Consider a world where there are only two risky stocks, $A$ and $B$, whose details are listed in the table below:

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Furthermore, the correlation between the returns of stocks $A$ and $B$ is $\rho_{A B} = \frac{1}{3}$. There is also a risk-free asset and in this world the CAPM is satisfied exactly.

a.) What is the expected rate of return of the market portfolio?

b.) What is the standard deviation of the market portfolio?

c.) What is the beta of stock A?

d.) What is the risk-free rate in this world?

Solution to a.): We have

$$Cov(r_A, r_B) = \frac{1}{3}(.15)(.09) = .0045$$

Recall that

\begin{align*} \sigma^{2}_{P} &= \sigma^{2}_{A}W_A^{2} + \sigma^{2}_{B}W_B^{2} + 2\sigma_A \sigma_B cov(r_A,r_B)\\ &= \sigma^{2}_{A}W_A^{2} + \sigma^{2}_{B}(1 - W_A)^{2} + 2\sigma_A \sigma_B cov(r_A,r_B)\\ &= .0306 W_A^{2} - .0162 W_A + .0082215\\ \frac{\partial \sigma^{2}_{P}}{\partial W_A} &= 0 \implies \boxed{W_A = .2647} \end{align*}

Note that

$$\frac{\partial^2 \sigma^{2}_{P}}{\partial W_A^2} = .0612 > 0$$

thus the variance is at a minimum. Hence

$$E[r_P] = W_A E[r_A] + W_B E[r_B] = .1279 \approx .13$$

Solution to b.): We have

$$\sigma^2{P} = .006074$$

$$\boxed{\sigma_P =.0779}$$

For some reason that I don't fully understand, my professor in the homework has $9\%$.

I am not sure how to get question part c.) or d.) because it seems that there isn't enough information.


a.) The market capitalization $m_{cap} = 100*\$1.50 + 150*\$2.0 = \$150 + \$300 = \$450$, so the weight of each asset is $1/3$ and $2/3$ respectively in the market portfolio. You don't need to find the minimum variance portfolio. If you plug in these values you get exactly $E[r_m] = 1/3*0.15 + 2/3*0.12 = 0.13.$

b.) The formula is wrong, as you multiply your covariance with the standard deviations of the assets. The correct formula is $$\sigma_P^2 = w_A^2\sigma_a^2 + w_B^2\sigma_b^2 + 2w_aw_b\sigma_a\sigma_b\rho_{ab} \\=1/3^2*0.15^2 + 2/3^2*0.09^2 + 2*1/3*2/3*0.0045 = 0.0080506$$ Taking the squareroot gives you a standard deviation of $9\%$.

c.) The Beta of an asset can be derived as $\beta_a = \frac{\sigma_a}{\sigma_{market}}\rho_{a,market}$. You thus need to find the correlation, or covariance, between the market and stock a.

d.) The risk-free rate of the market can be considered as implicitly defined in the CAPM formula, $$E[R_a] = R_f + \beta_a(E[R_m] - R_f).$$ When you know $\beta_a$, you get $$R_f = \frac{1}{1-\beta_a}(E[R_a]-\beta_aE[R_m])$$

  • $\begingroup$ For c.) thats what I thought we needed to find but I am not sure how to find the correlation or covariance between the market and stock $A$ $\endgroup$ – Wolfy Feb 5 '17 at 22:47
  • $\begingroup$ Use the definition of covariance and what you know about how the market is defined. Maybe the rule $\sigma(X,aX+bY) = a\sigma^2_X + ab\sigma(X,Y)$ will help. $\endgroup$ – Forgottenscience Feb 5 '17 at 22:57
  • $\begingroup$ I am still having trouble with part c and d could you help? $\endgroup$ – Wolfy Feb 12 '17 at 16:28
  • $\begingroup$ What are you having trouble with? Question d) is answered as soon as you have the answer in c). I have given you the covariance formula in the comment above. When you have the covariance, the correlation is $\rho_{a,market} = \frac{\sigma(a,market)}{\sigma_a \sigma_x}$, which you can insert directly into the formula I gave in the answer. $\endgroup$ – Forgottenscience Feb 12 '17 at 20:21
  • $\begingroup$ I don't know how to calculate the $\rho_{a,market}$ $\endgroup$ – Wolfy Feb 12 '17 at 22:09

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