The objective of this question is to examine how the variance of portfolios may be reduced as a result of diversification. Download through Bloomberg or Eikon or Yahoo! Finance monthly data of constituent stocks of Hang Seng Index covering the period June 2016 to June 2021

a. Form equal-weight and value-weight portfolios using 5, 10, 25, and all 50 stocks. Calculate the sample mean and standard deviation of the returns for each of the eight portfolios. Plot estimated standard deviations as a function of the number of stocks in the equal-weight portfolio. Comment on the shape of the function. Are the results consistent with what you would expect theoretically? â€œEyeballingâ€ the graph, does it look like adding more and more stocks will diversify away all the standard deviation? Why?

b. For all four equal-weight portfolios, decompose the estimated portfolio variance into its two components (the contributions of variances and covariances). Plot the percentage of the portfolioâ€™s variance due to the variances of individual security returns as a function of the number of stocks in the portfolio. Comment on the shape of the function. Are the results consistent with what you would expect theoretically? Use the relevant equations in your explanation. Hint: you do not have to estimate the pair-wise covariances in order to compute the decomposition.

c. Would you expect a 5-stock value-weight portfolio to exhibit more, less, or about the same variance as an equal-weight portfolio consisting of the same 5 stocks? Describe the factors that influence your decision. What if there were 1000 stocks? (Hint: Are large stocks typically more or less volatile than small stocks?