Corporate Finance

1. Corporate Finance Portfolio Theory Prof. André FarberSOLVAY BUSINESS SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

2. Portfolio selection • Objectives for this session • 1. Gain a better understanding of the rational for benefit ofdiversification • 2. Identify measures of systematic risk : covariance and beta • 3. Analyse the choice of an optimal portfolio A.Farber Vietnam 2004

3. Combining the Riskless Asset and asingle Risky Asset • Consider the following portfolio P: • Fraction invested • in the riskless asset 1-x(40%) • in the risky asset x(60%) • Expected return on portfolio P: • Standard deviation of portfolio : A.Farber Vietnam 2004

4. Relationship between expected return and risk • Combining the expressions obtained for : • the expected return • the standard deviation • leads to A.Farber Vietnam 2004

5. Risk aversion • Risk aversion : • For agiven risk, investor prefers more expected return • For agiven expected return, investor prefers less risk Expected return Indifference curve Risk A.Farber Vietnam 2004

6. Utility function • Mathematical representation of preferences • a: risk aversion coefficient • u=certainty equivalent risk-free rate • Example: a=2 • A6% 00.06 • B10% 10% 0.08 = 0.10 -2×(0.10)² • C15% 20% 0.07 =0.15 -2×(0.20)² • Bis preferred Utility A.Farber Vietnam 2004

7. Optimal choice with asingle risky asset • Risk-free asset :RFProportion =1-x • Risky portfolio S:Proportion =x • Utility: • Optimum: • Solution: • Example: a=2 A.Farber Vietnam 2004

8. Diversification A.Farber Vietnam 2004

9. A measure of systematic risk : beta • Consider the following linear model • RtRealized return on asecurity during period t • Aconstant :areturn that the stock will realize in any period • RMtRealized return on the market as awhole during period t • Ameasure of the response of the return on the security to thereturn on the market • utAreturn specific to the security for period t(idosyncratic returnor unsystematic return)- arandom variable with mean 0 • Partition of yearly return into: • Market related part ßRMt • Company specific part a+ut A.Farber Vietnam 2004

10. Beta - illustration • Example: • 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% A.Farber Vietnam 2004

11. Covariance and correlation • Statistical measures of the degree to which random variables movetogether • 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 –1and +1 A.Farber Vietnam 2004

12. Risk and expected returns for porfolios • In order to better understand the driving force explaining the benefitsfrom diversification, let us consider aportfolio of two stocks (A,B) • Characteristics: • Expected returns : • Standard deviations : • Covariance : • Portfolio: defined by fractions invested in each stockXA ,XBXA+ XB= 1 • Expected return on portfolio: • Variance of the portfolio's return: A.Farber Vietnam 2004

13. 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 x0.30 + 0.4 x 0.20 = 26% A.Farber Vietnam 2004

14. Diversification effect • Let us vary the correlation coefficient • Correlationcoefficient Expected return Standard deviation • -1 18 10.00 • -0.5 18 15.62 • 0 18 19.7 • 0.5 18 23.07 • 1 18 26.00 • Conclusion: • As long as the correlation coefficient is less than one, the standarddeviation of a portfolio of two securities is less than the weightedaverage of the standard deviations of the individual securities A.Farber Vietnam 2004

15. The efficient set for two assets: correlation = +1 A.Farber Vietnam 2004

16. The efficient set for two assets: correlation = -1 A.Farber Vietnam 2004

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

18. Marginal contribution to risk: some math • Consider portfolio M. What happens if the fraction invested in stock Ichanges? • Consider a fraction Xinvested in stock i • Take first derivative with respect to X: • Value the derivative for X=0 A.Farber Vietnam 2004

19. Marginal contribution to risk: conclusion • Risk of portfolio increase if and only if: • The marginal contribution of stock i to the risk is A.Farber Vietnam 2004

20. Choosing portfolios from many stocks • Porfolio composition : • (X1, X2, ... ,Xi, ... ,XN) • X1 + X2+... +Xi+... +XN=1 • Expected return: • Risk: • Note: • Nterms for variances • N(N-1) terms for covariances • Covariances dominate A.Farber Vietnam 2004

21. 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: A.Farber Vietnam 2004

22. Conclusion • 1. Diversification pays - adding securities to the portfolio decreases risk. This is because securities are not perfectly positively correlated • 2. There is a limit to the benefit of diversification : the risk of the portfolio can't be less than the average covariance (cov) between the stocks • The variance of a security's return can be broken down in the following way: • The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio: Portfolio risk Total risk of individual security Unsystematic or diversifiable risk A.Farber Vietnam 2004