A company sells precision grinding machines to four customers in four different countries. It has just signed a contract to sell, two months from now, a batch of these machines to each customer. The following table shows the number of machines (batch quantity) to be delivered to the four customers. The selling price of the machine is fixed in the local currency, and the company plans to convert the local currency at the exchange rate prevailing at the time of delivery. As usual, there is uncertainty in the exchange rates. The sales department estimates the exchange rate for each currency and its standard deviation, expected at the time of delivery, as shown in the table. Assume that the exchange rates are normally distributed and independent.

Question

1. Find the distribution of the uncertain revenue from the contract in U.S. dollars. Report the mean, the variance, and the standard deviation.

2. What is the probability that the revenue will exceed $2,250,000?

3. What is the probability that the revenue will be less than $2,150,000?

4. To remove the uncertainty in the revenue amount, the sales manager of the company looks for someone who would assume the risk. An international bank offers to pay a sure sum of $2,150,000 in return for the revenue in local currencies. What useful facts can you tell the sales manager about the offer, without involving any of your personal judgment?

5. What is your recommendation to the sales manager, based on your personal judgment?

6. If the sales manager is willing to accept the bank's offer, but the CEO of the company is not, who is more risk-averse?

7. Suppose the company accepts the bank's offer. Now consider the bank's risk, assuming that the bank will convert all currencies into U.S. dollars at the prevailing exchange rates. What is the probability that the bank will incur a loss?

8. The bank defines its value at risk as the loss that occurs at the 5th percentile of the uncertain revenue. What is the bank's value at risk?

9. What is the bank's expected profit?

10. Express the value at risk as a percentage of the expected profit. Based on this percentage, what is your evaluation of the risk faced by the bank?

11. Suppose the bank does not plan to convert all currencies into U.S. dollars, but plans to spend or save them as local currency or convert them into some other needed currency. Will this increase or decrease the risk faced by the bank?

12. Based on the answer to part 11, is the assumption (made in parts 7 to 10) that the bank will convert all currencies into U.S. dollars a good assumption?