Question 1
a) In deciding whether a long or short forward contract is the appropriate hedge, follow
the approach presented in the lecture and it’s hard to get it
wrong:
• What is the risk we face? Spot wool price is $14 per kilogram. In three months’
time, we will purchase wool. Our danger is that wool price will rise before we
make our purchase.
• What derivative position will make money if this unfavourable scenario
eventuates? Recall the payoff diagram to long positions. When prices rises, you
make money with a long position.
Hence, we will enter a long forward contract for the delivery of 500 kilograms of
wool in three months’ time at the quoted delivery price of $14.50 per kg.
b) With many commodities, physical delivery is possible. That is, a long forward contract
on wool is a commitment to buy 500 kilograms of wool at the $14.50 delivery price. In
most cases, however, derivative traders close-out just prior to maturity (this is mainly for
convenience). As the maturity time approaches, the forward contract has served its
hedging purpose. The contract can be closed out, then the wool is purchased on the
spot market.
(time 0) Enter long forward contract to purchase 500 kgs of wool $7,250 (3 mths)
Close-out by shorting forward contract for 500 kgs of wool $6,000 Loss on forward
contract $1,250
Purchase 500 kgs of wool on spot market $6,000
Total cost of 500 kgs of wool (6,000+1,250) $7,250
Aside: the question doesn’t explicitly state that the forward price when we close
out is $12 per kilogram. The calculation above assumes that, when we close out
the long forward by entering a short forward, the forward price ( F ) is $12. The
reason we can safely assume this is that, as the forward contract approaches
expiry, F converges to the spot price of the underlying wool. Tutorial 1 Question 6
demonstrated why this must be the case (to prevent arbitrage opportunities).
BFF3751 Tutorial 2 © Philip Gray 2020 1
c) Your partner is correct … had we not locked-in a $14.50 price for wool via the forward
contract, we could indeed have purchased 500 kgs of wool in the spot market for $6,000
(500 kgs × $12). But we had no way of knowing that wool price would fall. If we had
remained unhedged, and wool price had in fact risen, the wool would have cost us a lot
more than $7,250.
Your business partner needs to understand that hedging is not about guessing
which way price is going to move. Hedging is about removing the risk of an
unfavourable price movement by ‘locking in’ the price at which we can buy wool
at in three months’ time. In fact, had we not hedged using the long forward
contract – if we had done nothing – this is effectively amounts gambling that
prices won’t rise. And this is a risk many businesses are not prepared to take.
d) (time 0) Enter long forward contract to purchase 500 kgs of wool $7,250 (3 mths)
Close-out by shorting forward contract for 500 kgs of wool $8,000
Gain on forward contract $ 750
Purchase 500 kgs of wool on spot market $8,000
Total cost of 500 kgs of wool (8000 – 750) $7,250
What we see is that we were effectively hedged. Irrespective of whether the price of
wool falls (as it did in part b) or rises (as it did in part d), we know with complete
certainty that we will end up paying $7,250 to purchase the required 500 kgs of wool
(effectively $14.50 per kilogram).
BFF3751 Tutorial 2 © Philip Gray 2020 2
Question 2
a) On 1 September, the spot price of gold is USD 926 per ounce. The risk to the
business is that the price at which we will sell the gold late November will be lower than
this. This will have a negative impact on company profit. To summarise, we are exposed
to falling gold prices.
b) To eliminate the risk of a decrease in gold price, we need to enter a futures position
that makes money if gold price falls. It is important to know the payoff diagrams to long
and short positions! A short futures position makes money when the price of the
underlying asset falls. Hence, we enter a short futures contract covering 100,000
ounces of gold with delivery in November .
Another way to think about this is as follows. We can establish the correct hedge
position (either long or short) by doing today in the futures market what we will be
doing in the physical market in November. In November, we will be physically selling
100,000 ounces of gold, so we enter a short futures position today (since a short
futures position is a commitment to sell the underlying gold). No matter which way
you look at it, entering a short gold futures position is the correct hedge .
c) You have a short position covering 100,000 ounces of gold at a contracted futures
price of $935 per ounce. In other words, we are able to physically deliver 100,000
ounces of gold and will receive USD 93.5m (100,000 ounces × USD 935). The contract
specifications are likely to be very strict with respect to the quality of the gold delivered,
the time and place for delivery. At expiry in late November:
• Our short futures position means that we have an obligation to deliver (i.e., sell)
100,000 ounces of gold.
• Therefore, we physically deliver 100,000 ounces of gold to the person on the
other side of the contract, and.
• In exchange for this delivery, we receive 100,000 ounces × $935 = USD
93.5m.
d) To close-out the short position, we enter a futures position equal in magnitude and
opposite in direction t o the initial futures position. That is, since we initially entered a
short futures position covering the delivery of 100,000 ounces of gold on 30 November,
we close out by entering a long f utures contract for the delivery of 100,000 ounces of
gold on 30 November. The futures exchange effectively says: “he has an obligation to
sell 100,000 of gold on 30 November, and he also has an obligation to buy 100,000
ounces of gold on 30 November, so these two positions cancel each other out”.
As the clock ticks down to the expiry of this futures contract on 30 November, the
futures price will be exactly (or very close to) the spot price of gold at the end of
November ($920) – otherwise an arbitrage opportunity exists (see Tutorial 1
Question 6).
Since we went short at $935 per ounce and closed-out by going long at $920 per
ounce, we have a futures trading profit of $15 per ounce. On 100,000 ounces, this is
a total profit of
BFF3751 Tutorial 2 © Philip Gray 2020 3
$1,500,000. Closing out the initial short futures position is as simple as getting on
the computer and hitting “sell to close” on the Nov-expiry gold futures contract.
That closes-out the futures position, but we still have a truckload of gold to sell
(remember, we mine gold, so still have to sell our gold to someone). We physically
sell our 100,000 ounces of gold at the end-of-November spot price of $920 per
ounce and realise $92m. Add the $1,500,000 profit from the futures hedge and the
net monies received are $93.5m. Note that this is exactly the same outcome that
would be achieved if the gold was physically delivered under the futures contract.
e) Sometimes, futures contracts are cash settled . This happens when physical delivery
of the underlying asset is not possible. When contracts are cash settled, if we have not
closed out our initial short position by the time the November contract expires, then the
futures exchange will close it out automatically for us. Effectively, the futures exchange
calculates whether we made a profit or loss. If we made a profit, they amount is
transferred to our account. If we made a loss, they take it from our account. No physical
asset is exchanged. The ASX SPI200 futures contract are an example of contracts that
are cash settled.
In our case, we went short when the price was $935. At expiry, the price is $920.
Short positions make money when prices fall, so we receive the difference of $15
per ounce. On 100,000 ounces, this is a profit of $1,500,000.
As in part d), we still have to sell the truckload of gold at the spot price of $920,
raising $92m. Overall, our total cashflow remains $93.5m.
To summarise:
The purpose of this question is to illustrate the fact that it doesn’t matter whether a
futures/forward contract is closed out or physically delivered or cash settled
at expiry. The net effect on cash is identical (or very close to it). In most of the
tutorial questions in this unit, my calculations assume the futures position is
closed-out rather than physically delivered. In fact, the majority of futures contracts
in practice are closed-out rather than physically delivered.
BFF3751 Tutorial 2 © Philip Gray 2020 4
Question 3 () nb: These answers will round all exchange rates to 4 decimal places (recommended). Students might get slightly different answers if they use more dps or fewer dps. Before we get started: questions involving exchange rates often confuse students. This is because exchanges rates can be expressed in two ways (direct or indirect quotes). The fool- proof way to avoid confusion is as follows. In this question, we have to pay an invoice in Euros. So just treat the Euro as you would any other asset (e.g., gold, wool, share price, etc). If the question involved gold, you would ask “what is the price of gold?” If the question involved wool, you would ask “what is the price of wool?” This question involves Euros, so we need to focus on “what is the price of Euros?” Hence, using direct quotes is the way to go. a) At the spot (current) exchange rate of 0.6000, the ‘price’ of one EUR is AUD 1.6667 (1 ÷ 0.6000). Buying EUR 10,000 would cost AUD 16,667. b) We have to buy Euros one year from now (because the invoice for our purchase is in Euros). Currently, one Euro costs AUD 1.6667. The risk is that the price of EUR rises. If this happens, our purchase will cost more one year from now. The price of EUR rises when it strengthens relative to the AUD. c) In one year’s time, the spot rate is 0.6500. Therefore, the ‘price’ of one EUR is AUD 1.5385 (1 ÷ 0.6500). Thus, the Euro has weakened relative to the AUD. It used to be worth AUD 1.6667, but now it’s only worth AUD 1.5385. The cost of buying EUR 10,000 is AUD 15,385. We got lucky here. We were unhedged and therefore left ourselves exposed to a strengthening Euro. However, the Euro weakened relative to the AUD so we were able to purchase the EUR 10,000 a little cheaper. d) In one year’s time, the spot rate is 0.5700. The ‘price’ of one EUR is AUD 1.7544 (1 ÷0.5700). The EUR has strengthened. It used to be worth AUD 1.6667, but now it’s worth AUD 1.7544. The cost of buying EUR 10,000 is AUD 17,544. In this case, we will regret not having hedged our exposure. We feared that the Euro would strengthen and this it exactly what happened. Therefore, purchasing the rewquired EUR 10,000 has become more expensive. BFF3751 Tutorial 2 © Philip Gray 2020 5 e) The decision to take a long or short derivatives position can be addressed as follows: • What is the risk we face? Part (d) established that, if the Euro strengthens relative to the AUD, our purchase will cost more. Hence, our fear is that the price of the Euro will rise. • What derivative position will make money if this unfavourable scenario eventuates? Recalling the payoff diagrams for long and short positions, we know that long positions make money when prices rise. Hence, we hedge our exposure by entering a long f orward contract to purchase EUR 10,000 one year from now. This locks-in a price of AUD 1.6584 for our purchase (1 ÷ 0.6030). f) These calculations assume that we close-out the forward position, calculate whether we made a profit or loss, then buy the required EUR 10,000 on the spot market. Realistically, a forward contract on a currency would be physically delivered. Nonetheless, as we learned in Question 2, it makes no difference to the bottomline whether we close out of physically deliver. Think about this before you do the calcs. The Euro has actually weakened in this case. We took a long forward contract to profit when the Euro strengthened. Given that the Euro has weakened , our calcs must show a loss on closing out the forward position. (in July) Enter long forward contract to buy Euros 10,000 × 1.6584 16,584 (one year later)Close-out with short forward contract 10,000 × 1.5385 15,385 Loss 1,199 (one year later)Buy EUR 10,000 at spot rate 10,000 × 1.5385 15,385 Net Cost (15,385 + 1,199) 16,584 g) In this case, the EUR has strengthened, which was the thing that we feared and prompted us to enter a hedge. So our calcs must surely show a profit on closing out the forward position. (in July) Enter long forward contract to buy Euros 10,000 × 1.6584 16,584 (one year later)Close-out with short forward contract 10,000 × 1.7544 17,544 Gain 960 (one year later)Buy EUR 10,000 at spot rate 10,000 × 1.7544 17,544 Net Cost (17,544 – 960) 16,584 BFF3751 Tutorial 2 © Philip Gray 2020 6 Parts (f) and (g) illustrate that we had a perfect hedge. No matter which direction the EUR/AUD exchange rate moves, we have locked-in a total cost of AUD 16,584 for the purchase. Aside: you may notice that this question is just a counter exam from the lecture example. In the lecture example, we had made a sale and would receive GBP 10m. We were exposed to the GBP weakening so we hedged by entering a short forward covering GBP. In this question, we will be paying Euros. This exposed us to loss if the Euro strengthened. Hence we hedged by entering a long forward covering Euros. BFF3751 Tutorial 2 © Philip Gray 2020 7 Question 4 ()
a) By convention, the SPI200 futures contracts have a standard contract size of $A25
times the quoted index number. Since the November-maturity SPI200 contract is quoted
at 4530, the notional value of one contract is $113,250.
This does not mean that longing one of these SPI200 futures contracts costs $113,250.
Nor does it mean shorting one of these contracts raises $113,250. No money changes
hands upfront. It merely means that we have a commitment to trade an asset for a
locked-in price of $113,250.
b) Our portfolio is currently worth $5m. We will use the Nov-maturity SPI200 futures to
effectively lock-in a portfolio value of $5m. The number of SPI200 contracts to use also
depends on the beta of your portfolio:
No. of Contracts = β ⎛ │ ⎝ Value Value of to one be Hedged SPI200
⎞ │ ⎠ = 0.70
⎛ │ ⎝ 4530 5,000,000 ×
$25
⎞ │ ⎠ = 31 contracts Note: the number of contracts is rounded to the nearest whole
number; you can’t trade a fraction of a contract! This means our hedge will be slightly
less than perfect.
c) Calculating the gain/loss on the futures position is always trivial. Our initial position
was to enter 31 long SPI200 contracts when the quoted futures price was 4530. We
then close out by entering 31 short SPI200 contracts at 3600. The loss on SPI200
futures is:
(1 June) Enter long SPI200 futures 31 × $A25 × 4530 3,510,750 (30 Nov) Close-out
with short SPI200 futures 31 × $A25 × 3600 2,790,000
Loss 720,750
To calculate what our stock portfolio is worth after the sharp market decline, we will use
the CAPM. If the dividend yield on the market portfolio is 5% p.a., then over a 6 month
period (June-November), the market would have generated about 2.5% dividend yield.
Similarly, if the riskless interest rate is 3% p.a., this is roughly 1.5% over a 6- month
period.
If the market index fell from 4500 to 3600, this is a 20% fall (3600/4500 – 1). However,
the market portfolio generated a 2.5% dividend yield, so the overall return on the market
portfolio is -17.5%.
Plug these inputs into the CAPM:
R p
= R f
+ β p ( R m
R f ) = 0.015 + 0.70 ( – 0.175 –
0.015 ) = –
11.8%
BFF3751 Tutorial 2 © Philip Gray 2020 8
This makes sense. The market fell by -17.5% overall. Since our portfolio is a little less
risky than the market, our portfolio will fall by less than -17.5%. From our starting
portfolio value of $5m, an 11.8% drop is equivalent to $590,000. Hence, after the sharp
market decline, our portfolio will only be worth $4,410,000.
Combining the two calcs above, our net worth in November is $3,689,250 ($4,410,000 –
$720,750).
Hopefully, alarm bells are ringing LOUDLY for you here:
• It is very odd that we established a hedge to protect against a market drop, yet when
we calculated the gain/loss on the futures position, we lost $720,750 when the market
dropped.
• It is also odd that, despite hedging against price movements, our $5m portfolio is now
worth only $3,689,250. If the hedge was effective, we should be locked- in at around
$5m.
So we suffered a loss on our portfolio value and another loss on the futures trading.
What happened to offsetting one loss with a gain on the other position? Something is
very wrong here!
d) Our initial position was to enter 31 long SPI200 contracts when the quoted futures
price was 4530. We then close out by entering 31 short SPI200 contracts at 4950. The
gain on SPI200 futures is:
(1 June) Enter long SPI200 futures 31 × $A25 × 4530 3,510,750 (30 Nov) Close-out
with short SPI200 futures 31 × $A25 × 4950 3,836,250
Gain 325,500
If the market index rose from 4500 to 4950, this is a 10% rise (4950/4500 – 1). In
addition, the market portfolio generated a 2.5% dividend yield, so the overall return on
the market portfolio is +12.5%.
Plug these inputs into the CAPM:
R p
= R f
+ β p ( R m
R f ) = 0.015 + 0.70 ( 0.125 –
0.015 ) = +
9.2%
This makes sense. The market rose by 12.5% overall. Since our portfolio is a little less
risky than the market, our portfolio will rise by a little less than 12.5%. From our
BFF3751 Tutorial 2 © Philip Gray 2020 9
starting portfolio value of $5m, a 9.2% rise is equivalent to $460,000. Hence,
after the bull market, our portfolio will be worth $5,460,000.
Combining the two calcs above, our net worth in November is $5,785,500
($5,460,000 + $325,500).
Again, something is not right here! Hedging is about offsetting one position
against another. Here, we made a gain on our share portfolio and a gain on the
futures trading. This is not a hedged position.
e) Our advice to hedge using a long SPI200 position was incorrect. We have a long
share position (i.e., we own shares). An effective hedge is achieved when the
derivatives position is the opposite (i.e., short). We should have entered 31 short
Nov-expiry SPI200 futures. We must fire our advisor – she effectively doubled our risk!
f) If our original futures trade had been 31 short SPI200 contracts, we would make a
gain
when the market drops to
3600:
(1 June) Enter short SPI200 futures 31 × $A25 × 4530 3,510,750 (30 Nov)
Close-out with long SPI200 futures 31 × $A25 × 3600 2,790,000
Gain 720,750
Adding the futures trading gain ( $720,750) to the portfolio value (4,410,000)
gives a net value of $5,130,750. That makes more sense. We started with a
portfolio worth $5m. After hedging against a market drop, our finishing portfolio
value is still pretty close to $5m.
Similarly, if we had correctly entered short SPI200 contracts to hedge when the
market rises to 4950, we must make a loss on the short futures:
(1 June) Enter short SPI200 futures 31 × $A25 × 4530 3,510,750 (30 Nov)
Close-out with long SPI200 futures 31 × $A25 × 4950 3,836,250
Loss 325,500
Deduct the futures trading loss ( $325,500) from the portfolio value ($5,460,000)
gives a net value of $5,134,500). This is more like what we expect from hedging.
Irrespective of whether the underlying market index drops or rises, our ending
value does not change much.
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