ONLINE EXAM

Online Exam Full Course Code E261 A SPR24 Course Title Fixed Income Securities Faculty Suleyman Basak Date of Exam 17 March 2024 STUDENT DECLARATION By downloading this exam paper, you have: • • • • Read and understood the Online Exams Policy and Exam Instructions. Agreed to complete the exam individually without discussion or collaboration with others. Agreed to upload and submit the file(s) on Canvas before the final 15 minutes of the exam, and only use the final 15 minutes to resolve any technical issues. Understood that submissions received after the deadline will be subject to a late submission penalty as per section 5 of the Online Exams Policy. Document Name: Online Exams Cover Sheet (September 2021) Version: 2021/1.0 Approved by: Academic Policy Committee Date: Fixed Income Securities – Spring 2024 Final Examination Prof. Süleyman Başak Instructions • This is an open-book take-home exam which is being released to you online on Canvas and you are required to submit it through Canvas. The exam is an individual assignment. You may not discuss its content or the solutions with anyone. • Please write your LBS student id number legibly on the top of the first page of your solutions. • There are 4 questions on this examination with a total of 100 points. • The examination lasts for 2.5 hours, including the time to upload your solutions on Canvas. • Although you have access to your notes and other materials, it is highly recommended that you use your crib sheet, and a calculator, to solve the questions. • To receive full credit, you MUST show all the relevant algebraic steps in deriving your solutions. Partial credit is given to answers that are numerically incorrect but that show a correct understanding of the solution method. If the question is not clear, state your assumptions and if they are reasonable, you will be given credit. • Honour Code. Please note that by taking this exam, you comply with following honour code: “I confirm that I have had no prior knowledge of the content of this exam and the answers to the exam are all my own work. I also confirm that I will not knowingly disclose any information from this exam to others. I also understand that during the exam, I am not permitted in any way to communicate with any person.” 1 Questions 1. Forwards, Arbitrage, and Pricing – 20 points Today is year 0. Consider three bonds A, B and C. Bond A pays $9,231 in year 1, and has a current price of $8,693.43. Bond B pays $200 in year 2 and $200 in year 3, and has a current price of $346.01. Bond C pays $100 in year 3, and has a current price of $86.07. What is the implied continuously compounded forward rate between years 1 and 2 (i.e., 0 ṙ1,2 )? 2. Dollar Duration, Dollar Convexity, and Risk Management – 30 points After his retirement, Jerry Basak decides to invest $100 million of his fortune in fixed income markets. He hires you as his personal consultant. The current time t = 0 (continuously compounded) zero-coupon yield curve is flat at 15% across all maturities. Consider the following securities: • Security A – a 3-year zero-coupon bond with face value $100. • Security B – a 3-year bond with face value $100 and annual coupon 10%. Suppose you invest Jerry’s money in Securities A and B with the objective of currently achieving a (fully) dollar duration-hedged portfolio (i.e., ∆$ = 0). Moreover, suppose you believe that over the next instant, there is: • 60% probability that there will be a parallel upwards shift in the (continuously compounded) zero-coupon term structure to 25%. • 40% probability that there will be a parallel downwards shift in the (continuously compounded) zero-coupon term structure to 10%. (a) (15 points) What is the trading strategy in Securities A and B that will achieve the dollar duration-hedged portfolio at t = 0? (b) (7 points) What is your estimate of the expected value of the dollar durationhedged portfolio over the next instant? (c) (8 points) What is the expected value of the dollar duration-hedged portfolio over the next instant as predicted by the combined dollar duration-convexity model? 2 3. Exotic Bond Options and Risk Measurement – 25 points Consider the following CIR tree of the stochastic evolution of zero prices with a face value of $1. 1 d2 = 0.93 1 d3 = 0.88 3 d4 = 0.93 d = 0.95 1 d4 = 0.83 H 0 1 3 d5 = 0.88 H 1 d5 = 0.79 2 d3 = 0.95 H 0 d2 = 0.90 HH d = 0.90 0 d3 = 0.86 H 2 d4 = 0.86 HH HH 2 5 0 d4 = 0.82 d = 0.97 H HH d = 0.97 HH 1 2 0 d5 = 0.78 d = 0.93 1 3 H 3 4 H 3 d5 = 0.93 d = 0.89 d = 0.98 1 4 2 3 HH d = 0.95 1 d5 = 0.85 H 2 4 HH 2 d5 = 0.91 H HH d = 0.99 3 4 3 d5 = 0.96 t=0 t=1 t=2 HH HH H 4 d5 = 0.95 H HH H H 4 d5 = 0.98 H HH H H 4 d5 = 0.99 t=3 An innovative investment bank has decided to offer a new security called a Başaklet. This security is issued at t = 0 and matures at t = 1. A Başaklet gives its holder the right (the option) to pick the best performing security over the next period, out of 1-year, 3-year and 5-year zeros. In particular, the random payoff to one unit of a Başaklet at t = 1, is the highest payoff to a $1 investment in either the 1-year zeros, the 3-year zeros or the 5-year zeros purchased at t = 0. (a) (7 points) Order the 1-year, 3-year and 5-year zeros in terms of their “riskiness”, from the riskiest to the least risky. Explain your ranking. Hint: You may want to express the prices and payoffs of the zeros as a per dollar ($1) investment in the various zero coupon bonds. (b) (12 points) What is the price of one unit of a Başaklet at t = 0? Where does the riskiness of a Başaklet fall in the ranking above in part (a)? Explain your answer. (c) (6 points) What is the Cox-Ingersoll-Ross delta ∆CIR of one unit of a Başaklet at t = 0? 3 t=4 4. Hurricane Bonds, Risk Measurement and Reinsurance – 25 points The following is a CIR tree of the stochastic evolution of zero prices (with face value $1) with T = 2 years and h = 1 year. 0 d1 = 0.95 0 d2 = 0.90 0 d3 = 0.86 1 d2 = 0.93 1 d3 = 0.88 XXX XXX XXX 1 d2 = 0.97 X 1 d3 = 0.93 t=0 t=1 XX XXX XXX XX XXX XXX XXX X 2 d3 = 0.91 2 d3 = 0.95 2 d3 = 0.98 t=2 Today is year 0. IPG Insurance Company has just issued a 3-year Hurricane bond whose coupon payments are indexed to the annually compounded one year market rate plus 400 basis points. The market rate is the annually compounded one year interest rate set at the previous time period. The annual coupons, which are based on the face value of $1 million, are guaranteed. The principal repayment, however, is tied to the occurrence of major hurricanes. If IPG does not experience a hurricane which causes more than $1 billion in losses in the next 3 years, you will receive the entire face value of $1 million back. Otherwise, your recovery will depend on the severity of the hurricane. The following table presents the distribution of possible hurricane losses three years later. Event Hurricane Loss Probability Recovery value(∗) A 2 billion 0.005 0% B 1.5 billion 0.01 50% C 1.25 billion 0.005 75% D < 1 billion 0.98 100% (*) Recovery value is expressed as the fraction of the principal which you will get back at maturity. (a) (7 points) Determine an optimistic upper bound for the price of the Hurricane bond today. In other words, how much would the bond be worth today if you knew for sure that no hurricanes would occur over the next 3 years? (b) (6 points) Determine a pessimistic lower bound for the price of the Hurricane bond today. In other words, how much would the bond be worth if you knew for sure hurricane event A would occur? (c) (6 points) A risk neutral reinsurance company is willing to sell you a reinsurance contract, which will guarantee repayment of your principal 3 years from today. What is the price of this reinsurance contract today? Recall, risk-neutrality implies the company only cares about expected values. 4 (d) (6 points) You have decided to purchase the reinsurance contract from part (c), but rather than paying for it upfront, you have agreed to issue the reinsurance company a coupon bond with a 3-year maturity and a face value equal to 80% of the price of the reinsurance contract today. What will the coupon rate, x%, have to be for the reinsurer to agree? Assume the reinsurer believes your bond is default free. 5

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