CHAPTER SIX

THE PORTFOLIO SELECTION PROBLEM

INTRODUCTION

THE BASIC PROBLEM:•given uncertain outcomes, what risky

securities should an investor own?

INTRODUCTION

THE BASIC PROBLEM:•The Markowitz Approach

assume an initial wealtha specific holding period (one period)a terminal wealthdiversify

INTRODUCTION

Initial and Terminal Wealthrecall one period rate of return

where rt = the one period rate of return

wb = the beginning of period wealth

we= the end of period wealth

b

bet w

wwr

INITIAL AND TERMINAL WEALTH DETERMINING THE PORTFOLIO

RATE OF RETURN•similar to calculating the return on a

security

•FORMULA

0

01

w

wwrp

INITIAL AND TERMINAL WEALTH DETERMINING THE PORTFOLIO RATE

OF RETURNFormula:

where w0 = the aggregate purchase price at time t=0

w1 = aggregate market value at time t=1

0

01

w

wwrp

INITIAL AND TERMINAL WEALTH OR USING INITIAL AND TERMINAL

WEALTH

where w0 =the initial wealth

w1 =the terminal wealth

01 1 wrw p

THE MARKOWITZ APPROACH MARKOWITZ PORTFOLIO RETURN

•portfolio return (rp) is a random

variable

THE MARKOWITZ APPROACH MARKOWITZ PORTFOLIO RETURN

•defined by the first and second moments of the distributionexpected returnstandard deviation

THE MARKOWITZ APPROACH MARKOWITZ PORTFOLIO RETURN

•First Assumption:nonsatiation: investor always prefers a

higher rate of portfolio return

THE MARKOWITZ APPROACH MARKOWITZ PORTFOLIO RETURN

•Second Assumptionassume a risk-averse investor will

choose a portfolio with a smaller standard deviation

in other words, these investors when given a fair bet (odds 50:50) will not take the bet

THE MARKOWITZ APPROACH MARKOWITZ PORTFOLIO RETURN

•INVESTOR UTILITYDEFINITION: is the relative satisfaction

derived by the investor from the economic activity.

It depends upon individual tastes and preferences

It assumes rationality, i.e. people will seek to maximize their utility

THE MARKOWITZ APPROACH MARGINAL UTILITY

•each investor has a unique utility-of-wealth function

•incremental or marginal utility differs by individual investor

THE MARKOWITZ APPROACH MARGINAL UTILITY

•Assumesdiminishing characteristicnonsatiationConcave utility-of-wealth function

THE MARKOWITZ APPROACHUTILITY OF WEALTH FUNCTION

Wealth

Utility Utility of Wealth

INDIFFERENCE CURVE ANALYSIS INDIFFERENCE CURVE ANALYSIS

•DEFINITION OF INDIFFERENCE CURVES: a graphical representation of a set of

various risk and expected return combinations that provide the same level of utility

INDIFFERENCE CURVE ANALYSIS INDIFFERENCE CURVE ANALYSIS

•Features of Indifference Curves:no intersection by another curve“further northwest” is more desirable giving

greater utilityinvestors possess infinite numbers of

indifference curvesthe slope of the curve is the marginal rate of

substitution which represents the nonsatiation and risk averse Markowitz assumptions

PORTFOLIO RETURN

CALCULATING PORTFOLIO RETURN•Expected returns

Markowitz Approach focuses on terminal wealth (W1), that is, the effect various portfolios have on W1

measured by expected returns and standard deviation

PORTFOLIO RETURN

CALCULATING PORTFOLIO RETURN•Expected returns:

Method One:

rP = w1 - w0/ w0

PORTFOLIO RETURN

•Expected returns:Method Two:

where 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

N

tiip rXr

1

PORTFOLIO RISK

CALCULATING PORTFOLIO RISK•Portfolio Risk:

DEFINITION: a measure that estimates the extent to which the actual outcome is likely to diverge from the expected outcome

PORTFOLIO RISK

CALCULATING PORTFOLIO RISK

•Portfolio Risk:

where ij = the covariance of returns

between security i and security j

2/1

1 1

N

i

N

jijjiP XX

PORTFOLIO RISK

CALCULATING PORTFOLIO RISK•Portfolio Risk:

COVARIANCE– DEFINITION: a measure of the relationship

between two random variables– possible values:

• positive: variables move together• zero: no relationship• negative: variables move in opposite

directions

PORTFOLIO RISK

CORRELATION COEFFICIENT– rescales covariance to a range of +1 to -1

where

jiijij

jiijij /

END OF CHAPTER 6