Quantitative Finance for Physicists

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