Corporate Finance Risk and Return Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

A.Farber Vietnam 2004 |2|2 Risk and return Objectives for this session: 1. Review 2. Efficient set 3. Optimal portfolio 4. CAPM

A.Farber Vietnam 2004 |3|3 Review : 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:

A.Farber Vietnam 2004 |4|4 The efficient set for two assets: correlation = 0

A.Farber Vietnam 2004 |5|5 Example

A.Farber Vietnam 2004 |6|6 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

A.Farber Vietnam 2004 |7|7 Marginal contribution to risk: illustration

A.Farber Vietnam 2004 |8|8 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

A.Farber Vietnam 2004 |9|9 Some intuition

A.Farber Vietnam 2004 | 10 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

A.Farber Vietnam 2004 | 11 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

A.Farber Vietnam 2004 | 12 Optimal portofolio with borrowing and lending Optimal portfolio: maximize Sharpe ratio

A.Farber Vietnam 2004 | 13 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

A.Farber Vietnam 2004 | 14 Optimal portfolio: property Slope = M x j Slope = RFRF

A.Farber Vietnam 2004 | 15 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

A.Farber Vietnam 2004 | 16 CAPM - Illustration Expected Return Beta 1

A.Farber Vietnam 2004 | 17 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

A.Farber Vietnam 2004 | 18 Pratical implications Efficient market hypothesis + CAPM: passive investment Buy index fund Choose asset allocation