Risk and expected returns (2)

Transcription

1 Théorie Financière iè Risk and expected returns () Professeur André éfarber

2 Risk and return Objectives for this session: 1. Review: risky assets. any risky assets 3. Beta 4. Optimal portfolio 5. Equilibrium: CAP November 14, 009 Tfin 08 Risk and return ()

3 Review: The efficient set for two assets A eturn Expected r Optimal risky portfolio B Optimal asset allocation 5.00 Riskless rate Risk ( standard deviation) November 14, 009 Tfin 08 Risk and return () 3

4 Formulas Returns: normal distribution Expected return: R ~ N( R, P P P R P = + 1R 1 R ) Variance: P = = ρ P = 1 1 [ ] P = ' Ω November 14, 009 Tfin 08 Risk and return () 4

5 Choosing gportfolios from many stocks Porfolio o o composition o : ( 1,,..., i,..., N ) i N =1 Expected return: rn: Risk: P = R = R + R P j j j i = N i j ij j i i j R N i j ij Note: N terms for variances N(N-1) terms for covariances Covariances dominate November 14, 009 Tfin 08 Risk and return () 5

6 Using matrices 1 =... N R1 R =... R N Ω = N N... NN R P P = ' R = ' Ω November 14, 009 Tfin 08 Risk and return () 6

7 Some intuition Var Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Var Cov Cov Cov Cov Cov Var November 14, 009 Tfin 08 Risk and return () 7

8 Example Consider sde the risk of an equally yweg weighted tedpoto portfolio oof N "identical«stocks: s: R R, =, Cov( R, R ) = cov Equally weighted: Variance of portfolio: j = j i j 1 j = N P 1 = N If we increase the number of securities?: + (1 1 )cov N Variance of portfolio: P cov N November 14, 009 Tfin 08 Risk and return () 8

9 Diversification Risk Reduction of Equally Weighted Portfolios 35.00% 30.00% 00% ation 5.00% Unique risk Portfo olio standard devi 0.00% 15.00% 10.00% arket risk 5.00% 0.00% # stocks in portfolio November 14, 009 Tfin 08 Risk and return () 9

10 Conclusion 1. Diversification pays - adding securities es to the portfolio o o decreases eases risk. This is because securities are not perfectly positively correlated. 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: Total risk of Portfolio risk individual Unsystematic or security diversifiable ifi risk The proper definition of the risk of an individual security in a portfolio is the covariance of the security with the portfolio: November 14, 009 Tfin 08 Risk and return () 10

11 The efficient set for many securities Portfolio oto ochoice: oce:choose oosean efficient ce tpoto portfolioo 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 November 14, 009 Risk ik Tfin 08 Risk and return () 11

12 Optimal portofolio with borrowing and lending Optimal portfolio: maximize Sharpe ratio November 14, 009 Tfin 08 Risk and return () 1

13 Capital asset pricing model (CAP) Sharpe pe( (1964) Lintner (1965) Assumptions Perfect capital markets Homogeneous expectations ain conclusions: Everyone picks the same optimal portfolio ain implications: 1. is the market portfolio : a market value weighted portfolio of all stocks. 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 November 14, 009 Tfin 08 Risk and return () 13

14 arket equilibrium: illustration Wealth Risk free arket Firm 1 Firm Firm 3 asset Portfolio Optimal portfolio 100% 0% 50% 30% Alan Ben Clara arket November 14, 009 Tfin 08 Risk and return () 14

15 Capital Asset Pricing odel Expected return R j = RF + ( R RF ) β j R R j Risk free interest rate β j 1 Beta November 14, 009 Tfin 08 Risk and return () 15

16 Beta Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUELLES

17 easuring the risk of an individual asset The measure easueof risk of an individual dvdua assetin apoto portfolio ohas asto incorporate copoate the impact of diversification. The standard deviation is not an correct measure for the risk of an individual id 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 i i of fbeta is: Cov ( Ri, R ) i β i = = ( R ) November 14, 009 Tfin 08 Risk and return () 17

18 Beta Several interpretations etat of beta are possible: (1) Beta is the responsiveness coefficient of R i to the market () 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 November 14, 009 Tfin 08 Risk and return () 18

19 Beta as a slope 30 0, , , 15 Return on asset , -5-5 Slope = Beta = , , November 14, Return on market Tfin 08 Risk and return () 19

20 A measure of systematic risk : beta Consider the following ow linear model R = α + β R + u t t t R t Realized return on a security during period t α A constant : a return that the stock will realize in any period R t Realized return on the market as a whole during period t β A measure of the response of the return rn on the security to the return rn 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: arket related part ßR t Company specific part α + u t November 14, 009 Tfin 08 Risk and return () 0

21 Beta - illustration Suppose R t = % + 1. R t + u t If R t = 10% The expected return on the security given the return on the market E[R t R t ] = % + 1. x 10% = 14% If R t = 17%, u t = 17%-14% = 3% November 14, 009 Tfin 08 Risk and return () 1

22 easuring Beta Data: past returns for the security and for the market Do linear regression : slope of regression = estimated beta A B C D E F G H I 1 Beta Calculation - monthly data arket A B 3 ean.08% 0.00% 4.55% D3. =AVERAGE(D1:D3) 4 StDev 5.36% 4.33% 10.46% D4. =STDEV(D1:D3) 5 Correl 78.19% 71.54% D5. =CORREL(D1:D3,$B$1:$B$3) 6 R² 61.13% 51.18% D6. =D5^ 7 Beta D7. =SLOPE(D1:D3,$B$1:$B$3) 8 Intercept 0-1.3% 1.64% D8. =INTERCEPT(D1:D3,$B$1:$B$3) Data Date Rm RA RB % 0.81% 0.43% % -4.46% -7.03% % -1.85% % % -1.94% 6.91% % 3.49% 4.65% % 3.44% 7.64% % -4.7% 8.41% % -.70% -1.5% % -4.9% % % 3.75% 13.18% % 971% 9.71% 19.% % -1.67% 3.77% November 14, 009 Tfin 08 Risk and return ()

23 Decomposing of the variance of a portfolio p g p How much does each asset contribute to the risk of a portfolio? p The variance of the portfolio with risky assets can be written as B B AB B A A A P + + = B B AB A B AB B A A A B B AB B A AB B A A A P = = ) ( ) ( ) ( ) ( The variance of the portfolio is the weighted average of the covariances of hi di id l ih h f li BP B AP A B B AB A B AB B A A A + = ) ( ) ( each individual asset with the portfolio. November 14, 009 Tfin 08 Risk and return () 3

24 Example Exp.Return Sigma Variance Riskless rate A B Correlation 0 Prop. Variance-covariancecovariance A B Cov(Ri,Rp) Variance St.dev Exp.Ret. R November 14, 009 Tfin 08 Risk and return () 4

25 Beta and the decomposition of the variance The evariance a ceof the market etpoto portfolio ocan beexpressed pessedas: = i i n n To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio 1 i n = 1 i n or 1 β 1 + β i β i n β n = 1 November 14, 009 Tfin 08 Risk and return () 5

26 arginal contribution to risk: some math Consider sde portfolio poto o.. What happens if the fraction invested in stock I changes? Consider a fraction invested in stock i P = 1 ) + (1 ) i + Take first derivative with respect to for = 0 d P = ( d = 0 Risk of portfolio increase if and only if: ( i i i The marginal contribution i of stock i to the risk is i > ) November 14, 009 Tfin 08 Risk and return () 6

27 arginal contribution to risk: illustration tfolio Risk of port Fraction in B Cor = 0 Cor = 0.5 Cor = 0.50 Cor = 0.75 Cor = 1.0 November 14, 009 Tfin 08 Risk and return () 7

28 Beta and marginal contribution to risk Increase (sightly) the weight of i: The risk of the portfolio increases if: i > β = > 1 i i The risk of the portfolio is unchanged if: The risk of the portfolio decreases if: i i = < β i β i = = i i < 1 = 1 November 14, 009 Tfin 08 Risk and return () 8

29 Inside beta Remember e the relationship between the correlation o coefficient c e and the covariance: i ρ i = i Beta can be written as: β i = = Two determinants of beta the correlation of the security return with the market i the volatility of the security relative to the volatility of the market ρ i i November 14, 009 Tfin 08 Risk and return () 9

30 Properties of beta Two importants ts properties es of beta to remember e (1) The weighted average beta across all securities is 1 1 β1 + β iβi nβn = 1 () The beta of a portfolio is the weighted average beta of the securities β P = 1 P β 1 + P β ip β i np β n November 14, 009 Tfin 08 Risk and return () 30

31 Risk premium and beta 3. The expected return on a security is positively related to its beta Capital-Asset Pricing odel (CAP) : R = R + R R ) The expected return rn on a security equals: F ( F the risk-free rate plus β the excess market return (the market risk premium) times Beta of the security November 14, 009 Tfin 08 Risk and return () 31

32 CAP - Illustration Expected Return R R F 1 Beta November 14, 009 Tfin 08 Risk and return () 3

33 CAP - Example Assume: Risk-free rate = 6% arket risk premium = 8.5% Beta Expected Return (%) American Express BankAmerica Chrysler Digital Equipement Walt Disney Du Pont AT&T General ills Gillette Southern California Edison Gold Bullion November 14, 009 Tfin 08 Risk and return () 33

34 CAP two formulations Consider a future uncertain cash flow C to be received in 1 year. PV calculation based on CAP: V = 1+ r f ~ E(C ) cov( r, r ) β = + ( r rf )β C% V E ( C% ) Here: r = V = V cov( C%, r ) 1 + rf + ( r rf ) V Define : λ = V = See Brealey and yers Chap 9 r r f ~ cov( (CC, r ) ~ V ( 1+ rf + λ ) = E( C) V ~ ~ E ( C ) λ cov( C, r ) Certainty equivalent = 1+ r 1+ r f f November 14, 009 Tfin 08 Risk and return () 34

35 Risk-adjusted expected cash flow Using risk-adjusted discount rates is OK if you know beta. The adjusted risk-adjusted discount rate does not work for OPTIONS or projects with unknown betas. To understand how to proceed in that case, we need to go deeper into valuation theory. November 14, 009 Tfin 08 Risk and return () 35

36 Example (see Introduction) You observe the following data: Value Up market (u) Down market (d) Proba = Proba = Expected return Bond % arket Portfolio % What is the value of the following asset? What are its expected returns? NewAsset? ? November 14, 009 Tfin 08 Risk and return () 36

37 Valuation of project with CAP E( C) λ cov( C, r V = ) Certainty equivalent = 1+ r 1+ f r f Step 1: calculate statistics for the market portfolio: Up mkt Proba =.75 Down mkt Proba =.5 Return 0% -0% Expected return: r = (0.75)(0%) + (0.5)( 0%) = 10% arket risk premium: r r = 10% 5% = 5% Variance: = (0.75)(.0)² + (0.5)(.0)² (.10)² = Price of covariance: r rf.05 λ = = = F November 14, 009 Tfin 08 Risk and return () 37

38 Valuation of project with CAP () Step : Calculate statistics for the project Expected cash flow: EC ( %) = = 175 Covariance with market portfolio: cov( Cr %, % ) = (0.75)(00.) + (0.5)(100 (.)) (175)(0.10) = 7.5 (Reminder: cov( x, y) = Exy ( ) ExEy ( ) ( ) ) Step 3: Value the project V EC ( %) λ cov( Cr %,% ) 175 (1.67)(7.5) = = 1 + r F = = November 14, 009 Tfin 08 Risk and return () 38

39 Valuation of project with CAP (3) Once the value of the project is known, the beta can be calculated. Value Up mkt Proba Down mkt =.75 Proba =.5 Cash flow Returns 9.3% 3% % 38% Expected return: r = (0.75)(.93) + (0.5)( ) = 13.08% Beta: r r F 13.08% 5% β = = = 1.61 r r 10% 5% F November 14, 009 Tfin 08 Risk and return () 39