Thorie Financire 2004 2005 Risk and expected returns

Théorie Financière 2004 -2005 Risk and expected returns (2) Professeur André Farber Tfin 2004 08 Risk and return (2)

Risk and return • • • Objectives for this session: 1. Efficient set 2. Beta 3. Optimal portfolio 4. CAPM August 23, 2004 Tfin 2004 08 Risk and return (2) 2

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 August 23, 2004 Tfin 2004 08 Risk and return (2) 3

Choosing between 2 risky assets • Choose the asset with the highest ratio of excess expected return to risk: A Expected return B • • Example: RF = 6% Exp. Return A 9% B 15% • • • Asset Sharpe ratio A (9 -6)/10 = 0. 30 B (15 -6)/20 = 0. 45 ** August 23, 2004 Risk 10% 20% A Risk Tfin 2004 08 Risk and return (2) 4

Optimal portofolio with borrowing and lending M Optimal portfolio: maximize Sharpe ratio August 23, 2004 Tfin 2004 08 Risk and return (2) 5

Capital asset pricing model (CAPM) • • • Sharpe (1964) Lintner (1965) Assumptions • Perfect capital markets • Homogeneous expectations Main conclusions: Everyone picks the same optimal portfolio Main implications: – 1. M is the market portfolio : a market value weighted portfolio of all stocks – 2. The risk of a security is the beta of the security: Beta measures the sensitivity of the return of an individual security to the return of the market portfolio The average beta across all securities, weighted by the proportion of each security's market value to that of the market is 1 August 23, 2004 Tfin 2004 08 Risk and return (2) 6

Beta Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES Tfin 2004 08 Risk and return (2)

Measuring the risk of an individual asset • The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification. • The standard deviation is not an correct measure for the risk of an individual security in a portfolio. • The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification. • Remember: the optimal portfolio is the market portfolio. • The risk of an individual asset is measured by beta. • The definition of beta is: August 23, 2004 Tfin 2004 08 Risk and return (2) 8

Beta • Several interpretations of beta are possible: • (1) Beta is the responsiveness coefficient of Ri to the market • (2) Beta is the relative contribution of stock i to the variance of the market portfolio • (3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified August 23, 2004 Tfin 2004 08 Risk and return (2) 9

Beta as a slope August 23, 2004 Tfin 2004 08 Risk and return (2) 10

A measure of systematic risk : beta • Consider the following linear model • • Rt Realized return on a security during period t A constant : a return that the stock will realize in any period RMt Realized return on the market as a whole during period t A measure of the response of the return on the security to the return on the market • ut A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0 • Partition of yearly return into: – Market related part – Company specific part August 23, 2004 ß RMt + ut Tfin 2004 08 Risk and return (2) 11

Beta - illustration • Suppose Rt = 2% + 1. 2 RMt + ut • If RMt = 10% • The expected return on the security given the return on the market • E[Rt |RMt] = 2% + 1. 2 x 10% = 14% • If Rt = 17%, ut = 17%-14% = 3% August 23, 2004 Tfin 2004 08 Risk and return (2) 12

Measuring Beta • Data: past returns for the security and for the market • Do linear regression : slope of regression = estimated beta August 23, 2004 Tfin 2004 08 Risk and return (2) 13

Decomposing of the variance of a portfolio • How much does each asset contribute to the risk of a portfolio? • The variance of the portfolio with 2 risky assets • can be written as • The variance of the portfolio is the weighted average of the covariances of each individual asset with the portfolio. August 23, 2004 Tfin 2004 08 Risk and return (2) 14

Example August 23, 2004 Tfin 2004 08 Risk and return (2) 15

Beta and the decomposition of the variance • The variance of the market portfolio can be expressed as: • To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio August 23, 2004 Tfin 2004 08 Risk and return (2) 16

Marginal contribution to risk: some math • Consider portfolio M. What happens if the fraction invested in stock I changes? • Consider a fraction X invested in stock i • Take first derivative with respect to X for X = 0 • Risk of portfolio increase if and only if: • The marginal contribution of stock i to the risk is August 23, 2004 Tfin 2004 08 Risk and return (2) 17

Marginal contribution to risk: illustration August 23, 2004 Tfin 2004 08 Risk and return (2) 18

Beta and marginal contribution to risk • Increase (sightly) the weight of i: • The risk of the portfolio increases if: • The risk of the portfolio is unchanged if: • The risk of the portfolio decreases if: August 23, 2004 Tfin 2004 08 Risk and return (2) 19

Inside beta • Remember the relationship between the correlation coefficient and the covariance: • Beta can be written as: • Two determinants of beta – the correlation of the security return with the market – the volatility of the security relative to the volatility of the market August 23, 2004 Tfin 2004 08 Risk and return (2) 20

Properties of beta • Two importants properties of beta to remember • (1) The weighted average beta across all securities is 1 • (2) The beta of a portfolio is the weighted average beta of the securities August 23, 2004 Tfin 2004 08 Risk and return (2) 21

Risk premium and beta • 3. The expected return on a security is positively related to its beta • Capital-Asset Pricing Model (CAPM) : • The expected return on a security equals: the risk-free rate plus the excess market return (the market risk premium) times Beta of the security August 23, 2004 Tfin 2004 08 Risk and return (2) 22

CAPM - Illustration Expected Return 1 August 23, 2004 Tfin 2004 08 Risk and return (2) Beta 23

CAPM - Example • • • • Assume: Risk-free rate = 6% Beta American Express 1. 5 Bank. America 1. 4 Chrysler 1. 4 Digital Equipement 1. 1 Walt Disney Du Pont 1. 0 AT&T 0. 76 General Mills 0. 5 Gillette 0. 6 Southern California Edison 0. 5 Gold Bullion -0. 07 August 23, 2004 Market risk premium = 8. 5% Expected Return (%) 18. 75 17. 9 15. 35 0. 9 13. 65 14. 5 12. 46 10. 25 11. 1 10. 25 5. 40 Tfin 2004 08 Risk and return (2) 24

Pratical implications • Efficient market hypothesis + CAPM: passive investment • Buy index fund • Choose asset allocation August 23, 2004 Tfin 2004 08 Risk and return (2) 25

Théorie Financière 2004 -2005 Arbitrage Pricing Model Professeur André Farber Tfin 2004 08 Risk and return (2)

Market Model • Consider one factor model for stock returns: • Rj = realized return on stock j • = expected return on stock j • F = factor – a random variable E(F) = 0 • εj = unexpected return on stock j – a random variable • E(εj) = 0 Mean 0 • cov(εj , F) = 0 Uncorrelated with common factor • cov(εj , εk) = 0 Not correlated with other stocks August 23, 2004 Tfin 2004 08 Risk and return (2) 27

Diversification • Suppose there exist many stocks with the same βj. • Build a diversified portfolio of such stocks. • The only remaining source of risk is the common factor. August 23, 2004 Tfin 2004 08 Risk and return (2) 28

Created riskless portfolio • Combine two diversified portfolio i and j. • Weights: xi and xj with xi+xj =1 • Return: • Eliminate the impact of common factor riskless portfolio • Solution: August 23, 2004 Tfin 2004 08 Risk and return (2) 29

Equilibrium • No arbitrage condition: • The expected return on a riskless portfolio is equal to the risk-free rate. At equilibrium: August 23, 2004 Tfin 2004 08 Risk and return (2) 30

Risk – expected return relation Linear relation between expected return and beta For market portfolio, β = 1 Back to CAPM formula: August 23, 2004 Tfin 2004 08 Risk and return (2) 31