You're a portfolio manager at a hedge fund, meaning that you make investment decisions about other...

      

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You're a portfolio manager at a hedge fund, meaning that you make investment decisions about other people's money. Naturally enough, the people whose money you're investing want to know how risky your investment decisions are. To this end, a standard metric in the investment industry is the Value at Risk (VaR) of a portfolio. Letting the continuous random variable X represent the (unknown) change in value of the portfolio in question over a fixed time horizon (one month, for example), suppose that the PDF of X in your judgment, based on the best current information concentrates most of its probability on the positive part of the real number line R; in other words, in your judgment the portfolio will probably increase in value over the next month but may instead decrease. Let Y = -X, so that Y is the pessimistic side of the X coin (so to speak): if X> 0 with high probability then Y < 0 with the same high probability. To quantify the term "high," let a be a small positive number, so that (1-a) is close to 1; then the VaR of the portfolio is defined to be the (1-a) quantile of the distribution of Y. The tough part of implementing this idea is pinning down the PDF of X; in this problem you'll examine how sensitive the VaR is to this PDF specification. Let's take a = 0.01 in what follows; this is a frequent (and reasonably conservative) choice in calculating VaR values. It turns out that a portfolio based on sensible trading of stocks on the New York Stock Exchange will typically appreciate at a rate of about 7% per year, which translates to a rate of about 0.6% per month; this implies that, if the portfolio is expected to increase in value by about $10 million in the next month, which is consistent with the PDFs in part (b) below, the total value of the portfolio at the beginning of the month was about $1.7 billion. You're a portfolio manager at a hedge fund, meaning that you make investment decisions about other people's money. Naturally enough, the people whose money you're investing want to know how risky your investment decisions are. To this end, a standard metric in the investment industry is the Value at Risk (VaR) of a portfolio. Letting the continuous random variable X represent the (unknown) change in value of the portfolio in question over a fixed time horizon (one month, for example), suppose that the PDF of X in your judgment, based on the best current information concentrates most of its probability on the positive part of the real number line R; in other words, in your judgment the portfolio will probably increase in value over the next month but may instead decrease. Let Y = -X, so that Y is the pessimistic side of the X coin (so to speak): if X> 0 with high probability then Y < 0 with the same high probability. To quantify the term "high," let a be a small positive number, so that (1-a) is close to 1; then the VaR of the portfolio is defined to be the (1-a) quantile of the distribution of Y. The tough part of implementing this idea is pinning down the PDF of X; in this problem you'll examine how sensitive the VaR is to this PDF specification. Let's take a = 0.01 in what follows; this is a frequent (and reasonably conservative) choice in calculating VaR values. It turns out that a portfolio based on sensible trading of stocks on the New York Stock Exchange will typically appreciate at a rate of about 7% per year, which translates to a rate of about 0.6% per month; this implies that, if the portfolio is expected to increase in value by about $10 million in the next month, which is consistent with the PDFs in part (b) below, the total value of the portfolio at the beginning of the month was about $1.7 billion.