Interpreting the Internal Ratings-Based Capital Requirements in
Basel II
Hugh Thomas
Associate Professor of Finance
The Chinese University of Hong Kong
Shatin, NT, Hong Kong SAR, China
And
Zhiqiang Wang
Associate Professor of Finance
Dongbei University of Finance and Economics
Dalian, Liaoning, China
Draft of September 2004
We gratefully acknowledge the support of an Earmark Grant from the Hong Kong Research Grants
Council that made this research possible. Please address all correspondence regarding to this paper to
Hugh Thomas. [email protected].
Abstract
This paper describes the theoretical and institutional background to the formula
specified by the Bank for International Settlements Basel Committee on Banking’s
internal-ratings based (IRB) approach to Pillar 1 of Basel II: minimum capital
requirements. The IRB formula is based on the Vasicek formula and is the
conditional probability of default of a single borrower with normally distributed asset
returns. We discuss the assumptions of the Vasicek formula and the adjustments
made to it in the IRB formula. From these discussions, we make 10 observations
highlighting that the IRB formula does not correspond to industry best practice, but
represents a negotiated compromise to achieve simplicity, portfolio invariance and
bank acceptance of prescribed capital levels. The risk-measurement implications of
this compromise should be understood by regulators and bankers in order to
implement properly regulatory oversight, which constitutes Pillar 2 of Basel II and by
investors who will be should understand disclosure requirements of Pillar 3.
Key words:
Basel II
Internal Ratings-Based Approach
Vasicek Formula
Bank Risk Capital
Bank Regulatory Capital
1
Interpreting the Internal Ratings-Based Capital Requirements in Basel II
This article analyzes the internal ratings-based (IRB) approach of Basel II for
setting bank minimum capital requirements [1]. Bank managers frequently quantify
risk positions as dollars of equity capital put a risk (value at risk or risk capital) to
measure their bank’s internal performance and to set strategy [2]. A key goal of
Basel II is to make regulatory capital and risk capital definitions coincide more
closely than under Basel I [3] 1
. The Basel Committee worked from the late 1990s to
2004 on Basel II setting out first a general direction and then two detailed drafts
[4,5,6] responding to industry comments. The completed Basel II rests on three
pillars: minimum capital requirements, the supervisory review process and market
discipline, but among the three, the first is by far the most complex, taking up about
three quarters of the pages of Basel II. At the core of Pillar 1 is the IRB approach.
1 We refer to the 1988 Capital Accord, the amendment to incorporate market risks by the Basel Committee in 1996 and Basel Committee publications 9, 12, 18 and 36 collectively as “Basel I” [3].
2
The Basel Committee has explained the philosophy of the IRB approach
[1,7,8]. Economists at central banks have compared commercially available credit
risk models from a statistical point of view and have used them to calibrate the IRB
approach [9,10,11]. Wide briefly explains the IRB approach from a risk management
approach [12]. Yet bankers have expressed strong reservations about the opacity of
the equations and their coefficient values in the IRB approach [13]. To address these
reservations, we discuss the motives for the IRB approach, the basis of the IRB
formula, the derivation of the Vasicek formula that lies within IRB formula,
adjustments in the IRB formula and issues of concern to those who will apply it – the
single factor model, correlation, granularity, loss given default and maturity.
3
1. Implementing Basel II: a change in regulatory philosophy
Basel II in general and the IRB approach in particular are being implemented
to promote improved risk management in banking [1]. Among the top international
banks in the world, integrated quantitative risk measurement and management
systems are well developed. The advanced IRB approach2 is targeted only at top
international banks. The risk management capabilities of this select group of banks –
who have devoted considerable resources to analysis of risk capital – exceed those of
many bank regulators; hence, it is not surprising that Basel Committee notes that in
drafting Basel II it has gained substantially from industry interaction3. In fact a major
task of Basel II is to get regulators and bankers to speak the same language when
analyzing credit risk and its effect on required risk capital.
2 We focus discussion on the Advanced IRB approach. A major distinction between the Advanced and the Foundations IRB approaches is that in the Advanced, banks supply internal estimates of both probability of default and loss given default while in Foundations, regulators supply loss given default. 3 See [1] clause 15. The Committee sets as an objective that further movements towards accepting internal bank models for setting bank capital adequacy standards are possible in future. See [1] clause 18
4
Prior to Basel II, regulators have not needed to understand modern risk
management. But under Pillar 2, regulators review the processes by which bankers
assess their own capital adequacy. This involves much more than confirming that
bankers have placed risk positions in appropriate buckets for the purpose of
calculating capital adequacy formulae. Banks implementing the IRB approach have
existing risk capital assessment criteria which differ considerably from the regulatory
capital calculations in the IRB approach. Without understanding the theoretical and
practical justification of the IRB formula, neither bank supervisors nor bankers can
assess the relative validity of regulatory and internal capital adequacy approaches.
Understanding the IRB approach is also important to investors. Under Pillar
3, banks will disclose new information about risk measurement and regulatory capital
requirements, with the greatest volume of such disclosures centering on the IRB
approach. The Basel Committee anticipates that the additional disclosure will
facilitate the market’s assessment of bank capital adequacy. But such intelligent
assessment is possible only if investors understand that which is being disclosed.
2. Overview of the IRB Formula
The IRB formula is designed for the loan portfolios of large international
banks. To apply the IRB formula, each bank divides its assets into up to14 different
classes4. For thirteen of those of those classes – all but equity stock – the IRB
4 The 14 classes are composed of five major asset classes: corporate, sovereign, bank, retail and equity. The major class “corporate”, in addition to standard lending to corporations, includes five classes of specialized lending – i.e., lending to special purpose entities – including project finance, object finance, commodities finance, income producing real estate and high volatility commercial real estate. The major class “retail” is composed of three subclasses: secured by residential property, qualifying
5
formula applies. Each bank, based on its own internal ratings system, subdivides each
major asset class by borrower credit grades of relatively homogenous characteristics.
Banks may not select credit grades simply to minimize regulatory capital. They must
demonstrate to regulators that the credit grades provide appropriate predictive powers
and that the bank implements its credit grades for such internal functions as loan
pricing and monitoring in addition to simply meeting regulatory requirements.5 For
each credit grade, the bank provides key variables to plug into the IRB formula. We
simplify the IRB formula as follows:
[1]
KIRB = amount of capital required as a percent of exposure
LGD = loss given default as a percent of exposure
KV = total capital required as a percent of assets specified by the Vasicek
formula, assuming LGD=1.
PD = probability of default of the borrowers
MATA = an adjustment for the average maturity of the loans
The formula above calculates KIRB, being the amount of required capital expressed as a
percentage of exposure. Basel II finally specifies capital required as an amount of
currency. KIRB from equation [1] is multiplied by exposure at default (EAD),
expressed in currency (e.g., dollars, euros, yen, etc) to obtain the required capital
expressed in currency. EAD is gross exposure, is at least as large as current
exposure, and will be greater than current exposure in the case of committed but
undrawn lines of credit, where drawdown is likely to precede default.
revolving and all other retail. In addition, the major classes “retail” and “corporate” may contain “eligible purchased receivables”. 5 See [1] clause 444. Basel II requires a minimum of seven borrower grades for non-defaulting corporate, sovereign and bank borrowers and an unspecified number for retail borrowers in order to provide a “… meaningful distribution”. See [1] clauses 404 and 409.
6
Three underlying parameters in equation [1], loss given default (LGD),
probability of default (PD), and correlation between returns of the assets of the
obligor firms, , determine the value of KIRB. LGD is the maximum percent of a loan
that is actually at risk. If a borrower defaults but the lender realizes from sale of
assets of the borrower the full amount of its loan, then the loan is risk-less and no
capital is needed to cushion the bank against losses. Capital is only needed to the
extent that default is accompanied by loss.
PD is estimated for each grade of loans. Correlation, , does not appear
explicitly in equation [1] but is, together with PD, imbedded in the terms KV and
MATA and will be discussed below.
Observation 1: The IRB does not compute a capital charge for a bank based
on the bank’s own internal model. Banks use their internal ratings of
borrowers only to categorize borrowers’ credit classes and, within each class,
credit grades. Internal bank estimates of PD, LGD and EAD are plugged into
the IRB formula to obtain regulatory capital.
7
3. The Vasicek formula
The Vasicek formula forms the heart of the IRB formula [14]. Named after
Oldrich Vasicek, co-founder with Stephen Kealhofer and John Andrew McQuown of
KMV Corporation, a leading commercial developer of standards and procedures for
measuring and pricing bank credit risk6, the Vasicek formula, also called the
asymptotic single risk factor approach [11] is
[2]
Where:
N = cumulative standard normal distribution
N-1 = inverse standard normal distribution
q = the level of confidence with which one wishes to establish that the
bank capital is sufficient to sustain losses
= a measure of correlation between returns on the assets of the
borrowers in the portfolio.
The Vasicek formula specifies the level of capital (expressed as a percent of
EAD) that is required to prevent the bank from going bankrupt in one year with a
probability of no more than (1-q), assuming that the loan generates no income and,
when a loan goes into default, there is no recovery (i.e., LGD=1). In other words if
one set q=.999 (as specified in Basel II) and if the Vasicek formula were an accurate
reflection of reality, one would expect that a typical compliant bank would have an
approximately even chance of going bankrupt once in 693 years.7
6 The formula has not achieved the exposure it deserves partly because it was developed for commercial purposes. The abstract states “This is a highly confidential document that contains information that is the property of KMV Corporation…. This document is being provided to you under the confidentiality agreement that exists between your company and KMV. This document should only be shared on a need to know basis with other employees of your business….” [14] 7 Probability of survival in each year is 0.999. Observations are independent. 0.999693 = 0.4999, so 693 years elapse before the probability that the bank survives in all years first drops below 50%.
8
The Vasicek formula assumes that firm asset returns are normally distributed;
however, economic variables seldom exhibit normality. The Vasicek formula is a
single factor model: portfolio risk springs only from a single, economy-wide risk
factor. One might proxy such a single factor by real GDP growth. Yet if we examine
from mid-year 1921 through mid-year 2003 real US GDP growth for example (see
figure 1), we reject the normality assumption8.
[Insert figure 1 here]
Observation 2: Requiring a capital cushion that uses a q=.999 would be far
in excess of most regulators’ actual requirements if the Vasicek formula’s
statistical assumptions approximated reality. A high q is required because the
assumption of normality in the model is flawed.
In the formula, the credit risk of each borrower is expressed as an annual PD.
Each borrower in a credit grade of a class is assumed to be initially identical, with
assets whose values change through time as they are buffeted continuously by shocks
(uncorrelated through time but correlated across borrowers) drawn from a stable,
normal distribution. If the value of assets of a borrower falls below the borrower’s
debt, default occurs. Thus PD captures the leverage of the borrowers.
The Basel Committee refers to the Vasicek formula as “… a so-called Merton-
style model… ”9 following Merton [15], who shows that the value of the equity of a
firm equals the value of an in-the-money Black-Scholes model option to purchase the
assets of the firm for the face value of the firm’s debt. Like Merton’s formula, the
Vasicek formula models the assets of borrowing firms as random walks in continuous
time. Bankruptcy occurs when, at option expiry, the value of the assets of the
borrower is less than the face value of the debt of the borrower. But Merton is
8 The real GDP growth series is left skewed (-.94) and fat-tailed (kirtosis = 5.64). The Jarque-Bera statistic is 37 leading to rejection of normality at the 0.000000 level of confidence. 9 See [7] clause 172.
9
interested in the value of equity, whereas Vasicek is interested in the probability of
default on the debt. Merton looks at a single firm in isolation, whereas Vasicek is
interested in the properties of a portfolio of debt held by a bank.
4. The Statistical Meaning of the Vasicek Formula
Assume that a part of a loan portfolio (i.e., a credit risk grade within one of the
up to 14 different loan classes) is composed of n loans to n identical borrowers. The
number of borrowers is large, so the portfolio is perfectly diversified. Each of the
borrowers has a very simple capital structure: it has a single asset and a single
liability. The borrowers are all solvent with assets greater than liabilities at time t=0
but as time progresses the value of each of the n borrower’s assets changes, with the
values of each of the borrower’s assets changing by a different amount. At the end of
a given period of time (in the IRB formula, it is one year), some portion of the
portfolio’s n borrowers has slid into insolvency: those insolvent borrowers’ liabilities
exceed their assets. The objective of the Vasicek formula is to determine the
proportion of borrowers that, by the end of the year, have become bankrupt. Since, in
the Vasicek formula, LGD = 1, that proportion of borrowers constitutes Kv.
Critically, each borrower’s asset value change is positively correlated with
each other borrower’s asset value change: their correlation coefficient is . If =
0 , given that the portfolio is perfectly diversified, each portfolio would be riskless.
By riskless, we mean that there would be no variance around the expected proportion
PD of the portfolio that becomes insolvent over the year. However, the fact that >
0 means that the systematic risk component of each borrower’s asset returns cannot be
diversified away. We therefore wish to specify an amount Kv that constitutes the
proportion of n borrowers that are expected be insolvent after the end of one year in a
10
bad state of the economy, where we are 99.9% certain that such a bad state of the
economy will not occur.
Below, we solve for the probability that any given borrower will go bankrupt
in the bad economic state. This is the solution of the conditional probability of default
of a single borrower. Because all borrowers are the same, this conditional probability
is identical to the expected proportion of defaults in the portfolio, conditional on the
poor state of the economy. We then demonstrate that this expected proportion in the
poor state of the economy can be interpreted as the proportion of portfolio at risk in a
value at risk (VAR) sense.
The Conditional Probability of Default of a Single Borrower. We consider
a one-period model. Borrower assets are viewed at t = 0 and again, one year later10.
Let be the random element in the percent change in value of assets for a single
borrower over a one year horizon. This change is made up of two parts and ,
which are standard normally distributed random variables:
[3]
In equation [3], denotes economy-wide systematic random component and
denotes a company specific random component while is the risk weight of the
borrower on the systematic factor.
Although PD is the unconditional probability of default of a single borrower,
we assume that all borrowers are the same, so PD is also the proportion of borrowers
in a portfolio we would expect to default, given that we have no further information
about the state of the economy. is the probability that the percent
change in the value of the borrower’s y is less than the critical value , the point at
which a borrower becomes insolvent and defaults. We can determine the value of 10 The Merton model uses continuous time asset value change. For a demonstration that the continuous time model reaches the same conclusion as the simple, one period model presented here, contact the authors.
11
if we know the value PD by taking the inverse normal of the probability of default
to obtain the critical value of default. For example, if the borrower had
a probability of default of 2 percent, then critical value of y would be
.
[Insert figure 2 here]
PD gives the unconditional probability of default, which is a function of two
random variables. But we wish to focus on the conditional the probability of default
under the assumption that the economy is in a bad state. Figure 2 shows the normal
density function of the states of the economy over the (one year) period of interest,
where economic performance is measured only by the normalized random variable e.
The expected value of e is zero, but we set e = u, a bad state that would occur with
probability 0.001. Substituting for y from equation [3]:
[4]
Substituting u for e and rearranging terms
[5]
Substituting for and, remembering that the firm-specific normalized risk
factor is assumed to be normally distributed so that its probability can be expressed
using the cumulative standard normal distribution
[6]
The reader will note that equation [6] differs only slightly from [2] which we restate
below:
[2 (restated)]
12
The difference between the two is resolved if . In equation [2] q
was expressed as a percent confidence that the capital in the bank is sufficient. In the
case where Basel II specifies that q = 99.9%, as in figure 2. In
equation [6], u is directly reported as the number of standard deviations away from
the mean of a standard normal distribution going to the left (x moving in an
unfavorable direction.). With this interpretation, the two equations are the same.
The Conditional Probability of a Single Borrower Default and VAR.
Although we have shown that the Vasicek formula equals the probability of default of
a single borrower given that the bad state of the economy u has occurred, we have not
demonstrated its applicability to a large portfolio of (identical) loans. For such a
demonstration, one can reason as follows. Each has the same conditional probability
of default. As the number of identical loans in the portfolio rises, the variance around
the proportion of loans defaulting within the portfolio should drop by the law of large
numbers11. Vasicek [14] formalizes this intuition by calculating the probability
distribution of the proportion of loans in the portfolio in default. Although we will not
repeat his derivation here, he shows that if the bank’s loan portfolio is infinitely fine
grained (when the number of loans increased to an arbitrarily high number), the
probability that the actual proportion of loans in default and hence the proportion of
assets lost (assuming, as we have throughout, that loss given default is complete)
would exceed some agreed level of capital, KV ,has the cumulative distribution
function (CDF) as follows [16]
[7]
We plot this CDF for the unconditional PD of two percent (PD = 2%) and six
11 See Gordy [10] and Wilde [17] for conditions under which assumptions of identical PD and exposure amount can be dropped.
13
values of from 0.1 percent to 30 percent in figure 3. It shows that an 8 percent
capital ratio would only provide adequate capital, where “adequate” is defined as
being sufficient 99.9 percent of the time, if the asset correlation is 5 percent or less.
[Insert figure 3 here]
One can interpret the schedules in figure 3 as the VARs of the loan portfolio
(given on the x-axis) associated with a level of confidence given by the CDF function
on the y-axis.
5. The Single Factor Assumption, Granularity and Correlations
The Single Factor Assumption. The Vasicek formula is a single factor
model: borrowers’ assets change in value through the impact of only two types of
risk, borrower-specific idiosyncratic risks and a single economy-wide systematic risk.
The model derivation shows that, because of portfolio diversification in bank assets,
only the economy-wide systematic factor requires bank capital to protect the bank
against borrower risk.
The Capital Asset Pricing Model (CAPM) is also a single factor model, but
CAPM is used to calculate the expected returns of equity securities as a function of
risk, not to determine the VAR of a debt portfolio [18]. Like the Vasicek formula,
CAPM demonstrates that, because of portfolio diversification, only one component of
risk counts: economy-wide systematic risk, which CAPM measures with beta.
Unfortunately, empirical tests of realized stock returns show that the market prices
multiple risks: a single factor model such as CAPM performs poorly in explaining
observed stock returns [19]. If a single-factor fails to explain asset returns in the
relatively efficient public equity markets, it is not likely to be more appropriate in
private debt markets.
14
Commercial credit risk models typically do not make the single factor
assumption. CreditMetrics uses credit rating migration probabilities of each obligor’s
cash flows and tables of joint probability of migration [20]. Creditrisk+ uses a (small)
number of sectors with a single risk factor in each sector [21]. Moody’s-KMV in its
Portfolio ManagerTM software has about 110 factors that the user can specify in
determining obligor returns [11]. Few bankers would suggest that the only risk in
their loan portfolios arose from a single, undiversifiable, residual, leveraged, global
factor. Given this lack of theoretical and industry support, it is noteworthy that Basel
II received virtually no criticism of the single factor assumption. The paradox is
resolved by the need for portfolio-indifference. Neither regulators nor bankers would
accept a model that charged different banks different regulatory capital requirements
for the same loan. Yet, without the single factor assumption, an asset in one portfolio
would require (in terms of VAR) a different amount of capital from the same asset in
a different portfolio because the covariances of the asset with the two portfolios
would tend to differ. The only theoretically consistent basis for requiring that every
bank portfolio hold the same percent of risk capital for a given asset is a one-factor
risk model where all portfolios are assumed to be perfectly diversified with respect to
all but that one (undiversifiable) risk.
Observation 3: The theoretical justification for the single factor model is
weak and does not correspond to best practice in the banking industry. It
represents a compromise that allows identical capital charges for identical
risk positions in diverse banks to be theoretically justified.
Granularity. Lack of diversification historically has been one of the main
causes of bank distress, yet IRB assumes that banks have infinitely fine-grained
portfolios. Not only is this unlikely to be true in practice: it is not even desirable as a
15
bank objective. Banks profit from sector and industry expertise in pricing and
monitoring target-market borrowers. The Basel Committee recognized this initially
and introduced a granularity adjustment in 2001 [5,17,22]. Comprising the most
complex part of that first detailed draft of Basel II, it was dropped without comment
from subsequent drafts, most likely because of its complexity. The issue of
granularity remains in Basel II as a set of provisions under Pillar 2 for banks to review
their credit concentration.12
Observation 4: The complexity of IRB is far less than that deemed by risk
management professionals to be sufficient to capture imperfectly diversified,
portfolio-specific, aspects of credit risk.
Calculating Correlations. As figure 3 illustrates, the correlation between
changes in asset values over time of the representative firms in a portfolio is a critical
parameter in the IRB approach. As the correlation decreases, the VAR of the
portfolio drops. As approaches zero in a portfolio with PD = 2%, the CDF
approaches a vertical line at 2 percent. If =0, there would be no risk in the portfolio
and the only capital that would be required would be for expected losses, in this case,
2 percent of the portfolio. Since such losses, being expected, would already be
provided for, KIRB subtracts off PD from KV in equation [1], leaving no capital
required. Investigating larger correlations between asset returns of the representative
borrowers, if = 0.30 an 8 percent capital cushion would be woefully inadequate
since the CDF at 8 percent would be 94.6 percent: with a 5.4 percent probability in a
one year period, the bank’s capital would be insufficient to sustain losses.
Unfortunately, is a parameter that is impossible to observe directly:
theoretically it is the correlation expected to prevail in future; correlations evolve over
12 See [1] clauses 770–7.
16
time and actual correlations are likely to be multi-factored and varying between
borrowers [23]. Moreover, unlike publicly traded shares, the assets of most bank
borrowers have no available historical series from which to estimate correlations.
One can, however, use stock returns and the market model to estimate .
We show below that equation [3] is a standardized form of the market model:
[8]
where = the return rate of the i-th asset
= the return rate of market portfolio
= a random error (individual asset risk)
Normalizing the errors of equation [8], we have
[9]
where are the standard variance of i-th asset, the return rate of market
portfolio and the random error, respectively.
If , , , then the variables are
normalized random variables and their variances satisfy the following equation
[10]
Letting , we obtain equation [3] from the equations [9] and [10].
[3 (restated)]
In linear regression, and ; therefore, the term usually referred to as
the correlation coefficient , expressed in terms of is
[11]
17
In other words, the correlation between the separate borrows’ returns is the square of
the correlation between each of the borrower’s returns and the single, common,
systematic risk factor.
If one accepts the analogy of the market risk in CAPM with the systematic risk
in the Vasicek formula, one can roughly estimate using widely available data13:
= 20 percent per annum; the average stock volatility is = 46 percent per annum
and = 1. Solving for the asset correlation coefficient
; percent. [12]
Observation 5: Using the analogy with CAPM, = 19 is plausible. One
should note a potential confusion: is the correlation coefficient between
borrower asset returns and, therefore, is the square of the correlation of asset
returns with the systematic factor14.
The first complete draft of Basel II used a single correlation, = 20 percent, a
value very close to the 19 percent we calculated above [5]. The selection of 20
percent, however, occasioned immediate, intense debate. Banks protested that the
true correlations of retail and SME portfolios were considerably below 20 percent and
that, if implemented, such a high correlation coupled with the higher probabilities of
default in the retail and SME portfolios would lead banks to reduce their lending to
retail and SME sectors. Now Basel II’s correlations are lower and more flexible,
varying between values and according to equation [13] with calculated
values in table 1:
13 We obtain data from Damodaran’s website [24]. We use the standard deviation of equity market returns from 1928 through 2002 from the spreadsheet “Annual Returns on Stock, T.Bonds and T.Bills: 1928 – Current” and the average asset volatilities of from his spreadsheet “Firm Value and Equity Standard Deviations (for use in real option pricing models) market”.14 Some related models specify the risk process as correlation with the common risk factor where asset
returns are . See [25,26,27].
18
[13]
where SME is the small and medium sized borrower adjustment, applicable only to
firms with less than euros 50 million15. The SME adjustment is:
[14]
where S is the size of the borrower measured by annual sales in millions of euros. In
equation [13] increases in PD lead to decreases in . Figure 4 shows that, for PD
ranging from 0.01 percent to 7.00 percent, the weight on the lower value correlation
ranges between 0.5 percent and 97 percent.
[Insert figure 4 here]
The SME adjustment reduces by up to 4.8 percent. Effects of these adjustments
are given in Table 1 below.
Theory gives little guidance as to the value of [11]. Lopez calculates
values of implied by empirical observations of portfolio defaults. He finds
support for the ranges of suggested in Basel II, confirms that borrowers with
higher probabilities of default have lower , and concludes that asset return
correlations are higher between healthy firms in normal operations but that firms tend
to succumb to distress for idiosyncratic reasons, leading to falling as the
probability of default rises. He also finds that size affects correlations, with larger
firms’ asset returns being more correlated than smaller firms. His findings, then,
support Basel II’s parameterizing of .
15 In Basel II, equation [13] is actually expressed as
but as Credit Suisse points out, e-50 is
essentially zero on any computer calculation [28].
19
Observation 6: The values of IRB’s correlation coefficients have little
theoretical guidelines; however, modeling as an inverse function of PD
between prescribed upper and lower limits (adjusted for small firms) has
empirical support.
[Insert table 1 here]
6. Adjustments in IRB Formula for Loss Given Default, Expected Loss and
Maturity
Loss Given Default. One of the innovations of Basel II is its explicit
recognition of different levels of LGD in different portfolios. Banks under advanced
IRB calculate their own LGDs and input them into equation [1]. The product of [Kv –
PD] and LGD determines the percent of capital required. Calculating KV, PD and
LGD separately, however, implies that PD and LGD are independent. Yet experience
teaches that the value of collateral on a loan is inversely related to the probability of
default on the loan. This inverse relationship is borne out both theoretically and
empirically [26,27,29,30].
Observation 7: The assumption that LGD and PD are uncorrelated is made
to simplify the IRB model, and has neither theoretical nor empirical support.
Use of the assumption is likely to result in underestimation of required capital.
The Pro-cyclicality of Loss Given Default and Probability of Default. Pro-
cyclicality in setting capital requirements is the tendency for capital requirements to
increase during long recessions and decrease during long booms. In IRB, there are
two sources of pro-cyclicality. First, Basel II prescribes that estimates of PD, LGD
and EAD be obtained from five year horizons yet five years is less than a full
economic cycle. Second, pro-cyclicality may result from periodic credit reviews, if
20
downgrading of borrowers predominates in recessions and upgrading of borrowers
predominates in periods of strong economic growth.
Allen and Saunders review the literature concerning this problem, as well as
the related problem of correlations between PD, LGD and EAD and find near
unanimity that credit risk measurement models in banking accentuate pro-cyclicality,
increasing the volatility of economic cycles [31]. They observe, however that all
regulatory regimes that specify capital adequacy share the same problem. A PD
model can avoid pro-cyclicality only by being inconsistent over time, i.e., by failing
to assign higher PD's when, for example, the current credit quality of a corporate has
declined given economic forecasts. Borrowers have poorer prospects in recessions
and banks and regulators recognize this by implementing higher loan loss reserves,
regardless of the sophistication of the capital adequacy regime.
Observation 8: The use of recent historic data as inputs to the IRB and
periodically updated probabilities of default based on time consistent
assessments of creditworthiness of borrowers increase required capital during
recessions, potentially exacerbating the credit cycle.
Expected Loss. KIRB calculates capital required for unexpected losses by
taking the Vasicek formula’s calculation of KV (the capital required for both expected
and unexpected losses) and subtracting from it PD, the capital required for expected
losses. Banks provide for expected losses through the spread charged over cost of
funds and loan loss reserves.
Banks book loans in the expectation of profit but with the knowledge that a
portion of the loans will default. To avoid loss from a risky portfolio of loans, a bank
must set the total spread over cost of funds to be no less than the portion of the
portfolio that is expected to default. Roughly speaking, if PD = 2%, and the bank
21
charges two percent over the cost of funds and two percent of the portfolio defaults,
then the bank just breaks even (assuming LGD = 1). IRB’s treatment of capital for
expected losses is consistent with the statistical interpretation of expected losses being
covered by expected spread income16.
Banks also maintain general loan loss reserves which commonly deviate from
the statistical expectation of loan losses (EL):
EL = PD x LGD x EAD [15]
A bank may use loan loss reserves as a quasi-equity cushion, a means of reducing
taxes, a smoothing device for income or a provision for portfolios whose losses may
exceed the spread income described above. The use of loan loss reserves, moreover,
can differ considerably from bank to bank, country to country and time to time.
General loan loss reserves have traditionally been a part of Tier II capital (up
to a ceiling of 1.25 percent); however, under Basel II, IRB banks are not allowed this
use of general loan loss reserves. Instead they calculate EL as given above and, if
general loan loss reserves are less than EL, they must subtract the difference from
capital. If general loan loss reserves are more than EL, IRB banks can use the
difference as part of Tier II capital to the extent that it does not exceed 0.6 percent of
credit risk weighted assets 17. The calculations of EL and its netting from loan loss
reserves are made without reference to expected spread income.
Observation 9: The requirement that loan loss reserves be aligned with EL
institutionalizes a double counting, whereby EL are covered as a matter of
pricing by the spread over the cost of funds as well as by loan loss reserves
which under IRB must be no less than EL. This may penalize high EL lending
(such as retail and credit card lending).
16 In the initial drafts of Basel II only 75 percent of PD in only one class of loans, revolving retail loans, was added back in what was called the “future margin income adjustment” [5,6]. 17 See [1] clauses 43, 375, 380-383 and 386.
22
Maturity. The final adjustment in the IRB formula is the adjustment for the
average maturity, MATA in equation [1] above. That adjustments is given as
[16]
where b(PD) = a maturity adjustment function =
= effective weighted maturity of payments
Pt = payment made at time t
The Vasicek formula calculates capital for a one-year horizon. Since a longer
maturity loan to a borrower merits a higher capital charge than a shorter maturity loan
to the same borrower, IRB adjusts Vasicek for maturity. The adjustment accounts for
the potential for credit deterioration being larger for higher rated credits than for
lower rated credits that have a potential not only for deterioration but also for
improvement. Figure 5 shows how equation [14] adjusts for four different PDs.
In the consultation rounds of Basel II, the MATA adjustment incurred only
one consistent criticism from bankers: adjustments for very short-term facilities are
insufficient. Using the 0.5 percent probability of default, for example, MATA would
allow a risk adjustment of only 0.78 of for a loan with zero maturity when, in the view
of most bankers, such a facility would be essentially risk-less [32].
Observation 10: The adjustment for maturity is in accord with industry
practice; however, it allows insufficient reduction of required capital for very
short term facilities.
23
[Insert figure 5 here]
6. Conclusion
The IRB is a hybrid between a very simple statistical model of capital needs
for credit risk and a negotiated settlement. The statistical model used is not “best
practice” today and certainly will not become so in the future. Given the constraints
of the regulatory environment and the rapid development of risk management,
however, regulators would be unable impose a uniform “best practice” solution onto
all banks. Our observations reinforce the Basel Committee’s view that the IRB is
work-in-progress and will remain so long after the implementation of Basel II18.
18 [1] clause 18.
24
References
[1] Basel Committee on Banking Supervision. International convergence of capital measurement and capital standards: a revised framework. Basel; June 2004.
[2] Jorion, P. Value at risk. McGraw Hill, New York; 2001.
[3] Basel Committee on Banking Supervision. International convergence ofcapital measurement and capital standards. Basel Committee Publications No. 4, Bank for International Settlements, Basel; July 1988.
[4] Basel Committee on Banking Supervision. A new capital adequacy framework. Basel Committee Publications No. 50, Basel; June 1999.
[5] Basel Committee on Banking Supervision. The new Basel capital accord: Consultative document issued for comment by 31 May 2001. Basel; January 2001.
[6] Basel Committee on Banking Supervision. The new Basel capital accord: Consultative document issued for comment by 31 July 2003. Basel; April 2003.
[7] Basel Committee on Banking Supervision. The internal ratings based approach: Consultative document: supporting document to the new Basel capital accord issued for comment by 31 May 2001. Basel; January 2001.
[8] Basel Committee on Banking Supervision. Potential Modifications to the Committee’s Proposals. Press release. Basel; 5 November 2001.
[9] Gordy, M.A comparative anatomy of credit risk models. Journal of Banking and Finance 2000; 24 (1-2): 119-49.
[10] Gordy, M. A risk factor model foundation for ratings based capital rules. Board of Governors of the Federal Reserve System working paper 2002. [11] Lopez, JA. The empirical relationship between average asset correlation , firm probability of default and asset size. Economic Research Department, Federal Reserve Bank of San Francisco 2002.
[12] Wilde, T. IRB approach explained. Risk. May 2001: 87-90.
[13] British Bankers’ Association and the London Investment Banking Association. Response to the Basel Committee’s Second Consultation on a new Basel Accord. May 2001. Available from http://www.bis.org/bcbs/ca/bribanass.pdf.
[14] Vasicek, O. Probability of loss on loan portfolio. KMV Corporation. 1987Available from www.moodyskmv.com/research/portfoliotheory.html.
[15] Merton, RC. On the pricing of corporate debt: The risk structure of interest rates. The Journal of Finance 1974; 29(2): 449-70.
[16] Vasicek, O. Limiting loan loss probability distribution. KMV Corporation 1991
25
Available from www.moodyskmv.com/research/portfoliotheory.html.
[17] Wilde, T. Probing granularity. Risk August 2001: 103-6.
[18] Sharp, W. Capital asset prices: a theory of market equilibrium under conditions of risk. Journal of Finance 1964; 19: 425-42.
[19] Fama, E and French, K. The cross-section of expected stock returns.” The Journal of Finance 1992; 47(2): 427-65.
[20] Riskmetrics. CreditMetrics Technical Document. Riskmetrics. April 2, 1997. Available from http://www.riskmetrics.com/cmtdovv.html.
[21] Credit Suisse First Boston. Creditrisk+: a credit risk management framework. Available from http://www.csfb.com/creditrisk . [22] Gourieroux, C, Laurent, JP and Scaillet, O. Sensitivity analysis of values at risk. Journal of Empirical Finance 2000; 7: 225-45.
[23] Chan, L Karceski, J and Lakonishok, J. On portfolio optimization: forecasting covariances and choosing the risk model. Review of Financial Studies 1999; 12(5): 937-74.
[24] Damodaran, A. Damodaran Online. Available from http://pages.stern.nyu.edu/~adamodar
[25] Rosen, D and Sidelnikova, M. Understanding stochastic exposures and LGDs in portfolio credit risk. ALGO Research Quarterly 2002; 5(1): 43-5.
[26] Frye, J. Collateral damage. Risk April 2000: 91-4.
[27] Frye, J. Depressing recoveries. Risk. November 2000: 108-11.
[28] Credit Suisse. Credit Suisse Group CP3 Comments. 2003. Available from www.bis.org.
[29] Frye, J. A false sense of security Risk August 2003: 63-7.
[30] Altman, E, Resti, A and Sironi, A. Analyzing and explaining default recovery rates. ISDA Report December 2001. Also available as BIS working paper, no 113.
[31] Allen, L and Saunders, A. A survey of cyclical effects in credit risk measurement models. BIS Working Papers No. 126: January 2003.
[32] Risk Management Association. Response to the US Banking Agencies’ advance notice of proposed rulemaking regarding new risk-based bank capital rules. Board of Governors of the Federal Reserve 2003. Available from http://www.federalreserve.gov.
26
Figure 1
27
Frequency of US Annual Real Growth Rates
0
2
4
6
8
10
12
14
-16%
-15%
-14%
-13%
-12%
-11%
-10% -9
%-8
%-7
%-6
%-5
%-4
%-3
%-2
%-1
% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
11%
12%
13%
14%
15%
16%
Percent change per annum
fre
qu
en
cy
Figure 2: The State of the Economy
28
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-5.0
-4.8
-4.5
-4.3
-4.1
-3.9
-3.6
-3.4
-3.2
-2.9
-2.7
-2.5
-2.2
-2.0
-1.8
-1.6
-1.3
-1.1
-0.9
-0.6
-0.4
-0.2 0.1
0.3
0.5
0.7
1.0
1.2
1.4
1.7
1.9
2.1
2.4
2.6
2.8
3.0
3.3
3.5
3.7
4.0
4.2
4.4
4.7
4.9
Bad states Good states
Normalized economic growth rate exceeded with99.9% probability e = u = -3.09
Normalized average Growth rate e = 0
X-axis is e the normalized state of economic growth
Figure 3
29
Vasicek Formula Probability of Not Exhausting Bank Capital (Probability of Default = 2%)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.00
10.
003
0.00
50.
007
0.00
90.
011
0.01
30.
015
0.01
70.
019
0.02
10.
023
0.02
50.
027
0.02
90.
031
0.03
30.
035
0.03
70.
039
0.04
10.
043
0.04
50.
047
0.04
90.
051
0.05
30.
055
0.05
70.
059
0.06
10.
063
0.06
50.
067
0.06
90.
071
0.07
30.
075
0.07
70.
079
Capital as percent of loans
Cu
mu
lati
ve P
rob
abili
ty
ρ=30
%ρ=
20%
ρ=10
%ρ=
5%ρ=
2%
ρ=0.
1%
Figure 4
30
Scaling Factor for Maximum and Minimum Correlation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10.
000
0.00
2
0.00
5
0.00
7
0.00
9
0.01
1
0.01
3
0.01
6
0.01
8
0.02
0
0.02
2
0.02
4
0.02
6
0.02
9
0.03
1
0.03
3
0.03
5
0.03
8
0.04
0
0.04
2
0.04
4
0.04
6
0.04
9
0.05
1
0.05
3
0.05
5
0.05
7
0.06
0
0.06
2
0.06
4
0.06
6
0.06
8
Probability of Default
Wei
gh
t o
n t
he
min
imu
m r
ho
Table 1Correlations and maturity adjustments
Exposure Y: maturity adjustment
Sovereign, corporate and bank
0.12 0.24 Yes
SMEs 0.12 minus 0.0 to 0.04
0.24 minus 0.0 to 0.04
Yes
Highly Volatile Commercial Real Estate
0.12 0.30 Yes
Residential Mortgage
0.15 0.15 No
Revolving Retail (eg., credit cards)
0.04 0.04 No
Other Retail Exposures
0.03 0.16 --
31
Figure 5
Maturity Adjustment Y as a function of maturity
0
0.5
1
1.5
2
2.5
3
3.5
00.
5 11.
5 22.
5 33.
5 44.
5 55.
5 66.
5 77.
5 88.
5 99.
5 10
maturity in years
ad
jus
tme
nt
va
lue
PD = 0.005
PD = 0.01
PD = .03
PD = .05
32
