Question 3 [2] (a) Define the Value at Risk (VaR) for a time period T and a confidence level 1 - a. (b) Define the log returns on an investment on day t as It = log Yt Yt-1 where yt is the value of the investment at the end of day t. We decide to model the distribution of log-returns on an investment as a N(0,02) random variable with unknown variance o2 Assume that the log-returns on the investment for days 1,..., n are independent and identically distributed. [4] [4] (i) Derive the maximum likelihood estimator (MLE) 2 for o? and find its asymptotic variance. (ii) Show that the cumulative distribution function (cdf) of the N(0,0%) distribution is given by Fo(x) = F(x/o), where F(x) is the cdf of the standard normal distribution. Furthermore, show that the quantile function of the N(0,02) distribution is given by Qo(p) = oQ(p), where Q(p) is the quantile function of the standard normal distribution. (iii) Derive an expression for the VaR over 1 day for a