91Ó°ÊÓ

Stocks A and B have the following historical returns: $$\begin{aligned} &\begin{array}{lcc} \text { Year } & \text { Stock A's Returns, } r_{A} & \text { Stock B's Returns, } \\ \hline 2001 & (18.00 \%) & (14.50 \%) \\ 2002 & 33.00 & 21.80 \\ 2003 & 15.00 & 30.50 \\ 2004 & (0.50) & (7.60) \\ 2005 & 27.00 & 26.30 \end{array}\\\ &\mathbf{r}_{B} \end{aligned}$$ a. Calculate the average rate of return for each stock during the period 2001 through 2005. b. Assume that someone held a portfolio consisting of 50 percent of Stock A and 50 percent of Stock B. What would the realized rate of return on the portfolio have been in 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?

Step by step solution

01

Calculate Average Returns for Each Stock

To find the average return for Stock A and Stock B, we need to sum up their annual returns from 2001 to 2005 and divide by the number of years, which is 5. For Stock A: \[ r_{A, avg} = \frac{(-18 + 33 + 15 - 0.5 + 27)}{5} \]For Stock B: \[ r_{B, avg} = \frac{(-14.5 + 21.8 + 30.5 - 7.6 + 26.3)}{5} \]

02

Calculate Realized Portfolio Returns Each Year

The portfolio return is calculated by averaging the returns of Stock A and Stock B for each year, given equal weights of 50% each. For any year, say 2001, calculate it as follows:\[ r_{p, 2001} = 0.5 \times r_{A, 2001} + 0.5 \times r_{B, 2001} \]Repeat this calculation for each year from 2001 to 2005.

03

Calculate Average Portfolio Return

Sum up the calculated portfolio returns for each year and divide by 5 to find the average portfolio return:\[ r_{p, avg} = \frac{(r_{p, 2001} + r_{p, 2002} + r_{p, 2003} + r_{p, 2004} + r_{p, 2005})}{5} \]

04

Calculate Standard Deviation of Returns

Standard deviation is a measure of return volatility. For Stock A, use:\[ \sigma_{A} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (r_{A, i} - r_{A, avg})^{2}} \]Do the same for Stock B and the portfolio using their respective average returns.

05

Calculate Coefficient of Variation

The coefficient of variation (CV) is the ratio of the standard deviation to the average return. For Stock A, use:\[ CV_{A} = \frac{\sigma_{A}}{r_{A, avg}} \]Repeat for Stock B and the portfolio with their standard deviations and average returns.

06

Determine Investment Preference

A risk-averse investor typically prefers options with lower CV, implying less risk per unit of return. Compare the CVs for Stocks A, B, and the portfolio to see which is preferable for such an investor.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Historical Returns

Historical returns refer to the past performance of an investment over a specific period. These returns are crucial in portfolio management as they help assess how stocks have behaved in various market environments. Past performance, however, does not guarantee future results, but understanding historical returns provides a foundation for calculating averages and assessing potential risks and returns.

When looking at the historical returns of Stocks A and B, the data spans five years from 2001 to 2005 as follows: - Stock A: -18.00%, 33.00%, 15.00%, -0.50%, 27.00% - Stock B: -14.50%, 21.80%, 30.50%, -7.60%, 26.30% These figures show how each stock performed yearly, reflecting their volatility and overall performance. Observing such trends over previous periods supports informed decision-making on potential investments in these stocks.

Average Rate of Return

The average rate of return provides a simple measure of how much an investment has gained or lost on average over a given period. Calculating it involves summing up all returns over the period and dividing by the number of years.

For Stock A, the formula looks like this: \[ r_{A, avg} = \frac{(-18 + 33 + 15 - 0.5 + 27)}{5} = 11.10\% \]For Stock B, it is:\[ r_{B, avg} = \frac{(-14.5 + 21.8 + 30.5 - 7.6 + 26.3)}{5} = 11.30\% \]This metric is crucial for assessing the performance without the noise of annual fluctuations. It helps investors understand the overall growth or decline trajectory of an asset across different market conditions.

Standard Deviation

Standard deviation is a vital statistical measure used in portfolio management to quantify the amount of variation or dispersion of a set of returns. A higher standard deviation indicates a higher volatility and thus, higher risk.

To calculate the standard deviation for an investment, you must determine the average return and then compute the differences from the mean for each year's return. You then square those differences, average them, and take the square root of that average:- Stock A:\[ \sigma_{A} = \sqrt{\frac{1}{5} ((-18 - 11.1)^2 + (33 - 11.1)^2 + (15 - 11.1)^2 + (-0.5 - 11.1)^2 + (27 - 11.1)^2)} \]- Stock B: Follow the same procedure using Stock B's returns and its average.Understanding standard deviation helps investors gauge the consistency and reliability of returns, thereby significantly influencing risk assessment.

Coefficient of Variation

The coefficient of variation (CV) is a sophisticated financial metric used to evaluate the risk per unit of return. It is calculated as the ratio of the standard deviation to the average return, offering a relative measure of risk.

For Stock A, it is calculated by:\[ CV_{A} = \frac{\sigma_{A}}{r_{A, avg}} \]Similarly for Stock B and the portfolio, compute their CVs using their respective standard deviations and average returns.The CV provides investors a clearer picture of which stock or portfolio offers a better risk-reward profile. A lower CV indicates that a stock offers lower risk for the additional unit of return, making it a preferable choice for risk-averse investors.