Question

For Dell, suppose its mean and standard deviation are 10% and 20%. For IBM, suppose its mean and standard deviation are 6% and 15%. Their correlation coefficient is 0.25. The risk-free rate is 2%. The optimal risky portfolio consists of 40% in IBM and 60% in Dell. [Note: This is an important exercise to obtain some intuition about CAPM. However, neither the formulas in Part 2 nor the algebraic proof in Part 5 are required for this course. If I were to test these in an exam, I will give you the formulas that you need to use.) 1. If all market participants have homogeneous beliefs, they hold Dell and IBM in the same proportion in their optimal risky portfolios. Assume that Dell stock sells for $30 per share and IBM stock sells for $50 dollars per share. What do homogeneous beliefs imply about the ratio of the number of shares that are available on the market? 2. Treat the optimal risky portfolio as the market portfolio. What is the beta of Dell (BD) with respect to the market? (Hint: Beta of asset A with respect to asset B equals AB, where Cov denotes Var(TB) the covariance between the two assets. Additionally, if B is composed of A and some other asset C, we have the following formula: Cov(raro) = Cov(ra, WATA + wcrc) = wĄVar(ra) + wcCov(rar), where wa and wc are weights for A and C in the composite portfolio.) 3. Does CAPM hold for Dell? 4. Redo Part 2 and 3 for IBM. What do you find? 5. See if you can prove CAPM in its abstract form. That is, without plugging in the numbers, confirm CAPM algebraically (for two given stocks).

Answer 1

Answer:

1. If all market participants have homogeneous beliefs , they hold Dell and IBM in same proportion in their optimal risky portfolios. If assume that Dell stock Sells for $ 30 and IBM for $ 50 per share. Then the ratio of number of shares available in the market for Dell and IBM be :

Dell : IBM

50 X 60% : 30 X 40%

30 : 12

5 : 2

That means 5 stocks of Dell to 2 stocks of IBM

2. First we need to calculate covariance_{D I  :}

Here we use D for Dell and I for IBM

covariance_{D I = Correlation coefficient of D ,I X  $\sigma _{_{D}} X \sigma _{_{ I }}$} = 0.25 X 0.20 X 0.15 = 0.0075

Beta of Dell ( $\beta _{D}$ ) = covariance_{D I / Variance of D = 0.0075 / ( 0.20 - 0.10 ) = 0.075 if round off then 0.08}

3. CAPM for Dell =

= $R_{f} + ( ( \beta ) X (R _{D}+ R _{f} ))$ = 2% + ( 0.08 x ( 10% -2% ) = 2.64 %

CAPM does not hold for Dell as Market expects the 10% return at 60% that means 6% return ( 10% x 60% ) and as per CAPM model the return is 2.64% which is lesser

4. First we calculate Beta of IBM

Beta of IBM ( $\beta _{I}$ ) = covariance_{D I / Variance of I = 0.0075 / ( 0.15 - 0.06 ) = 0.083}

_{CAPM for IBM =}

= $R_{f} + ( ( \beta ) X (R _{I}+ R _{f} ))$ = 2% + ( 0.083 x ( 16% -2% ) = 2.332 %

CAPM may hold for IBM as Market expects the 6% return at 40% that means 2.4% return ( 6% x 40% ) and as per CAPM model the return is 2.332% which is closer to expected return