The aim of this project is to prepare, evaluate and analyse stock market data and to recommend an optimal portfo-lio consisting of two stocks. You have been assigned three stocks, all three must be included in the analysis which works towards your recommendation of a final optimal portfolio. The project requires a deep understanding of both the statistics and the mathematics components of this unit. It is recommended that you work on this on a weekly basis
Import the adjusted stock prices for the three stocks which you have been assigned. See the Markdown file for hints.
Plot prices over time
Plot the prices of each asset over time separately. Succinctly describe in words the evolution of each asset over time.
Calculate returns and plot returns over time
Calculate the daily percentage returns of each asset using the following formula:
Where Pt is the asset price at time t. Then plot the returns for each asset over time.
Histogram of returns
Create a histogram for each of the returns series (explain how you determined the number of bins to use).
Summary table of returns
Report the descriptive statistics in a single table which includes the mean, median, variance, standard deviation, skewness and kurtosis for each series. What conclusions can you draw from these descriptive statistics?
Are average returns significantly di erent from zero?
Under the assumption that the returns of each asset are drawn from an independently and identically distributed normal distribution, are the expected returns of each asset statistically di erent from zero at the 1% level of signif-icance? Provide details for all 5 steps to conduct a hypothesis test, including the equation for the test statistic. Calculate and report all the relevant values for your conclusion and be sure to provide an interpretation of the results.
Are average returns di erent from each other?
Assume the returns of each asset are independent from each other. With this assumption, are the mean returns statistically di erent from each other at the 1% level of significance? Provide details for all 5 steps to conduct each of the hypothesis tests using what your have learned in the unit. Calculate and report all the relevant values for your conclusion and be sure to provide and interpretation of the results. (Hint: You need to discuss the equality of variances to determine which type of test to use.)
Calculate and present the correlation matrix of the returns. Discuss the direction and strength of the correlations.
Testing the significance of correlations
Is the assumption of independence of stock returns realistic? Provide evidence (the hypothesis test including all 5 steps of the hypothesis test and the equation for the test statistic) and a rationale to support your conclusion.
Advising an investor
Suppose that an investor has asked you to assist them in choosing two of these three stocks to include in their portfolio. The portfolio is defined by
Where r1 and r2 represent the returns from the first and second stock, respectively, and w1 and w2 represent the proportion of the investment placed in each stock. The entire investment is allocated between the two stocks, so w + 1 + w2 = 1.
The investor favours the combination of stocks that provides the highest return, but dislikes risk. Thus the investor’s happiness is a function of the portfolio, r:
Where E(r) is the expected return of the portfolio, and Var(r) is the variance of the portfolio.
Given your values for E(r1), E(r2), Var(r1), Var(r2) and Cov(r1, r2) which portfolio would you recommend to the investor? What is the expected return to this portfolio?
Provide evidence to support your answer, including all the steps undertaken to arrive at the result. (*Hint: review your notes from tutorial 6 on portfolio optimisation. A complete answer will include the optimal weights for each possible portfolio (pair of stocks) and the expected return for each of these portfolios.)
The impact of financial events on returns
Two significant financial events have occurred in recent history. On September 15, 2008 Lehman Brothers declared bankruptcy and a Global Financial Crisis started. On March 11, 2020 the WHO declared COVID-19 a pandemic. Use linear regression to determine if
- Any of the stocks in your data exhibit positive returns over time.
- Either of the two events had a significant impact on returns.
Report the regression output for each stock and interpret the results to address these two questions. How would you interpret this information in the context of your chosen portfolio?