Tutorial Questions
Question 1 () In each of the following cases, calculate the correct (no-arbitrage) delivery price for the forward contract: a) A forward contract for the delivery of one BHP share one year from today. The current (i.e., spot) price for BHP is $25. Assume that BHP is not expected to pay any dividends over the next year. b) A forward contract for the delivery of one BHP share one year from today. The current price of BHP is $25. BHP is expected to pay a $2 dividend three months from today. c) A forward contract written on the Small Ordinaries market index for delivery in nine months. The Small Ords is currently sitting at 2020. The expected dividend yield on the Small Ords index is 4% p.a. continuously compounded. d) A forward contract to buy one ounce of gold in two years’ time. The spot gold price is $926 per ounce. The storage/security costs for gold bullion are 4% p.a. continuously compounded. e) A forward contract to sell one ounce of gold in two years’ time. All other details the same as (d). In each case above, assume that the riskless interest rate is 6% p.a. continuously compounded. Question 2 ()
The current (i.e., spot) price of gold is $900 per ounce. The riskless interest rate is 10% per annum. For simplicity, assume that there are no storage/security costs of gold.
a) What is the arbitrage-free forward price for the delivery of gold in 8 months’ time?
b) If you see an 8-month forward price for gold quoted at $980 per ounce, explain what trades you would make to capture an arbitrage profit. Explain what trades occur now, what happens in 8 months’ time, and therefore quantify the magnitude of the profit.
c) If you see an 8-month forward price of gold quoted at $940 per ounce, explain what trades you would make to capture an arbitrage profit.
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Question 3
It is 3 August 1987 and the All Ordinaries Index (AOI) is at 2030. SPI futures written on the AOI with a Dec-1987 maturity are quoted at 2127. Using a combination of numerology, astrology and your trusty old crystal ball, you are predicting a market crash sometime during the latter part of 1987.
You have great confidence in your prediction and decide to put your money where your mouth is. You will use the SPI futures to place a bet on a market crash.
a) Describe how you will use the futures market to place your ‘bet’. Specifically, tell me whether you will take a long or short futures position in the SPI. Don’t be too greedy – limit yourself to trading 10 futures contracts.
For a while it looked like you had made a catastrophic mistake. By 1 September, the AOI had risen to 2150. And it kept rising – by 15 October, it was 2184. Then, on Tuesday 20 October 1987, the AOI dropped more than 500 points in a single day. Your crystal ball is in fine form! As the end of December 1987 approached, the AOI was around 1288, and the December maturity SPI futures were quoted at 1323. At this point, you close-out your futures position.
b) Describe what futures transaction you will enact to close-out your position. Specifically, will you take long or short SPI futures? What is your profit?
c) Imagine now, that your prediction had been wrong and that the SPI futures were quoted at 2250 at expiry. What is your profit/loss?
nb: the details portrayed in this question reflect what actually happened surrounding the famous stock market crash in October 1987.
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Question 4
Lecture 3 shows that, for any forward or futures contract, there is just one price that does not allow arbitrage profits to be made. If the quoted forward/futures price were to deviate from this no-arbitrage price, savvy traders will detect the mispricing and profit from it.
The concept of mispricing for forward/futures contracts is very different to mispricing of something like shares. I might have the view that Commonwealth Bank shares are overpriced at $84, but this is just my opinion. The ‘true’ value of CBA shares is never revealed so I never know whether or not they were overpriced. However, with forward/futures contracts, our formulae give us the correct/fair value for a forward/futures contract. And it is unlikely that quoted prices diverge from the correct price, because there are many smart traders who are constantly on the lookout for mispriced derivatives.
In fact, these days, computers can be programmed to scour the markets for mispriced derivatives. The computer can constantly monitor spot prices, forward prices, interest rates, storage costs etc. and plug these inputs into our formulae. If and when the computer identifies a mispricing, it can be programmed to automatically initiate the sorts of trades we do in this question to capture the arbitrage profit.
For each of the following, calculate the no-arbitrage futures/forward price, establish whether the contract is over or under valued, and explain what trades you would implement to capture the arbitrage profit on offer:
a) A forward contract for the delivery of wheat in five months trades at $150 per tonne. The spot price for wheat is $140 per tonne, the riskless rate is 6% p.a. and storage costs are 2% p.a..
b) Assume the same information as part a), except that the traded forward price is $130 per tonne.
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Question 5 (*)
You operate a business in the United States that frequently trades with partners in Europe. As a result, you often receive payments denominated in Euros, and occasionally you pay bills in Euros. This exposes you to foreign exchange risk, therefore you make use of forward contracts to manage this risk. The spot exchange rate between US dollars and Euros is USD 1.0000 = EUR 0.8750. The riskfree rates of interest in the US and Europe are 2% and 5% respectively.
For hedging purposes, you visit your local bank and make an enquiry about the 6-month forward rate for Euros. That is, you are interested in a forward contract for the delivery of Euros six months’ from today.
a) Using the appropriate formula, calculate what the 6-month forward rate must be to prevent arbitrage.
I am embarrassed to admit this, but the formula for calculating forward rates on currencies often confuses me. I can never remember whether I should be plugging in direct quotes or indirect quotes for S0. In the exponent, I struggle to remember whether the local or foreign interest rate should come first. For this reason, I like to do a quick exercise that allows me to check whether I plugged numbers into the formula correctly. It goes like this.
b) Assume that you borrow USD 10,000 today. With 2% p.a. interest continuously compounded, how much will you owe in six months’ time?
c) Assume that, after borrowing USD 10,000, you immediately convert them into Euros at the spot exchange rate. How many Euros will you have?
d) Next, assume that you take those Euros and invest them in a bank account that pays 5% p.a. interest in Euros. How much will this investment grow to in six months?
Now, unless there are crazy arbitrage opportunities that allow smart people to make riskless profits, it just has to be the case that the loan balance in the US is tied to the balance of the bank account in Europe. The relation is the correct (no-arbitrage) 6-month forward rate.
e) Divide your answer to part (b) by your answer to part (d). It should match with your calculation of the 6-month forward rate in part (a).
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Question 6
You are the Head of Risk Management for a major Australian gold mine. Today is 1st August and your production schedule suggests that you will have 100,000 ounces of gold to ready for sale by 30th September.
All of your gold sales are made to foreign buyers with sales denominated in US dollars. Gold is currently priced at USD 931 per ounce. The spot exchange rate is AUD 1.00 = USD 0.7964.
a) Identify the risks that your company faces in relation to the September gold sale. There are two obvious risks.
b) Despite the inherent risks, you decide to do nothing to manage your exposures. At the end of September, the spot price of gold is USD 861 per ounce, and the spot exchange rate is AUD 1.00 = USD 0.90. Calculate the AUD revenue from the sale of 100,000 ounces of gold (then start looking for a new job, preferably not in risk management).
Ignore (b) and go back to 1st August. In addition to the spot price information quoted above, the following forward rates are relevant. September-delivery gold forward contracts are being quoted at USD 940 per ounce. Foreign exchange forward contracts with September delivery are quoted at AUD 1.00 = USD 0.80.
c) Outline the derivative contract/s that you would enter to manage the risks identified in part (a).
Now roll forward to September and assume that the spot price for gold and USD are as suggested in part (b). Of course, the futures price for September-delivery gold and USD will have converged to these end-of-September spot price (else arbitrage).
d) Assume that you close out your derivative positions, then sell the gold on the spot market. Calculate the AUD revenue from the sale of 100,000 ounces of gold, net of the gains/losses on the forward contracts.
nb: this question looks a little like Question 2 from Tutorial 2. However, you will find it has an added layer of complexity.
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