1. CHAPTER SIX THE PORTFOLIO SELECTION PROBLEM

2. INTRODUCTION n THE BASIC PROBLEM: given uncertain outcomes, what risky securities should an investor own?

3. INTRODUCTION n THE BASIC PROBLEM: The Markowitz Approach 3 assume an initial wealth 3 a specific holding period (one period) 3 a terminal wealth 3 diversify

4. INTRODUCTION n Initial and Terminal Wealth 3 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

5. INITIAL AND TERMINAL WEALTH n DETERMINING THE PORTFOLIO RATE OF RETURN similar to calculating the return on a security FORMULA

6. INITIAL AND TERMINAL WEALTH n 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

7. INITIAL AND TERMINAL WEALTH n OR USING INITIAL AND TERMINAL WEALTH where w 0 =the initial wealth w 1 =the terminal wealth

8. THE MARKOWITZ APPROACH n MARKOWITZ PORTFOLIO RETURN portfolio return (r p ) is a random variable

9. THE MARKOWITZ APPROACH n MARKOWITZ PORTFOLIO RETURN defined by the first and second moments of the distribution 3 expected return 3 standard deviation

10. THE MARKOWITZ APPROACH n MARKOWITZ PORTFOLIO RETURN First Assumption: 3 nonsatiation: investor always prefers a higher rate of portfolio return

11. THE MARKOWITZ APPROACH n MARKOWITZ PORTFOLIO RETURN Second Assumption 3 assume a risk-averse investor will choose a portfolio with a smaller standard deviation 3 in other words, these investors when given a fair bet (odds 50:50) will not take the bet

12. THE MARKOWITZ APPROACH n MARKOWITZ PORTFOLIO RETURN INVESTOR UTILITY 3 DEFINITION: is the relative satisfaction derived by the investor from the economic activity. 3 It depends upon individual tastes and preferences 3 It assumes rationality, i.e. people will seek to maximize their utility

13. THE MARKOWITZ APPROACH n MARGINAL UTILITY each investor has a unique utility-of- wealth function incremental or marginal utility differs by individual investor

14. THE MARKOWITZ APPROACH n MARGINAL UTILITY Assumes 3 diminishing characteristic 3 nonsatiation 3 Concave utility-of-wealth function

15. THE MARKOWITZ APPROACH UTILITY OF WEALTH FUNCTION Wealth Utility Utility of Wealth

16. INDIFFERENCE CURVE ANALYSIS n INDIFFERENCE CURVE ANALYSIS DEFINITION OF INDIFFERENCE CURVES: 3 a graphical representation of a set of various risk and expected return combinations that provide the same level of utility

17. INDIFFERENCE CURVE ANALYSIS n INDIFFERENCE CURVE ANALYSIS Features of Indifference Curves: 3 no intersection by another curve 3 “further northwest” is more desirable giving greater utility 3 investors possess infinite numbers of indifference curves 3 the slope of the curve is the marginal rate of substitution which represents the nonsatiation and risk averse Markowitz assumptions

18. PORTFOLIO RETURN n CALCULATING PORTFOLIO RETURN Expected returns 3 Markowitz Approach focuses on terminal wealth (W 1 ), that is, the effect various portfolios have on W 1 3 measured by expected returns and standard deviation

19. PORTFOLIO RETURN n CALCULATING PORTFOLIO RETURN Expected returns: 3 Method One: r P = w 1 - w 0 / w 0

20. PORTFOLIO RETURN Expected returns: 3 Method Two: where r P = the expected return of the portfolio X i = 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

21. PORTFOLIO RISK n CALCULATING PORTFOLIO RISK Portfolio Risk: 3 DEFINITION: a measure that estimates the extent to which the actual outcome is likely to diverge from the expected outcome

22. PORTFOLIO RISK n CALCULATING PORTFOLIO RISK Portfolio Risk : where  ij = the covariance of returns between security i and security j

23. PORTFOLIO RISK n CALCULATING PORTFOLIO RISK Portfolio Risk: 3 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

24. PORTFOLIO RISK 3 CORRELATION COEFFICIENT – rescales covariance to a range of +1 to -1 where

25. END OF CHAPTER 6