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THE PORTFOLIO SELECTION PROBLEM

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INTRODUCTION

• THE BASIC PROBLEM:– given uncertain outcomes, what risky securities

should an investor own?

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INTRODUCTION

• THE BASIC PROBLEM:– The Markowitz Approach

• assume an initial wealth

• a specific holding period (one period)

• a terminal wealth

• diversify

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INTRODUCTION

• Initial and Terminal Wealth• recall 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

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INITIAL AND TERMINAL WEALTH

• DETERMINING THE PORTFOLIO RATE OF RETURN– similar to calculating the return on a security– FORMULA

0

01

w

wwrp

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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

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INITIAL AND TERMINAL WEALTH

• OR USING INITIAL AND TERMINAL WEALTH

where

w0 =the initial wealth

w1 =the terminal wealth

01 1 wrw p

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THE MARKOWITZ APPROACH

• MARKOWITZ PORTFOLIO RETURN

– portfolio return (rp) is a random variable

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THE MARKOWITZ APPROACH

• MARKOWITZ PORTFOLIO RETURN– defined by the first and second moments of the

distribution• expected return

• standard deviation

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THE MARKOWITZ APPROACH

• MARKOWITZ PORTFOLIO RETURN– First Assumption:

• nonsatiation: investor always prefers a higher rate of portfolio return

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THE MARKOWITZ APPROACH

• MARKOWITZ PORTFOLIO RETURN– Second Assumption

• assume 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

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THE MARKOWITZ APPROACH

• MARKOWITZ PORTFOLIO RETURN– INVESTOR UTILITY

• DEFINITION: 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

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THE MARKOWITZ APPROACH

• MARGINAL UTILITY– each investor has a unique utility-of-wealth

function– incremental or marginal utility differs by

individual investor

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THE MARKOWITZ APPROACH

• MARGINAL UTILITY– Assumes

• diminishing characteristic

• nonsatiation

• Concave utility-of-wealth function

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THE MARKOWITZ APPROACH

UTILITY OF WEALTH FUNCTION

Wealth

Utility Utility of Wealth

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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

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INDIFFERENCE CURVE ANALYSIS

• INDIFFERENCE CURVE ANALYSIS– Features of Indifference Curves:

• no intersection by another curve• “further northwest” is more desirable giving greater

utility• investors possess infinite numbers of indifference

curves• the slope of the curve is the marginal rate of

substitution which represents the nonsatiation and risk averse Markowitz assumptions

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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

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PORTFOLIO RETURN

• CALCULATING PORTFOLIO RETURN– Expected returns:

• Method One:

rP = w1 - w0/ w0

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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 iN = the number of securities in the portfolio

N

tiip rXr

1

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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

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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

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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

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PORTFOLIO RISK

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

where

jiijij

jiijij /