Assignment1-F08
AK/ADMS 4503 3.0 Derivative Securities
Fall 2008
Assignment #1 Solutions
Instructions:
(1) This assignment is to be done individually . You must sign and submit the standard cover page supplied as the last page of this assignment.
(2) This assignment is due on October 19, 2008.
(3) The work can be typed or handwritten. If it is handwritten and too difficult
to read due to messiness and poor handwriting, it will receive zero credit.
(4) You must show your work to receive full credit.
(5) This assignment contains 5 questions and carries a total of 30 points .
Question 1 (6 marks)
The NASDAQ-100 futures trade at the CME, and each contract is on $100 times the index. The NASDAQ-100 spot is 1,670 points, and is expected to pay a dividend yield of 1% per annum continuously compounded. The risk-free rate is 2% per annum continuously compounded.
(a) What is the theoretical 1-year futures price? (2 marks)
(b) The 1-year futures price is 1,701 points. Show that there is an arbitrage and show how to benefit from it? Show all details. (4 marks)
Solution
(a) The theoretical 1-year futures price given by:
()()78
.686,11%)1%2(exp 670,1)(exp 0=×?×=×?×=T T F T q r S F
(b) Since the market overprices the contract, there is an arbitrage that consists in shorting the 1-year futures contract, borrow the money at the risk-free rate to buy the index today. By doing this, you lock in a net profit of $1,421.62. Here are the details:
Today, you must: - Short one 1-year contract to sell the index at 1,701 in 1 year - Borrow exactly exp(-1%) x 1,670 = 1,653.38 , at the risk-free rate 2% for
1 year (to buy 0.99005 units of the index)
In 1 year:
- The dividend yield paid on your holdings will make you having exactly 1
unit of index
- You deliver the index for 1,701 according to your short contract
- You pay back your loan at 1,653.38 x exp(2%) = 1,686.78 - Your profit is then (1,701 – 1,686.78) = 14.2162
Your net dollar profit is $100 x 14.2162 = $1,421.62
Question 2 (6 marks)
Consider a coupon-bearing bond selling at $950 and paying coupons in 5 months and 11 months from today. The risk-free interest is 2% per annum continuously compounded. The face value of the bond is $1,000.
(a) If the 1-year forward contract on this bond is selling at a fair price of $949.05, what is the coupon rate? (2 marks)
(b) What is the theoretical 6-month forward price? (1 mark)
(c) The 6-month forward contract is selling at $955 in the market. Is there any arbitrage opportunity? If yes, show how to benefit from it. Show all details.
(3 marks)
Solution
(a) We know that %)2exp()950($05.949$×?=I where:
())12/11%2exp()12/5%2exp(×?+×?×=Coupon I
Solving for the coupon, we find that the coupon is $10, which means that the coupon rate 2% APR semi-annually compounded.
(b) Given that the coupon is $10, the theoretical price for the 6-month contract must be:
()[]53.949$)5.0%2exp(12/5%2exp 10$950$0=×××??=F
(c) Since the contract is overpriced by the market, an arbitrageur can lock in a profit by borrowing/buying the bond and taking a short 6-month forward.
Here are the details of the strategy:
Today
- Borrow a total amount of $950 to buy the bond:
o Borrow $9.92 at 2% today to be reimbursed at $10 in 5 months (this
is exactly the coupon that you would receive in 5 months)
o Borrow ($950 - $9.92 = $940.08) at 2% today to be reimbursed at
$949.53 in 6 months
-Short one forward contract today to sell the bond at $955 in 6 months
In 5 months
-Receive the coupon of $10 and pay back the small part of the loan
In 6 months
-Deliver the bond at $955
-Pay back the large part of the loan at $949.53
-The net profit is $5.47
Question 3 (6 marks)
Consider a Canadian company that is planning to buy some equipment from a
British manufacturer in Oct 1, 2009 (that is in one year). The cost of this machinery is GBP 10 million. You have been asked to analyze the consequences
of entering into a futures contract to reduce the company’s exposure to foreign exchange risk. The current quotes are available from the market:
Spot CAD/GBP 1.90
Canadian TBill Rate 2%
UK TBill Rate 3%
(a) What is the theoretical 1-year (October 2009 contract) forward CAD/GBP?
(2 marks)
(b) Based on the following scenarios for the spot exchange rate one year from
now, CAD/GBP = 1.7 or 2.1, explain why the company should hedge its
currency risk exposure. Explain which strategy may be appropriate for the
company and what will be the total cost (for the equipment) in CAD in one
year. (2 marks)
(c) Assume now that you enter into the strategy proposed in (b) and that after
six months, i.e. April 1, 2009, the management of the company decides to
buy immediately the equipment from the British manufacturer and to close
out the forward position. Assume that on April 1, 2009, the spot CAD/GBP
= 1.95, the forward (with six months remaining to the maturity) rate
CAD/GBP = 1.93, what is the effective total cost in CAD for the
equipment? (2 marks)
Solution
(a) The 1-year forward contract is given by:
()
()8811
.11%)3%2(exp 90.1)(exp 000=×?×=×?×=F T r r S F f
(b) CAD/GBP 1.70 2.10
Cost CAD 17 million CAD 21 million
This shows that the company has a considerable exchange rate risk exposure. If the exchange rate moves from 1.9 to 1.95 over one year, the company makes a loss of CAD 500,000! The company will face a loss if the exchange rate CAD/GBP increases dramatically since it will have to buy GBP 10 million.
To hedge against any dramatic increase, the firm must take a long position in the 1-year forward to buy GBP at CAD 1.8811. The total cost will be CAD 18.811 million (= GBP 10 million x 1.8811) whatever the spot exchange rate will be in one year.
(c) The company decides to buy the machinery after six months for GBP 10 million at CAD/GBP = 1.95, that is a cost of CAD 19.5 million. On the other hand, the company makes a net profit of CAD 0.4841 million on the forward position, (1.93 – 1.8811) x exp(-2% x 0.5) x 10 = 0.4841. Therefore, the effective total cost is 19.5 – 0.4841 = CAD 19.0159 million, or equivalently a CAD/GBP of 1.9016.
Question 4 (6 marks)
The Aluminum sells at $1.20 per pound and it has a convenience yield y = 2%, and a storage cost of 1% (per annum continuously compounded). The risk-free interest rate is 4% per annum continuously compounded. An investor takes a short position in a 1-year futures contract on Aluminum today. Assume that each contract is on 44,000 pounds.
(a) What is the 1-year futures price per pound? (2 marks)
(b) Suppose that the investor closes out her position 9 months from now and makes a total profit of $2,200. What is the spot price of Aluminum 9 months from now if there is no arbitrage? (4 marks)
Solution
(a) The theoretical 1-year futures price is $1.20 x exp(4% + 1% - 2%) = $1.2365
(b) If the investor makes a $2,200 total profit on her short position in 9 months, or a $0.05 (= 2,200 / 44,000) profit per pound, this means that the futures price 9 months from now is $1.1865 and that the contract has 3 months remaining. The spot price in 9 months is then given by:
S = 1.1865 x exp(-3% x 0.25) = $1.1777
Question 5 (6 marks)
A portfolio manager has sold short a portfolio of stocks worth $100 million for three months. The beta of the portfolio is 2. The manager would like to use the CME futures contract on the S&P 500 index to hedge the portfolio over the next three months. The index is currently 1,200 points, and each contract is on $250 times the index. The S&P 500 dividend yield is 1% and the risk-free rate is 2%.
(a) How many long or short positions should the manager take? (4 marks)
(b) What would be the net gain/loss in three months in the following scenarios? (2 marks)
Scenario Portfolio value In 3 months S&P 500 Futures In 3 months 1 $110 million 1260
2 $90 million 1140
Solution
(a) First, we need to calculate the theoretical S&P 500 futures price today:
1200 x exp[(2%-1%) x .25] = 1203.
Since the manager is short selling the portfolio, any rise in the stock market would cause big losses. The manager should then take long S&P 500 futures contracts, exactly 665 contracts:
6651203
250$000,000,100$2*=××==F P N β
(b) The net gain/loss for each scenario is given by the gain/loss on the short
position on the portfolio added to the gain/loss on the futures position.
Scenario 1: (100 – 110) million + 665 x 250 x (1260 – 1203) = –$523,750 Scenario 2: (100 – 90) million + 665 x 250 x (1140 – 1203) = –$473,750
Atkinson Faculty of Liberal and Professional Studies
YORK UNIVERSITY
Toronto, Ontario
ADMS 4503 3.0
Derivative Securities
Tahani
Professor Nabil
Sections A and B
Assignment #1
Due Date: October 19, 2008
Personal Work Statement
I, the undersigned:
?warrant that the work submitted herein is my work and not the work of others ?acknowledge that I have read and understood the Senate Policy on Academic Honesty
?acknowledge that it is a breach of the University Regulations to give and receive unauthorized assistance on a graded piece of work
Name (typed or printed) York Student # Signature