Portfolio Problem and The Market Model

Transcription

1 Portfolio Problem and The Market Model 1

2 Building Portfolios Let s assume we are considering 3 investment opportunities 1. IBM stocks 2. ALCOA stocks 3. Treasury Bonds (T-bill) How should we start thinking about this problem? 2

3 Building Portfolios Let s first learn about the characteristics of each option by assuming the following models: IBM N(µ I, σi 2) ALCOA N(µ A, σa 2 ) and The return on the T-bill is 3% After observing some return data we can came up with estimates for the means and variances describing the behavior of these stocks 3

4 Building Portfolios The data tells us that... IBM ALCOA T-bill ˆµ I = 12.5 ˆµ A = 14.9 µ Tbill = 3 ˆσ I = 10.5 ˆσ A = 14.0 σ Tbill = 0 corr(ibm, ALCOA) = 0.33 Now what?? What should I do? 4

5 Building Portfolios How about combining these options? Is that a good idea? Is it good to have all your eggs in the same basket? Why? What if I place half of my money in ALCOA and the other half on T-bills... Remember that: E(aX + by ) = ae(x ) + be(y ) Var(aX + by ) = a 2 Var(X ) + b 2 Var(Y ) + 2ab Cov(X, Y ) 5

6 Building Portfolios So, by using what we know about the means and variances we get to: ˆµ P = 0.5ˆµ A + 0.5µ Tbill ˆσ 2 P = 0.5 2ˆσ 2 A ˆµ P and ˆσ P 2 portfolio refer to the estimated mean and variance of our What are we assuming here? 6

7 Building Portfolios What happens if we change the proportions... ALCOA Average Return T-bill Standard Deviation 7

8 Building Portfolios What about investing in IBM and ALCOA? ALCOA Average Return IBM Standard Deviation How much more complicated this gets if I am choosing between 100 stocks? 8

9 The Market Model Empirical version of the CAPM Frequently used application of the regression model Provides us with a simple way to estimate covariances between all stocks Is often used as a benchmark to gauge performance of portfolio managers (Windsor fund example in Section 1) 9

10 The Market Model The model takes the following form: R i = α + βr M + ɛ with ɛ N(0, σ 2 i ) where R i is the return on stock i, and R M is the return on the market index That implies: E(R i ) = α + βe(r M ) Var(R i ) = βi 2Var(R M) + σi 2 Cov(R i, R j ) = β i β j Var(R M ) What about 100 stocks? All we need to do is to run 100 regressions and choose our portfolio!! 10

11 The Market Model - Portfolio Implications Let R p be the return on a portfolio with N securities so that R p = w i R i i=1 where w i is the weight assigned to security i. That implies: E(R p ) = = w i E(R i ) i=1 w i α i + i=1 i=1 w i β i E(R M ) 11

12 The Market Model - Portfolio Implications...and Var(R p ) = = = = i=1 i i=1 ( N i=1 w 2 i Var(R i ) + w 2 i β 2 i Var(R M ) + i i=1 w i w j Cov(R i, R j ) j,i j w i w j β i β j Var(R M ) + j w i β i ) = β 2 pvar(r M ) + w 2 i σ 2 i + i=1 i w 2 i σ 2 i w j β j Var(R M ) + j=1 i=1 w 2 i σ 2 i w i w j β i β j Var(R M ) j,i j i=1 w 2 i σ 2 i 12

13 The Market Model - Portfolio Implications Now, assume that w i = 1 N, i.e, we place equal amount of money in each security... Var(R p ) = β 2 pvar(r M ) + = β 2 pvar(r M ) + i=1 i=1 = β 2 pvar(r M ) + 1 N w 2 i σ 2 i 2 1 σi 2 N i=1 = β 2 pvar(r M ) + 1 N σ2 1 N σ2 i 13

14 The Market Model - Portfolio Implications as N grows, i.e, we add more and more securities to our portfolio, 1 N σ2 0 and therefore, Var(R p ) = β 2 pvar(r M ) The red term is what we call the non-diversifiable risk! The blue term is diversifiable, ie, we can make it go away by spreading our wealth into many stocks. The Market Model (CAPM) implies that there exists only ONE SOURCE of risk in the market! 14

15 beta # of stocks 15

16 Testing the CAPM Let s try to test weather or not the CAPM works in reality... in order to so we will focus on the returns of 25 portfolios constructed by sorting firms by their market cap and book-to-price ratio. If the CAPM is true, all the α s have to be zero and we should find no other systematic source of common variation in these portfolios... How can we evaluate these implications? Let s first run a market model regression for each portfolio... that gives us estimates of α s and β s for each security. 16

17 Testing the CAPM This plot shows the estimated α along with its 95% confidence interval for each regression... Does it look good? alpha Portfolios 17

18 Testing the CAPM Also, if the CAPM is correct, E(R i ) = β i E(R M ), right? We can look at this by plotting, for each portfolio, the R i vs the estimated ˆβ i RM CAPM Rbar Beta*Rm What should the red line be? 18

19 Testing the CAPM Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) ** Fit Signif. codes: 0 *** ** 0.01 * Residual standard error: on 23 degrees of freedom Multiple R-squared: ,Adjusted R-squared: F-statistic: on 1 and 23 DF, p-value: > confint(model0) 2.5 % 97.5 % (Intercept) Fit

20 Testing the CAPM Maybe there s is some systematic variation that we are ignoring in the returns... something else might be responsible for the variability in these portfolios... Size (B/M fixed) Rbar Beta*Rm 20

21 Testing the CAPM B/M (Size Fixed) Rbar Beta*Rm 21

22 Testing the CAPM It looks like there s a systematic association between the size of a firm and its average returns... also true for book-to-price... These are known as the size effect and value effect... small firms tend to earn returns too high for their β s and the same is true for firms with high book-to-price ratio! 22

23 Fama-French 3 Factors Fama and French proposed an extension to the CAPM to incorporate these effects... they basically added 2 factors that capture the common source of variation related to size and book-to-price... We now have the following regressions: R i = α i + β i R M + β SMB i SMB + βi HML HML + ɛ You can think one this model as a generalization to the market model that implies 3 different sources of risk... all else can be diversified away! 23

24 Fama-French 3 Factors The results... alpha Portfolios 24

25 Fama-French 3 Factors Fama-French Rbar Much nicer, right?! 25 Fit

26 Fama-French 3 Factors Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) Fit e-06 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 23 degrees of freedom Multiple R-squared: 0.608,Adjusted R-squared: F-statistic: on 1 and 23 DF, p-value: 4.333e-06 > confint(model1) 2.5 % 97.5 % (Intercept) Fit

27 Fama-French 3 Factors And, we have no more systematic variation... Size (B/M fixed) Rbar Fit 27

28 Fama-French 3 Factors And, we have no more systematic variation... B/M (Size Fixed) Rbar Fit 28