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4. The Uris & Warren Ratings Agency rates bonds on a simplified scale with just three categories: A, B, and C. The Professors Pension Fund has all its money invested in two bonds, X and Y, both of which are currently rated B. Over the course of the next year the ratings of the bonds may change; the end-of-year value (in millions) of each bond depends on its end-of-year rating as in the following table: Rating X Value Y Value A 100 100 75 75 50 50 The joint distribution of the end-of-year ratings of the two bonds is given by the following table: 17 A B A 0.20 0.05 X B 0 0.40 0.10 0 0.05 0.20 (a) Find the probability that bond X will be rated C at the end of one year. (b) Find the expected value of the year-end value of bond X. (c) The Pension Fund buys an insurance contract that will pay 20 million if either bond is downgraded to C. If both bonds are downgraded the contract still pays 20; if neither is downgraded the contract pays nothing. Find the expected value of the payoff of the insurance contract.1. An inverse floater is a type of security whose payments move in the opposite direction of short-term interest rates. The security is ordinarily structured so that no matter how high interest rates rise, the payment cannot be negative, and no matter how low interest rates drop, the payment cannot exceed some specified cap, e.g., 7%. Consider the following specific case: if the prevailing short-term rate is X, the inverse floater pays Y = 100 x max(0.07 - X, 0) on $100 of face value. (The notation "max(0.07 - X,0)" means use whichever is larger, 0.07 - X or 0.) Suppose the distribution of the short-term rate X at the next payment date is given by the following table: |0.04 0.06 0.07 0.08 0.09 P(X = r) 0.30 0.20 0.20 0.15 0.15 Find the expected payment E[Y]. 2. A mining company plans to develop two potential gaussite reserves. Each reserve has probability 0.30 of successfully yielding usable gaussite, and the success of each reserve is independent of the other. If either of the two reserves is successful, it will generate $4 million in profit; if both are successful, profits will be $7 million because excess supply will lower prices. If neither is successful, profits will be 0. Let X be the company's profit. Find E[X]. 16 3. The Gourmet Cafe serves the exotic Bernoulli Salmon at lunch and dinner. The number of customers ordering the salmon at lunch and dinner are given by the following distributions: Lunch demand | 0 1 2 probability 0.3 0.5 0.2 Dinner demand | 0 probability |0.2 0.4 0.4 Assume the lunch and dinner demands are independent of each other so the joint distri- bution of the lunch and dinner demands is given by the following table: Lunch 0 2 0 0.06 0.10 0.04 0.2 Dinner 0.12 0.20 0.08 0.4 2 0.12 D.20 0.08 0.4 0.3 0.5 0.2 Each entry of the table is just the product of the marginal probabilities at the end of the corresponding row and the bottom of the corresponding column.) The chef orders the fish in advance at a cost of $3.50 per serving. Any fish left over at the end of the day is discarded. (a) What is the expected total demand for the fish in a day? (b) Suppose the chef orders three servings. What is the breakeven selling price (i.e., the price at which the expected revenue from sales of the fish equals the cost of the fish ordered)? Assume that a customer who would have ordered the fish but finds it sold out simply leaves rather than order something else. (Hint: Expected revenue = price times expected number of units sold.)11. Cluster I had exams in Finance and Marketing last week. All 60 students in the cluster took both exams. The results were as follows: o Finance: mean = 25, standard deviation = 2 o Marketing: mean = 75, standard deviation = 12 o Correlation between score in Finance and same student's score in Marketing = 0.84 Mary, a student in Cluster Y, scored a 30 in Finance and a 90 in Marketing. We are interested in comparing her performance on the two exams relative to the rest of the class. In particular, we would like to make a statement about which of her scores ranked higher compared to the other scores on the same exam. Select one of the choices below and complete the statement you select. 11 $ observations 10 Figure 4: Histogram for Problem 10 (i) Mary's score in Finance probably ranks higher than her score in Marketing because (i) Mary's score in Marketing probably ranks higher than her score in Finance because (iii) Mary's scores on the two exams probably rank about equally high because (iv) We cannot make any comparison between the two scores because 12. Seven students from the 1998 MBA class took jobs in brain surgery after graduation. Five of the students reported their starting salaries: $55,000, $90,250, $90,250, $95,500, and $105,000. Choose one of the following: (a) Based on the information given, the largest possible value of the median starting salary for all seven students is (b) Based on the information given, it is not possible to put an upper limit on the median starting salary for all seven students.6. The observations X1,..., An have a mean of 50 and a standard deviation of 7. Which of the following statements is guaranteed to be true according to Chebyshev's rule? (Write "True" or "False" next to each.) (i) At least 75% of the observations are between 36 and 64. (ii) At least 80%% of the observations are between 34 and 66 (iii) At least 88.9%% of the observations are between 31 and 73 (iv) Fewer than 15% of the observations are below 30 7. Suppose the observations X1, X2,..., Xn have mean 10. Suppose that exactly 75%% of the observations are less than or equal to 15. According to Chebyshev's rule, what is the smallest possible value of the population standard deviation of these observations? 10 0.2 0.15 Frequency 0.05 0 10 20 30 4 50 60 70 10 90 180 Price Figure 3: Histogram of bond prices at default, 1974-1995. (Source: Moody's Investor Services.) 8. Which of the following best describes the data in Figure 3? (Base your answer on the appearance of the histogram. You do not need to do any calculations. Select just one statement below and complete the one you select.) (a) The mean is greater than the median because . (b) The median is greater than the mean because (c) The mean and median are roughly equal because. 9. One proposal that has received little attention from Major League Baseball is to pay pitchers according to the following rule: each pitcher receives a base salary of $4.25 million, minus $0.25 million times his earned run average (ERA). (A lower ERA is associated with better performance.) If this rule were adopted, what would be the correlation between a pitcher's earnings and ERA? (Assume that the ERA cannot exceed 17, so this rule never results in negative earnings. You may also assume a standard deviation of 1.2 for ERA.) 10. Using the data in Figure 4, answer both (a) and (b) below, providing a numerical value for each. (a) The mean of the data in the histogram is (b) The median of the data in the histogram is 11. Cluster I had exams in Finance and Marketing last week. All 60 students in the cluster took both exams. The results were as follows: o Finance: mean = 25, standard deviation = 2 o Marketing: mean = 75, standard deviation = 12 o Correlation between score in Finance and same student's score in Marketing = 0.84 Mary, a student in Cluster V, scored a 30 in Finance and a 90 in Marketing. We are interested in comparing her performance on the two exams relative to the rest of the class. In particular, we would like to make a statement about which of her scores ranked higher compared to the other scores on the same exam. Select one of the choices below and complete the statement you select
