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regression equation for the excess return Zi Â¼ Ri Ã€ Rf

Zi (t) Â¼ ai Ã¾ bi ZM (t) Ã¾ ei (t) (10:2:4)

It is usually assumed that ei (t) is a normal process and the S&P 500

Index is the benchmark for the market portfolio return RM (t). More

details on the CAPM validation and the general results for the mean-

variance efficient portfolios can be found in [2, 3].

As indicated above, CAPM is based on the belief that investing in

risky assets yields average returns higher than the risk-free return.

Hence, the rationale for investing in risky assets becomes question-

able in bear markets. Another problem is that the asset diversification

116 Portfolio Management

advocated by CAPM is helpful if returns of different assets are

uncorrelated. Unfortunately, correlations between asset returns may

grow in bear markets [4]. Besides the failure to describe prolonged

bear markets, another disadvantage of CAPM is its high sensitivity to

proxy for the market portfolio. The latter drawback implies that

CAPM is accurate only conditionally, within a given time period,

where the state variables that determine economy are fixed [2]. Then

it seems natural to extend CAPM to a multifactor model.

10.3 ARBITRAGE PRICING THEORY (APT)

The CAPM equation (10.2.1) implies that return on risky assets is

determined only by a single non-diversifiable risk, namely by the risk

associated with the entire market. The Arbitrage Pricing Theory

(APT) offers a generic extension of CAPM into the multifactor

paradigm.

APT is based on two postulates. First, the return for an asset i

(i Â¼ 1, . . . , N) at every time period is a weighed sum of the risk factor

contributions fj (t) (j Â¼ 1, . . . , K, K < N) plus an asset-specific com-

ponent ei (t)

Ri (t) Â¼ ai Ã¾ bi1 f1 Ã¾ bi2 f2 Ã¾ . . . Ã¾ biK fK Ã¾ ei (t) (10:3:1)

In (10.3.1), bij are the factor weights (betas). It is assumed that the

expectations of all factor values and for the asset-specific innovations

are zero

E[f1 (t)] Â¼ E[f2 (t)] Â¼ . . . Â¼ E[fK (t)] Â¼ E[ei (t)] Â¼ 0 (10:3:2)

Also, the time distributions of the risk factors and asset-specific

innovations are independent

0

Cov[fj (t), fj (t0 )] Â¼ 0, Cov[ei (t), ei (t0 )] Â¼ 0, t 6Â¼ t (10:3:3)

and uncorrelated

Cov[fj (t), ei (t)] Â¼ 0 (10:3:4)

Within APT, the correlations between the risk factors and the asset-

specific innovations may exist, that is Cov[fj (t), fk (t)] and

Cov[ei (t), ej (t)] may differ from zero.

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Portfolio Management

The second postulate of APT requires that there are no arbitrage

opportunities. This implies, in particular, that any portfolio in which

all factor contributions are canceled out must have return equal to

that of the risk-free asset (see Exercise 3). These two postulates lead to

the APT theorem (see, e.g., [5]). In its simple form, it states that there

exist such K Ã¾ 1 constants l0 , l1 , . . . lK (not all of them equal zero)

that

E[Ri (t)] Â¼ l0 Ã¾ bi1 l1 Ã¾ . . . Ã¾ biK lK (10:3:5)

While l0 has the sense of the risk-free asset return, the numbers lj are

named the risk premiums for the j-th risk factors.

Let us define a well-diversified portfolio as a portfolio that consists

P

N

wi Â¼ 1, so that wi < W=N

of N assets with the weights wi where

iÂ¼1

and W % 1 is a constant. Hence, the specific of a well-diversified

portfolio is that it is not overweighed by any of its asset components.

APT turns out to be more accurate for well-diversified portfolios

than for individual stocks. The general APT states that if the return of

a well-diversified portfolio equals

R(t) Â¼ a Ã¾ b1 f1 Ã¾ b2 f2 Ã¾ . . . Ã¾ bK fK Ã¾ e(t) (10:3:6)

where

X X

N N

aÂ¼ wi a i , b i Â¼ wk bik (10:3:7)

kÂ¼1

iÂ¼1

then the expected portfolio return is

E[R(t)] Â¼ l0 Ã¾ b1 l1 Ã¾ . . . Ã¾ bK lK (10:3:8)

In addition, the returns of the assets that constitute the portfolio

satisfy the simple APT (10.3.5).

APT does not specify the risk factors. Yet, the essential sources of

risk are well described in the literature [6]. They include both macro-

economic factors including inflation risk, interest rate, and corporate

factors, for example, Return on Equity (ROE).4 Development of

statistically reliable multifactor portfolio models poses significant

challenges [2]. Yet, multifactor models are widely used in active

portfolio management.

118 Portfolio Management

Both CAPM and APT consider only one time period and treat the

risk-free interest rate as an exogenous parameter. However, in real

life, investors make investing and consumption decisions that are in

effect for long periods of time. An interesting direction in the port-

folio theory (that is beyond the scope of this book) describes invest-

ment and consumption processes within a single framework. The risk-

free interest rate is then determined by the consumption growth and

by investor risk aversion. The most prominent theories in this direc-

tion are the intertemporal CAPM (ICAPM) and the consumption

CAPM (CCAPM) [2, 3, 7].

10.4 ARBITRAGE TRADING STRATEGIES

The simple investment strategy means â€˜â€˜buy and holdâ€™â€™ securities

of â€˜â€˜goodâ€™â€™ companies until their performance worsens, then sell them

and buy better assets. A more sophisticated approach is sensitive to

changing economic environment and an investorâ€™s risk tolerance,

which implies periodic rebalancing of the investor portfolio between

cash, fixed income, and equities. Proponents of the conservative

investment strategy believe that this is everything an investor should

do while investing for the â€˜â€˜long run.â€™â€™ Yet, many investors are not

satisfied with the long-term expectations: they want to make money

at all times (and who could blame them?). Several concepts being

intensively explored by a number of financial institutions, particu-

larly by the hedge funds, are called market-neutral strategies.

In a nutshell, market-neutral strategy implies hedging the risk of

financial losses by combining long and short positions in the port-

folio. For example, consider two companies within the same industry,

A and B, one of which (A) yields consistently higher returns. The

strategy named pair trading involves simultaneously buying shares

A and short selling shares B. Obviously, if the entire sector rises,

this strategy does not bring as much money as simply buying

shares A. However, if the entire market falls, presumably shares B

will have higher losses than shares A. Then the profits from short

selling shares B would more than compensate for the losses from

buying shares A.

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Portfolio Management

Specifics of the hedging strategies are not widely advertised for

obvious reasons: the more investors target the same market ineffi-

ciency, the faster it is wiped out. Several directions in the market-

neutral investing are described in the literature [8].

Convertible arbitrage. Convertible bonds are bonds that can be

converted into shares of the same company. Convertible bonds

often decline less in a falling market than shares of the same company

do. Hence, the idea of the convertible arbitrage is buying convertible

bonds and short selling the underlying stocks.

Fixed-income arbitrage. This strategy implies taking long and short

positions in different fixed-income securities. By watching the correl-

ations between different securities, one can buy those securities

that seem to become underpriced and sell short those that look

overpriced.

Mortgage-backed securities (MBS) arbitrage. MBS is actually a

form of fixed income with a prepayment option. Yet, there are so

many different MBS that this makes them a separate business.

Merger arbitrage. This form of arbitrage involves buying shares of a

company that is being bought and short selling the shares of the buying

company. The rationale behind this strategy is that companies are

usually acquired at a premium, which sends down the stock prices of

acquiring companies.

Equity hedge. This strategy is not exactly the market-neutral one, as

the ratio between long and short equity positions may vary depending

on the market conditions. Sometimes one of the positions is the stock

index future while the other positions are the stocks that constitute

this index (so-called index arbitrage). Pair trading also fits this

strategy.

Equity market-neutral strategy and statistical arbitrage. Nicholas

discerns these two strategies by the level of constraints (availability of

resources) imposed upon the portfolio manager [8]. The common

feature of these strategies is that (in contrast to the equity hedge),

they require complete offsetting of the long positions by the short

positions. Statistical arbitrage implies fewer constraints in the devel-

opment of quantitative models and hence a lower amount of the

portfolio managerâ€™s discretion in constructing a portfolio.

Relative value arbitrage. This is a synthetic approach that may

embrace several hedging strategies and different securities including

120 Portfolio Management

equities, bonds, options, and foreign currencies. Looking for the arbi-

trage opportunities â€˜â€˜across the boardâ€™â€™ is technically more challenging

but potentially rewarding.

Some academic research on efficiency of the arbitrage trading

strategies can be found in [9â€“12] and references therein. Note that

the research methodology in this field is itself a non-trivial problem

[13].

10.5 REFERENCES FOR FURTHER READING

A good introduction into the finance theory, including CAPM, is

given in [1]. For a description of the portfolio theory and investment

science with an increasing level of technical detail, see [5, 14].

10.6 EXERCISES

1. Consider a portfolio with two assets having the following

returns and standard deviations: E[R1 ] Â¼ 0:15, E[R2 ] Â¼ 0:1,

s1 Â¼ 0:2, s2 Â¼ 0:15. The proportion of asset 1 in the portfolio

g Â¼ 0:5. Calculate the portfolio return and standard deviation.

The correlation coefficient between assets is (a) 0.5; (b) Ã€0.5.

2. Consider returns of stock A and the market portfolio M in three

years:

A Ã€7% 12% 26%

M Ã€5% 9% 18%

Assuming the risk-free rate is 5%, (a) calculate b of stock A; and

(b) verify if CAPM describes pricing of stock A.

3. Providing the stock returns follow the two-factor APT:

Ri (t) Â¼ ai Ã¾ bi1 f1 Ã¾ bi2 f2 Ã¾ ei (t), construct a portfolio with

three stocks (i.e., define w1 , w2 , and w3 Â¼ 1 Ã€ w1 Ã€ w2 ) that

yields return equal to that of the risk-free asset.

4. Providing the stock returns follow the two-factor simple APT,

derive the values of the risk premiums. Assume the expected

returns of two stocks and the risk-free rate are equal to R1 , R2 ,

and Rf , respectively.

Chapter 11

Market Risk Measurement

The widely used risk measure, value at risk (VaR), is discussed in

Section 11.1. Furthermore, the notion of the coherent risk measure is

introduced and one such popular measure, namely expected tail losses

(ETL), is described. In Section 11.2, various approaches to calculating

risk measures are discussed.

11.1 RISK MEASURES

There are several possible causes of financial losses. First, there is

market risk that results from unexpected changes in the market prices,

interest rates, or foreign exchange rates. Other types of risk relevant

to financial risk management include liquidity risk, credit risk, and

operational risk [1]. The liquidity risk closely related to market risk is

determined by a finite number of assets available at a given price (see

discussion in Section 2.1). Another form of liquidity risk (so-called

cash-flow risk) refers to the inability to pay off a debt in time. Credit

risk arises when one of the counterparts involved in a financial

transaction does not fulfill its obligation. Finally, operational risk is

a generic notion for unforeseen human and technical problems, such

as fraud, accidents, and so on. Here we shall focus exclusively on

measurement of the market risk.

In Chapter 10, we discussed risk measures such as the asset return

variance and the CAPM beta. Several risk factors used in APT were

121

122 Market Risk Measurement

also mentioned. At present, arguably the most widely used risk meas-

ure is value at risk (VaR) [1]. In short, VaR refers to the maximum

amount of an asset that is likely to be lost over a given period at a

specific confidence level. This implies that the probability density

function for profits and losses (P=L)1 is known. In the simplest case,

this distribution is normal

1

PN (x) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€(x Ã€ m)2 =2s2 ] (11:1:1)

2ps

where m and s are the mean and standard deviation, respectively.

Then for the chosen confidence level a,

VAR(a) Â¼ Ã€sza Ã€ m (11:1:2)

The value of za can be determined from the cumulative distribution

function for the standard normal distribution (3.2.10)

Ã°

za

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€z2 =2]dz Â¼ 1 Ã€ a

Pr(Z < za ) Â¼ (11:1:3)

2p

Ã€1

Since za < 0 at a > 50%, the definition (11.1.2) implies that positive

values of VaR point to losses. In general, VaR(a) grows with the

confidence level a. Sufficiently high values of the mean

P=L (m > Ã€sza ) for given a move VaR(a) into the negative region,

which implies profits rather than losses. Examples of za for typical

values of a Â¼ 95% and a Â¼ 99% are given in Figure 11.1. Note that

the return variance s corresponds to za Â¼ Ã€1 and yields a % 84%.

The advantages of VaR are well known. VaR is a simple and

universal measure that can be used for determining risks of different

financial assets and entire portfolios. Still, VaR has some drawbacks

[2]. First, accuracy of VaR is determined by the model assumptions

and is rather sensitive to implementation. Also, VaR provides an

estimate for losses within a given confidence interval a but says

nothing about possible outcomes outside this interval. A somewhat

paradoxical feature of VaR is that it can discourage investment

diversification. Indeed, adding volatile assets to a portfolio may

move VaR above the chosen risk threshold. Another problem with

VaR is that it can violate the sub-additivity rule for portfolio risk.

According to this rule, the risk measure r must satisfy the condition

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Market Risk Measurement

0.45

0.4

0.35

VaR at 84%

z = âˆ’1

0.3

0.25

VaR at 95%

z = âˆ’1.64

0.2

0.15

VaR at 99%

z = âˆ’2.33

0.1

0.05

0

âˆ’5 âˆ’4 âˆ’3 âˆ’2 âˆ’1 0 1 2 3 4 5

Figure 11.1 VaR for the standard normal probability distribution of P/L.

r(A Ã¾ B) r(A) Ã¾ r(B) (11:1:4)

which means the risk of owning the sum of two assets must not be

higher than the sum of the individual risks of these assets. The

condition (11.1.4) immediately yields an upper estimate of combined

risk. Violation of the sub-additivity rule may lead to several problems.

In particular, it may provoke investors to establish separate accounts

for every asset they have. Unfortunately, VaR satisfies (11.1.4) only if

the probability density function for P/L is normal (or, more generally,

elliptical) [3].

The generic criterions for the risk measures that satisfy the require-

ments of the modern risk management are formulated in [3]. Besides

the sub-additivity rule (11.1.4), they include the following conditions.

r(lA) Â¼ lr(A), l > 0 (homogeneity) (11:1:5)

r(A) r(B), if A B (monotonicity) (11:1:6)

r(A Ã¾ C) Â¼ r(A) Ã€ C (translation invariance) (11:1:7)

In (11.1.7), C represents a risk-free amount. Adding this amount to

a risky portfolio should decrease the total risk, since this amount is

124 Market Risk Measurement

not subjected to potential losses. The risk measures that satisfy the

conditions (11.1.4)â€“(11.1.7) are called coherent risk measures. It can

be shown that any coherent risk measure represents the maximum of

the expected loss on a set of â€˜â€˜generalized scenariosâ€™â€™ where every such

scenario is determined with its value of loss and probability of occur-

rence [3]. This result yields the coherent risk measure called expected

tail loss (ETL):2

ETL Â¼ E[LjL > VaR] (11:1:8)

While VaR is an estimate of loss within a given confidence level,

ETL is an estimate of loss within the remaining tail. For a given

probability distribution of P/L and a given a, ETL is always higher

than VaR (cf. Figures 11.1 and 11.2).

ETL has several important advantages over VaR [2]. In short, ETL

provides an estimate for an average â€˜â€˜worst case scenarioâ€™â€™ while VaR

only gives a possible loss within a chosen confidence interval. ETL

has all the benefits of the coherent risk measure and does not discour-

age risk diversification. Finally, ETL turns out to be a more conveni-

ent measure for solving the portfolio optimization problem.

0.45

0.4

0.35

ETL at 84%

z = âˆ’1.52

0.3

0.25

ETL at 95%

z = âˆ’2.06

0.2

0.15

ETL at 99%

z = âˆ’2.66

0.1

0.05

0

âˆ’5 âˆ’4 âˆ’3 âˆ’2 âˆ’1 0 1 2 3 4 5

Figure 11.2 ETL for the standard normal probability distribution of P/L.

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Market Risk Measurement

11.2 CALCULATING RISK

Two main approaches are used for calculating VaR and ETL [2].

First, there is historical simulation, a non-parametric approach that

employs historical data. Consider a sample of 100 P/L values as a

simple example for calculating VaR and ETL. Let us choose the

confidence level of 95%. Then VaR is the sixth smallest number in

the sample while ETL is the average of the five smallest numbers

within the sample. In the general case of N observations, VaR at the

confidence level a is the [N(1 Ã€ a) Ã¾ 1] lowest observation and ETL is

the average of N(1 Ã€ a) smallest observations.

The well-known problem with the historical simulation is handling

of old data. First, â€˜â€˜too oldâ€™â€™ data may lose their relevance. Therefore,

moving data windows (i.e., fixed number of observations prior to

every new period) are often used. Another subject of concern is

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