Stock A and Stock B have the following historical returns:

$$\begin{array}{lcc} \text{Year} & \text{Stock A's Returns, {r}}_{\text {A }} & \text{Stock B's Returns, {r}}_{\text {B }} \\ \hline 2011 & -18.00 \% & -14.50 \% \\ 2012 & 33.00 & 21.80 \\ 2013 & 15.00 & 30.50 \\ 2014 & -0.50 & -7.60 \\ 2015 & 27.00 & 26.30 \end{array}$$

Calculate the average rate of return for each stock during the period 2011-2015.

Stock A and Stock B have the following historical returns:

$$\begin{array}{ccc} \text{Year} & \text{Stock A's Returns, r}_{\text {A }} & \text{Stock B's Returns, r}_\text{B} \\ \hline 2011 & -10.00 \% & -3.00 \% \\ 2012 & 18.50 & 21.29 \\ 2013 & 38.67 & 44.25 \\ 2014 & 14.33 & 3.67 \\ 2015 & 33.00 & 28.30 \end{array}$$

Calculate the standard deviation of returns for each stock and for the portfolio.

Stocks A and B have the following historical returns:

$$\begin{array}{ccc} \textbf{Year} & \textbf{Stock A's Returns}, \mathbf{r}_{\mathbf{A}} & \textbf{Stock B's Returns}, \mathbf{r}_{\mathbf{B}} \\ \hline 2011 & (24.25 \%) & 5.50 \% \\ 2012 & 18.50 & 26.73 \\ 2013 & 38.67 & 48.25 \\ 2014 & 14.33 & (4.50) \\ 2015 & 39.13 & 43.86 \end{array}$$

Looking at the annual returns on the two stocks, would you guess that the correlation coefficient between the two stocks is closer to $+0.8$ or to $-0.8$ ?

Stocks A and B have the following historical returns:

$$\begin{array}{ccc} \textbf{Year} & \textbf{Stock A's Returns}, \mathbf{r}_{\mathbf{A}} & \textbf{Stock B's Returns}, \mathbf{r}_{\mathbf{B}} \\ \hline 2011 & (24.25 \%) & 5.50 \% \\ 2012 & 18.50 & 26.73 \\ 2013 & 38.67 & 48.25 \\ 2014 & 14.33 & (4.50) \\ 2015 & 39.13 & 43.86 \end{array}$$

Calculate the average rate of return for each stock during the period $2011$ through $2015$. Assume that someone held a portfolio consisting of $50 \%$ of Stock A and $50\%$ of Stock B. What would the realized rate of return on the portfolio have been in each year from $2011$ through $2015$? What would the average return on the portfolio have been during that period?

Stocks A and B have the following historical returns:

$$\small{ \begin{array}{ccc} \textbf{Year} & \textbf{Stock A's Returns}, \mathbf{r}_{\mathbf{A}} & \textbf{Stock B's Returns}, \mathbf{r}_{\mathbf{B}} \\ \hline 2010 & (24.25 \%) & 5.50 \% \\ 2011 & 18.50 & 26.73 \\ 2012 & 38.67 & 48.25 \\ 2013 & 14.33 & (4.50) \\ 2014 & 39.13 & 43.86 \end{array}}$$

a. Calculate the average rate of return for each stock during the period $2010$ through $2014$. Assume that someone held a portfolio consisting of $50 \%$ of Stock A and $50 \%$ of Stock B. What would the realized rate of return on the portfolio have been in each year from $2010$ through $2014$? What would the average return on the portfolio have been during that period?

b. Calculate the standard deviation of returns for each stock and for the portfolio. Use the following Equation:

$$\text { Estimated } \sigma=\sqrt{\frac{\sum_{t=1}^N\left(\bar{r}_t-\bar{r}_{\text {Avg }}\right)^2}{N-1}}$$

c. Looking at the annual returns on the two stocks, would you guess that the correlation coefficient between the two stocks is closer to $+0.8$ or to $-0.8$ ?

d. If more randomly selected stocks had been included in the portfolio, which of the following is the most accurate statement of what would have happened to $\sigma_{\mathrm{p}}$ ?

• $\sigma_{\mathrm{p}}$ would have remained constant.

• $\sigma_p$ would have been in the vicinity of $20 \%$.

• $\sigma_{\mathrm{p}}$ would have declined to zero if enough stocks had been included.

Stocks A and B have the following historical returns:

$$\begin{array}{ccc} \textbf{Year} & \textbf{Stock A's Returns}, \mathbf{r}_{\mathbf{A}} & \textbf{Stock B's Returns}, \mathbf{r}_{\mathbf{B}} \\ \hline 2011 & (24.25 \%) & 5.50 \% \\ 2012 & 18.50 & 26.73 \\ 2013 & 38.67 & 48.25 \\ 2014 & 14.33 & (4.50) \\ 2015 & 39.13 & 43.86 \end{array}$$

If more randomly selected stocks had been included in the portfolio, which of the following is the most accurate statement of what would have happened to $\sigma_p$ ?

1. $\sigma_{\mathrm{p}}$ would have remained constant.

2. $\sigma_{\mathrm{p}}$ would have been in the vicinity of $20 \%$.

3. $\sigma_{\mathrm{p}}$ would have declined to zero if enough stocks had been included.

A mutual fund manager has a $20 million portfolio with a beta of 1.7. The risk-free rate is 4.5%, and the market risk premium is 7%. The manager expects to receive an additional$5 million, which she plans to invest in a number of stocks. After investing the additional funds, she wants the fund’s required return to be 15%. What should be the average beta of the new stocks added to the portfolio?

Stocks A and B have the following historical returns:

$$\begin{array}{ccc} \textbf{Year} & \textbf{Stock A's Returns}, \mathbf{r}_{\mathbf{A}} & \textbf{Stock B's Returns}, \mathbf{r}_{\mathbf{B}} \\ \hline 2011 & (24.25 \%) & 5.50 \% \\ 2012 & 18.50 & 26.73 \\ 2013 & 38.67 & 48.25 \\ 2014 & 14.33 & (4.50) \\ 2015 & 39.13 & 43.86 \end{array}$$

Calculate the standard deviation of returns for each stock and for the portfolio. Use Equation

$$\text { Estimated } \sigma=\sqrt{\frac{\sum_{\mathrm{t}=1}^{\mathrm{N}}\left(\overline{\mathbf{r}}_{\mathrm{t}}-\overline{\mathbf{r}}_{\text {Avg }}\right)^2}{\mathrm{~N}-1}}$$

For risk averse investors, how much return is expected for taking on more risk?

Based on the following information, calculate the expected return and standard deviation for Stock A and Stock B:

State of	Probability of	Rate of Return If State Occurs
Economy	State of Economy	Stock A	Stock B
Recession	.10	.04	-.17
Normal	.60	.09	.12
Boom	.30	.17	.27

Answer the following.

The probability distribution of a less risky expected return is more peaked than that of a riskier return. What shape would the probability distribution be for (a) completely certain returns and (b) completely uncertain returns?

Stocks R and S have the following probability distributions of returns:

$$\begin{array}{ccc} \text{Probability} & \text{Stock R} & \text{Stock S} \\ \hline 0.5 & -2 \% & 20 \% \\ 0.1 & 10 & 12 \\ 0.4 & 15 & 2 \end{array}$$

Calculate the standard deviation of returns for each stock and for the portfolio. Which stock is considered riskier with respect to total risk?

Suppose you held a diversified portfolio consisting of a $7,500 investment in each of 20 different common stocks. The portfolio’s beta is 1.25. Now suppose you decided to sell one of the stocks in your portfolio with a beta of 1.0 for$7,500 and use the proceeds to buy another stock with a beta of 0.80. What would your portfolio’s new beta be?

Stocks A and B have the following historical returns:" $$ \begin{matrix} \text{Year} & \text{Stock A's Returns, }{r_A} & \text{Stock B's Returns, }{r_B}\\ \text{2011} & \text{(18.00\\%)} & \text{(14.50\\%)}\\ \text{2012} & \text{33.00} & \text{21.80}\\ \text{2013} & \text{15.00} & \text{30.50}\\ \text{2014} & \text{(0.50)} & \text{(7.60)}\\ \text{2015} & \text{27.00} & \text{26.30}\\ \end{matrix} $$ a. Calculate the average rate of return for each stock during the period 2011 through 2015. b. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would the realized rate of return on the portfolio have been each year? What would the average return on the portfolio have been during this period? c. Calculate the standard deviation of returns for each stock and for the portfolio. d. Calculate the coefficient of variation for each stock and for the portfolio. e. Assuming you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or the portfolio? Why?