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332 11. Risk Capital

Table 11.1. Performance numbers of two regional divisions of a bank

Eastern Western

Assets 1,000 1,000

Income 10 11

Return on Assets 1.0 % 1.1 %

Economic Capital 75 51

RORAC 13.3% 21.6%

In our simpliﬁed framework where we deliberately neglect investment bud-

gets, risk-adjusted capital equals risk capital. There is a considerable ﬂexibil-

ity in the deﬁnition of the terms in (11.9). Capital allocated for investments,

e.g. in infrastructure modernization, may be included in the denominator. In

the numerator, return may be corrected by expected losses from the risky

business, may be understood before or after taxes, etc. Basically, these sub-

tleties are quite irrelevant so long as a system is consistently rolled out in the

entire bank.

RORAC is a standard measure of bank performance. The kind of insight

it provides for senior management is best illustrated by an example [257].

Suppose that a bank has an Eastern and a Western Division, and that they

report the ﬁgures summarized in Table 11.1 to their board of directors.

Standard performance measures such as income, or the return on assets,

are pretty comparable for both divisions. Economic capital analysis changes

this simple picture. Economic capital reﬂects the diﬀerent level of risk as-

sociated with both divisions, and leads to strikingly diﬀerent numbers for

the return over risk-adjusted capital. The Western division earns an excel-

lent 21.6% RORAC while the Eastern division sticks at 13.3% not too far

above common values for the hurdle rate where senior management starts

wondering about the future of the business.

Given the diﬀerence in RORAC between the two divisions, one can (i)

inquire about its origins in terms of business, (ii) perform a similar analysis

within the Eastern division, perhaps in terms of districts, to understand if

there is a similar heterogeneity of performance, (iii) change the capital allo-

cation between the two divisions (and/or within the divisions), and (iv) give

guidelines for the managers of the badly performing division/districts on how

to improve their results.

From Table 11.1, it is clear that the Western division earns about 50%

more from each dollar of capital invested than the Eastern division. If the

primary aim of the bank is to maximize its return over risk-adjusted capital,

a transfer of capital from the Eastern to the Western division should be

considered. If additional capital is available for investment, it should only be

invested in the Western division.

11.3 The Regulatory Framework 333

Assume that both divisions are only active in the credit sector. An analy-

sis of their portfolios might show that the Eastern division has a higher

fraction of commercial lending and a lower fraction of retail lending, a higher

average probability of default, and a higher average maturity than the West-

ern division. These observations shed light on the diﬀering RORAC numbers.

Commercial lending is signiﬁcantly more risky than retail lending, in general.

A commercial portfolio of the same size as a retail portfolio contains a smaller

number of loans with higher notional amounts, and thus higher exposures at

default. In addition, the risk is increased by the observed higher default fre-

quency, and the longer maturity. The longer the maturity of a loan, the more

likely is a default of the borrower, all other things being equal. This analysis

shows how the business of the Eastern division could be changed in order to

raise its performance numbers: more retail lending, shorter maturities, only

well-rated debtors acceptable.

The RORAC analysis can be taken one level deeper, in addition. If per-

formance diﬀerences similar to those between the divisions are uncovered at

the level of districts, too, similar measures (capital allocation changed, busi-

ness focus changed) can be set up by the managers of the Eastern division.

Ultimately, the system can be extended down the entire hierarchy of the

bank to the level of the individual transaction. Every transaction then can

be analyzed to check if is adds value to the bank.

Other performance measures are constructed using diﬀerent formulae.

They may diﬀer in details of emphasis on speciﬁc factors. However, they

all follow the basic principle of comparing risk and return in a single number,

illustrated by RORAC. Finally, they all serve the same purpose of quantita-

tively supporting management decisions.

11.3 The Regulatory Framework

11.3.1 Why Banking Regulation?

Banking is one of the most heavily regulated activies in the economy ri-

valled, perhaps, only by air traﬃc. From birth to death, a bank is subject

to a plethora of regulation acts. The founding of a bank is subject to reg-

ulation. Its operations are subject to regulation (One purpose of regulation

is to prevent the elimination, by competition, of badly performing banks

from the marketplace). When the “unthinkable” happens despite regulation,

regulation also governs the closing down of a banking operation.

It is not our task to discuss if regulation to the extent practised today

is reasonable. Banking regulation follows two main purposes. An immediate

purpose is to protect the deposits of customers, and thereby the stability

of the economy. Unlike other industries, an important part of the ﬁnancial

resources of a bank is contributed by the deposits of a very large number of

people who mostly are inexperienced in ﬁnancial matters. The depositors are

334 11. Risk Capital

unable both conceptually and economically, to supervise the bank in its role

as a borrower of money, and to protect themselves against business practices

of banks not in their interest. A regulatory institution thus steps in to ensure

that banks operate in the interest of their depositors.

A second purpose of banking regulation is to ensure the safety and stabil-

ity of the ﬁnancial system by limiting the risks a bank can take. In fact, new

developments in banking regulation often have followed the breakdown of ﬁ-

nancial institutions is the wake of excessive risk taking. While the regulatory

acts are decreed and enforced by national regulators, the globalization of the

ﬁnancial industry has also led to an increasing international harmonization

of regulatory frameworks.

Banking regulation is imposed along two avenues. One is direct rule writ-

ing, describing what is permissible and what is not. The other is the setting

of certain capital requirements which, explicitly or implicitly, depend on the

riskiness of the banks’ businesses. The ﬁrst avenue is the ﬁeld of lawyers and

internal and external auditors. The second avenue is tightly related to risk

management, and we will discuss it in the following.

11.3.2 Risk-Based Capital Requirements

Capital plays a signiﬁcant role in the risk-return tradeoﬀ at banks. Increasing

capital reduces the risk of default of the bank by increasing its cushion against

losses or, more generally, earnings volatility. Firms with greater capital can

take more risk.

Capital also inﬂuences growth opportunities, proﬁts, and the returns to

shareholders. Banks with more capital can borrow at lower interest rates and

can make larger loans. Both normally yield higher income. With more capital,

a bank can more easily invest in growth and acquisitions, creating the seeds

for increase future proﬁts. On the other hand, the holding of greater capital

decreases the returns to shareholders. Finding the optimal capital level is

an important task of bank management. We will not pursue these topics

here. We also discard a number of further important questions on the use of

capital to cover risk, such as “What constitues capital?”, “How do capital

requirements impact a bank’s policies and business practices?”, or “What are

the advantages or disadvantages of various sources of external and internal

capital?”. These important topics are treated in the standard literature on

bank management [256].

Instead, we focus on the important quantitative problem of risk man-

agement: “How much capital is adequate given the exposure of the bank?”,

rephrased here as “How much capital do regulators require to hold given the

exposure of the bank?”. With reference to both the main body of this book

in general, and to the preceding chapter in particular, one quickly might

come up with the suggestion to tie regulatory capital requirements to the

unexpected losses of a bank at a certain conﬁdence level. While economically

reasonable (cf. the discussion in the preceding section on economic capital),

11.3 The Regulatory Framework 335

bank regulators apparently do not trust the ability of banks to accurately

and reliably determine the capital numbers in question. Consequently, the

internal determination of capital requirement (“internal models”) is allowed

only in the often less important domains of market and (in the future) oper-

ational risk. The procedures which regulators impose on banks for the most

important area of credit risk work according to a very diﬀerent logic: divide

your assets into certain classes, and attach to them risk weights and capital

numbers ﬁxed in advance by the regulators. These regulations have been in

practice for about two decades and are being loosened somewhat in the near

future with the arrival of the new Basel II Capital Accord. Abandoning them

completely in favor of an bank-wide internal model covering all risky assets

will certainly take another one or two decades.

Banking regulation by risk-based capital requirements is responsible for

many job openings in the ﬁnancial industry. For this reason, a discussion

of these practices, though not comprehensively based on rigorous scientiﬁc

methods, is mandatory.

Historically, in many countries in the 1970s, national regulators imposed

capital requirements on certain assets held by a bank, resp. ﬁxed limits on the

volume of assets a bank was allowed to hold, depending on its capital. Inter-

national harmonization of supervisory rules was one of the tasks of the Basel

Committee on Banking Supervision, located at the Bank of International

Settlements (BIS) in Basel, Switzerland. In 1988, a ﬁrst international capital

accord (now dubbed “Basel I”) [258] was reached by the Committee which

represents members of the Group of Ten Countries’ (G 10) central banks (i.e.

the most important countries of western Europe plus the United States and

Canada) and and their regulatory authorities. Despite the limited represen-

tativity of the Basel Committee, in the years following its publication, the

accord has been implemented in the national legislation and rule making of

more than 100 countries worldwide, in particular in all countries members of

the Organization of Economic Cooperation and Development (OECD).

A second round of many years of negotiation by the Basel Committee has

led to the publication of a ﬁnal version of a new international capital accord,

Basel II, in July 2004. Its purpose is to set rules for more a risk-sensitive

determination of regulatory capital and to create incentives for the imple-

mentation of better risk management procedures in banks. The rules agreed

upon in Basel II are scheduled to become eﬀective on January 1, 2007, and

January 1, 2008, depending on the sophistication of the procedures adopted

by a bank. In the meanwhile, the accord must be transferred into national

legislation in the countries represented in the Basel Committee. Based on

the experience with Basel I, it is expected that Basel II will set the risk

management and capital standards for the ﬁnancial industry worldwide, for

the one or two decades to come. By early 2005, more than 100 countries had

committed themselves to the implementation of Basel II in the years to come.

336 11. Risk Capital

11.3.3 Basel I: Regulation of Credit Risk

The ﬁrst Basel Accord in 1988 marked the birth of risk-based capital stan-

dards in banking regulation [258]. The Basel I agreement only covers credit

risk. For many banks except investment banks, the biggest of their risks is

credit risk. The basic procedure to determine the bank’s capital involves four

steps.

1. Classify all your loans in one of ﬁve risk categories appropriate to the

obligor, the collateral, or the guarantor of the asset. These ﬁve asset

categories are described below and are distinguished according to the

order of magnitude of the default probability of the assets. A bank carries

a big risk from the default of a debtor, i.e. her failure to correctly deliver

all payments due, cf. Sect. 10.4.2 on credit risk. However, the notion of a

default probability has not been used in Basel I. It is introduced in Basel

II, and reverse engineering can be done to estimate the numerical default

probabilities of the ﬁve classes.

2. Convert oﬀ-balance sheet commitments to their on-balance sheet equiv-

alents, and proceed as in 1.

We will not dive into the practices of moving assets oﬀ the balance sheet

of a bank, nor into the conversion procedure required here. It is suﬃcient

to mention, at this point, that when a bank can generate income from

assets which are “expensive” to hold (e.g. in terms of risk capital), it

may be tempted to keep part of this income while avoiding the cost of

the risk. Securitization is one way of doing this. Assets, e.g. loans, may be

packaged into a new kind of security (e.g., collaterized debt obligations,

CDOs) and sold to the capital markets. The counterposition opened by

this security makes the loans disappear from the bank’s balance sheet. A

reader of the ﬁnancial statement will not be able to correctly assess the

riskiness of the bank’s business practices based on that information alone.

Many derivative positions do not appear in a bank’s ﬁnancial statement.

Long term loan commitments are another example of oﬀ-balance sheet

activities. When such a commitment is pending, the bank has not yet

given out a loan to be covered by capital. Nevertheless the bank carries

a risk because the obligor may take the loan, in particular when its cred-

itworthiness declines. Dramatic examples of these practices have been

given by Enron and Kmart in late 2001/early 2002, just before ﬁling for

bankruptcy.

3. Multiply the amount of assets (in home currency) in each risk category

by an appropriate risk weight factor. The sum over all ﬁve risk categories

gives the risk-weighted assets.

4. Mulitply the risk-weighted assets by a minimum capital percentage to

obtain the capital required to hold against the assets. The capital ratio is

8%. (Here, this rule is simpliﬁed somewhat to avoid a discussion related

to subtleties of the deﬁnition of capital.)

11.3 The Regulatory Framework 337

Asset category 1 contains assets of the best quality available: direct oblig-

ations from the US government or other OECD governments, currencies and

coins, gold, government securities, and unconditional government guaranteed

claims. These assets do not carry a default risk – the US government is not

considered to default, and neither are the other OECD governements. The

risk weight of this category is zero. No capital must be held against these

assets.

Category 2 contains claims on public sector entities excluding central

government, and loans guaranteed by such entities. At national discretion, a

risk weight of 0, 10, 20, or 50% is attached to these assets.

Category 3 consists of obligations of multilateral development banks or

guaranteed by these banks, obligations of banks incorporated in the OECD

and loans guaranteed by these banks, obgligations of banks incorporated out-

side the OECD with a residual maturity of less than one year and loans with

residual maturity up to one year guaranteed by these institutions, and oblig-

ations by non-domestic OECD public sector entities and loans guaranteed by

such entities. Assets in this category carry a risk weight of 20%. The capital

to be held against these assets is 1.6% of the asset value.

Category 4 contains loans fully secured by mortgage on residential prop-

erty. Its risk weight is 50%, implying a capital charge of 4% eﬀectively.

Category 5 contains, among others, obligations of the private sector, of

banks outside the OECD with residual maturities of more than a year, real

estate loans other than ﬁrst mortgages, premises and other ﬁxed assets, cap-

ital instruments issued by other banks, etc. The risk weight of this category

is 100%, i.e. all assets carry a capital requirement of 8%. Oﬀ-balance sheet

activities are converted into on-balance sheet assets with conversion factors

similar to the risk weights, and then entered into category 5. Some national

supervisors chose more conservative risk weights for the ﬁve categories.

The main diﬀerence between the the ﬁve categories is the likelihood of

default of the assets. Category-1 assets are approximated as risk-free. Their

interest rates do not contain an adjustment to compensate for the possibility

of a default. When, e.g., in the Black–Scholes equation, (4.85), the risk-free

interest rate is sought, the interests paid by these Category-1 assets should

be used. Assets in the other categories are risky and can default. The capital

ratio of 8% on risk-weighted assets has not been derived from a model or a

theoretical framework. Most likely, it is a result of both good guessing and

political bargaining.

There is no direct relation of the capital numbers determined to the risk

of a bank’s credit portfolio. This, in fact, has been the main criticism of the

Basel I framework: The capital charge levied on a portfolio is independent of

its risk.

It is not permissible to estimate a default probability of the assets in the

ﬁve categories from their risk weights and the overall capital ratio. Strictly

speaking, capital is used only to cover unexpected losses in the sense of

338 11. Risk Capital

(10.37). Expected losses should be contained in the credit spread, the dif-

ference in interest earned by the asset and the risk-free rate. Notice that, in

the Basel I framework, capital scales linearly with asset volume. This is not a

usual property of risk measures which, except in special circumstances, scale

sublinearly with asset volume. In an uncorrelated Gaussian world, risk scales

as the square root of asset volume, cf. Sect. 10.3. The failure of the regulatory

capital requirement to scale sublinearly with asset volume points to its two

major shortcomings: (i) the lack of a scientiﬁc basis for its determination,

and (ii) the love of regulators to assume worst-case scenarios. When perfect

correlation is assumed, i.e. all assets in a portfolio default at the same time,

a linear dependence of risk on asset volume is expected. Apparently, such a

scenario is at the origin of the regulatory credit risk capital determination.

The capital determination process in the Basel I Accord appears rudi-

mentary and gross. Apparently, it ignores all the ﬁne statistics and physics-

inspired analysis presented in the main body of this book. It has been pre-

sented here to give an impression of the current state of banking regulation,

and of the kind of details risk management and accounting experts in banks

have to go into.

Basel I should not be blamed, though, for its rudimentary character in

terms scientiﬁc credit risk modeling. Market risk modeling using advanced

statistics was well developed at the time Basel I was negotiated. Advanced

credit risk modeling, on the other hand, only developed during the 1990s. To-

day, sophisticated ﬁnancial institutions are able to manage their credit risk

according to an internal (statistical) model. However, even in developed coun-

tries, many banks limit their formal treatment of credit risk to a framework

such as that set out by Basel I.

11.3.4 Internal Models

Market risk has not been regulated by Basel I. A 1996 paper by the Basel

Committee deﬁnes market risk and both sets up a standardized framework for

regulatory capital requirements for market risk and allows the recognition of

an internal model for the determination of market risk capital [259]. Market

risk is subdivided into interest rate risk, equity position risk, foreign exchange

risk, and commodities risk and includes the risk from derivative positions

with these assets as underlyings. The standardized measurement method for

market risk follows a philosophy similar to the Basel I treatment of credit risk

discussed in the preceding section, and is not covered here. A capital charge

is imposed only on the market risk in the trading book, i.e. for those assets

which the bank holds for short-term trading purposes. There is no capital

charge for market risk in the banking book.

Instead of the standardized risk-weighting procedure, a bank can elect

to use an internal model to determine its regulatory capital requirement

[249, 255, 259]. In some countries, depending on the size of its trading book,

it may be obliged to do so. An internal model is an internally built risk

11.3 The Regulatory Framework 339

measurement model which has received supervisory approval. Banks with

important trading activities will develop such a model for their risk manage-

ment and economic capital allocation, anyway. The point here is that, when

implementing regulatory restrictions and parameter settings, this model may

be used to determine the regulatory capital. It is expected that the capital

numbers based on such an internal model will come out lower than those

from the standardized risk-weighting procedure. As some of the regulatory

settings for the internal model may be overly conservative, banks often run

two structurally similar models, one with the regulatory settings to determine

regulatory capital, and one for economic capital and risk management with

the settings which internally are deemed most appropriate.

A bank’s capital charge for market risk essentially is the value at risk

of its trading assets as well as foreign exchange and commodity positions,

whether or not they are in the trading book. The regulators do not prescribe

a particular type of model nor a speciﬁc computational methodology. The

internal model must, however, satisfy a number of general requirements:

1. Value at risk should be computed on each business day and should be

based on a one-sided 99% conﬁdence level.

2. The holding period underlying the value-at-risk calculation is ﬁxed to ten

days.

3. The model must measure all material risks of the institution.

4. The model may utilize historical correlations within broad categories

of risk factors (equity and commodity prices, foreign exchange, interest

rates), but not among these categories. The consolidated value at risk

is the sum of the value-at-risk numbers of the categories, i.e. a perfect

correlation is assumed between the categories.

5. The nonlinear price characteristics of options must be adequately ad-

dressed.

6. The historical observation period used to estimate future price and rate

changes must have a minimum length of one year.

7. The data history must be updated at least once every three months, and

more frequently if market conditions require.

8. Each yield curve in a major currency must be modeled using at least

six risk factors appropriate to the interest-rate sensitivity of the traded

assets. The model must also include spread risk.

The modeling is further complicated by the distinction made between

general and speciﬁc market risk, and event risk. General market risk refers to

all changes in the market value of assets resulting from broad market move-

ments. It is approximated, e.g., by the variation of a representative market

index. Speciﬁc market risk is the residual risk associated with individual se-

curities, not reﬂected by broader market moves. It is related to the β-factors

(10.71) of the Capital Asset Pricing Model discussed in Sect. 10.5.5 and

measured the return dynamics of an asset relative to a broad market index.

Event risk denotes rare events aﬀecting an individual security. An example

340 11. Risk Capital

often cited is the rating downgrade of a bond issuer. The distinction of gen-

eral and speciﬁc market risk and event risk certainly is somewhat arbitrary

(just think about the reﬂection of a rating downgrade of a listed company in

its share price and the ramiﬁcations this may have on the stock market as a

whole). It may become important, though, when approximations are used in

the model-building process.

In summary, the value at risk Λ

var

(0.01, 10d) as deﬁned in (10.25) must

be calculated every business day based on the preceding prescriptions. Reg-

ulatory capital for market risk is related to this value at risk by a number

of add-ons [259]. Firstly, the capital to be held on day t is the higher of the

value at risk on the preceding business day t − 1, and the moving average

of the value at risk over the last 60 business days. Secondly, a multiplication

factor smaller than 5 is applied to this number based on the regulator’s “as-

sessement of the bank’s risk management system” (i.e. a somewhat subjective

quantity), and the model’s performance in backtesting. To be speciﬁc, in the

implementation of internal models in Germany, the multiplication factor is

decomposed into a ﬁxed basic value of 3, and two add-ons for backtesting

and the subjective evaluation by the supervisors which both vary between 0

and 1 [249]. Moreover, additional add-on charges are implemented for banks

which do not include explicitly speciﬁc risks and event risks into their internal

model [249, 259].

Notice that regulatory capital is determined with reference to value at risk,

and not with reference to the unexpected losses (10.38), as economic capital

would be. Likely, for a ﬁnancial institution with a good pricing framework,

where expected losses (10.37) are included in the prices of products and

services, there is some double counting of the expected losses in capital and

in the prices.

Stress testing and backtesting are important steps in the introduction of

an internal model. Stress testing makes a model using parameters estimated

from historical time series, more forward-looking. Stress testing is the study

of model behavior under extreme scenarios which have not been realized in

that past time used for estimating the model’s parameters. To formulate

these scenarios, one may recur to past events such as the crashes described

in Chap. 9. Such scenarios are either given by the supervisors or developed

by the bank itself. In the end, the suﬃciency of the bank capital with respect

to the losses incurred is evaluated.

Backtesting is the process of running a completed model on long historical

time series, before going live. In this way, one can check that the model

performs according to expectation before it is actually used in day-to-day risk

management. E.g., when the value at risk of the entire bank is determined at

the 99% conﬁdence level, the actual frequency of losses bigger than the 99%

value at risk is expected be 0.01. The time series must be long enough that

this frequency, as well as its uncertainty can be estimated with acceptable

precision.

11.3 The Regulatory Framework 341

One may only speculate on the reasons why internal models have not been

recognized for the credit risk capital determination in Basel I (and continue

not to be recognized under Basel II). Credit risk by far needs most capital in

almost all banks. Moreover, the data situation in credit risk is not as good

as for market risk: no bank would reevaluate their credit portfolio on a daily

basis (there is simply not enough new information to warrant such an action).

In addition, the 8% capital ratio has been determined quite arbitrarily. Most

likely, in view of these uncertainties, regulators did not, and still do not have

enough conﬁdence in these models to allow banks to determine the biggest

portion of their capital requirements by an internal model, independently of

the quality and intensity of the regulatory examination.

11.3.5 Basel II: The New International Capital

Adequacy Framework

The ﬁnancial world has changed enormously during the 15 years since the im-

plementation of the Basel I Accord. Financial instruments have become more

complex, perhaps more so in the important area of credit risk than in mar-

ket risk. Financial operations and technology have increased in complexity.

In parallel, methods in risk management have become more sophisticated.

Consequently, a new, more risk-sensistive framework for regulatory capital

is called for. Moreover, it has been realized that important risks or aspects

of risk had been left out of Basel I. Thus at the same time, such a new

framework could be formulated to include broader risk categories into the

regulatory capital calculation.

After more than ﬁve years of negotiation, the second Basel Capital Accord

(“Basel II”) was ﬁnalized in summer 2004. It is scheduled to be implemented

in the G10 countries by 2007 (some of the more sophisiticated approaches by

2008 only) and will be adopted by many other countries subsequently. Basel II

essentially reﬁnes the treatement of credit risk and introduces operational risk

as a new risk type to be covered by a capital charge. Moreover, it formalizes

criteria for the supervisory review process of banks as well as criteria for the

disclosure of risk information towards the capital markets.

There are a few common principles underlying the Basel II accord.

• Basel II has been conceived as a compensation approach, i.e., on the aver-

age, banks should hold the same regulatory capital after the implementa-

tion of Basel II as before, when they use capital determination methods of

comparable sophistication.

• Good risk management processes are most important in a bank, perhaps

more important than the actual amount of risk taken by a bank.

• There should be incentives for banks to improve their risk management sys-

tems despite the investments necessary. Good risk management therefore

should be rewarded by a signiﬁcant capital reduction.