Question 1 (*)
Answering this question doesn’t require any understanding of why the forward prices
are what they are. It is simply an exercise of plugging the numbers given into the
formulae. Students are not expected to memorise the formulae – they will be provided
on an exam if required. Surely, I wouldn’t ask a question this easy!
The lecture worked through a series of examples for different underlying assets. In each
case:
• We identified how to create a synthetic long forward contract (most often, a ‘borrow
and buy’ strategy can replicate the outcome of a long forward),
• We then combined borrow and buy trades with a short forward contract.
• Doing this creates a strategy that is riskless and has zero cash flow at time 0. In
essence, the short forward cancels out the synthetic long forward (i.e., borrow and buy).
• Consequently, the strategy must also have zero cashflow at time T, otherwise there
would be an arbitrage opportunity.
By following this approach, we were able to derive formulae that give the correct
(no-arbitrage) forward price. The formulae below differ a little depending on what the
underlying asset is.
What is the underlying asset? Correct forward price
Base case 0 F = S e rT Asset with known discrete income (I) ( 0 ) F = S − I e rT Asset with
continuous dividend yield (q) 0
( ) F = S e r − qT Asset with continuous storage cost (u) 0
F = S e
( r + uT
) An exchange rate F =
S 0
e
( r − f
) T
BFF3751 Tutorial 3 Philip Gray 2020 1
a) This example uses the base case. It is a very basic asset (a share) and we are
assuming there
are no other cashflows (i.e., dividends) involved.
F = S 0
e rT =25 e 0.06 1 = $26.55
b) What differs in this case is that the stock will pay a $2 dividend during the life of the
forward contract. This would have implications if we were creating a synthetic long
forward with a ‘borrow and buy’ strategy. Hence, the formula includes the present value
of expected dividends.
First, calculate the present value of the dividend payment expected in three months’
time:
I = 2 e − 0.06 3/12 = $1.97 Second, plug into the formula for an underlying asset that generates
a known discrete income:
F = ( S 0
−
I )
e rT = ( 25 − 1.97
)
e 0.06 1 =
$24.45
c) In this case, the underlying asset is the Small Ordinaries market index. The concepts
are similar to a forward contract written on a single stock. It’s just more complicated
because there are hundreds of stocks in this index, all paying dividends at various
times. For this reason, rather than calculate the present value of expected dividends
individually for each stock, we sort of aggregate all dividends and represent them with
an average dividend yield (q):
F = S e
( r − qT ) = e ( − )
As we have seen repeatedly, creating a synthetic long forward requires borrowing
money and buying the underlying asset. The loan incurs an interest cost (r). However, if
the underlying asset was a portfolio of hundreds of stocks, most of them paying
dividends, the dividend yield (q) would help offset the interest cost. This is why the
exponent of the formula has ‘r-q’.
0
2020 0.06 0.04 9/12 2050.52
BFF3751 Tutorial 3 Philip Gray 2020 2
d) What differs in this case is that the underlying asset (gold) doesn’t pay a dividend or
generate an income. Instead, there will be a cost to safely store it. That is, if we created
a synthetic long forward – borrow money and buy gold – we would have to pay storage
costs to keep our gold safe. So unlike part (c), where dividends offset some of the
interest cost, here we need to add the storage cost to the interest cost in the exponent
‘r+u’:
F = S e ( r +
uT )
= e ( + )
e) All of the parts to Question 1 are asking you to calculate correct (no arbitrage)
forward price. There is one and only one forward price that precludes arbitrage
opportunities. Of course, it doesn’t matter if you intend to enter a long forward contract
or a short forward contract, the correct price is the same. So this is a trick question
(sorry:). The answer is the same as part (d).
Aside: In practice, there are slightly different ‘correct’ prices for forward contracts;
specifically, there are often bid/ask spread around quoted prices. That is, there is a
slightly different price for buyers and sellers (anyone who has traded shares online will
be accustomed to seeing bid/ask spreads. Similarly, when we exchange currency at the
airport, we see slightly different prices depending on whether we are buying or selling
the currency in question. While, in theory, there is one and only one correct price,
bid/ask spreads allow the market maker to make a small profit on the transaction.
In each of these questions, you have calculated the correct forward price – the one and
only price that precludes arbitrage. In every case, if the quoted forward price differs from
the correct price there is an arbitrage opportunity.
I could write a bunch of additional questions saying ‘if the quoted forward price was
different, explain what trades you would enter to capture the arbitrage opportunity?’ For
example, if the forward price on BHP in part (a) was $27 (instead of the correct price of
$26.55), what trades would you enter? And so on for each part.
Students who are keen to learn how to make arbitrage profits might like to challenge
themselves by doing this. If you make an attempt at this, feel free to email me to check
whether your answer is correct (philip.gray@monash.edu).
0 926 0.06 0.04 2 $1131.02
BFF3751 Tutorial 3 Philip Gray 2020 3
Question 2 (*)
nb: this question assumes that there are no storage costs ( u ) for the gold. This is
unrealistic, but it makes it a little easier for you to articulate the trading strategy required
to capture the arbitrage opportunity.
a) 0.10 8/12 900 $962.05 per ounce. F e = =
b) Compared to the correct forward price ($962.05), a quoted delivery price of $980 per
ounce is too high – the forward contract is way overpriced. What do we do when
something is overpriced? We sell it! Hence, we would enter a short forward position
on gold (we know $980 is too high so we are very happy have an arrangement that
allows us to sell gold at that price in 8 months’ time).
What else do we need to do? Conscious that our short forward represents an
obligation to sell gold in 8 months’ time, we can prepare for this right now by
simply buying some actual gold on the spot market. We can borrow the required
money, so that we are not out of pocket. If we buy some gold and keep it for 8
months, we are well prepared to deliver on our obligation to sell under the short
forward:
(today) Enter short forward contract on gold nil (today) Borrow money
+$900 (today) Buy gold at spot price -$900
(8 mths later) Repay loan and interest 900 exp(0.108/12) -$962.05 (8
mths later) Deliver our gold under terms of forward contract +980
Arbitrage profit $17.95 per ounce
Thus, we see that if forward contracts for delivery of gold in 8 months’ time are
priced at anything other than $962.05, there is arbitrage opportunity.
In Lecture 3, we repeatedly saw this ‘borrow and buy’ strategy, which we referred
to as a ‘synthetic long forward’. It essentially generates the same outcome as a
real long forward position (i.e., the outcome is that we own gold in 8 months’
time). When you combine a synthetic long forward with a real short forward, they
pretty much cancel each other out, meaning there is no risk and zero cost
upfront.
BFF3751 Tutorial 3 Philip Gray 2020 4
c) A quoted delivery price of $940 is too low – the forward contract is underpriced. What
do you do when something is underpriced? You buy it! Hence, we would take a long
forward position on gold (we know $940 is too cheap and we are happy to have an
arrangement that allows us to buy gold at that price 8 months from now).
What else do we need to do? Conscious that our long forward position
represents an obligation to buy gold in 8 months’ time, we can prepare for this
right now by short selling gold in the spot market. Short selling entails borrowing
someone else’s gold and selling it today. There is an underlying promise to return
the gold to the original owner at some time in the future (this will be no problem
because we know we will be receiving gold thanks to the long forward).
(today) Enter long forward contract on gold nil (today) Short sell gold
raises money $900 (today) Invest proceeds from short sale at
riskless rate -$900
(8 mths later) Proceeds have grown to 900 exp(0.10 8/12) $962.05
(8 mths later) Buy gold under terms of forward contract -$940.00
Arbitrage Profit +$ 22.05 per ounce
(8 mths later) Use the gold purchased under forward contract to settle the short
sale.
Once again, we see that if forward contracts for delivery of gold in 8 months’ time
are priced at anything other than $962.05, there is arbitrage opportunity.
Aside: in the lecture, we talked a lot about ‘borrow and buy’ creating a synthetic
long forward. In this case, we didn’t borrow and buy. We “short sold the
underlying gold and invested the proceeds”. This is effectively a ‘synthetic short
forward’. When you combine a synthetic short forward with a real long forward,
they pretty much cancel each other out, meaning there is no risk and zero cost
upfront.
BFF3751 Tutorial 3 Philip Gray 2020 5
Question 3
In Lecture 2 + Tutorial 2, we have used SPI futures to hedge against a falling market.
You can also use SPI futures to speculate on a movement in a particular direction. This
is a risky strategy, since a wrong prediction will generate a loss.
The quoted numbers in this question are the actual market figures surrounding the 1987
stock market crash. If you had a crystal ball to predict the 1987 crash, this is how easily
you could have become rich.
a) We are predicting a falling share market. What futures strategy makes money in a
falling market? A short futures position. We short December-maturity SPI futures at their
current quote 2127. If we guess right and the market falls, the quoted price for Dec SPI
futures will also fall. We can close-out by entering long Dec SPIs at a lower price and
therefore make a profit. That’s the plan – let’s see how it plays out.
b) Since we initially shorted ten Dec-1987 SPI futures back in August, we close-out now
by trading ten long Dec-1987 SPI futures at the quoted price of
1323.
(in Aug) Short ten December maturity SPI futures 10$A252127 $531,750 (in Dec)
Long ten December maturity SPI futures 10$A251323 $330,750 Futures trading
profit $201,000
c) As noted above, the danger with speculation is that, if your prediction is wrong, you
will suffer a loss. There is nothing ‘locked-in’ about
speculating!
(in Aug) Short ten December maturity SPI futures 10$A252127 $531,750 (in Dec)
Long ten December maturity SPI futures 10$A252250 $562,500
Futures trading loss $ 30,750
Note: the standard contract size for SPI futures is $A25 times whatever the quoted
delivery price is.
BFF3751 Tutorial 3 Philip Gray 2020 6
Question 4
First of all, we need to calculate the correct/fair forward price for the delivery of wheat
five months from now. Since wheat is a commodity that involves a storage cost ( u ), we
use the formula for an asset with a continuous storage cost:
F = S e
( r +
uT )
= e ( + )
a) Conscious that the no-arbitrage forward price for delivery of wheat in five months is
$144.75 per tonne, the quoted forward price of $150 is too high. As an algorithmic
trader, I can program my computer to figure this out very easily. And I can tell my
computer what trades to enter when it finds an overpriced forward contract. For starters,
I will definitely enter a short forward contract for the delivery of wheat in five months’
time.
Comparing the quoted forward price to the correct price, identifying an overpriced
contract and deciding to short it – that’s the easy part. Often, the more difficult task is to
figure out what other trades to implement today to capture the arbitrage profit. There are
a couple of ways to approach a problem like this (they are essentially equivalent). I’ll
talk through both and you can decide which is easier for you to understand.
The first approach focuses on the fact that the short forward contract commits us to
selling wheat in five months’ time. That’s definitely going to happen because I entered a
short forward contract on wheat. Next, I start thinking about what I can do right now to
facilitate this sale of wheat in five months’ time. I could just buy the wheat right now on
the spot market. Put it into storage for five months (which will incur some storage costs).
When it comes time to deliver under the forward contract, I retrieve the wheat from
storage and sell it as per the short forward contract. Of course, buying wheat right now
will cost money. As an arbitrageur, I don’t want to be out of pocket, so I’ll just borrow the
necessary money (which will incur some interest costs).
A second, and clearly equivalent, way to think about it is as follows. Since the forward is
overpriced, we will short it. As an arbitrageur, I know I have to cancel out that position
synthetically. Lecture 3 demonstrated over and over that a ‘borrow and buy’ strategy is
a synthetic long forward. Without thinking too much about the underlying mechanics, my
recipe would be (i) enter the short forward contract, (ii) borrow and buy wheat to create
a synthetic long forward, and (iii) the two cancel each other out leaving me with the
arbitrage profit.
0
140 0.06 0.02 5/12 $144.75 per tonne
BFF3751 Tutorial 3 Philip Gray 2020 7
So, either way you think about it, here’s what to do. We borrow $140 to buy one tonne
of wheat today on the spot market. We put this wheat into storage. Over the next five
months, we are incurring interest (6%) on the loan and also paying storage costs (2%)
for the wheat. The time T column shows the combined effect of both these charges.
Action
Cash Flows Time 0
Cash Flows Time T Short forward position Buy wheat Borrow
nil -140 +140
+150
-144.75 = -140 exp((0.06+0.02)5/12) Net cash flow $0.00 +$5.25 per tonne
Hence, we are able to make $5.25 per tonne. It is a totally riskless arbitrage profit.
Naturally, I would like to do this for millions of tonnes!
b) Part (a) is arguably easier because the trading strategy involved buying wheat. This
part
may be more conceptually difficult because it involves short selling the wheat.
Again, we know the no-arbitrage price for delivery of wheat in five months is $144.75.
So at $130, the forward contract is way underpriced. When something is undervalued,
we would buy it (i.e., enter a long forward on wheat). This commits us to buying wheat
in five months’ time.
What else do we do to capture the arbitrage profit? Knowing that I will be taking delivery
of wheat five months from now, I need to make arrangements that facilitate getting rid of
that wheat in five months. Short selling the wheat is the answer. If we short sell wheat,
that means someone gives us wheat today and we sell it on the spot market, with an
agreement that we return some wheat to them in five months’ time. We will have no
trouble fulfilling this obligation, because we will be receiving wheat under the long
forward contract.
So we do the following: (i) enter a long forward to buy wheat in five months’ time, (ii)
short sell wheat at today’s spot price and receive $140, and (iii) invest the proceeds
($140) to earn interest.
Obviously, we receive interest (6%) on the money invested. Under the short-selling
arrangement, we also receive money (2%) in relation to storage costs. In effect, the
person who gave us the wheat to short sell is saving 2% p.a. because they do not have
to store the wheat. Action
Cash Flows Time 0
Cash Flows Time T Long forward position Short-sell wheat Invest proceeds
nil +140 -140
-130
+144.75 = -140 exp((0.06+0.02) 5/12) Net cash flow $0.00 +$14.75 per tonne
BFF3751 Tutorial 3 Philip Gray 2020 8
Question 5 (*)
In my opinion, forward contracts involving exchange rates are the most confusing. This
is because an exchange rate between two currencies can be quoted in two ways (direct
and indirect quotes). This question is designed to remove much of the confusion
surrounding exchange rates.
a) This question portrays you as a business person based in the United States, dealing
with clients in Europe. For that reason, I would express the exchange rate in a way that
reflects what Euros cost me in the US (i.e., the price of one Euro):
EUR 1.0000 = USD 1.1429
Therefore, based in the US, the cost of a Euro is USD 1.1429. In the formula, this is
what we use for S 0 . The exponent then takes the local US interest rate (2%) and
subtracts the foreign interest (5% in Europe). This is multiplied by the time (in years)
until the expiry of the forward contract:
F = S 0
e
( r −
f ) T =1.1429 e
( 0.02 − 0.05 )
0.5 =
1.1259
Hence, the correct/fair 6-month forward rate between USD and EUR is:
EUR 1.0000 = USD 1.1259 EUR 0.8882 = USD 1.0000
Parts (b)-(e) of this question walk you through the replicating strategy for a long forward
contract on a currency. [We pretty much did this back in Tutorial 1 Question 5] It looks a
little more complicated than some of the things we did in Lecture 3, but it is still
essentially a ‘borrow and buy’ strategy. We borrow USD today and then buy (i.e.,
convert them) into Euros at the spot exchange rate. We pay interest on the loan in the
US and receive interest on the investment in Europe.
b) If we borrow USD 10,000 today at 2% p.a. interest, the balance of the loan in six
months’
time will be USD 10,100.50 (=10,000 exp(0.02 × 0.5)).
c) If we convert USD 10,000 into Euros at the spot exchange rate, this gives us EUR
8,750
(=USD 10,000 × 0.8750).
d) If we invest EUR 8,750 at 5% interest for six months, it will grow to EUR 8,971.51.
e) Taking the ratio of the answers in part (b) and part (d) gives 1.1259. This is the
correct/fair
6-month forward rate we got from the formula in part (a).
BFF3751 Tutorial 3 Philip Gray 2020 9
nb: the choice of USD 10,000 for the loan is completely arbitrary. We would get the
same answer no matter what we choose.
Aside:
The answer above uses the ‘direct’ method of quoting an exchange rate. Given that the
question frames the problem from the perspective of someone in the United States, we
used a direct quote: EUR 1.0000 = USD 1.1429. That is, in the US, it costs USD 1.1429
to buy one Euro.
• It was this direct quote (1.1429) that we used in the formula for S 0 .
• “r” in the formula was the local US interest rate (2%)
• “f” in the formula was the interest rate in the foreign country (5%).
Now, you might ask the question: what if we had taken the indirect quote (USD 1.0000 =
EUR 0.8750) and inserted it into the formula for S 0 ? That’s OK, we should still be able
to get to the same answer. To do it that way is essentially doing it from the perspective
of someone based in Europe:
• Taking the quote USD 1.0000 = EUR 0.8750, we can insert S 0 = 0.8750.,
• Given that we are doing this from the perspective of someone based in Europe, “r” will
now be the local European interest rate 5%.
• And the foreign rate “f” will be the US interest rate of 2%.
F = S 0
e
( r −
f ) T =0.8750 e
( 0.05 − 0.02 )
0.5 =
0.8882
This is the same 6-month forward rate we saw at the outset of the question. So, if you
understand this properly, there are two ways to get to the answer. Having said that,
there is a lot of potential for confusion in these sorts of questions. I recommend that you
find one approach that you understand and stick to that.
BFF3751 Tutorial 3 Philip Gray 2020 10
Question 6
a) Clearly, there are two risks associated with the September sale of gold. First , there is
a risk that the price of gold will fall between now and September. This will have a
negative impact on company profits when it comes time to sell our 100,000 ounces of
gold. Second , since our gold sales are denominated in USD, we are also exposed to
exchange rate risk – if the USD weakens relative to the AUD, we will receive fewer AUD
from the sale, again hurting profits.
Aside: there are other risks, that we won’t worry too much about here. For example,
our production schedule forecasts that we will have 100,000 ounces of gold ready
for sale in late September. We might enter a forward contract to hedge 100,000
ounces, but then production might end up being less (or more) than 100,000 ounces
(hence, the hedge will be incomplete). Or production delays may mean the gold is
not ready for sale until say October or November. These are real-world complexities.
b) In late September, we sell 100,000 ounces of gold at the spot price of USD 861 per
ounce, realising USD 86.1m. When we convert these USD into AUD (i.e., we sell the
USD) at the exchange rate, we realise AUD 95,666,667 (= 86.1m 0.90). When you
analyse this amount, you realise that you have suffered from both a falling gold price
and a weakening USD. As Head of Risk Management, you are not doing a very good
job!
c) Hedging these exposures is not rocket science. First, we hedge the gold price risk.
Since we are exposed to a falling gold price, we enter a forward contract that makes
money if gold price falls (i.e., enter a short forward contract covering 100,000 ounces).
This locks in the sale price of our September gold at USD 940 per ounce, guaranteeing
that we will receive USD 94m for our gold.
Second, while we know for certain that we will receive USD 94m late September, we
are still exposed exchange rate risk. Currently, one US dollar is worth AUD 1.2557
(=1/0.7964). However, if the USD weakens relative to AUD, when we convert the
USD 94m into Aussie dollars, we get less.
Hence, we enter a forward contract that makes money if the USD weakens (i.e., a
short forward contract on USD). The quoted forward rate for September delivery is
AUD 1.00 = USD 0.80. In other words, each USD we sell generates AUD 1.25. Our
contract size would be USD 94m (or equivalently AUD 117.5m = USD 94m 0.80).
BFF3751 Tutorial 3 Philip Gray 2020 11
d) Even before I start to answer this, I know the conclusion will be revenue of AUD
117.5m.
This was the figure I ‘locked in’ back in August when I entered the two
hedges.
This answer assumes that the two forward contracts are closed out (rather than
physically delivered), although the final revenue would be identical if we assumed
physical delivery.
First, close out the gold forward. We know we will have made money on this contract
– we specifically took a short gold forward to make money if the gold price fell (which
it has). We entered a short gold forward contract at USD 940 per ounce, and now
close it out by entering a long September-delivery gold forward contract at USD 861
per ounce. The profit on closing is USD 7.9m ((940 – 861) 100,000 ounces).
Convert this to AUD at the spot exchange rate and we get AUD 8.777778m (USD
7.9m 0.90).
Second, we close out the currency forward contract. Again, we must make money
on this – we specifically entered a short forward on USD to make money if the USD
weakened (which it did). We entered a short forward at USD 1.00 = AUD 1.25, and
now close out by entering a long forward contract at USD 1.00 = AUD 1.1111. The
profit on closing is AUD 13.0566m ((1.25 – 1.1111) USD 94m).
So these two derivatives hedges realise AUD 21,834,378 in profit (8,777,778 +
13,056,600). All that remains to do is to sell the actual gold that we produced. We
sell the 100,000 ounces at the end-of-September spot price of USD 861 per ounce.
This realises USD 86.1m. Convert to AUD at spot exchange rate and realise AUD
95,666,667 (USD 86.1m 1.1111).
All up, we have AUD 117,501,044 (95,666,667 + 8,777,778 + 13,056,600), which is
what we knew the answer must be. [nb: there is a little bit of rounding error in here]
Aside: questions involving foreign exchange are potentially confusing. In this question,
back in part (c), once I knew I would receive USD 94m from the short gold forward, I
decided to enter a short currency forward that allows me to sell USD 94m at a locked
in exchange rate of 0.80.
It would have been equally valid to say I will hedge the forex risk by entering a long
forward that allows me to buy AUD 117.5m . If the forward rate is USD 1.0000 = AUD
1.25000, then a short forward on USD 94m is identical to along forward on AUD
117.5m.
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