Théorie Financière Risk and expected returns (2) Professeur André Farber

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

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

August 23, 2004 Tfin Risk and return (2) |4 Choosing between 2 risky assets Choose the asset with the highest ratio of excess expected return to risk: Example: R F = 6% Exp.Return Risk A 9% 10% B 15% 20% Asset Sharpe ratio A (9-6)/10 = 0.30 B (15-6)/20 = 0.45 ** A B A Risk Expected return

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

August 23, 2004 Tfin Risk and return (2) |6 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

Beta Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

August 23, 2004 Tfin Risk and return (2) |8 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 Risk and return (2) |9 Beta Several interpretations of beta are possible: (1) Beta is the responsiveness coefficient of R i 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 Risk and return (2) |10 Beta as a slope

August 23, 2004 Tfin Risk and return (2) |11 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

August 23, 2004 Tfin Risk and return (2) |12 Beta - illustration Suppose R t = 2% R Mt + u t If R Mt = 10% The expected return on the security given the return on the market E[R t |R Mt ] = 2% x 10% = 14% If R t = 17%, u t = 17%-14% = 3%

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

August 23, 2004 Tfin Risk and return (2) |14 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 Risk and return (2) |15 Example

August 23, 2004 Tfin Risk and return (2) |16 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 Risk and return (2) |17 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 Risk and return (2) |18 Marginal contribution to risk: illustration

August 23, 2004 Tfin Risk and return (2) |19 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 Risk and return (2) |20 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 Risk and return (2) |21 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 Risk and return (2) |22 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 Risk and return (2) |23 CAPM - Illustration Expected Return Beta 1

August 23, 2004 Tfin Risk and return (2) |24 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

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

Théorie Financière Arbitrage Pricing Model Professeur André Farber

August 23, 2004 Tfin Risk and return (2) |27 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

August 23, 2004 Tfin Risk and return (2) |28 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.

August 23, 2004 Tfin Risk and return (2) |29 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:

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

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