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uted. Then the OLS method can be used for estimating bi in the
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
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.
117
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
119
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
123
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.
125
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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