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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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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.

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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

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