FINANCE 10. Risk and expected returns Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2006

MBA 2006 Risk and return (2) |2 Measuring the risk of an individual asset La mesure du risque dun titre dans un portefeuille doit tenir compte de limpact de la diversification. Lécart type nest donc pas la bonne mesure. Le risque se mesure par la contribution du titre au risque du portefeuille. Remember: the optimal portfolio is the market portfolio. The risk of an individual asset is measured by beta. The definition of beta is:

MBA 2006 Risk and return (2) |3 Beta Plusieurs interprétations du beta: (1) Beta mesure la sensibilité de R i par rapport au marché (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

MBA 2006 Risk and return (2) |4 Beta as a slope

MBA 2006 Risk and return (2) |5 A measure of systematic risk : beta Consider the following linear model R t Realized return on a security during period t A constant : a return that the stock will realize in any period R Mt 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 u t 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 ß R Mt –Company specific part + u t

MBA 2006 Risk and return (2) |6 Measuring Beta Data: past returns for the security and for the market Do linear regression : slope of regression = estimated beta

MBA 2006 Risk and return (2) |7 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

MBA 2006 Risk and return (2) |8 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

MBA 2006 Risk and return (2) |9 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

MBA 2006 Risk and return (2) |10 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

MBA 2006 Risk and return (2) |11 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

MBA 2006 Risk and return (2) |12 CAPM - Illustration Expected Return Beta 1

MBA 2006 Risk and return (2) |13 CAPM - Example Assume: Risk-free rate = 6% Market risk premium = 8.5% Beta Expected Return (%) American Express BankAmerica Chrysler Digital Equipement Walt Disney Du Pont AT&T General Mills Gillette Southern California Edison Gold Bullion

MBA 2006 Risk and return (2) |14 Pratical implications Efficient market hypothesis + CAPM: passive investment Buy index fund Choose asset allocation

Arbitrage Pricing Model Professeur André Farber

MBA 2006 Risk and return (2) |16 Market Model Consider one factor model for stock returns: R j = realized return on stock j = expected return on stock j F = factor – a random variable E(F) = 0 ε j = unexpected return on stock j – a random variable E(ε j ) = 0 Mean 0 cov(ε j,F) = 0 Uncorrelated with common factor cov(ε j,ε k ) = 0 Not correlated with other stocks

MBA 2006 Risk and return (2) |17 Diversification Suppose there exist many stocks with the same β j. Build a diversified portfolio of such stocks. The only remaining source of risk is the common factor.

MBA 2006 Risk and return (2) |18 Created riskless portfolio Combine two diversified portfolio i and j. Weights: x i and x j with x i +x j =1 Return: Eliminate the impact of common factor riskless portfolio Solution:

MBA 2006 Risk and return (2) |19 Equilibrium No arbitrage condition: The expected return on a riskless portfolio is equal to the risk-free rate. At equilibrium:

MBA 2006 Risk and return (2) |20 Risk – expected return relation Linear relation between expected return and beta For market portfolio, β = 1 Back to CAPM formula: