7
The Portfolio Selection Problem
A portfolio is a collection of securities. Portfolio selection problem is equivalent to
the investor selecting the optimal portfolio from a set of possible portfolios.
Markowitz (1952) was the pioneer of the Modern Portfolio Theory.
Holding period: Markowitz’s approach begins by assuming that an investor has a
given sum of money to invest at the present time. This money will be invested
for a particular length of time known as the investor’s holding period.
Expected holding period return: The investor should recognize that security
returns in the forthcoming holding period are unknown. However, the investor
could estimate the expected holding period return on the various securities
under consideration and then invest in the one with the highest expected
return.
Risk: The investment decision would be generally unwise because the typical
investor, although wanting “returns to be high”, also wants “returns to be as
certain as possible”. The risk is referred as the uncertainty in receiving the
expected return.
7.1
Initial and Terminal Wealth
Return = (End-of-period wealth – Beginning-of-period wealth) / (Beginning-of-period Wealth)
7.2
Determining the Rate of Return on a Portfolio
 Portfolio return:
Rp = (W 1 – W 0) / W 0
 Initial wealth: The wealth in the beginning of the investment period.
 Terminal wealth: The wealth at the end of the investment period
 Random variable: A variable that takes on alternative values as described
by a particular probability distribution. For example expected value and
standard deviation.
 Expected value (Mean): A measure of central tendency of the probability
distribution of a random variable that equals the weighted average of all
possible outcomes using their probabilities as weights.
 Standard deviation: A measure of the dispersion of possible outcomes
around the expected value of a random variable.
7.3 Nonsatiation and Risk Aversion
Two assumptions are implicit in the portfolio selection problem. Markowitz used two
specific assumptions:
 Nonsatiation: The assumption in the Markowitz approach in which the
investors are assumed to always prefer higher levels of terminal wealth to
lower levels of terminal wealth. Thus, given two portfolios with the same
standard deviation, the investor will choose the portfolio with the higher
expected return. However, it is not quite so obvious what the investor will do
when he has to choose between two portfolios with the same level of expected
return but different levels of standard deviation.
 Risk-aversion: It is assumed that investors are risk averse, which means
they will choose the portfolio with the smaller standard deviation.
7.4 Utility
The exact relationship between utility and wealth is called the investor’s utility of
wealth function.
 Utility: The relative enjoyment or satisfaction that people drive from
economic activity such as work, consumption, or investment. People are
presumed to be rational and to allocate their resources (such as time and
money) in ways that maximize their own utilities. The Markowitz portfolio
selection problem can be viewed as an effort to maximize the expected utility
associated with the investor’s terminal wealth.
 Marginal utility: A unique increment of utility from an extra dollar of
wealth. The marginal utility may differ among investors. Further, that marginal
utility may depend on the level of wealth that the investor possesses before
receiving the extra dollar.
 Diminishing Marginal Utility: A common assumption is that investors
experience diminishing marginal utility of wealth. Each extra dollar of wealth
always provides positive additional utility, but the added utility produced by
extra dollar becomes successively smaller. The assumption of nonsatiation
requires that the utility of wealth function is always positively sloped no matter
what the level of wealth. However, this utility of wealth function is concave. An
investor with diminishing marginal utility is necessarily risk-averse. This riskaverse investor is unwilling to accept a fair bet. The utility of wealth function
explains that preference.
 Certainty equivalent: Moving horizontally from the utility axis at the
expected utility of the risky investment to the utility of wealth function and then
moving down to the terminal wealth axis indicates certainty equivalent wealth
associated with this risky investment.
 Risk premium: The additional dollar amount in expected terminal wealth
offered by the risky investment over the certain investment.
7.5
Indifference Curves
 Indifference curve: A set of risk and expected return combinations that
provide an investor with the same amount of utility. The investor is indifferent
about the risk – expected return combinations on the same indifference curve.
Because indifferent curves indicate an investor’s preferences for risk and
expected return, they can be drawn on a two-dimensional figure where the
horizontal axis indicates risk as measured by standard deviation, and the
vertical axis indicates reward as measured by expected return. Because all
portfolios that lie on a given indifference curve are equally desirable to the
investor, by implication indifference curves cannot intersect.
7.6
Calculating Expected Returns and Standard Deviation for Portfolios
 The expected return on a portfolio consisting of N securities:
_
N
_
rp   X i r i
i 1
_
rp = the expected return of the portfolio
Xi = the proportion of the portfolio’s initial value invested in security i
_
ri = the expected return of security i
N = the number of securities in the portfolio
 Expected return vector: Used to calculate the expected return for any
portfolio formed from the N securities. This vector consists of one column of
numbers, where the entry in row i contains the expected return of security i.
 Probability distribution: A model describing the relative frequency of
possible values that a random variable can assume.
 Normal distribution: A symmetrical bell-shaped probability distribution,
completely described by its mean and standard distribution.
 The clearest example arises when the probability distribution for a
portfolio’s return can be approximated by the familiar bell-shaped curve known
as a normal distribution.
 Standard deviation of a portfolio:
p 
N
N
i 1
j 1
 X
i
X j  ij
 Covariance: A statistical measure of the relationship between two random
variables and of how to random variables, such as the returns on securities i
and j, “move together”.
 Correlation coefficient: Rescales the covariance to facilitate comparison
with corresponding values for other pairs of random variables and lies
between –1 and +1.
ij = ij / (i j)
 Variance: The standard deviation of security i squared.
 Variance – covariance matrix:
Column1 Column2 Column3
Row1
146
187
145
Row2
187
854
104
Row3
145
104
289
Cell (i,j) : Covariance between security i and j
Cell (i,i) : Variance of security i