Fall 2012 Solution Keys for an Additional Practice Exam

Transcription

1 Derivatives Professor Michel A. Robe Fall 2012 Solution Keys for an Additional Practice Exam

2 Derivatives Professor Michel A. Robe MT Exam: Spring 2010 Data for Questions 1 to 4. Consider the following term structure of futures prices and trading data for natural gas (NG) on the NYMEX. All prices are quoted in US dollars per million British thermal units or $/MMBtu; the standard contract size is for 10,000 MMBtu). 3/6/2008 Session Overview Open High Low Settle Change April May June July Aug Sep Question 1 (2.5 points) At what prices did the July 2008 settle on Thursday, March 6, 2008? On March 5? (i) (ii) (iii) (iv) (v) (vi) and 9.875, respectively; and 9.884, respectively; on both days; and 9.880, respectively; None of the above four combinations; Not enough information. Answer: (v) is the answer. The price at the Thursday close ( settle ) was $ Since the change from the previous settle was -$0.004, the March 5 settle must have been $9.888=9.884-(-0.004).

3 Question 2 (10 points) Consider again the same data. At the market open on March 6, you took a long position in a July 2008 futures. Your FCM, a clearing member of the NYMEX, asked for an initial margin of $3,000; for simplicity, assume that the maintenance margin is also $3,000. His brokerage fees are $25 for a round-trip, i.e., to buy and sell a contract. (a) What delivery price have you locked in? (2 points) (i) (ii) (iii) (iv) (v) None of the above; (vi) Not enough information. Answer: (iv) is the answer the open price was $ (b) If you decided to close out your position at the end of the day, what was your profit or loss on March 6? Explain briefly. Remember that the contract size is 10,000 MMBtu. (3 points) Answer: From the open to the close, the price rose by $0.004 = $( )/MMBtu. Since you went long and the contract size is 10,000 MMBtu, you made $40 that day. Net of the round-trip brokerage fee, you earned $15. (c) Assuming that there was no intra-daily marking to market (intra-day marking to market would only happen on an extremely volatile day, which was clearly not the case on March 6), please detail all your cash-flows. Does your profit or loss (computed in part b) match your total cash-inflow or cash-outflow at the end of the day? Explain briefly why or why not. (5 points) Answer: Yes. Since you closed out your position, you recovered your initial margin of $3,000. Hence, the sum total of your cash-flows that day (-$3,000 - $25 + $40 + $3,000) also matches your profit ($40). Note that, if you had not closed out your position, then the net cash flow would have been -$3,000 - $25 + $40 = -$2,985.

4 Question 3 (10 points) Consider again the same data. Please use the settlement prices on March 6 to answer the following questions. (a) What is the price of the nearby futures? (2.5 points) (i) (ii) (iii) (iv) (v) None of the above; Answer: The nearby contract is the nearest-expiring contract in the present case, it is the April contract (which, somewhat confusingly, actually expires in late March!). (b) On March 6, a practitioner would say that the term structure of futures prices for natural gas was: (1.5 points) (i) (ii) (iii) in contango backwardated Not enough information Explain briefly. (1 point) Answer: The market is contangoed, as the term structure of futures prices is upward sloping (i.e., the settlement prices are higher the further off is the expiration date of the contract: < < < < < (c) Assume that NG storage costs are 1% (annualized) and that the spot price of natural gas is $ 9.70 per MMBtu. U.S. interest rates, for maturities up to 3 months, stand at 2.4% (LIBOR). Using the May 2008 contract (which matures in about 2 months), can you compute the convenience yield? If so, explain and show your work. If not, explain why not. (5 points) Answer: F 2 = S (1 + r/6 + u/6 y/6) so y = 6 * [ % % 9.776/9.70] = -1.3% Intuitively, the 2-month futures should have traded at about 3.4%/6 = 0.567% more than the spot, or $ But, as we see from the Bloomberg prices, it was trading well over that, at $9.776 which is 0.21%, or 1.3% on an annualized basis, more than what it should be. This negative convenience yield indicates the futures price is too high relative to spot (given the cost of carry). In other words, there seems to have been an arb opportunity in which one could have bought spot and carried while shorting the May futures.

5 Question 4 (10 points) Consider again the same data. Please use the settlement prices on March 6 to answer the following questions. (a) What should be the OTC (over-the-counter) price of 10,000 MMBtu of NG for forward delivery in late April i.e., 6 weeks from now. Explain briefly, and state any assumptions you make to arrive at your conclusion. (2.5 points) (Hint: Don t make the question more complicated than it is) Answer: Insofar as the futures and forward prices are very, very close you can use the May futures price of $9.776 / MMBtu (May is the contract expiring in late April) as a substitute for the OTC 6-weeks out OTC price. (b) Use your answer in part (a) to answer this question (I won t penalize you twice if that answer was incorrect). If you sell 10,000 MMBtu of natural gas OTC, for delivery in late April, what will be your cash-flow today? At delivery? Between today and the contract s delivery date? Assume that this OTC forward contract is commodity-settled. (2.5 points) Answer: Today or any time before delivery, no cash-flow takes place (it s a forward), assuming away any collateral requirement to which you might agree with your OTC counterparty. At delivery, since you sold forward, you d receive $97,760 from your OTC counterparty. (c) Would your answer change if the OTC contract were cash-settled (i.e., if it were an NDF or non-deliverable forward )? If it does, please detail your cash-flows using the NDF. (5 points) (Hint: At this point, you don t know the spot price at maturity; so, just call it ST.) Answer: Today or any time before delivery, same thing as in part b it s a forward, so there is no cash-flow until maturity (I am again assuming away any collateral requirement to which you might agree with your OTC counterparty). At delivery, however, you d receive $97,760-10,000*S T from your OTC counterparty (where S T is the as-yet-unknown spot price of natural gas on the May-futures delivery day).

6 Question 5. (10 points) (a) The 3-month euro-currency rates (used for interbank transactions) on March 6, 2007 were: US$ 5.57%, Japanese Yen 0.57%. On the basis of those rates, would you expect the Yen to have been at a forward discount or premium against the $? Explain briefly. Answer: This is a Covered IRP question. The interest-rate differential is 5% (annualized) in favor of the USD, so the Yen should trade at an annualized forward premium of about 5% to compensate. Otherwise, there would be scope for covered interest-rate arbitraging. (b) You are provided with the following Bloomberg (mid-point) quotes on the same day (3-6-07). rate ($/1 ) swap rate annualized fwd premium spot N-A N-A 30-day fwd / * 12 = 5.22% 90-day fwd / * 4 = 5.18% 180-day fwd / * 2 = 5.32% What swap rates (in points) are implied by those quotes? What are the corresponding annualized forward premia or discounts (in percents)? Place your answers in the table above, and explain briefly (i.e., show your work) in the space provided below. Answer: See completed table above. (c) Should there be a relationship between one of the percentage premia or discounts you computed in part (b) and your answer in part (a)? Which one? Why? Explain briefly. Answer: Yes. According to covered IRP, the 90-day annualized premium in part (b) should be close to the premium implied by the annualized 3-month (i.e., 90-day) rates in part (a).

7 Question 6 (10 points) (a) Sallie Mae borrows long-term with bonds and lends shorter-term. It enters into a hedge position using 3 month FRAs (e.g., a 9 against 12 FRA) at 5%, and holds this position until the expiration date when the contract expires with a spot rate of 4%. Assuming that the FRA notional is $1m, Sallie Mae will a. profit by receiving cash of $2,500. b. own a 5% dollar deposit. c. lose by having to pay cash of $2,500. d. none of the above. Answer: Not Mid-Term Exam Material, Fall 2012 (b) You observe the following prices. Spot market palladium is selling for $800, the one-year T- bill rate is 1%, and palladium futures for delivery in one year are priced at $812. Assuming away storage costs and convenience yields, which of the following transactions would you expect an arbitrageur to be most likely to undertake? Explain briefly. a. Short T-bills and spot market palladium, go long palladium futures. b. Go long spot market palladium and short palladium futures. c. Go long T-bills and palladium futures, short spot market palladium. d. Short T-bills and palladium futures, go long spot market palladium. Answer: The answer is d. Given spot-futures parity, we should have F= S (1 + r + u y) = $808. The futures, in other words, is overpriced so short it, and cover by buying palladium spot. To do so, borrow i.e., short T-bills. (c) If a speculator wants to bet that interest rates are going to fall by more than the market consensus would indicate, (s)he should a. go long T-bill or T-bond futures b. short stock index futures c. not enough information d. go long the distant gold contracts Answer: Not Mid-Term Exam Material, Fall 2012 (a would be the answer)

8 Bonus Question 1. (5 points) (a) Which of the following is the best example of an "inter-commodity" spread position a. being long Ru2000 (Russel 2000) futures and short S&P 500 futures. b. being long an S&P 500 index fund and short S&P 500 futures. c. being both long and short March T-bill futures. d. being short both June T-bond futures and June T-bill futures. Answer: a is an example of cross- or inter-commodity spread. (b is a hedge; c is nothing net zero; d is a directional position). (b) Short positions in Euro$ futures are more likely to make money when I) and short positions in stock index futures are more likely to make money when II) a. I) interest rates rise...ii) stock prices rise b. I) interest rates fall...ii) stock prices fall c. I) interest rates rise...ii) stock prices fall d. I) interest rates fall...ii) stock prices rise Answer: Not Mid-Term Exam Material, Fall 2012 Bonus Question 2. (5 points) You are a speculator. The S&P 500 stock index futures for June delivery is currently trading at 1,320. The spot value of the index is 1,300, but you expect the S&P to rise to 1,340 by June. If the risk-free rate is 2%, and the risk-adjusted discount rate on stock investments is 12%, should you go long or short futures? The dividend yield on S&P stocks is 2% per year, but you are NOT allowed to buy the S&P spot. Explain. Answer: It s a bonus!