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### portfolio selection problem modified

1. 1. CHAPTER SEVENTHE PORTFOLIO SELECTION PROBLEM
2. 2. INTRODUCTION• Portfolio is a collection of securities.• With a given amount of wealth and securities, an investor can design innumerable portfolios.• THE BASIC PROBLEM: – given uncertain outcomes, what risky securities should an investor own?4/19/2012 Syed Karim Bux Shah 2
3. 3. INTRODUCTION• THE BASIC PROBLEM: – The Harry Markowitz Approach • assume an initial wealth • a specific holding period (one period) • a terminal wealth • diversify4/19/2012 Syed Karim Bux Shah 3
4. 4. INTRODUCTION• Initial and Terminal Wealth • recall one period rate of return we wb rt wb where rt = the one period rate of return wb = the beginning of period wealth we= the end of period wealth4/19/2012 Syed Karim Bux Shah 4
5. 5. INITIAL AND TERMINAL WEALTH• DETERMINING THE PORTFOLIO RATE OF RETURN – similar to calculating the return on a security – FORMULA w1 w0 rp w04/19/2012 Syed Karim Bux Shah 5
6. 6. INITIAL AND TERMINAL WEALTH• DETERMINING THE PORTFOLIO RATE OF RETURN w1 w0 Formula: rp w0 where w0 = the aggregate purchase price at time t=0 w1 = aggregate market value at time t=14/19/2012 Syed Karim Bux Shah 6
7. 7. INITIAL AND TERMINAL WEALTH• OR USING INITIAL AND TERMINAL WEALTH w1 1 rp w0where w0 =the initial wealth w1 =the terminal wealth4/19/2012 Syed Karim Bux Shah 7
8. 8. THE MARKOWITZ APPROACH• MARKOWITZ PORTFOLIO RETURN – portfolio return (rp) is a random variable – defined by the first and second moments of the distribution • expected return • standard deviation4/19/2012 Syed Karim Bux Shah 8
9. 9. THE MARKOWITZ APPROACH• MARKOWITZ PORTFOLIO RETURN – defined by the first and second moments of the distribution • expected return (mean returns) • standard deviation (dispersion of returns about mean)4/19/2012 Syed Karim Bux Shah 9
10. 10. THE MARKOWITZ APPROACH • MARKOWITZ PORTFOLIO RETURN – First Assumption: • Non-satiation: investor always prefers a higher rate of portfolio return/higher terminal wealth. • This leads to a conclusion “Given two portfolios with similar risk, investor would prefer the portfolio with higher returns.Preferable Portfolio Portfolio Returns Risk A 12% 10% B 8% 10% 4/19/2012 Syed Karim Bux Shah 10
11. 11. THE MARKOWITZ APPROACH • MARKOWITZ PORTFOLIO RETURN – Second Assumption • Risk aversion: assume a risk-averse investor will choose a portfolio with a smaller standard deviation Portfolio Returns RiskPreferable A 12% 10% Portfolio B 12% 08% • in other words, these investors when given a fair bet (odds 50:50) will not take the bet, i.e. \$5 if head, and \$-5 if tail. Note expected return on this is 0=(5*0.5)+(-5*0.5). 4/19/2012 Syed Karim Bux Shah 11
12. 12. THE MARKOWITZ APPROACH• MARKOWITZ PORTFOLIO RETURN – INVESTOR UTILITY – DEFINITION: is the relative satisfaction derived by the investor from the economic activity- work, consumption, investment. – It depends upon individual tastes and preferences-One individual may not seek same satisfaction/utility from same activity. – It assumes rationality, i.e. people will seek to maximize their utility – Utility wealth function: shows relationship between utility and wealth.4/19/2012 Syed Karim Bux Shah 12
13. 13. THE MARKOWITZ APPROACH• MARGINAL UTILITY – each investor has a unique utility-of-wealth function – incremental or marginal utility differs by individual investor and depends upon the amount of wealth one already possesses. – Richer investor value marginal \$ less than a poor investor does.4/19/2012 Syed Karim Bux Shah 13
14. 14. THE MARKOWITZ APPROACH• MARGINAL UTILITY – Assumes • diminishing characteristic: As one has more of wealth, additional/marginal unit of wealth will add positive utility but on decreasing rate i.e. utility derived from marginal unit will keep on decreasing with successive units. • An investor with diminishing marginal utility is risk averse and such an investor rate certain investment higher than riskier one. • nonsatiation • Concave utility-of-wealth function4/19/2012 Syed Karim Bux Shah 14
15. 15. THE MARKOWITZ APPROACHUTILITY OF WEALTH FUNCTION Utility Risk Utility of premium Uc Wealth Ur Certainty equivalent Wealth 103 110 100 1054/19/2012 Syed Karim Bux Shah 15
16. 16. Conclusions• Uc=Utility from certain investment• Ur=Utility from risky investment• Uc > Ur• The amount of positive utility derived from an additional \$1 < the amount of negative utility (disutility) resulted from loss of \$1.• Note: This is evident from the slope of utility wealth function which is increasing on decreasing rate. At any point on curve slope towards right is lower than the slope to left (Concavity).4/19/2012 Syed Karim Bux Shah 16
17. 17. Understanding Certainty Equivalents and Risk Premiums• Suppose you are given two options A and B for investing \$100. A: that you will earn Rs.105 with certainty. B: that you will earn either Rs.110 or nothing, probability of both events is 50:50.Note: Both options have same expected pay off i.e. Rs.105.• Which option would you choose?• Your decision depends upon your attitude to risk. You are: Risk indifferent, if both options are equally attractive to you. Risk averse: if you choose option A, preferring safe \$ to risky \$. Risk taker: if you choose plan B.4/19/2012 Syed Karim Bux Shah 17
18. 18. Understanding Certainty Equivalents and Risk Premiums• A risk averse investor will choose option B only if: ~ ceteris paribus, he receives lesser pay off in riskless investment e.g. Rs.101 ~ ceteris paribus, he receives even higher pay off in risky investment (Option B) e.g. Rs.120.Note there must be an amount, where the investor regard both investments equally. For example in the Option A, if instead of certain \$105, you are offered \$103 and as a result you now regard both certain and risky investments equal, i.e. you derive same level of expected utility from both options. We call \$103 Certainty Equivalent (CE). And the difference between expected payoff and CE is called Risk Premium (RP), a compensation to investor for additional risk taking.The more risk averse you are the higher risk premium you demand and hence the lower CE, you have.Risk averse have positive RP, risk neutral have zero risk premium, and risk takers have negative risk premium.Expected payoff (EP)= Risk Premium (RP) + Certainty Equivalent (CE)CE = EP-RPRP = EP-CE4/19/2012 Syed Karim Bux Shah 18
19. 19. 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 utility4/19/2012 Syed Karim Bux Shah 19
20. 20. 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 assumptions4/19/2012 Syed Karim Bux Shah 20
21. 21. Indifference Curves Analysis Return further northwest A risk averse investor will A choose Portfolio A, which offers highest returns, with B C relatively lower risk. Risk4/19/2012 Syed Karim Bux Shah 21
22. 22. 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 deviation4/19/2012 Syed Karim Bux Shah 22
23. 23. PORTFOLIO RETURN• CALCULATING PORTFOLIO RETURN – Expected returns: • Method One: rP = w1 - w0/ w04/19/2012 Syed Karim Bux Shah 23
24. 24. PORTFOLIO RETURN – Expected returns: • Method Two: N rp X i ri t 1 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 portfolio4/19/2012 Syed Karim Bux Shah 24
25. 25. Expected returns• Portfolio expected return is a weighted average of expected returns of its constituents securities, i.e. each security contributes to portfolio by its expected return and its proportion in portfolio.4/19/2012 Syed Karim Bux Shah 25
26. 26. 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 outcome4/19/2012 Syed Karim Bux Shah 26
27. 27. PORTFOLIO RISK• CALCULATING PORTFOLIO RISK – Portfolio Risk: 1/ 2 N N P Xi X j ij i 1 j 1 where ij = the covariance of returns between security i and security j4/19/2012 Syed Karim Bux Shah 27
28. 28. 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 directions4/19/2012 Syed Karim Bux Shah 28
29. 29. PORTFOLIO RISK CORRELATION COEFFICIENT – rescales covariance to a range of +1 to -1 Note: Covariance between two ij ij i j securities i and j = correlation between i and j x Standard deviation of I x Standard deviation of j. where ρ i j = +1: denotes perfectly positive relationship between i and j’s returns, implying that as returns ij ij / i j of i increase so does j’s. ρ i j = -1: denotes perfectly negative relationship. ρ i j = 0: indicate no identifiable relationship. Note:-1 ≤ ρ i j ≤ +14/19/2012 Syed Karim Bux Shah 29
30. 30. 4/19/2012 Syed Karim Bux Shah 30
31. 31. Graphical representation of correlation B’s return B’s return A’s return A’s returna) Perfectively Positively b) Perfectively negativelycorrelated returns correlated returns4/19/2012 Syed Karim Bux Shah 31
32. 32. Calculating Portfolio RiskExp: Given the following variance-covariance matrix for three securities A, B, and C, as well as the percentage of the portfolio for each security, calculate the portfolio’s risk (standard deviation σp. Variance-covariance Matrix Security A Security B Security C (50%) (30%) (20%) Security A 459 -211 112 Security B -211 312 215 Security C 112 215 1794/19/2012 Syed Karim Bux Shah 32
33. 33. Calculating Portfolio Risk 1/ 2 N N Solution: We know PF risk equals P Xi X j ij i 1 j 1(.5x.5x459) = (.5x.3x-211)= (.5x.2x112)=114.75 -31.65 11.2(.3x.5x-211)= (.3x.3x312)= (.3x.2x215)=-31.65 28.08 12.9(.2x.5x112)= (.2x.3x215)= (.2x.2x179)=11.2 12.9 7.16 Note: This reduces to ((.5x.5x459) + (.3x.3x312) + (.2x.2x179) + ½ ½ 2 (.5x.3x-211) + 2 (.5x.2x112) + 2 (.3x.2x215)) = (134.89) = 11.61% 4/19/2012 Syed Karim Bux Shah 33
34. 34. Calculating Portfolio Risk % of PF in each stock Sec A 0.5 Variance-covariance Matrix Sec B 0.3 Sec A Sec B Sec C Sec C 0.2Sec A 459 -211 112Sec B -211 312 215Sec C 112 215 179 Some important points aboutSolution: Variance-covariance Matrix: 114.75 -31.65 11.2 -31.65 28.08 12.9 11.2 12.9 7.16 1. It is Square Matrix, having N2 elements for N securities. 2. Variance appear on thePortfolio Variance = 134.9 diagonal of matrix. 3. The matrix is symmetric.Portfolio SD = 11.61 % 4/19/2012 Syed Karim Bux Shah 34
35. 35. Risk-seeking Investor• Risk seeking investor will prefer: – a gamble when presented a choice. – Large gambles over small gambles, because utility gained from winning is greater for him than disutility gained from loosing. – on indifference curve position of Farthest northeast• Risk seeking investors utility functions will be convex and their indifference curves will be negatively sloped.4/19/2012 Syed Karim Bux Shah 35
36. 36. Risk-neutral investors Risk neutral investors: • are indifferent to risk. • Have horizontal indifference curves (IC). • Will prefer farthest north position on IC. Note that as risk-neutral investor just for 1% additionalReturn Preferable A IC expected returns (from portfolio 15% B Portfolio A compared to B) is willing to 14% take 10% additional risk. Such an investor consider the return factor only, ignoring risk altogether. 10% 20% Risk 4/19/2012 Syed Karim Bux Shah 36