In finance, VaR does not stand for variance but for "value at risk." It is another measure of risk. It was introduced by J.P. Morgan in the 1980s as a way to answer a question that was often asked by investors: "How much might I loose?" Let X be a random variable representing the profit on an investment at the end of some time horizon, such as a year. If X is negative the investment results in a loss. Choose a confidence level a (a = 95% is a commonly used value). The a-VaR is the value z for which P(X z) = 1-a. This means that the profit on the investment will be greater than a with probability a, and it will be r or smaller with probability 1-a. In particular, if z is negative and a = 95%, a loss of r or greater will occur with probability 5%. (1.1) Easy Investment A has a profit at the end of a year that is normally distributed with mean $2 million and standard deviation $1.5 million. What is the 95%-VaR of investment A? (1.2) Moderate The investor owning A considers the potential loss calculated in (1.1) to be excessive. Risk might be reduced using a hedging strategy. Also available is investment B, with a profit at the end of a year that has mean -0.3 million dollars and standard deviation $ 1 million. The correlation between the profits on investments A and B is -0.8. What is the expected profit of investment A+B at the end of a year? What is the standard deviation of the profit on investment A+B at the end of a year? Assuming that the profit on investment A+B is normally distributed, compute the 95%-VaR of investment A+B. (1.3) Challenging The investor is considering hedged portfolios of the form A + t B for t> 0. Can you find the value of t that minimizes the variance of this portfolio? Assuming that the profit on investment A + t B is normally distributed, compute the 95%-VaR of investment A + t B for the value of t found above.