CHAPTER SIX

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CHAPTER SIX

• 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

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INITIAL AND TERMINAL WEALTH

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

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INITIAL AND TERMINAL WEALTH

• DETERMINING THE PORTFOLIO RATE OF RETURN
• Formula
• where w0 the aggregate purchase
  price at time t0
• w1 aggregate market value at
  time t1

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INITIAL AND TERMINAL WEALTH

• OR USING INITIAL AND TERMINAL WEALTH
• where
• w0 the initial wealth
• w1 the terminal wealth

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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 5050) 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

Utility
Utility of Wealth
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 portfolios initial
  value invested in security i
• ri the expected return of security i
• N the number of securities in the
  portfolio

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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 sij the covariance of returns
  between security i and security j

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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

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• END OF CHAPTER 6