FINANCE 9. Optimal Portfolio Choice

1. FINANCE9. Optimal Portfolio Choice Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007

2. Introduction: random portfolios MBA 2007 Portfolio choice

3. Covariance and correlation • Statistical measures of the degree to which random variables movetogether • Covariance • Like variance figure, the covariance is in squared deviation units. • Not too friendly ... • Correlation • covariance divided by product of standard deviations • Covariance and correlation have the same sign • Positive :variables are positively correlated • Zero :variables are independant • Negative :variables are negatively correlated • The correlation is always between –1and +1 MBA 2007 Portfolio choice

4. Risk and expected returns for porfolios • In order to better understand the driving force explaining the benefitsfrom diversification, let us consider aportfolio of two stocks (A,B) • Characteristics: • Expected returns : • Standard deviations : • Covariance : • Portfolio: defined by fractions invested in each stockXA ,XBXA+ XB= 1 • Expected return on portfolio: • Variance of the portfolio's return: MBA 2007 Portfolio choice

5. Example • Invest $ 100 m in two stocks: • A $ 60 m XA = 0.6 • B $ 40 m XB = 0.4 • Characteristics (% per year) A B • • Expected return 20%15% • • Standard deviation 30%20% • Correlation 0.5 • Expected return = 0.6 × 20% + 0.4 × 15% = 18% • Variance = (0.6)²(.30)² + (0.4)²(.20)²+2(0.6)(0.4)(0.30)(0.20)(0.5) s²p = 0.0532 Standard deviation = 23.07 % • Less than the average of individual standard deviations: • 0.6 x0.30 + 0.4 x 0.20 = 26% MBA 2007 Portfolio choice

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8. Combining the Riskless Asset and asingle Risky Asset • Consider the following portfolio P: • Fraction invested • in the riskless asset 1-x(40%) • in the risky asset x(60%) • Expected return on portfolio P: • Standard deviation of portfolio : MBA 2007 Portfolio choice

9. Relationship between expected return and risk • Combining the expressions obtained for : • the expected return • the standard deviation • leads to MBA 2007 Portfolio choice

10. Risk aversion • Risk aversion : • For agiven risk, investor prefers more expected return • For agiven expected return, investor prefers less risk Expected return Indifference curve Risk MBA 2007 Portfolio choice

11. Utility function • Mathematical representation of preferences • a: risk aversion coefficient • u=certainty equivalent risk-free rate • Example: a=2 • A6% 00.06 • B10% 10% 0.08 = 0.10 -2×(0.10)² • C15% 20% 0.07 =0.15 -2×(0.20)² • Bis preferred Utility MBA 2007 Portfolio choice

12. Optimal choice with asingle risky asset • Risk-free asset :RFProportion =1-x • Risky portfolio S:Proportion =x • Utility: • Optimum: • Solution: • Example: a=2 MBA 2007 Portfolio choice

13. Choosing portfolios from many stocks • Porfolio composition : • (X1, X2, ... ,Xi, ... ,XN) • X1 + X2+... +Xi+... +XN=1 • Expected return: • Risk: • Note: • Nterms for variances • N(N-1) terms for covariances • Covariances dominate MBA 2007 Portfolio choice

14. Some intuition MBA 2007 Portfolio choice

15. Example • Consider the risk of an equally weighted portfolio of N"identical« stocks: • Equally weighted: • Variance of portfolio: • If we increase the number of securities ?: • Variance of portfolio: MBA 2007 Portfolio choice

16. Diversification MBA 2007 Portfolio choice

17. The efficient set for many securities • Portfolio choice: choose anefficient portfolio • Efficient portfolios maximiseexpected return for a given risk • They are located on the upperboundary of the shaded region (each point in this regioncorrespond to a given portfolio) Expected Return            Risk MBA 2007 Portfolio choice

18. Optimal portofolio with borrowing and lending M Optimal portfolio: maximize Sharpe ratio MBA 2007 Portfolio choice

19. Efficient markets

20. Notions of Market Efficiency • An Efficient market is one in which: • Arbitrage is disallowed: rules out free lunches • Purchase or sale of a security at the prevailing market price is never a positive NPV transaction. • Prices reveal information • Three forms of Market Efficiency • (a) Weak Form Efficiency • Prices reflect all information in the past record of stock prices • (b) Semi-strong Form Efficiency • Prices reflect all publicly available information • (c) Strong-form Efficiency • Price reflect all information MBA 2007 Portfolio choice

21. Realization Expectation Efficient markets: intuition Price Price change is unexpected Time MBA 2007 Portfolio choice

22. Weak Form Efficiency • Random-walk model: • Pt -Pt-1 = Pt-1 * (Expected return) + Random error • Expected value (Random error) = 0 • Random error of period t unrelated to random component of any past period • Implication: • Expected value (Pt) = Pt-1 * (1 + Expected return) • Technical analysis: useless • Empirical evidence: serial correlation • Correlation coefficient between current return and some past return • Serial correlation = Cor (Rt, Rt-s) MBA 2007 Portfolio choice

23. Random walk - illustration MBA 2007 Portfolio choice

24. Semi-strong Form Efficiency • Prices reflect all publicly available information • Empirical evidence: Event studies • Test whether the release of information influences returns and when this influence takes place. • Abnormal return AR : ARt = Rt - Rmt • Cumulative abnormal return: • CARt = ARt0 + ARt0+1 + ARt0+2 +... + ARt0+1 MBA 2007 Portfolio choice

25. Strong-form Efficiency • How do professional portfolio managers perform? • Jensen 1969: Mutual funds do not generate abnormal returns • Rfund - Rf =  + (RM - Rf) • Insider trading • Insiders do seem to generate abnormal returns • (should cover their information acquisition activities) MBA 2007 Portfolio choice