FINANCE 9 Optimal Portfolio Choice Professor Andr Farber

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

Introduction: random portfolios 06 June 2021 MBA 2007 Portfolio choice 2

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 06 June 2021 MBA 2007 Portfolio choice 3

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 XA , XB XA + X B = 1 • Expected return on portfolio: • Variance of the portfolio's return: 06 June 2021 MBA 2007 Portfolio choice 4

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 x 0. 30 + 0. 4 x 0. 20 = 26% 06 June 2021 MBA 2007 Portfolio choice 5

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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%) Riskless asset Risky asset Expected return 6% 12% Standard deviation 0% 20% • Expected return on portfolio P: • Standard deviation of portfolio : 06 June 2021 MBA 2007 Portfolio choice 8

Relationship between expected return and risk • Combining the expressions obtained for : • the expected return • the standard deviation • leads to 06 June 2021 MBA 2007 Portfolio choice 9

Risk aversion • Risk aversion : • For a given risk, investor prefers more expected return • For a given expected return, investor prefers less risk Expected return Indifference curve Risk 06 June 2021 MBA 2007 Portfolio choice 10

Utility function • Mathematical representation of preferences • a: risk aversion coefficient • u = certainty equivalent risk-free rate • Example: a = 2 • • A 6% 0 B 10% C 15% 20% B is preferred 06 June 2021 Utility 0. 06 0. 08 = 0. 10 - 2×(0. 10)² 0. 07 = 0. 15 - 2×(0. 20)² MBA 2007 Portfolio choice 11

Optimal choice with a single risky asset • Risk-free asset : RF • Risky portfolio S: Proportion = 1 -x Proportion = x • Utility: • Optimum: • Solution: • Example: a = 2 06 June 2021 MBA 2007 Portfolio choice 12

Choosing portfolios from many stocks • Porfolio composition : • (X 1, X 2, . . . , Xi, . . . , XN) • X 1 + X 2 +. . . + Xi +. . . + XN = 1 • Expected return: • Risk: • Note: • N terms for variances • N(N-1) terms for covariances • Covariances dominate 06 June 2021 MBA 2007 Portfolio choice 13

Some intuition 06 June 2021 MBA 2007 Portfolio choice 14

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: 06 June 2021 MBA 2007 Portfolio choice 15

Diversification 06 June 2021 MBA 2007 Portfolio choice 16

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) Expected Return Risk 06 June 2021 MBA 2007 Portfolio choice 17

Optimal portofolio with borrowing and lending M Optimal portfolio: maximize Sharpe ratio 06 June 2021 MBA 2007 Portfolio choice 18

Efficient markets MBA 2007 Portfolio choice

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 06 June 2021 MBA 2007 Portfolio choice 20

Efficient markets: intuition Price Realization Expectation Price change is unexpected Time 06 June 2021 MBA 2007 Portfolio choice 21

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) 06 June 2021 MBA 2007 Portfolio choice 22

Random walk - illustration 06 June 2021 MBA 2007 Portfolio choice 23

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 = ARt 0 + ARt 0+1 + ARt 0+2 +. . . + ARt 0+1 06 June 2021 MBA 2007 Portfolio choice 24

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) 06 June 2021 MBA 2007 Portfolio choice 25