CHAPTER 6 POWERPOINT PRESENTATION

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 wealth



a specific holding period (one period)



a terminal wealth



diversify

INTRODUCTION



Initial and Terminal Wealth



recall one period rate of return

where r_t = the one period rate of return w_b = the beginning of period wealth w_e= the end of period wealth

b

b e

t

w

w r w 



INITIAL AND TERMINAL WEALTH



DETERMINING THE PORTFOLIO RATE OF RETURN

• similar to calculating the return on a security

• ^FORMULA

0

0 1

w

w r

_p

w 



INITIAL AND TERMINAL WEALTH



DETERMINING THE PORTFOLIO RATE OF RETURN

Formula:

where w

₀

= the aggregate purchase price at time t=0

w

₁

= aggregate market value at time t=1

0

0 1

w

w r

_p

w 



INITIAL AND TERMINAL WEALTH



OR USING INITIAL AND TERMINAL WEALTH

where

w

₀

=the initial wealth

w =the terminal wealth

  ₀

1 1 r w

w   _p

THE MARKOWITZ APPROACH



MARKOWITZ PORTFOLIO RETURN

• portfolio return (r

p

) is a random variable

THE MARKOWITZ APPROACH



MARKOWITZ PORTFOLIO RETURN

• defined by the first and second moments of the distribution



expected return



standard 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 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

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

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

• ^Assumes



diminishing characteristic



nonsatiation



Concave utility-of-wealth function

THE MARKOWITZ APPROACH

UTILITY 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 utility



investors possess infinite numbers of indifference curves



the slope of the curve is the marginal rate of

substitution which represents the nonsatiation and

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:

r

_P

= w

₁

- w

₀

/ w

₀

PORTFOLIO RETURN

• Expected returns:



Method Two:

where r_P = the expected return of the portfolio Xi = the proportion of the portfolio’s initial value invested in security i

r_i = the expected return of security i

N = the number of securities in the portfolio







^N

t

i i

p

X r

r

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 j

ij j

i

P

X X 



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

j i

ij

ij   

 

j i

ij

ij   

  /

END OF CHAPTER 6