Exam 24 September 2017, questions and answers PDF

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Venue Student Number Research School of Finance, Actuarial Studies Statistics EXAMINATION Semester 2 2017 COMMENTS SUGGESTED SOLUTION KEY FINM2002 DERIVATIVES FINM7041 APPLIED DERIVATIVES Writing Time: 2 hours Reading Time: 15 minutes Exam Conditions: Central Examination Students must return the exa...

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Venue

____________________

Student Number

____________________

Research School of Finance, Actuarial Studies & Statistics EXAMINATION Semester 2 - Mid-Semester, 2017 COMMENTS + SUGGESTED SOLUTION KEY

FINM2002 DERIVATIVES / FINM7041 APPLIED DERIVATIVES Writing Time: 2 hours Reading Time: 15 minutes

Exam Conditions: Central Examination Students must return the examination paper at the end of the examination This examination paper is not available to the ANU Library archives

Materials Permitted In The Exam Venue: (No electronic aids are permitted e.g. laptops, phones) Paper Based Dictionary Non-Programmable Calculator Materials to Be Supplied To Students: Script books (20 page) Instructions to Students: 1. This exam paper comprises a total of 7 pages. Please ensure your paper has the correct number of pages. 2. The exam includes a total of 4 questions with sub-parts. The questions are of unequal value, with marks indicated for each question. You must attempt to answer all questions. 3. Do not round calculations until providing your final answer to each question. Final answers should be rounded up to 4 decimal places. Penalty will apply. 4. Include all workings for each question. Marks will not be awarded for answers that do not include workings. 5. Ensure to include your student number on your answer book. 6. You should start your solutions to each question on a new page. 7. Organize your answers to sub-parts in the same order as they appear on the exam paper.

Total Marks = 100 This exam counts towards 25% of your final assessment.

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Question 1 (30 marks) For the following sub-parts within this question, assume the treasury yield curve for all maturities is flat at 4% p.a, continuously compounded. Parts (a) to (d) involve direct calculation of forward/futures prices, covered in Lecture 2. Part (e) involves the mechanics of closing out a futures position and valuation of forward/futures contracts, covered in Workshop 2/Extension to Lecture 2 during Week 3’s Tuesday class. Part (f) involves computing the fair price of forward/futures prices (as in part (a)), and exploiting arbitrage opportunities if market price is not the fair price – Lecture 2 slides and covered in Week 2’s Friday class from the 1h11’30” mark to the 1h15’20” mark. The nature of this part is identical to exploiting arbitrage opportunities when the lower bound of option prices are violated (Q3a of this exam). a) Shares on Woolworths Ltd. (WOW) is trading at $25.30 per share today. What should be the price on a forward contract that delivers one WOW share 6 months from now? Assume dividends won’t be paid within the period. (2 marks) Solution: F0 = S0e(rt) (1 mark) = 25.30 e(0.04x6/12) (0.5 mark) = 25.8110939 ≈ $25.8111 (0.5 mark) b) Shares on Woolworths Ltd. (WOW) is trading at $25.30 per share today. What should be the price on a forward contract that delivers one WOW share a year from now? Woolworths is expected to pay $1 in cash dividend 4 months from now. (3 marks) Solution: F0 = [S0 – PV(dividend)]e(rt) (1 mark) = [25.30 – 1 e(-0.04x4/12)]e(0.04x1) (1 mark) = [25.30 – 0.986755162]e(0.04x1) (0.5 mark) = 25.30548718 ≈ $25.3055 (0.5 mark) c) Consider a long forward contract to purchase a coupon bond whose current price is $950. The forward contract matures in ten months, a coupon payment of $40 is expected after 8 months. What is the forward price? (3 marks) Solution: F0 = [S0 – PV(coupon)]e(rt) (1 mark) = [950 – 40 e(-0.04x8/12)]e(0.04x10/12) (1 mark) = [950 – 38.94742997]e(0.04x10/12) (0.5 mark) = 941.9328003 ≈ $941.9328 (0.5 mark)

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d) Consider a 1-year futures contract on the ASX S&P200. Suppose that the stocks underlying the index provide a dividend yield of 2% p.a. continuously compounded, that the current value of the index is 5690. What should be the futures price? (3 marks) Solution: F0 = S0e(r-q)t (1 mark) = [5690]e((0.04-0.02)x1) (1 mark) = 5804.945625 ≈ 5804.9456 index points Each contract will hedge a dollar value of = 5804.945625 x 25 = $145123.6406 (1 mark) e) Gold was trading at $1600/oz on May 1. You had been following market events closely and you felt that given the combination of uncertain stock market outlook, geopolitical tensions, and demand for gold, you expect gold prices to rise for the latter part of 2017. You are optimistic of your view and you are confident it’s the time to implement a strategy right away (i) The date is May 1, describe how you would use futures to trade on your outlook for gold. Assume you can take position in one November gold contract that matures on November 30 and the market price is at the no-arbitrage level. (1 mark) Solution: Since you expect prices will go up, you will take a LONG position in the futures contract today. If gold price does move up later, you can make a profit by closing out your position with a short position in the November gold contract. (ii) Fast forward to July 1, the price of gold had dropped to $1500/oz. You started to doubt yourself and decided close out your existing position from May. Describe what you need to do to close out your position. Assume quoted futures prices are at their no-arbitrage levels. Calculate the value of your position. (5 marks) Solution: On May 1, when you took a long position in the November gold, the price should’ve been FMay = SMaye(rt) = 1600 e(0.04x7/12) (1 mark) = 1637.772296 On July 1, the price would have been instead FJuly = SJulye(rt) = 1500 e(0.04x5/12) (1 mark) = 1525.209496 You had a long position that had a delivery price of FMay and is closing out by taking a short position in the November futures, now with a delivery price of FJuly, (1 mark) so your payoff on the futures position would be (FJuly – FMay) e-rt (1 mark) = (1525.209496 – 1637.772296) e-0.04x5/12 = -110.7023014. So you realize a loss of $110.7023 (1 mark). (iii) Assume today is September 1, the price of gold has climbed to $1700/oz. If you didn’t lose confidence in July and had waited until today, what would have been the value of your Semester 2 - Mid-Semester, 2017

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position today if you kept the position you took in May (i.e., in part (i))? Assume quoted futures prices are at their no-arbitrage levels. Calculate the value of your position. (4 marks) Solution: On July 1, the price would have been instead FSept = SSepte(rt) = 1700 e(0.04x3/12) (1 mark) = 1717.085284 You had a long position that had a delivery price of FMay and is marked to market against/ or closed out by taking a short position in the November futures, now with a delivery price of FJuly, so the value on the futures position would be (FSept – FMay) e-rt (1 mark) = (1717.085284 – 1637.772296) e-0.04x3/12 = 78.52381023. So you realize a gain of $78.5238 (2 marks). f) The spot price of silver is $18/oz. Assume zero storage cost: (i) What should be the forward price on silver for delivery in one year? (1 mark) Solution: F0 = S0e(rt) = 18 e(0.04x1) = 18.73459394 ≈ $18.7346 (1 mark) (ii) If you see a one-year forward contract on silver that is quoted at $19/oz, explain what you would do to capture an arbitrage profit and show all cash flow involved. (8 marks) Solution: The quoted price for the forward is too high relative to the no-arbitrage level (1 mark). We can set up an arbitrage strategy – buy-low-sell-high: (Total of 7 marks, 1 mark for each point) Today Short the quoted forward contract Borrow $18 (to buy an ounce of silver) Buy silver at spot price $18 Net cash-flow = 0 At maturity, Repay 18 e(0.04x1) = 18.73459394 Deliver the gold by the terms of the forward contract for $19 The arbitrage profit is $19 – 18.73459394 = $0.265406065

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Question 2 (27 marks) You work in FINM Bank on the swaps desk. Your boss has instructed that you set up an interest rate swap for two of your clients, on a notional principal of $100M. Acting as the intermediary, your company is going to net 25 basis points per annum. The details of the requirements are the following: Optin Inc wishes to borrow at a floating rate of interest. Rigidt Ltd wishes to borrow at a fixed rate of interest. After accounting for the 25 bps p.a. fee for FINM Bank, the swap should be set up such that 60% of the remaining gains goes to Optin and 40% goes to Rigidt. a) On their own, Optin and Rigidt have been quoted the following rates per annum (adjusted for differential tax effects). Please set up the interest rate swap. You must draw a diagram illustrating all cash flows with directions, as well as show clearly how FINM Bank receives its 25bps per annum fee. In your diagram and calculations, the borrowed floating rate must be paid all the way across the swap. (11 marks)

Floating Rate

Optin Inc.

Rigidt Ltd.

LIBOR + 1.1%

LIBOR + 3.85%

2.25%

6.25%

Fixed Rate

Solution: This question is almost identical to Q3 of Tutorial 3, as well as Quiz 3. The original example is from Lecture 3 slides (slide 20 to 29) and covered in Week 3’s Friday lecture. Optin has a comparative advantage in fixed (by 4%) (1 mark) Rigidt has c.adv. in floating (by 2.75%) (1 mark) => total gains from trade = 4-2.75=1.25%=125bps (1 mark) Of the 125bps, 0.25%(25bps) should go to FINM bank, leaving 100bps to be divided between Optin and Rigidt, so: Optin should be 60bps (0.6%) better off (1 mark) => so it should end up paying LIBOR + 0.5% Rigidt would be 40bps (0.4%) better off (1 mark) => so it should pay fixed 5.85% The question asked for a diagram of cash flow and directions: The arrows of the borrowing by both Optin and Rigidt from their outside lenders must be going outwards, as they need to pay the outside lender, cash flow goes out. Pays Rigidt floating, LIBOR+3.85%

Borrow from outside lender, fixed 2.25%

Optin Receives from Rigidt, via FINM and after FINM deducts fee, 5.6%

FINM Passes Optin’s payment to Rigidt in full. Passes Rigidt’s payment to Optin but keeps 0.25% as fee.

Semester 2 - Mid-Semester, 2017

Borrow from outside lender, LIBOR +3.85%

Receives from Optin, via FINM, LIBOR+3.85%

Rigidt Pays Optin fixed, 5.85%

FINM2002/7041 Derivatives Page 5 of 16

Optin Borrow at comparative advantage Payment terms

Payment terms

Net borrowing cost

Borrow from outside lender @ fixed 2.25% (1 mark) Pays Rigidt LIBOR + 3.85% (1 mark) Receives from Optin 5.6%

FINM Investment

Rigidt

Passes, in full, to Rigidt

Borrow from outside lender @ floating LIBOR +3.85% (1 mark) Gets from Optin LIBOR + 3.85% (1 mark)

Keeps 0.25% and passes 5.6% to Optin (1 mark)

Pays LIBOR +0.5%

Pays Optin fixed 5.85% (1 mark)

Pays fixed 5.85%

b) What do the rates that Optin and Rigidt could obtain on their own reflect on their individual credit rating status? Discuss the reasons for why might Optin and Rigidt find this interest rate swap attractive. (5 marks) Solution: The question covers the intuition & function of IRS, which was extensively discussed in Lecture 3 while I explained why two companies would enter into an IRS, Lecture 3 (slides 20 & 21) and available in Week 3’s Friday lecture recording (from the 1h20’10’” mark to the 1h24’32” mark). Compared to Rigidt, Optin can borrow at lower rates in both markets (fixed and floating). This indicates Optin has a better credit rating relative to Rigidt (1 mark). However, Optin still wishes to enter into the swap in part because it believes interest rates may decrease in the future, so it wants to have exposure to the floating rate (1 mark). By entering into the swap, it has both exposure to the floating rate, and is able to borrow at an even lower rate than on its own (1 mark). For Optin, it may be expecting interest rates to increase in the future, hence it wishes to lock into a fixed rate right now (1 mark). It enters the swap and gets the exposure to the fixed rate as well as a better rate than on its own (1 mark). Both finds this swap attractive because of their opposing market views on the interest rate.

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c) Rigidt has also entered into another interest rate swap one year ago with a different counterparty, which has 14 months remaining. In this contract, Rigidt has agreed to pay 6.0% per annum and to receive three-month LIBOR in return on a notional principal of $100M with payments being exchanged every three months. The three-month LIBOR rate one month ago was 9% per annum. The spot LIBOR rate is 8.5% per annum for all maturities. All rates are compounded quarterly. What is the value of this swap to Rigidt now? (11 marks) Solution: This question is almost identical to Q5 of Tutorial 3, as well as Quiz 4. Class content is in Lecture 3 slides (slide 20 to 41) and covered in the first 39 minutes of Week 4’s Friday lecture. Since Rigidt is paying fixed and receiving floating, the swap can be regarded as a long position in a floating-rate bond and a short position in the fixed-rate bond. So the value of the IRS would be Value (Floating bond) – Value (Fixed bond) (1 mark) The applicable discount rates are the spot LIBOR rate for different maturities at present. Right now, the LIBOR curve is flat at 8.5% for all maturities, so the discount rate is 8.5% per annum with quarterly compounding (1 mark), and in terms of continuously compounded rate, is: ฀฀฀฀ �= 4xln(1+0.085/4) = 8.4109469% (1 mark, subtract 0.5 if decimal places are not ฀฀฀ ฀ = ฀ ฀ ฀฀฀฀ �1 + ฀฀

kept as instructed)

The fixed-rate coupon is 6%/4 = 1.5% or 1.5% x 100M = $1.5M per period (1 mark) Value of the fixed-rate bond is 5

2

8

11

1.5฀฀ −0.084109469×12 + 1.5฀฀ −0.084109469×12 + 1.5฀฀ −0.084109469×12 + 1.5฀฀ −0.084109469×12 + 14

(100 + 1.5)฀฀ −0.084109469×12 = 97.74747367 (3 marks total: 0.5 for each correct calculation of the expected four coupon payments & 0.5 for the final coupon + principal payments. 0.5 mark for the correct answer. Subtract 0.5 if decimal places are not kept as instructed. The next floating-rate coupon, to be paid in 2 months, was set in advance last month (immediately after the previous coupon payment date), so it was set according to the last 3-month LIBOR rate: 9%/4 x 100M = 2.25% or $2.25M per period (1 mark) Since the floating rate bond will have a value of PAR immediately after the next floating-rate coupon payment, the value of the floating-rate bond is PV(PAR + next coupon) (1 mark): (100 + 2.25)฀฀ −0.084109469×2/12 =100.8266343 (1 mark for correct numerical answer, subtract 0.5 if decimal places are not kept as instructed) So the value of the IRS is 100.8266343 - 97.74747367 = 3.07916065 ≈ 3.0792M Correct final numerical answer 1 mark, -0.5 if rounding is not appropriate)

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Question 3 (27 marks) a) PNN Ltd. issues a non-dividend paying stock. The current stock price is $6.50. Suppose there is a European put option on PNN’s stock with an expiry date in 9 months, with strike price $7. The put is currently selling for $0.23. The risk-free rate is 3.00%, compounded semi-annually. Identify all possible arbitrage opportunities, describe in words and show with cash flows for all periods how you would obtain the arbitrage profit. (15 marks) Solution: The question is almost identical to Q2a of the 2009 past final. A similar problem was originally covered in Lecture 4 – please refer to “Handwritten Notes - Option bounds + Arbitrage” on Wattle, page 4, and lecture recording (towards the end of Week 4’s Friday lecture, and also re-emphasized in Week 5’s Tuesday lecture/workshop). Convert to continuously compounded rate: mln(1+rm/m) = 2ln(1+0.03/2)= 0.029777225 (1 mark) Since the question asks to identify ALL possible strategies, both the upper and lower bounds need to be checked: Upper bound still holds as p < X (p < 7). (1 mark) Lower bound has been violated. Lower bound must satisfy Max(Xe-rT-S0, 0), So compare: p ?> Xe-rT-S0 (1 mark) LHS = P=0.23 with RHS = Xe-rT-S0 = 0.345402303 => LHS7

Buy put

-0.23

Exercise, +7(1 mark)

Leave to expire (1 mark)

Buy the underlying stock

-6.50

(Sold off to close the put)

Sell at the spot for ST>7 (1 mark)

Borrow the PV(X)

+7e-rT=6.845402303

-7 (1 mark)

-7 (1 mark)

Net cashflow

0.115402303

0 (1 mark)

S T-7 > 0 (1 mark)

We are going to have immediate payoff of $0.115402303, with the potential for more if ST>7 at maturity. (1 mark)

ALTERNATIVELY: If we borrowed $6.73 initially (8 marks total for the below) t=0 t=T, ST≤7

t=T, ST>7

Buy put

-0.23

Exercise, +7 (1 mark) Leave to expire (1 mark)

Buy the underlying stock

-6.50

(Sold off to close the put)

Borrow enough to cover cost of long positions Net cashflow

+6.73

-6.881991432 -6.881991432 (Repay loan, 1 mark) (Repay loan, 1 mark)

0

0.118008568 (1 mark)

Sell at the spot for ST>7 (1 mark)

> 0.118008568 (1 mark)

At maturity, we will earn $0.118008568 if ST≤7, and the potential for more if ST>7 at maturity. (1 mark)

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b) Using two portfolios, derive the put-call parity relationship for European options on nondividend-paying assets. Please ensure to describe the composition of each portfolio, and the initial costs and terminal values of the portfolios. (12 marks) Solution: This question was covered in full based on Lecture 4 slides, please see relevant recording from the beginning of Week 5’s Friday lecture (the 4’51” mark onwards). Take the following portfolios: A: c + Xe-rT (long on call + invest cash) B: put + underlying share (long put + long share) 10 marks in total for contents in the table t=0 (init. Costs) A

B

Long call c option Invest cash Xe-rT Total c+ Xe-rT Long put p option Long share S0 Total p+S0 Compare portfolio values

t=T (final payoffs) ST ≥ X ST...