FINANCE 9. Optimal Portfolio Choice Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007

June 13, 2015 MBA 2007 Portfolio choice |2 Introduction: random portfolios

June 13, 2015 MBA 2007 Portfolio choice |3 Covariance and correlation Statistical measures of the degree to which random variables move together 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 –1 and + 1

June 13, 2015 MBA 2007 Portfolio choice |4 Risk and expected returns for porfolios In order to better understand the driving force explaining the benefits from diversification, let us consider a portfolio of two stocks (A,B) Characteristics: –Expected returns : –Standard deviations : –Covariance : Portfolio: defined by fractions invested in each stock X A, X B X A + X B = 1 Expected return on portfolio: Variance of the portfolio's return:

June 13, 2015 MBA 2007 Portfolio choice |5 Example Invest $ 100 m in two stocks: A $ 60 m X A = 0.6 B $ 40 m X B = 0.4 Characteristics (% per year) A B Expected return 20% 15% Standard deviation 30% 20% Correlation 0.5 Expected return = 0.6 × 20% × 15% = 18% Variance = (0.6)²(.30)² + (0.4)²(.20)²+2(0.6)(0.4)(0.30)(0.20)(0.5)  ²p =  Standard deviation = % Less than the average of individual standard deviations: 0.6 x x 0.20 = 26%

June 13, 2015 MBA 2007 Portfolio choice |6

June 13, 2015 MBA 2007 Portfolio choice |7

June 13, 2015 MBA 2007 Portfolio choice |8 Combining the Riskless Asset and a single 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 : Riskless asset Risky asset Expected return 6%12% Standard deviation 0%20%

June 13, 2015 MBA 2007 Portfolio choice |9 Relationship between expected return and risk Combining the expressions obtained for : the expected return the standard deviation leads to

June 13, 2015 MBA 2007 Portfolio choice |10 Risk aversion Risk aversion : For a given risk, investor prefers more expected return For a given expected return, investor prefers less risk Expected return Risk Indifference curve

June 13, 2015 MBA 2007 Portfolio choice |11 Utility function Mathematical representation of preferences a: risk aversion coefficient u = certainty equivalent risk-free rate Example: a = 2 A 6% B 10% 10% 0.08 = ×(0.10)² C 15% 20% 0.07 = ×(0.20)² B is preferred Utility

June 13, 2015 MBA 2007 Portfolio choice |12 Optimal choice with a single risky asset Risk-free asset : R F Proportion = 1-x Risky portfolio S: Proportion = x Utility: Optimum: Solution: Example: a = 2

June 13, 2015 MBA 2007 Portfolio choice |13 Choosing portfolios from many stocks Porfolio composition : (X 1, X 2,..., X i,..., X N ) X 1 + X X i X N = 1 Expected return: Risk: Note: N terms for variances N(N-1) terms for covariances Covariances dominate

June 13, 2015 MBA 2007 Portfolio choice |14 Some intuition

June 13, 2015 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:

June 13, 2015 MBA 2007 Portfolio choice |16 Diversification

June 13, 2015 MBA 2007 Portfolio choice |17 The efficient set for many securities Portfolio choice: choose an efficient portfolio Efficient portfolios maximise expected return for a given risk They are located on the upper boundary of the shaded region (each point in this region correspond to a given portfolio) Risk Expected Return

June 13, 2015 MBA 2007 Portfolio choice |18 Optimal portofolio with borrowing and lending Optimal portfolio: maximize Sharpe ratio M

Efficient markets

June 13, 2015 MBA 2007 Portfolio choice |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

June 13, 2015 MBA 2007 Portfolio choice |21 Efficient markets: intuition Expectation Time Price Realization Price change is unexpected

June 13, 2015 MBA 2007 Portfolio choice |22 Weak Form Efficiency Random-walk model: –P t -P t-1 = P t-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 (P t ) = P t-1 * (1 + Expected return) –Technical analysis: useless Empirical evidence: serial correlation –Correlation coefficient between current return and some past return –Serial correlation = Cor (R t, R t-s )

June 13, 2015 MBA 2007 Portfolio choice |23 Random walk - illustration

June 13, 2015 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: CAR t = AR t0 + AR t0+1 + AR t AR t0+1

June 13, 2015 MBA 2007 Portfolio choice |25 Strong-form Efficiency How do professional portfolio managers perform? Jensen 1969: Mutual funds do not generate abnormal returns R fund - R f =  +  (R M - R f ) Insider trading Insiders do seem to generate abnormal returns (should cover their information acquisition activities)