Midterm winter 2023

APS 502 Financial Engineering I March. 2, 2023 Instructions: Closed book and closed notes except for one side of a 3 by 5 inch notecard. Only a simple scienti°c non-°nancial calculator with no programming capability is allowed. Please write neatly as this will aid in providing maximum partial credit. Show All Work. IMPORTANT: Interpretation of the exam questions is part of the exam and so no questions will be taken DURING the exam that ask for clari°cation of the question. You MUST turn in this ques- tion sheet with your answer booklet or else your exam will NOT be marked. Problem 1 (21 points, 7 points each) Suppose that you lend a client $500,000 and the client will pay you back over the next 30 years using equal monthly payments. Interest will be compounded monthly with a nominal rate of 8.5%. (a) What is the e/ective annual rate on this loan? (b) What is the monthly payment? (c) How much does the client owe you immediately after the 60th monthly payment? Problem 2 (24 points, 8 points each) Suppose you have the following three bonds. Assume all coupon payments are annual and that the face value for all bonds is $100. Bond Maturity (years) Yield Coupon rate 1 1 4% 4% 2 2 4.2% 5% 3 3 4.8% 5% (a) Determine the price of each bond. (b) Determine the 1 year, 2 year, and 3 year spot rates. (Hint: The price of each bond that you computed in (a) is also equal to the present value of the cash ±ows of the bond where the present value of each cash ±ow is discounted using the appropriate spot rate.) (c) Using the spot rates from (b) and assuming expectation dynamics illus- trate the Invariance Theorem. Problem 3 (24 points, 6 points each) You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 6%. (a) What is the present value and duration of your tuition obligation? 1
(b) Suppose that you will invest in a single zero coupon bond to o/set the tuition obligation. What maturity and face value for a zero-coupon bond would immunize your obligation? (c) Suppose you buy the zero coupon bond in (b) to immunize your tuition obligation. What happens to your net position (that is, to the di/erence between the value of the zero coupon bond and that of your tuition obligation) if yields immediately increases to 9%. (Answer this by computing the new net position.) (d) Repeat (c) but instead yields decrease immediately to 7%. Problem 4 (16 points) Recall the °shing problem from lecture i.e. you will make a decision to °sh or not °sh at the start of three consecutive periods. If you °sh then you will get 50% of the °sh in the lake and assume that you will get the °sh at the start of the period. If you °sh then the population of the °sh goes back to 90% of what it was at the start of the next period. If you don²t °sh then the population of the lake doubles at the start of the next period. Assume that the per period interest rate is 15% and assume that there are 20 °sh in the lake at the start of the °rst period. Solve for the optimal °shing strategy using dynamic programming. Draw all relevant °gures and show your work. Problem 5 (15 points) Steven is a bond trader that is currently trying to maximize his pro°t in the bond market. Four bonds are available for purchase and sale (see Table 1). The bid and ask price of each bond is shown in Table 2. Steve can buy up to 1000 units of each bond at the ask price or sell up to 1000 units of each bond at the bid price. During each of the next three years, the person who sells a bond will pay the owner of the bond the cash payments in Table 1. For example, if you sell one unit of Bond 1 then you must pay the owner 100, 110, and 1100 at the end of the °rst, second, and third years, respectively. (Of course, if you buy one unit of Bond 1 then you will receive from the seller 100, 110, and 1100 at the end of the °rst, second, and third years, respectively.) Steve²s goal is to maximize his revenue from selling bonds less his payment for buying bonds subject to the constraint that after each year²s payments are received, his current cash position which is amount that is the di/erence between cash (only coupons and face values) received from owning bonds and cash (only coupons and face valiues) paid out due to selling bonds is non-negative. Assume that cash payments are discounted, with a payment of $1 one year from now being equivalent to a payment of 90 cents or 0.90 dollars now. The selling and purchasing of bonds occurs only at the start of the °rst year and there is no selling or purchasing of bonds in later periods. Formulate an LP that maximizes net pro°t from buying and selling bonds subject to the constraints previously described. De°ne all variables in your LP model and explicitly use actual numerical values for all parameters. Why do you think we limit the number of units of each bond that can be bought or sold? 2
Year Bond 1 Bond 2 Bond 3 Bond 4 1 100 80 70 60 2 110 90 80 50 3 1100 1120 1090 1110 Table 1 Bond Bid Price Ask Price 1 980 990 2 970 985 3 960 972 4 940 954 Table 2 3
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