Suppose a hedge fund owns a $1,000,000 position in a portfolio and used $50,000 of its own capital and $950,000 in borrowed money for the purchase. If the value of the portfolio falls below $950,000 at the end of any trading day, then the hedge fund must liquidate and repay the loan. The portfolio was selected by cointegration analysis and its price is an AR(1) process, (Pt u) = (Pt-1 u) + t, where Pt is the price of the portfolio at the end of trading day t, u $1,030,000, 0 = 0.99, and the standard deviation of t is $5000. The hedge fund knows that the price will eventually revert to $1,030,000 (assuming that the model is correct and, of course, this is a big assumption). It has decided to liquidate its position on day t if Pt > $1,020,000. This will yield a profit of at least $20,000. However, if the price falls below $950,000, then it must liquidate and lose its entire $50,000 investment plus the difference between $950,000 and the price at liquidation. In summary, the hedge fund will liquidate at the end of the first day such that the price is either above $1,020,000 or below $950,000. In the first case, it will achieve a profit of at least $20,000 and in the second case it will suffer a loss of at least $50,000. Presumably, the probability of a loss is small, and we will see how small by simulation. Run a simulation experiment similar to the one in Sect. 2.4.2 to answer the following questions. Use 10,000 simulations. Problem 9 What is the expected profit? Suppose a hedge fund owns a $1,000,000 position in a portfolio and used $50,000 of its own capital and $950,000 in borrowed money for the purchase. If the value of the portfolio falls below $950,000 at the end of any trading day, then the hedge fund must liquidate and repay the loan. The portfolio was selected by cointegration analysis and its price is an AR(1) process, (Pt u) = (Pt-1 u) + t, where Pt is the price of the portfolio at the end of trading day t, u $1,030,000, 0 = 0.99, and the standard deviation of t is $5000. The hedge fund knows that the price will eventually revert to $1,030,000 (assuming that the model is correct and, of course, this is a big assumption). It has decided to liquidate its position on day t if Pt > $1,020,000. This will yield a profit of at least $20,000. However, if the price falls below $950,000, then it must liquidate and lose its entire $50,000 investment plus the difference between $950,000 and the price at liquidation. In summary, the hedge fund will liquidate at the end of the first day such that the price is either above $1,020,000 or below $950,000. In the first case, it will achieve a profit of at least $20,000 and in the second case it will suffer a loss of at least $50,000. Presumably, the probability of a loss is small, and we will see how small by simulation. Run a simulation experiment similar to the one in Sect. 2.4.2 to answer the following questions. Use 10,000 simulations. Problem 9 What is the expected profit
