108156.pdf

5. Portfolio Management 93

This illustrates the following general rule:

Var(aK)=a^2 Var(K),

σaK=|a|σK

for any real numbera.

Remark 5.1

Another natural way to quantify risk would be to use the variance Var(k)(or
the standard deviationσk) of the logarithmic returnk. The choice betweenK
andkis dictated to a large extent by the properties needed to handle the task
in hand. For example, if one is interested in a sequence of investments following
one another in time, then the variance of the logarithmic return may be more
useful as a measure of risk. This is because of the additivity of risks based on
logarithmic returns:

Var(k(0,n)) = Var(k(1)) +···+Var(k(n)),

wherek(i) is the logarithmic return in time stepi=1,...,nandk(0,n)isthe
logarithmic return over the whole time interval from 0 ton, provided that the
k(i) are independent. The above formula holds becausek(0,n)=k(1) +···+
k(n) by Proposition 3.2, and the variance of a sum of independent random
variables is the sum of their variances. (This is not necessarily so without
independence.)
However, in the present chapter we shall be concerned with a portfolio of
several securities held simultaneously over a single time step. The properties
ofE(K)andVar(K), whereK is the ordinary return on the portfolio (see
formulae (5.4) and (5.5) below), are much more convenient for this purpose
than those for the logarithmic return.

Exercise 5.3

Consider two risky securities with returnsK 1 andK 2 given by

Scenario Probability ReturnK 1 ReturnK 2

ω 1 0. 510 .53% 7 .23%

ω 2 0. 513 .87% 10 .57%

Compute the corresponding logarithmic returnsk 1 andk 2 and compare

Var(k 1 ) with Var(k 2 )andVar(K 1 ) with Var(K 2 ).