4. (25 pts) Suppose you have 1000 daily returns. The returns some from a stationary distribution prepared as follows. With probability 1/ 2, the return is normal with mean zero and stande deviation 1%. With prob- ability 1/ 2 the return is normal with mean zero and standard deviation 2%. (a) What is the mathematical expression for the probability density of the returns? (b) What is the expected daily return? (0) What is the median daily return? (d) What is the standard deviation of the expected returns? (you may want to use the denition of standard deviation as the square root of the variance, where VAR(X) = E[(X E(X))2], using your result from part a) to simplify EOE). Then integrate. (e) What is the 95% one day VaR of the returns? (f) What is 1.65* the standard deviation of returns from part c)? (g) Why is your answer to part e) not the same as your answer to part f)? (h} What if, instead of the return being equally likely to be from the high volatility and the low volatility state, it was from the high volatility state for the rst 500 returns and from the low volatility stage for the second 500 returns. What would your historical estimate for VaR be if you used the last year of daily return data? The last 4 years? [1) Download as long a time series of TSX index prices as you can, compute daily log returns, and compute 90 day look back volatility estimates. (j) Comment on what this example tells you about VaR, both in terms of your answer to Question 3 and in general