Voit J. The Statistical Mechanics of Financial Markets

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322 10. Risk Management



i,j=1

. (10.69)

In order to optimize the portfolio of M correlated assets, one can now follow

the same strategy as in the absence of correlations, Sect. 10.5.5. The only

diﬀerence is the replacement of the variance by the covariance matrix. As

an example, the q



determining the optimal fractions of the assets in the

portfolio are given by





j=1

−1

(µ

− µ

) , (10.70)

in analogy to (10.62).

This simplicity is due to the fact that the covariance matrix C

can

be diagonalized [17]. One can therefore formulate, from the outset, a new

set of stochastic processes obtained by linear combination of the original

ones, so that they are uncorrelated. Their variances are the eigenvalues of

the covariance matrix, and the transformation from the original to the new

stochastic process is mediated by the matrix built from the eigenvectors, as in

any standard eigenvalue problem. In this way, a portfolio of correlated assets

is transformed into one of uncorrelated assets which, unfortunately, do not

exist on the market but are constructed with the only purpose of simplifying

the portfolio optimization problem. The procedure can also be generalized to

correlated L´evy distributed assets [17].

In a Gaussian world, all optimal portfolios are proportional so long as

they refer to the same market. Equation (10.62) shows why this is so. The

optimal asset fractions in the portfolio q



∝ λ,andonlyλ depends (linearly)

on the required excess return of the portfolio over the risk-free investment

¯µ −µ

. Linear combinations of optimal portfolios are optimal, too. A market

portfolio which contains all assets according to their market capitalization,

is also an optimal portfolio. Of course, the returns and the risks of all these

optimal portfolios may be diﬀerent – but they all have the same risk-return

relation, i.e. satisfy (10.63).

The practical deﬁnition of “the market” itself is not a trivial issue. In

the US, the S&P500 index is generally taken as a benchmark for portfolio

managers, indicating that it is taken as a proxy for “the market”. With 500

stock included, it certainly is well diversiﬁed. Some argue, however, that the

limitation to the 500 biggest stocks gives it a bias, and that small caps which

often generate the biggest returns, are ignored. They would advocate that the

Russell 1000 or Wilshire 5000 indices are much better representations of “the

US market” [3]. The Dow Jones Industrial Average with 30 blue chips only

certainly is not representative of the broader US market. In the same way,

the Dow Jones Stoxx 50, DAX, CAC40, etc. indices are not representatives of

the European, German, and French markets. For investors with world-wide

portfolios, the MSCI World index is virtually the only benchmark available.

10.5 Risk Management 323

A market portfolio can therefore be taken to measure the performance of

an individual ﬁnancial asset, or of entire portfolios, by relating their returns

to that of the market portfolio (value Π,return¯µ)

− µ

= β

(¯µ − µ

) ,β

2

δS

− µ



δΠ

− ¯µ

3





δS

− µ





δΠ

− ¯µ



, (10.71)

where β

is the covariance of the asset or portfolio j with the market portfolio.

This is the basis of the Capital Asset Pricing Model (CAPM) which relates

the returns of assets to their covariance with a market portfolio. It cannot be

generalized to non-Gaussian markets.

10.5.6 Strategic Risk Management

Risk management, in the ﬁrst place, starts with a selection of those asset

classes whose risk is deemed acceptable given the risk appetite, resp. risk

tolerance, of an investor. There may be major diﬀerences, e.g., in the risk

tolerance of the trading operations of a universal bank, a focussed investment

bank, an insurance company or an industrial corporation. There is thus no

clear distinction between strategic risk management and asset management,

in general.

The diﬀerent classes of assets: bonds, stocks, commodities, currencies,

real estate, private equity, etc. carry diﬀerent risk and return expectations.

Strategic risk management will select those classes of assets which can be

used for investment, and those which are excluded. This selection usually is

followed by more detailed rules which may set limits on the fraction of assets

to be held as stocks (in the insurance business, this fraction may even be set

by a regulator), the use of derivatives for speculation, or their use for hedging

purposes. Strategic risk management may diﬀer between the assets held for

trading purposes, and the positions entered for business purposes [254], i.e.

in the case of banks, between the trading book and the banking book.

In many industrial sectors, though not in banking, natural hedging of

foreign exchange risk is an important strategic consideration. Foreign ex-

change risk comes from buying goods in one currency and selling products

in another currency. The exposure to currency ﬂuctuations for a corporation

is less when products are manufactured and sold in areas using the same

currency in which most of the raw materials bought are billed. For many

corporations deciding on the opening of new plants in foreign countries, the

opportunities for natural hedging are an important consideration.

Strategic risk management is all the more important the less tradable

products are available for use in risk management. Although credit deriva-

tives have been created and now are traded regularly, an important part of

324 10. Risk Management

credit risk management simply consists in deﬁning how many loans can be ex-

tended to speciﬁc classes of clients, in order to optimize the risk-return proﬁle.

Also the participation in credit pooling initiatives by which several ﬁnancial

institutions swap parts of their credit risks, requires extensive preparation

and thus strategic decisions. As mentioned earlier, once the loans have been

given, there are only limited options for acting on the portfolio. In the area of

operational risk, the operational risk associated with all new products should

be assessed systematically, before the ﬁnal decision on the introduction of the

product is taken.

11. Economic and Regulatory Capital

for Financial Institutions

Suppose that at time t, a speculator invests a capital amount of S(t) dollars

in the stock market. He expects a return δS

(t) on a time scale τ.Inthe

preceding chapter, we discussed the risk associated with this position, i.e. to

what extent the actual return δS

(t) may deviate from the expected outcome

δS

(t).

The present chapter is concerned with the inverse problem. Given a cer-

tain risk of a bank, how can the bank ensure that it can safely take this risk,

i.e. that the risk poses no threat to the prosperity or even the survivial of

the bank. This is all the more important as the risk may not necessarily arise

from speculative proprietary trading but simply from the bank’s day-to-day

business with its customers.

11.1 Important Questions

In this chapter, we discuss the following important questions:

• What is the relation between risk and capital requirements for a ﬁnancial

institution?

• How much capital does a bank need?

• Which factors determine the capital requirement of a bank?

• How much capital does a business line in a bank need?

• What is the relation between the capital requirement of a bank and those

of its various business lines?

• Can capital be used as a tool for risk management?

• Are banks free in the determination of their capital requirements?

• What is the diﬀerence between economic and regulatory capital?

• What is the current framework for regulatory capital calculations?

• To what extent is regulatory capital risk-sensitive?

• What is “Basel II”, and how will it aﬀect the determination of regulatory

capital in the future?

• What is to come after “Basel II”?

326 11. Risk Capital

11.2 Economic Capital

When Nick Leeson’s positions on the Japanese derviatives market blew up

Barings, the bank did not have enough money to cover the losses, and went

bankrupt. When Long Term Capital Management got out of control, more

than three billion US$ were provided by a consortium of banks to cover the

losses and unwind LTCM in an orderly fashion. These examples show that a

capital cushion is needed to protect a bank (or any other business) against

unexpected events – risk. Capital, or better economic capital, turns out to be

the central concept determining how much risk a bank can take. Capital allo-

cation, i.e. the attribution of certain fractions of the total capital available to

business units, is an important tool in bank management. Capital allocation

sets limits on how much risk individual business lines, departments, or trad-

ing desks can take. Here, we do not go into the subtleties of deﬁning capital.

We take its existence for granted, and discuss its use in bank management.

11.2.1 What Determines Economic Capital?

Every time risk strikes, bank capital is used to cover up the losses. Both

the times of the losses and amounts lost, are stochastic variables. How much

capital should a bank put aside to cover its losses? A look at the Barings

case helps to give an answer. A bank needs enough capital to guarantee its

survival when the worst case within the management horizon hits. We opened

Sect. 10.3.4 with the observation that a good manager needs a clear deﬁnition

of the realm of his management activity, i.e. what is to be managed, and what

should not be managed. This boundary determines the capital requirement

of a bank, resp. an individual business line within a bank.

More formally, the economic capital requirements of a bank therefore are

determined by the survival probability which its management targets. In the

language of credit risk, the complement of the survival probability is the

default probability which, itself, is indicated by a bank’s creditworthiness

rating. If, e.g., a bank is rated by Moody’s as A1, its implied annual default

probability is estimated to be about 0.05% (cf. Table 11.2 below). Its survival

probability for the next year is 99.95%. It sets the conﬁdence level on the

evaluation of the risks which must be covered by capital, and thus on the

amount of capital required. When senior management wants to conserve the

A1 rating, economic capital must equal at least the 0.05% value at risk of

the bank. If senior management wishes to improve further the bank’s rating,

an even higher conﬁdence level should be set.

While individual realizations of losses are unpredictable, statistics on the

loss histories provides the expected losses deﬁned in (10.37). In a statistical

sense, losses of this order of magnitude are predictable over the time horizon

used, and a prudent banker will build up loss provisions for these events.

These loss provisions are approximately constant in time and are better bal-

anced by the income generated by the bank’s operations, rather than taken

11.2 Economic Capital 327

out of the bank’s capital base regularly. Ultimately, the expected losses should

be included in the pricing of the bank’s products and services.

The actual losses diﬀer almost always from their expectation values. When

they exceed the expected losses, capital indeed must be used to cover them.

However, if loss provisions and the pricing of products and services are made

correctly, economic capital is only used to cover the unexpected losses, deﬁned

in (10.38) at the conﬁdence level set by the bank. More capital than this value

may be held in practice, e.g., to take into account possible stress scenarios

(e.g. reduced liquidity) associated with catastrophic events.

11.2.2 How Calculate Economic Capital?

The principle of an economic capital calculation is simple: calculate the value

at risk at the chosen conﬁdence level, and subtract the expected losses. The

practice of economic capital calculation, however, presents almost unsur-

mountable challenges. All risk types (market, credit, operational, etc.) for

all portfolios in all businesses of the bank must be aggregated to a single

number. Aside many other challenges, one important issue is the estima-

tion of the relevant correlation matrices between the various assets held. An

impression of the consequences of correlations can be gained by comparing

(10.34) and (10.35).

In practice, the problem is solved only partially, and at a very low level of

aggregation. Economic capital may be determined systematically for individ-

ual portfolios, and individual risk types. A variety of approximate techniques

is available to estimate the (mostly market) value at risk (or related risk

measures) for reasonably complex portfolios [245, 246, 255].

Current bank research is focussed on the integration of market risk and

credit risk into an overarching risk model, and thus capital framework. The

integration of operational risk has not been attempted to date. The impor-

tance of correlations is seen easily when thinking about an economic down-

turn: stock market prices fall, and at the same time, due to the bad economic

conditions, the number of credit defaults rises. Also, there may be correla-

tions between the variation of interest rates, and the number of defaulting

loans. These correlations are an important driver of economic capital needs.

Another fundamental challenge becomes apparent when attempting to in-

tegrate market and credit risk: the widely disparate time scales of the data

sets used. Market data are available at high frequency. Credit risk data are

certainly available on an annual basis, for large-volume credits perhaps quar-

terly. The standard time scale for an economic capital determination is one

year. On the other hand, the time scale of risk management non-defaulted

credits is somewhere between the quarter and a year, depending on the ex-

posure. The time scale of market risk management ﬁnally varies from the

intraday range to ten trading days, perhaps. Often, approximations such as

the

√

T -law, exact only for uncorrelated Gaussian assets, are used to relate

the diﬀerent time scales involved. Similarly, in some instances capital ﬁgures

328 11. Risk Capital

may simply be added, implying perfect correlation between the assets in the

various classes, to produce the total economic capital. Much research still

needs to be done in order to develop accurate economic capital numbers for

realistic situations.

11.2.3 How Allocate Economic Capital?

The inverse problem to risk aggregation, capital allocation, is as important

from a practical point of view and as much unsolved from a fundamental

standpoint. Moreover, capital allocation is a problem in its own right, and

often a practical necessity even when the risk aggregation problem has not

been solved satisfactorily. Capital allocation can be understood as an invest-

ment or budgeting process. It is done in any industry, enterprise and even

private households more or less consciously. Business administration provides

concepts for capital allocation resp. budgeting from an investment perspec-

tive.

Risk-based capital allocation attempts to allocate the capital of the bank

to its businesses, portfolios and risk drivers. Let us ﬁrst assume a stationary

environment. Then the capital for the next period can be allocated on the

basis of the present risk proﬁle. The challenge is that capital is an additive

quantity whereas risk is a subadditive quantity.

Assume that risk has been aggregated over all portfolios, businesses, and

risk drivers. Unless all assets are perfectly correlated, subadditivity resp.

diversiﬁcation will guarantee that the total risk is less than the sum of all

partial risks. This is independent of the risk measure used provided that it is

coherent. The total risk of the bank therefore is known, and we assume that

it is balanced by the bank’s capital. How much capital should be allocated

to each of the bank’s businesses?

To be more speciﬁc, we assume that there are three businesses only: A,

B, and C, and that the bank has ﬁxed target rating of A1 with a 99.95%

conﬁdence level. At that conﬁdence level, the capital requirement (unexpected

losses) of business A is assumed to be 2 × 10

$, that of business B is 10

and business C claims 5×10

$. With the additional assumptions of vanishing

correlation and normal distribution, the total capital required by the bank is

5.48 × 10

$. The eﬀect of diversiﬁcation is clearly visible. With this amount

of capital available, the bank as a whole can safely balance the risk of its

businesses. However, the capital is not suﬃcient to give every business the

amount required so that it could, on its own, balance its risk at the desired

conﬁdence level. This would require 8 ×10

Several ways out of this dilemma are conceivable.

• Every business receives 68.5% (=5.48/8) of its initial capital request. In

this case, there are two options for proceeding:

– Every business reduces its operations by the amount necessary to make

the capital allocated appropriate for its risk at the 99.95% conﬁdence

11.2 Economic Capital 329

level. While this makes the risk management of every individual business

safe, bank capital is wasted, as the aggregate reduced risk only requires

3.75 × 10

$ of capital. With an assumed return on capital of 10%, the

bank wastes 1.73 ×10

$ of income – a disadvantageous strategy indeed.

– Business operations are not reduced following the reduced capital allo-

cation. The full amount of risk continues to be managed at the 99.95%

conﬁdence level in every business. Each individual business is undercap-

italized but the bank as a whole is capitalized correctly. A system of risk

sharing agreements must be elaborated between the businesses because

in some years, one business may need more capital than it has to cover

its losses. However then, the other businesses are expected to have excess

capital with respect to their realized risks which can be transferred to

the suﬀering unit.

• Capital allocation is used as a book-keeping device only but capital is

not allocated physically. Each business unit must behave as if it had been

given the amount of capital requested. Business A, e.g., must follow its

management strategy based on a capital of 2 × 10

$, include the cost of

this capital amount in its proﬁt and loss statement, etc. However, the sum

rule on capital is no longer operational, and the risk is balanced against

physical capital only at the bank level, not on the business unit level.

• By extending this idea, one can set up a central “insurance” function which

takes over the unexpected losses of the various businesses against an insur-

ance premium. Operating within a bank, a fair price plus a proﬁt margin

can be charged for such a service. Business A, e.g., can “sell” its unexpected

losses up to a cap set by the 99.95% conﬁdence level to this insurance func-

tion. In exchange it pays a premium equal to the cost of this capital, say

5% + margin, to the insurance department. Here, all risks are aggregated

eﬀectively, and balanced by capital.

• The rules of the risk management game can be changed so that risk mea-

sures become additive. Then a strict proportionality between risk and cap-

ital can be implemented. We discuss this path in the following.

The preceding discussion, and that in Chap. 10 started from the ﬁxing

of a common conﬁdence level for all businesses, portfolios, and risk drivers.

Then risk was aggregated bottom-up, referring at every aggregation layer to

the common conﬁdence level. The rating of the bank, say A1, attached to

this conﬁdence level, implicitly was transferred to all business units.

One can take a diﬀerent approach, though, and not require the same con-

ﬁdence level for each of the bank’s units. Instead, one can allocate capital

based only on the contribution made by each business unit to the aggregate

risk of the bank at the chosen conﬁdence level. Here, the reference is made

to the numerical value of the the risk measure, e.g. value at risk or expected

shortfall, at the appropriate conﬁdence level on the bank level. In the above

example, with an A1 target rating and a 99.95% conﬁdence level, the unex-

pected losses of the bank are 5.48 ×10

$. The capital allocation scheme then

330 11. Risk Capital

is based on the contribution of the individual businesses to bank-wide losses

of precisely this order or magnitude.

In the following, we take expected shortfall (Sect. 10.3.6) as the risk mea-

sure of choice because the scheme can be implemented straightforwardly only

with a risk measure which can be represented as a mathematical expectation

value. Value at risk is not suitable for this purpose. Moreover, to keep the

discussion simple, we neglect expected losses. For a continuous distribution,

the expected shortfall for a portfolio Π, (10.52), simpliﬁes to

ES(P

var

; Π)=−

var

Π | Π ≤−Λ

var

) . (11.1)

The speciﬁc expression appropriate to our example is

ES(0.05%; Π)=−2 × 10

Π | Π ≤−Λ

var

(5 × 10

−4

) . (11.2)

In the present context, Π is taken to be the entire bank. In terms of its three

businesses A, B, and C, and their I

, J

,andK

respective subportfolios,

the bank portfolio is

Π(t)=Π

(t)+Π

(t) (11.3)



i=1

(t)+



j=1

(t)+



k=1

(t) . (11.4)

Now simulate a very large number of scenarios at least at the level of the

business units and determine the 0.05% value at risk Λ

var

(5 × 10

−4

)ofthe

bank. Next calculate the expected shortfall of the bank, ES(5 × 10

−4

; Π)by

summing over those scenarios whose losses exceed Λ

var

(5 × 10

−4

ES(5 × 10

−4

; Π)=5.48 × 10

$=ES



var

(5 × 10

−4

); Π



. (11.5)

The ﬁrst equality in (11.5) emphasises the deﬁnition of expected shortfall in

terms of a preselected default probability (resp. conﬁdence level), whereas the

second equality relates it to the dollar value of the bank-wide 0.05% value at

risk. This relation to the bank-wide value at risk is important in the following.

Π(t) is additive over businesses, and the expectation value ... is additive

over scenarios. We therefore can write

ES [Λ

var

); Π]=−

var

Π | Π ≤−Λ

var

)

= −

var

Π

+ Π

| Π ≤−Λ

var

)

= −

var

{Π

| Π ≤−Λ

var

)

+ Π

| Π ≤−Λ

var

)

+ Π

| Π ≤−Λ

var

)} . (11.6)

11.2 Economic Capital 331

The three terms in (11.6) are sometimes called risk contributions and have

the desired property of being an additive decomposition of the bank’s risk,

as measured by expected shortfall. Based on these risk contributions, an easy

capital allocation is possible.

Notice, however, that

−2×10

Π

| Π ≤−Λ

var

(5×10

−4

) = ES(5×10

−4

; Π

)=2×10

$ , (11.7)

because Λ

var

(5 ×10

−4

) is the bank’s value at risk and not the 0.05% value at

risk of business A. The risk contribution sums the contributions of business

A to the most catastrophic scenarios for the bank as a whole, no matter what

their relevance for business A. For this same reason,

−2 × 10

Π

| Π ≤−Λ

var

(5 × 10

−4

) =ES



var

(5 × 10

−4

); Π



, (11.8)

although, most likely, scenarios contributing to the right-hand side of the

inequality will also contribute on the left-hand side.

Quite generally, the risk contribution of business A to the bank-wide ex-

pected shortfall is diﬀerent from the stand-alone expected shortfall of busi-

ness A, both when computed at the bank-wide conﬁdence level of 99.95%,

and when computed at the bank-wide value at risk Λ

var

(5 × 10

−4

). Once

the risk contribution of business A has been determined, the process can be

iterated to reallocate business A’s capital to its I

subportfolios Π

11.2.4 Economic Capital as a Management Tool

Economic capital is an important management tool. In the preceding section,

a stationary environment was assumed for capital allocation, and capital al-

location was discussed with the focus of balancing actual risk by economic

capital. The argument can easily be turned around: When there is an im-

balance between allocated capital and current risk, a powerful incentive for

change is created. When capital allocated to a business is reduced, the busi-

ness is forced to either reduce its operations, or to engage in less risky or

better diversiﬁed operations. When economic capital is increased, a business

may take more risk, either from expansion, or from trading riskier products,

etc.

How can one come to a decision about increasing or decreasing the capital

cushion of individual businesses? The central question is: Which of the three

businesses A, B, or C of the bank generates the highest return from the

capital allocated? Models for bank performance can support such decisions.

One such model (among many others) is the RORAC system [256]. RORAC

is the abbreviation of Return Over Risk-Adjusted Capital, and is deﬁned as

RORAC =

return

allocated risk capital

. (11.9)