# Question: The relevance of variance and covariance developed for random variables

The relevance of variance and covariance developed for random variables in this chapter carries over to the analysis of returns on real stocks. The data file â€śdaily _ stocksâ€ť includes the daily percentage changes in the value of stock in Apple and stock in McDonaldâ€™s in 2010 and 2011. Use these data to define two random variables, one for the daily percentage change in Apple stock, and the other for McDonaldâ€™s. The scatterplot shows these returns.
Motivation
(a) An investor is considering buying stock in Apple and McDonaldâ€™s. Explain to the investor, in nontechnical language, why it makes sense to invest some of the money in both companies rather than putting the entire amount into one or the other.
Method
(b) Why is the Sharpe ratio useful for comparing the performance of two stocks whose values donâ€™t have the same variances?
(c) Based on the observed sample correlation in these data, does it appear sensible to model percentage changes in the values of these stocks as independent?
Mechanics
(d) Find the Sharpe ratio for a daily investment concentrated in Apple, concentrated in McDonaldâ€™s or equally split between Apple and McDonaldâ€™s. Use the same interest rate as in the text (0.015). Which choice seems best?
(e) Form a new column in the data that is the average of the columns with the percentage changes for Apple and McDonaldâ€™s. Find the mean and SD of this new variable and compare these to the values found in part (d).
(f) Find the Sharpe ratio for an investment that puts three-quarters of the money into McDonaldâ€™s and one-quarter into Apple. Compare this combination to the mix considered in part (d). Which looks better?
Message
(g) Explain your result to the investor. Does it matter how the investor divides the investment between the two stocks?

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