%.
Table 3.1 Risk
Weights from the Basel
|
Obligor
|
Risk Weight
|
|
Domestic
public sector entities
(excluding
central governments)
|
0%, 10%, 20%
or 50%
Percentage Set
by domestic regulator
|
|
OECD banks
and regulated firms
|
20%
|
|
Loans fully
secured by residential property
|
50%
|
|
Counterparties
in derivatives transactions
|
50%
|
|
100%
|
Exposure
The type of instrument determines
the Exposure:
-
For fully funded loans or bonds, the exposure is the
face amount.
-
For unfunded commitments; the exposure is 50% of the
commitment for undrawn commitments with maturity over
on year and 0% of the commitment for undrawn commitments with maturity less
than one year.
-
For credit products (e.g. guarantees) the exposure is
100% of the notional value of the contract.
-
For derivatives the exposure is determined by the
equation: Replacement Costs +(Add-On Percentage ´ Notional Principal). In Table 3.2 the add-on
percentages are given.
Table 3.2 Add-on Percentages for Derivative Contracts
under the Basel
|
Maturity
|
Interest
Rate
|
Exchange
Rate and Gold
|
Equity
|
Precious
Metals Except Gold
|
Other
Commodities
|
|
One year or
less
|
0.0%
|
1.0%
|
6.0%
|
||
|
More than
one year to five years
|
0.5%
|
5.0%
|
8.0%
|
7.0%
|
12.0%
|
|
More than
five years
|
1.5%
|
7.5%
|
10.0%
|
8.0%
|
15.0%
|
Shortcomings of Basel
In the last part of this Chapter
several shortcomings of the Basel I Capital Accord are discussed.
Although Basel
- The Basel
I Capital Accord provides inconsistent treatment of credit risks. For
example, a relatively risky bank in an OECD country requires less
regulatory capital than a relatively less risky corporation.
- The Basel
I Capital Accord does not measure risk on a portfolio basis. It does not
take account of diversification or concentration and there is no provision
for short positions. For example the amount of regulatory capital on a
mortgage loan in a portfolio consisting only out of mortgages is the same
as when the mortgage loan is part of a portfolio consisting of a variety
of loan products.
- The Basel
I Capital Accord provides for no regulatory relief as models and
management of capital improve.
4. Basel
4.1 Introduction
In January 2001 the BCBS released
its proposal for a new Accord. This new Accord, the Basel II Capital Accord, is
the successor of the Basel I Capital Accord.
The Basel II Capital Accord attempts to improve the Basel I Capital
Accord in the following points:
-
In the Basel II Capital Accord banks are granted a
greater flexibility to determine the appropriate level of capital to be held in
reserve according to their risk exposure.
-
The Basel II Capital Accord focuses on the enhancement
of the stability and reliability of the international financial system.
-
The Basel II Capital Accord stimulates the improvement
of risk management.
In the remainder of this chapter
the Basel II accord is described.
4.2 The three pillars of the new Accord
The
Basel I Capital Accord focused only on minimum regulatory capital requirements.
The Basel II Capital Accord broadens this focus by describing the supervisory
process in the Basel II Capital Accord by
“three pillars”:
-
Pillar 1 - Minimal
regulatory capital requirements.
-
Pillar 2 -
Supervisory review of capital adequacy.
-
Pillar 3 - Market
discipline and disclosure.
The
focus of the paper lies in the measurement of credit risk, which is extensively
described in Pillar 1. Below the contents of the Pillars are shortly discussed
and in Chapter 6 the credit risk measurement is broadly discussed.
Figure 4.1 The Three
Pillars of Basel
Pillar 1- Minimum Regulatory Capital Requirements
For the first pillar of the Basel
II Capital Accord the Basel Committee proposed capital requirements associated
with three categories of risk:
- Market
Risk
Market
Risk is the risk that the value of an investment will decrease due to
moves in market factors. Within the
Basel II Capital Accord, there are two methods to measure market risk: The
Standardized Approach and the Internal Models Approach.
- Operational
Risk
Operational risk is defined in the Basel
II as the risk of loss resulting from inadequate or failed internal processes,
people and systems, or from external events. Three different methods can be
used to measure operational risk: The Basic Indicator Approach, the
Standardized Approach and the Advanced Measurement Approach.
- Credit
Risk
Credit risk is the possibility of a loss as a result of a
situation that those who owe money to the bank may not fulfil their obligation.
The following methods can be used to determine credit risk: The Standardized
Approach, The Foundation Internal Rating Based Approach and the Advanced Rating
Based Approach. The Standardized Approach provides improved risk sensitivity
compared to Basel
Pillar 2 - Supervisory review of capital adequacy
The second pillar of Basel II is
a supervisory review of capital adequacy. The second pillar notes that national
supervisors must ensure that banks develop an internal capital assessment
process and set capital targets consistent with their risk profiles.
Furthermore it encourages the bank’s management to develop risk management
techniques and their use within capital management. The supervisors are responsible for evaluating
how well banks are assessing their capital adequacy needs relative to their
risks. Internal processes of the bank are subject to supervisory review and
intervention. In the Netherlands
Pillar 3 – Market discipline and disclosure
The third pillar of the Basel II
Capital Accord is about market discipline and disclosure. The main goal of this
pillar is to promote the development of financial reporting about risks. In
this way market participants can get a better understanding of banks risks
profiles and the adequacy of their capital position by disclosure. Pillar 3 in the Basel II Capital
Accord sets out disclosure requirements and recommendations in several areas.
These requirements apply to all banks and when a bank cannot meet these
requirements it can be constrained in the way it manages capital. For example
the bank may not use any of the advanced techniques under Pillar 1.
5. Value at Risk and other financial risk measures
5.1 Introduction
VaR expresses the expected
maximum loss that may be incurred over a defined time horizon and within a
specified confidence interval. Value at Risk (VaR) is one of the most popular
tools used in financial risk measurement. The concept of VaR has now been
incorporated in the Basel II Capital Accord. Understanding VaR and its
application in risk management is essential for understanding the Basel II
Capital Accord. In this Chapter, first some methods for the computation of VaR
are described. Then some other risk measures like Expected Loss and Expected
Shortfall and the limitations of VaR are discussed.
5.2 VaR Approaches
A variety of approaches exist for
estimating Value At Risk. The most common assumption in these models is that
historical data is the best estimator for future changes. However, every
approach has besides this assumption its own specific set of assumptions. The
following models can be distinguished:
- The
Variance-Covariance (VCV) Approach:
The VCV approach assumes that risk
factor returns are always (jointly) normally distributed and that the change in
portfolio value is linearly dependent on all risk factor return.
- The
Historical Simulation (HS) Approach:
The HS approach assumes that asset
returns in the future will have the same distribution as they had in the past.
In the MCS
approach future asset returns are simulated.
In the following paragraphs these methods are discussed
in more detail.
Variance-Covariance
model
The Variance-Covariance method
is based on the following two assumptions:
- The changes in the value of the portfolio
are linearly dependent on all the changes in the value of the assets.
- The asset returns are jointly normally
distributed
In order to measure the
portfolio’s volatility, a covariance matrix must be estimated in advance. If a
portfolio consists of N assets, the portfolio return is a linear combination of
the returns on the underlying assets. The VaR over the portfolio is then
calculated in the following way:
VaRP = α x σP x
P0
With,
P0 = Portfolio Value; the sum of all asset values,
Pi = Value of asset I,
wi =
Pi / P0 ; the relative weight of asset i with
respect to the total portfolio,
w = column vector of weights; w = {w1, w2,
w3, … , wN},
R = column vector of the rate of return; every asset of the
portfolio has a different rate of return,
∑ = covariance matrix (mentioned above),
RP = wTR,
σP = 
, the standard deviation of the portfolio,
, the standard deviation of the portfolio,
α = the
chosen VaR-percentile. Where an α of 1,96 and 2,58 respectively indicating a
VaR-percentile of 95% and 99%.
Advantages:
-
The methodology is based on well-known techniques.
-
If input variables such as the covariance matrix are
known, VaR calculation can be done in a simple Excel spreadsheet.
-
VaR calculation is fast with the covariance variance
method because it is not simulation-based but analytical.
Disadvantages:
-
The portfolio is composed of assets whose changes are
linear.
-
The assumption that the asset returns are normally
distributed is rarely true (parametric).
Historical
Simulation
In the historical simulation approach the distribution of future shifts
in risk factors of a portfolio is treated in the same way as prior-period
distributions of shifts. This means that past shifts are used to revalue the
portfolio and simulate its profits and losses. VaR is then obtained by reading
the relevant quantile (e.g. 95% quantile
or 99% quantile) off the histogram of
simulated returns.
Advantages:
-
The historical simulation method
is very simple to implement.
-
It is non-parametric; i.e. it
does not assume a normal distribution of asset returns
Disadvantages:
-
The method requires a large
database containing historical data; the quality of data management must be
high.
-
The method is computationally
intensive.
Monte Carlo Simulation
Estimating VaR via Monte Carlo simulations is similar in some respects to
applying the historical and the parametric methods. First, the joint
distribution of risk factors is specified and used to generate a large number
of risk-factor variation scenarios. These scenarios are then used to compute the
hypothetical results of the portfolio. Last, VaR is determined in the same way
as in the historical simulation approach but on the basis of the simulated
sample.
In this Section some other
financial risk measures besides Value at Risk are discussed. Furthermore the
concept of ‘coherent risk measure’ is explained. A financial risk measure is a
metric measuring the uncertainty of portfolio loss. Several risk measures are
defined for portfolio loss. The most important risk measures are:
-
Expected Loss (EL): The mean
of the portfolio loss distribution.
-
Unexpected Loss (UL): The
standard deviation of the portfolio loss distribution
-
Value At Risk (VaR) or Economic
Capital (EC); where EC is defined as the 99,95% VaR - EL
-
Expected Shortfall or Conditional
Value at Risk (ES or CvaR): CVaR or ES is the portfolio loss one expects to
suffer, given that the portfolio loss is equal to or larger than its VaR.
In the literature researchers
have extensively criticized the use of VaR as a measure of risk. For example,
Artzner, Delbaen, Eber and Heath (1999) show that VaR is not a ‘coherent’
measure of risk. A coherent risk measure needs to satisfy five properties. If A
en B denote portfolios, then f is said to be a coherent risk measure if:
-
Translation invariance: f(A + ar) = f(A) – r , where r is a
reference risk free investment
-
Sub-additivity: f(A+B) £ f(A) + f(B)
-
Positive homogeneity: for all l ³ 0, f (lA) = lf(A)
-
Monotoneity: if A £ B then f(A) £ f(B)
-
Relevance: if A ¹ 0 then f(A) > 0
The risk measures UL and VaR are
not coherent risk measures. UL is not monotonous and VaR does not satisfy the
sub-additivity property, because of that the following situation may arise:
VaR(A+B) > VaR(A) + VaR(B). This means that in this situation
diversification has no effect. Expected Shortfall is on the other hand a
coherent risk measure and therefore guarantees that the portfolio
diversification is always positive. Although
VaR is now widely used in financial risk measurement, it is probably not the
risk measure of the future!
6. Basel
6.1 Introduction
Table 6.1 General comparison between the three
approaches
6.2 The Standardized Approach
The standardized approach is
similar to the approach in the Basel I Capital Accord in that the risk weights
are determined by the category of borrower (sovereign, bank, corporate). The SA
is in principal the same as the current approach. However, it is more risk
sensitive than the current approach. Every asset class or derivative is
assigned a risk weight. These risk weights are based on the external credit
ratings provided by the credit rating agencies.
Table 6.2
provides the risk weights proposed by the Basel Committee. The Standardized
Approach is clearly an improvement on the Basel I Capital Accord:
-
The SA eliminates
the OECD club preference in the Basel I Capital Accord.
-
The SA provides
greater differentiation for corporate credits.
-
Higher risk
categories (150%) are introduced in the SA.
-
The SA contains
the option to allow higher risk weights for equities.
Table 6.2 Risk Weight in Standardized Approach of Basel
|
Rating of Entity
|
Risk
Weights
|
||
|
Sovereigns
|
Banks
|
Corporates
|
|
|
AAA to AA-
|
0%
|
20%
|
20%
|
|
A+ to A-
|
20%
|
50%
|
50%
|
|
BBB+ to BBB-
|
50%
|
100%
|
|
|
BB+ to BB-
|
100%
|
100%
|
100%
|
|
B+ to B-
|
100%
|
100%
|
100%
|
|
Below B-
|
150%
|
150%
|
150%
|
|
Unrated
|
100%
|
50%
|
100%
|
For Residential Mortgages the
risk weights are set to 35% and for all other retail loans the risk weights are
75%. The following equation is used to calculate the minimum regulatory capital
for credit risk and is the same as in the Basel I Capital Accord:
Regulatory
Capital = Risk Weight ´
Exposure ´ 8%
6.3 Internal
Ratings Based Approach
The SA was targeted at banks
desiring a simplified capital framework. For the more sophisticated banks the
Basel Committee proposed two IRB approaches, foundation and advanced. The
difference between these approaches is subtle.
6.3.1 Minimal Regulatory Capital Requirements (K)
The following formula is used
to calculate the minimal capital requirements, K, under the Internal
Ratings Based Approach:
R =
Asset Correlation; the correlation between an individual loan and the global
state of the world economy,
N[ …] = the cumulative distribution for a standard normal variable,
G[ …] = the inverse cumulative distribution for a standard normal variable,
LGD = Loss Given Default,
PD = Probability of Default,
M = Maturity of the loan,
b(PD) = Smoothed
regression maturity function. The slope of the adjustment function with respect
to M decreases as the PD increases.
In Section 6.4 the parameters
of the capital requirements formula are further explained. For calculating the
regulatory capital one now simply multiplies the capital requirements with the
exposure at default:
Regulatory
Capital = K x EAD
6.3.2 Economic Concepts behind the IRB Capital
Requirements Function
The
IRB approach focuses on the frequency of bank insolvencies arising from credit
losses that supervisors are willing to accept. It is possible to estimate the
amount of loss which will be exceeded with a predefined probability by means of
a stochastic credit portfolio model. The area under the right hand site of the
curve is the likelihood that a bank will not be able to meet its own credit
obligations by its profits and capital. The percentile or confidence
level is then defined as 100% minus this likelihood and the corresponding
threshold is called the Value At Risk (VaR).
6.4 Interpreting the IRB Capital Requirements function
The Basel II IRB approach is
built on the following credit risk drivers.
-
Probability of Default (PD): The Probability of Default quantifies the risk that
the obligor will default in the coming 12 months.
-
Maturity (M): The maturity of a loan expresses the amount of time
until the loan is fully due and payable.
Then, the Expected Loss
can be written as:
EL = PD x EAD x LGD
In the Basel I Capital Accord
(Chapter 3) and the Standardized Approach of Basel II (Section 6.2) these main
drivers for credit risk are not incorporated in the calculation of regulatory
capital. However in the IRB approach they play a central role. In the following
paragraph is explained on what mathematical basis the capital requirements
function has been developed.
The model behind the Capital Requirements function
The Basel Capital Requirements
function used for the derivation of supervisory capital charges are based on a
specific model developed by BCBS. An important restriction was made to fit
supervisory needs; The capital
required for any given loan should only depend on the risk of that loan and
must not depend on the portfolio it is added to. In other words; the model
underlying the Basel Capital Requirements function should be portfolio
invariant.
Gordy (2003) has shown that
essentially only so called Asymptotic Single Risk Factors (ASRF) models are
portfolio invariant. ASFR models are derived from “traditional” credit
portfolio models by the law of large numbers. When a portfolio consists of a
large number of relatively small exposures, idiosyncratic risks associated with
individual exposures tend to cancel out one-another. Systematic risks that
affect many exposures have than only an effect on portfolio losses. The
systematic risks that affect all borrowers to a certain degree (e.g. industry
and region risks) are modelled in the ASFR model by one systematic risk factor.
Banks are encouraged to use other credit risk models to fit their internal risk
management needs.
It is possible to estimate the
sum of the expected and unexpected losses associated with each credit exposure.
This is accomplished by calculating the conditional expected loss (CEL)
for an exposure given an appropriately conservative value of the single
systematic risk factor. The CEL is in the ASRF model expressed as the product
of a conditional PD and a downturn LGD. The conditional PDs are
derived from bank-reported average PDs, which reflect expected default rates
under normal business conditions, using a supervisory mapping function. The
downturn LGD reflects the LGD under economic downturn conditions. During
cyclical economic downturns the LGDs are higher than during “average” economic
conditions. Adding up the total economic resources (provisions and capital)
that a bank must hold to cover all its exposures conditional expected losses,
will lead to a portfolio-wide VaR. The mapping function used to derive
conditional PDs from average PDs is derived from an adaptation of Merton’s
model to credit portfolios. According to the Merton Model, borrowers default if
they cannot completely meet their obligations at a fixed assessment horizon of
one year because the value of their assets is lower than the due amount.
Vasicek (2002) showed that under certain conditions the Merton Model can
naturally be extended to a specific ASRF credit portfolio model. On the basis
of the work of Merton and Vasicek, the BCBS decided to adopt the assumptions of
a normal distribution for the systematic and idiosyncratic risk factors.
The BCBS determined that given
the evolving nature of bank practices in the area of LGD quantification, it would
be inappropriate to apply a single supervisory LGD mapping function similar to
the PD mapping function. In the advanced IRB approaches banks are required to
estimate their own downturn LGDs. These LGDs should reflect the fact that
during economic downturn conditions the LGD is higher than during average
business conditions.
Figure
6.2 Decomposition of the Capital Requirements Formula
As can be seen in Figure 6.2
the capital requirements formula can roughly be viewed as subtracting the conditional
expected loss with the expected loss and than multiply this difference with a
maturity factor. In the next paragraph the asset correlations (R) are described
in detail.
Asset Correlations
The asset correlations show
how the asset value of one borrower depends on the asset value of another
borrower. In other words it could be described as the dependence of the asset
value of a borrower on the general state of the economy. In this way all
borrowers are linked to each other by this single risk factor. The asset
correlations are asset class dependent because different borrowers show
different degrees of dependency on the overall economy.
The asset correlations of the
capital requirements formula for corporate, bank and sovereign exposures have
been derived by analysis of data sets from G10 supervisors. The analysis of
this data has revealed two systematic dependencies:
- Asset
correlations decrease with increasing PDs: The higher the PD the higher
the idiosyncratic risk of a borrower.
- Asset
correlations increase with firm size: Idiosyncratic risks are higher for
smaller firms.
The asset correlation functions
used in the Basel
Figure 6.3
Asset Correlations for different type of Assets
Wholesale
(Corporates, Banks and Sovereigns) Asset Correlation
The asset correlation for
wholesale exposures is given by the following formula:
,
The asset correlation assumptions
are different for retail exposures than for corporate exposures. The difference
relates to the actual calibration of the asset correlation curves. For retail
exposure the curves have been derived from economic capital figures from large
internationally active banks and from historical loss data from supervisory
databases fo the G10 countries. A consequence of this derivation of the retail
asset correlations is that for determining the capital requirements for retail
exposures the maturity adjusment are not applied. This is because the economic
capital data and the supervisory loss data implicity contain the maturity
effects.
-
Residential Mortgages: Asset
Correlation (R) = 0.15
-
Qualifying Revolving Retail Exposures: Asset
Correlation (R) = 0.04
-
Other Retail Exposures: 
Figure
6.5 Other Retail Asset Correlation
In comparison with Figure 6.4 it can be seen that the
asset correlation for other retail exposures declines slower than the asset
correlation for corporate exposures when PD increases. This can be explained by
the fact that the pace in which the asset
correlation function decreases is determined by the so-called “k-factor”. For corporate exposures this k-factor is set
to 50 and for other retail exposures this k-factor is set to 35. In the following
paragraph the last part of the capital requirements function is discussed,
namely the maturity adjustment.
Maturity
Adjustment
The capital requirements should
increase with maturity. Both intuition and empirical evidence indicate that
long-term credits are riskier than short-term credits. The form of the Basel Basel
Figure 6.6 Smoothed regression maturity adjustment
The
“full” maturity adjustment that is incorporated in the capital requirements
function is as follows:
The maturity adjustment is
introduced in the formula to reflect the potential credit quality deterioration
of loans with longer maturities. The average portfolio effective maturity is
set by the Basel Committee at 2.5 years.
In the last paragraph of this Section the confidence level that is set by
the supervisory authorities is discussed.
Determination
of the confidence level
The confidence level is set by
the supervisory authorities at 99,9%. It was first set to 99,5% but a higher
confidence level was chosen to protect against model uncertainties as well as
estimation errors occurring from the PD, LGD and EAD estimation by banks. This
confidence level is incorporated in the capital requirements formula by the
term G(0.999). In Figure 6.7 can be seen that changing the confidence level
from 99,95% to 99,99% has had a big effect on the capital requirements
function.
Figure
6.7 Impact of changing the confidence level
In the last Section of this
Chapter the differences between the two IRB approaches are discussed.
6.5
Differences between the Foundation and Advanced IRB approaches
The main difference between the
foundation and advanced IRB approaches lies in the definition of the input
variables. Both approaches rely on banks’ PD estimates, but banks’ internal
estimates of LGD, EAD and loan maturity are only taken account of in the
advanced IRB approach. In Table 6.3 these differences are summarized.
Although banks rely on their own
PD estimates, they are obliged to use a minimum probability of default of 0.03
percent PD. The transition from an Foundation IRB status to an Advanced IRB
status can only be made if the internal risk measurement systems and data
management systems are of high quality.
Table 6.3 Difference between determinations of Risk
Weights
|
Determinants of Risk Weights
|
Foundation
|
Advanced
|
|
Probability
of Default (PD)
|
Bank
determines
|
Bank
determines
|
|
Loss
Given Default (LGD)
|
Bank
determines
|
|
|
Exposure
At Default (EAD)
|
Supervisor
determines
|
Bank
determines
|
|
Maturity
(M)
|
-
|
Maturity
adjustment
|
Conclusion
In this paper, the credit risk
measurement under the Basel II Capital Accord was discussed. First a general
introduction was given of the Basel II Capital Accord and then the treatment of
credit risk under this new capital accord was discussed in more detail. Below,
some conclusions are drawn with respect to credit risk measurement and Basel
II.
The main goal of the Basel II
Capital Accord is to achieve more tractability and uniformity in the banking
world. Therefore, the Basel II Capital Accord is based on some of the same risk
measurement concepts as in the economic capital models that more sophisticated
banks use internally. However, in its goal of achieving tractability and
uniformity are a number of supervisory parameters and simplifying assumptions
embedded. The structure of the Basel II Capital Accord is therefore inevetibly
different from the internal structure of banks’ economic capital models.
Therefore, It is important to look for cases in which the Basel II measures are
more conservative than a bank’s economic model, which might create potential
incentives for banks to engage in new forms of regulatory capital arbitrage.
One of the most important
assumptions of the Basel II Capital Accord is that the bank’s portfolio is well
diversified and does not contain any significant concentrations of individual
borrowers. Given this assumption the IRB Capital Requirements functions is not
designed to be sensitive to variations in concentration risk across banks. The
underlying risk of a bank’s portfolio can therefore not be fully reflected in
the IRB charges. Take for example a portfolio that is highly concentrated in a
particular geographic market. The correlation between these assets in the portfolio
is for sure higher than the correlation values provided by Basel II. Although
the BCBS had proposed a granularity adjustment to address single-borrower
concentrations, it dropped this proposal later on. The IRB capital requirements
function is therefore insensitive to single-borrower concentrations and the
total amount of regulatory capital is linear dependent on the EAD. If now two
loans have the same risk profile (PD, LGD and M), but their EAD is different,
they still get the same outcome of the IRB Capital Requirements function. Therefore,
it can be concluded that the credit risk of a loan can only be measured
accurately if concentration risk is taken into account. In the future, this
risk must be included in the calculation of regulatory capital, if we want to
bring credit risk measurement to a higher level.
Finally, it is for a bank important
to explore it’s assumptions around correlation within and across different
credit portfolio’s. A related issue is the way a bank reflects the potential risk
posed by concentrations in certain type of exposures. For a bank risk and data
management have become a vital part of their business. It has been shown by
several impact studies that achieving a high level of risk and data management
and thereby earning the A-IRB status leads to a decrease in the minimal
regulatory capital requirements. The Basel II Capital Accord is definitely a
step forward, but for achieving a financial world without regulatory arbitrage
several steps surely still need to be made!
Abbreviations
References
Risk measures + VaR
Philipe Artzner, Freddy Dalbaen, Jean-Marc Eber, David
Heath. Coherent measures of Risk.
Mathematics of Finance, no. 3, p. 203-228. 1999.
Alexander Kurth, Dirk Tasche. Credit Risk Contributions to Value-at-Risk and Expected Shortfall. DP
TU München. 2002.
Carlo Acerbi, Dirk Tasche. On
the coherence of expected shortfall. Journal of Banking & Finance 26(7),
p.1487-1503. 2002.
Carlo Acerbi, Claudio Nordio, Carlo Sirtori. Expected
Shortfall as a tool for financial risk measurement. Working Paper. 2001.
Tor Jacobsen, Jesper Linde, Kasper Roszbach. Credit risk versus capital requirements under Basel II: are SME loans
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[*] Belgium, Canada, France, Germany, Italy, Japan,
Luxemburg, Netherlands, Spain, Sweden, Switzerland, United Kingdom, United
States of America
[*] Organisation for Economic Co-operation and Development
(OECD) is an international organisation of those developed countries that
accept the principles of representative democracy and a free market economy.
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