1. Lottery payments A major lottery advertises that it pays the winner $10 million. However this prize money is paid at the rate of $500,000 each year (with the first payment being immediate) for a total of 20 payments. What is the present value of this prize at 10% interest compounded annually? Report your answer in $millions, rounded to two decimal places. So, for example, if you compute the answer to be 5.7124 million dollars then you should submit an answer of 5.71. 1 point 2. Sunk Costs (Exercise 2.6 in Luenberger) A young couple has made a deposit of the first month's rent (equal to $1,000) on a 6-month apartment lease. The deposit is refundable at the end of six months if they stay until the end of the lease. The next day they find a different apartment that they like just as well, but its monthly rent is only $900. And they would again have to put a deposit of $900 refundable at the end of 6 months. They plan to be in the apartment only 6 months. Should they switch to the new apartment? Assume an (admittedly unrealistic!) interest rate of 12% per month compounded monthly. Stay Switch 1 point 3. Relation between spot and discount rates Suppose the spot rates for 1 and 2 years are s1=6.3% and s2=6.9% with annual compounding. Recall that in this course interest rates are always quoted on an annual basis unless otherwise specified. What is the discount rate d(0,2)? Please submit your answer rounded to three decimal places. So, for example, if your answer is 0:4567 then you should submit an answer of 0:457. 1 point 4. Relation between spot and forward rates Suppose the spot rates for 1 and 2 years are s1=6.3% and s2=6.9% with annual compounding. Recall that in this course interest rates are always quoted on an annual basis unless otherwise specified. What is the forward rate, f1,2 assuming annual compounding? Please submit your answer as a percentage rounded to one decimal place so, for example, if your answer is 8.789% then you should submit an answer of 8.8. 1 point 5. Forward contract on a stock The current price of a stock is $400 per share and it pays no dividends. Assuming a constant interest rate of 8% per year compounded quarterly, what is the stock's theoretical forward price for delivery in 9 months? Please submit your answer rounded to two decimal places so for example, if your answer is 567.1234 then you should submit an answer of 567.12 1 point 6. Bounds using different lending and borrowing rate Suppose the borrowing rate rB=10% compounded annually. However, the lending rate (or equivalently, the interest rate on deposits) is only 8% compounded annually. Compute the difference between the upper and lower bounds on the price of an perpetuity that pays A=10,000$ per year. Please submit your answer rounded to the nearest dollar so if your answer is 23,456.789then you should submit an answer of 23457. 1 point 7. Value of a Forward contract at an intermediate time Suppose we hold a forward contract on a stock with expiration 6 months from now. We entered into this contract 6 months ago so that when we entered into the contract, the expiration was T=1 year. The stock price$ 6 months ago was S0=100, the current stock price is 125 and the current interest rate is r=10% compounded semi-annually. (This is the same rate that prevailed 6 months ago.) What is the current value of our forward contract? Please submit your answer in dollars rounded to one decimal place so if your answer is 42.678 then you should submit an answer of 42.7. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Term structure of interest rates and swap valuation Suppose the current term structure of interest rates, assuming annual compounding, is as follows: s1 s2 s3 s4 s5 s6 7.0% 7.3% 7.7% 8.1% 8.4% 8.8% What is the discount rate d(0,4)? (Recall that interest rates are always quoted on an annual basis unless stated otherwise.) Please submit your answer rounded to three decimal places. So for example, if your answer is 0.4567 then you should submit an answer of 0.457. 1 point 2. Swap Rates Suppose a 6-year swap with a notional principal of $10 million is being configured. What is the fixed rate of interest that will make the value of the swap equal to zero. (You should use the term structure of interest rates from Question 1.) Please submit your answer as a percentage rounded to two decimal places. So for example, if your answer is 4.567% or equivalently 0.04567, then you should submit an answer of 4.57. 1 point 3. Hedging using futures Suppose a farmer is expecting that her crop of oranges will be ready for harvest and sale as 150,000 pounds of orange juice in 3 months time. Suppose each orange juice futures contract is for 15,000 pounds of orange juice, and the current futures price is F0=118.65 cents-per-pound. Assuming that the farmer has enough cash liquidity to fund any margin calls, what is the risk-free price that she can guarantee herself. Please submit your answer in cents-per-pound rounded to two decimal places. So for example, if your answer is 123.456, then you should submit an answer of 123.47. 1 point 4. Minimum variance hedge Suppose a farmer is expecting that her crop of grapefruit will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3 months time. She would like to use futures to hedge her risk but unfortunately there are no futures contracts on grapefruit juice. Instead she will use orange juice futures. Suppose each orange juice futures contract is for 15,000 pounds of orange juice and the current futures price is F0=118.65 cents-per-pound. The volatility, i.e. the standard deviation, of the prices of orange juice and grape fruit juice is 20% and 25%, respectively, and the correlation coefficient is 0.7. What is the approximate number of contracts she should purchase to minimize the variance of her payoff? Please submit your answer rounded to the nearest integer. So for example, if your calculations result in 10.78 contracts you should submit an answer of 11. 1 point 5. Call Options Consider a 1-period binomial model with R=1.02, S0=100, u=1/d=1.05. Compute the value of a European call option on the stock with strike K=102. The stock does not pay dividends. Please submit your answer rounded to two decimal places. So for example, if your answer is 3.4567 then you should submit an answer of 3.46. 1 point 6. Call Options II When you construct the replicating portfolio for the option in the previous question how many dollars do you need to invest in the cash account? Please submit your answer rounded to three decimal places. So for example, if your answer is 43.4567 then you should submit an answer of 43.457. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Quiz instructions Compute the price of an American call option with strike K=110 and maturity T=.25years. 1 point 2. Quiz instructions Compute the price of an American put option with strike K=110 and maturity T=.25years. 1 point 3. Quiz instructions Is it ever optimal to early exercise the put option of Question 2? Yes No 1 point 4. Quiz instructions If your answer to Question 3 is "Yes", when is the earliest period at which it might be optimal to early exercise? (If your answer to Question 3 is "No", then you should submit an answer of 15 since exercising after 15 periods is not an early exercise.) 1 point 5. Quiz instructions Do the call and put option prices of Questions 1 and 2 satisfy put-call parity? Yes No 1 point 6. Quiz instructions Compute the fair value of an American call option with strike K=110 and maturity n=10 periods where the option is written on a futures contract that expires after 15 periods. The futures contract is on the same underlying security of the previous questions. 1 point 7. Quiz instructions What is the earliest time period in which you might want to exercise the American futures option of Question 6? 1 point 8. Quiz instructions Compute the fair value of a chooser option which expires after n=10 periods. At expiration the owner of the chooser gets to choose (at no cost) a European call option or a European put option. The call and put each have strike K=100 and they mature 5 periods later, i.e. at n=15. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1 point 1. Quiz instructions Compute the price of a zero-coupon bond (ZCB) that matures at time t=10 and that has face value 100. Submission Guideline: Give your answer rounded to 2 decimal places. For example, if you compute the answer to be 73.2367%, submit 73.24. 1 point 2. Quiz instructions Compute the price of a forward contract on the same ZCB of the previous question where the forward contract matures at time t=4. Submission Guideline: Give your answer rounded to 2 decimal places. For example, if you compute the answer to be 73.2367%, submit 73.24. 1 point 3. Quiz instructions Compute the initial price of a futures contract on the same ZCB of the previous two questions. The futures contract has an expiration of t=4. Submission Guideline: Give your answer rounded to 2 decimal places. For example, if you compute the answer to be 73.2367%, submit 73.24. 1 point 4. Quiz instructions Compute the price of an American call option on the same ZCB of the previous three questions. The option has expiration t=6 and strike =80. Submission Guideline: Give your answer rounded to 2 decimal places. For example, if you compute the answer to be 73.2367%, submit 73.24. 1 point 5. Quiz instructions Compute the initial value of a forward-starting swap that begins at t=1, with maturity t=10 and a fixed rate of 4.5%. (The first payment then takes place at t=2 and the final payment takes place at t=11 as we are assuming, as usual, that payments take place in arrears.) You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.) Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be -220,432.23, submit -220432. 1 point 6. Quiz instructions Compute the initial price of a swaption that matures at time t=5 and has a strike of 0. The underlying swap is the same swap as described in the previous question with a notional of 1 million. To be clear, you should assume that if the swaption is exercised at t=5 then the owner of the swaption will receive all cash-flows from the underlying swap from times t=6to t=11 inclusive. (The swaption strike of 0 should also not be confused with the fixed rate of 4.5% on the underlying swap.) Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be -220,432.23, submit -220432. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Quiz instructions Assume b=0.05 is a constant for all i in the BDT model as we assumed in the video lectures. Calibrate the ai parameters so that the model term-structure matches the market term-structure. Be sure that the final error returned by Solver is at most 108. (This can be achieved by rerunning Solver multiple times if necessary, starting each time with the solution from the previous call to Solver. Once your model has been calibrated, compute the price of a payer swaption with notional $1M that expires at time t=3 with an option strike of 0. You may assume the underlying swap has a fixed rate of 3.9% and that if the option is exercised then cash-flows take place at times t=4,,10. (The cash-flow at time t=i is based on the short-rate that prevailed in the previous period, i.e. the payments of the underlying swap are made in arrears.) Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457. 1 point 2. Quiz instructions Repeat the previous question but now assume a value of b=0.1. Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be 10,456.67, submit 10457. 1 point 3. Quiz instructions Construct a n=10-period binomial model for the short-rate, ri,j. The lattice parameters are: r0,0=5%, u=1.1, d=0.9 and q=1q=1/2. This is the same lattice that you constructed in Assignment 5. Assume that the 1-step hazard rate in node (i,j) is given by hij=abji2 where a=0.01 and b=1.01. Compute the price of a zero-coupon bond with face value F=100 and recovery R=20%. Submission Guideline: Give your answer rounded to two decimal places. For example, if you compute the answer to be 73.2367, submit 73.24. 1 point 4. Quiz instructions The true price of 5 different defaultable coupon paying bonds with non-zero recovery are specified in worksheet ???????????????????????????????????????????? in the workbook ????????????????????????????????????????????_????????????.????????????????. The interest rate is r=5% per annum. Calibrate the six month hazard rates ???????? to ???????????? to by minimizing the ???????????????????????????????? ensuring that the term structure of hazard rates are non-decreasing. You can model the non-decreasing hazard rates by adding constraints of the form ????????????????,,????????????????????????. Report the hazard rate at time 0 as a percentage. Submission Guideline: Give your answer in percent rounded to two decimal places. For example, if you compute the answer to be 73.2367%, submit 73.24. 1 point 5. Quiz instructions Modify the data on the ???????????????????????????????????????? worksheet in the workbook ????????????????????_????????????_????????????.???????????????? to compute a par spread in basis points for a 5yr CDS with notional principal N=10 million assuming that the expected recovery rate R=25%, the 3-month hazard rate is a flat 1%, and the interest rate is 5% per annum. Submission Guideline: Give your answer in basis points rounded to two decimal places (1 bps = 0.01%). For example, if you compute the answer to be 73.2367 bps, submit 73.24. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Quiz Instructions (Level-Payment Mortgages) Compute the monthly payment on a 30-year level payment mortgage assuming an annual mortgage rate of 5% and an initial mortgage principal of $400,000. Submission Guideline: Give your answer rounded to two decimal places. For example, if you compute the answer to be $73.2367, submit 73.24. 1 point 2. Quiz Instructions (Mortgage Pass-Throughs) Consider a $400 million pass-through MBS that has just been created (so the 'seasoning' of the pass-through is equal to 0). The underlying pool of mortgages each has a maturity of 20 years and an annual mortgage coupon rate of 6%. The pass-through rate of the mortgage pool is 5%. Assuming a prepayment multiplier of 100 PSA what is the total amount of interest paid to the pass-through investors? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. 1 point 3. Quiz Instructions (Mortgage-Pass Throughs) Referring to the same mortgage pass-through of the previous question, what is the total amount of the prepayments? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. 1 point 4. Quiz Instructions (Mortgage-Pass Throughs) Referring to the same mortgage pass-through of the previous question, what is the total amount of the prepayments if the rate of prepayments increases to 200 PSA? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. 1 point 5. Quiz Instructions (Principal-Only MBS and Interest-Only MBS) Suppose we construct principal-only (PO) and interest-only (IO) mortgage-backed securities (MBS) using the mortgage pass-through of the previous questions. Assume a prepayment multiplier of 100 PSA. What is the present value of the PO MBS if we use an annual risk-free rate of 4.5% to value the cash-flows? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. 1 point 6. Quiz Instructions (Principal-Only MBS and Interest-Only MBS) Referring to the previous question, what is the value of the IO MBS? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. 1 point 7. Quiz Instructions (Principal-Only MBS and Interest-Only MBS) Referring to the previous question, what is the average life of the IO MBS? Submission Guideline: Give your answer in years rounded to two decimal places. For example, if you compute the answer to be 12.1234 years, submit 12.12. 1 point 8. Quiz Instructions (Principal-Only MBS and Interest-Only MBS) Suppose now that you purchased the IO MBS of the previous question and that the price you paid was the same price that you calculated in the previous question. The risk-free interest rate suddenly changes from 4.5% to 3.5%. Everything else stays the same. How much money have you made or lost on your investment? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. 1 point 9. Quiz Instructions (Principal-Only MBS and Interest-Only MBS) Referring to the previous question, suppose the risk-free interest rate suddenly changes from 4.5% to 3.5% and that the pre-payment multiplier changes from 100 PSA to 150 PSA. How much money have you made or lost on your investment in the IO MBS? Submission Guideline: Give your answer in millions rounded to two decimal places. For example, if you compute the answer to be $123,456,789,12, submit 123.46. Best regards