Question 3: Download stock prices (adjusted close) for the following six equities on the first day of each month from January 1, 2018 to December 31, 2022: MSFT, AAPL, ORCL, GOOG,

EBAY, INTC. Data can be downloaded, for instance, at https://finance.yahoo.com/. For each equity, compute the time series of arithmetic and logarithmic returns and report the sample mean, variance, skewness, and kurtosis for each. Consider a portfolio in which the same number of dollars is invested into each equity on January 1, 2018. After this date, the number of shares held in each equity is held constant. Compute the above statistical quantities for both arithmetic and logarithmic returns for this portfolio. Lastly, make a plot of the expected annual (logarithmic) return versus volatility. The plot will consist of seven points, one for each equity and one for the portfolio described above. Give some brief comments on the results./nQuestion 4: Download stock prices (adjusted close) for MSFT on the first day of each month from January 1, 2018 to December 1, 2022. Compute the sample mean, variance, and kurtosis for the arithmetic returns. In this question you will estimate parameters of a two component normal mixture via moment matching (the parameters have some restrictions to make the process easier). Recall that a two component normal mixture is a model of the following form: r = Zr₁ + (1 - Z)r₂ T₁ ~ N(μ1,01) T₂ ~ N(H₂,0₂) P(Z = 1) = P, 1 where Z, r₁, and r2 are independent. The restrictions we will consider are that #₁ = 2 (which will be denoted by μ) and p is fixed beforehand. The only parameters that remain to be estimated are 01 and 02. This is done be solving: V[r] =² K[r] = k where V and K are the variance and kurtosis operators. Plot the estimated values of 01 and 02 as a function of p for p [0.8,0.999]. Note: You will need to derive expressions for V[r] and K[r] for this mixture model in terms of its parameters. Also, sometimes there will be more than one possible pair (01,02) which match the moments. In this case, choose the pair such that 02 > 01 since state 2 is supposed to represent the low probability distressed market state.