Chapter 10
Portfolio Management
This chapter begins with the basic ideas of portfolio selection.
Namely, in Section 10.1, the combination of two risky assets and
the combination of a risky asset and a risk-free asset are considered.
Then two major portfolio management theories are discussed: the
capital asset pricing model (Section 10.2) and the arbitrage pricing
theory (Section 10.3). Finally, several investment strategies based on
exploring market arbitrage opportunities are introduced in Section
10.4.
10.1 PORTFOLIO SELECTION
Optimal investing is an important real-life problem that has been
translated into elegant mathematical theories. In general, opportun-
ities for investing include different assets: equities (stocks), bonds,
foreign currency, real estate, antique, and others. Here portfolios
that contain only financial assets are considered.
There is no single strategy for portfolio selection, because there is
always a trade-off between expected return on portfolio and risk of
portfolio losses. Risk-free assets such as the U.S. Treasury bills guar-
antee some return, but it is generally believed that stocks provide
higher returns in the long run. The trouble is that the notion of ‘‘long
run’’ is doomed to bear an element of uncertainty. Alas, a decade of
111
market growth may end up with a market crash that evaporates a
significant part of the equity wealth of an entire generation. Hence,
risk aversion (that is often well correlated with investor age) is an
important factor in investment strategy.
Portfolio selection has two major steps [1]. First, it is the selection
of a combination of risky and risk-free assets and, secondly, it is the
selection of risky assets. Let us start with the first step.
For simplicity, consider a combination of one risky asset and one
risk-free asset. If the portion of the risky asset in the portfolio is
a(a 1), then the expected rate of return equals
E[R] ¼ aE[R
r
] þ (1 a)R
f
¼ R
f
þ a(E[R
r
] R
f
) (10:1:1)
where R
f
and R
r
are rates of returns of the risk-free and risky assets,
respectively. In the classical portfolio management theory, risk is
characterized with the portfolio standard deviation, s.
1
Since no
risk is associated with the risk-free asset, the portfolio risk in our
case equals
s ¼ as
r
(10:1:2)
Substituting a from (10.1.2) into (10.1.1) yields
E[R] ¼ R
f
þ s(E[R
r
] R
f
)=s
r
(10:1:3)
The dependence of the expected return on the standard deviation is
called the risk-return trade-off line. The slope of the straight line
(10.1.3)
s ¼ (E[R
r
] R
f
)=s
r
(10:1:4)
is the measure of return in excess of the risk-free return per unit of
risk. Obviously, investing in a risky asset makes sense only if s > 0,
that is, E[R
r
] > R
f
. The risk-return trade-off line defines the mean-
variance efficient portfolio, that is, the portfolio with the highest
expected return at a given risk level.
On the second step of portfolio selection, let us consider the port-
folio consisting of two risky assets with returns R
1
and R
2
and with
standard deviations s
1
and s
2
, respectively. If the proportion of the
risky asset 1 in the portfolio is g(g 1), then the portfolio rate of
return equals
E[R] ¼ gE[R
1
] þ (1 g)E[R
2
] (10:1:5)
112 Portfolio Management

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