Introduction to Financial
Futures Markets
F520 Asset Valuation and Strategy
F520 – Futures
1
Derivatives and Spot Prices
• Derivative
A security whose value depends on the values of other
“underlying” securities (also know as “contingent claims”)
• Spot Price
Rate or price quoted for delivery in one or two business
days from the date of the transaction
F520 – Futures
2
Derivative Securities
• Forward Contracts -- A forward market is a market in
which a commodity/exchange rate for a future exchange of
commodities or financial contracts is fixed today.
• Futures Contracts -- An agreement reached at one point in
time calling for the delivery of some commodity at a
specified later date at a price established at the time of
contracting.
• Options Contracts -- The right (but not the obligation) to
buy or sell a particular good at a specified price
• Swap Contract – An agreement where two parties agree to
exchange periodic cash flows.
F520 – Futures
3
Roles of Derivative Markets
• Risk management and risk transference: Firms can take on
projects which otherwise might be rejected due to their
high levels of risk
– Examples are expansion of facilities in other countries, your choice
of obtaining a fixed or adjustable rate mortgage, ect.
– Due to the existence of speculators risk averse individuals are able
to transfer their risk more easily through a more active market
• Derivatives provide another market that investors can use
to alter their risk exposure to an asset when new
information is acquired. Competition between markets
should increase efficiency of the financial markets
• Trading financial derivatives generates publicly observable
prices
F520 – Futures
4
Hedging versus Speculating
• Speculating -- In contrast to hedging, the purpose of speculation
is to profit from a change in a future rate or price. Speculating
involves the assumption of additional risk
• Hedging -- Hedging is typically defined as utilizing financial
instruments or contracts to reduce or eliminate the risk from
future changes in rates or prices. The purpose of hedging a
transaction is to replace an uncertain future cash flow, rate, or
price, with a fixed and certain cash flow, rate, or price
– Microhedging -- Using futures (forward) contract to hedge a
specific asset or liability
– Macrohedging -- Hedging the entire exposure of an FI
– Routine Hedging versus Selective Hedging
• Routine Hedge -- Seeking to hedge all exposure
• Selective Hedge -- Only partially hedging the gap or hedging
occasionally
F520 – Futures
5
Forward Contracts
• An agreement to buy or sell an asset at a specified future
time at a specified price
• The specified price is the delivery price and it stays fixed
• The party which agrees to buy has a “long position”; the
seller has a “short position”
• A rise in the spot price generates a profit for those holding
a long position and a loss for those holding a short
position. The net effect is a “zero sum” game.
• Cash flows are exchanged only on the maturity date.
F520 – Futures
6
Spot
0
1
2
3 Months
1
2
3 Months
Price agreed / paid
Between buyer / seller
Commodity Delivered
Forward
0
Price agreed between
buyer / seller
Time of delivery / payment
Agreed between buyer / seller
(3 months)
Buyer pays forward
price / seller delivers
commodity
F520 – Futures
7
Key Elements of Forward Contracts
• Forward contracts involve no exchange of cash initially
• Forward contracts cannot be traded on an exchange
• Forward contracts typically are used in foreign exchange
markets, but there is also a large forward market now in
interest-rate contracts
• Forwards have credit risk and are illiquid (do not trade
frequently)
F520 – Futures
8
Pricing Forwards
• Risk-Free Arbitrage Opportunity arises when an
investment is identified that requires no initial outlays yet
guarantees nonnegative payoffs in the future
– Zero investment
– Zero risk
– Guaranteed return
F520 – Futures
9
Given information
• Following are current prices and information
– In the cash market Asset XYZ is selling for $100
– Asset XYZ pays the holder (with certainty) $12 per
year in four quarterly payments of $3, and the next
quarterly payment is 3 months from now.
– The forward contract requires delivery 3 months from
now.
– The current three month interest rate at which funds can
be loaned or borrowed is 8% per year (2 percent
quarterly).
F520 – Futures
10
Assume the forward price is 107.
Is this a fair price?
F520 – Futures
11
Assume the forward price is 107.
Is this a fair price?
Cash-and-Carry Arbitrage Transaction
Prices for Analysis
Spot price of asset
Forward price of asset
Financing Rate
Cash Flow Yield
100
107
2%
3%
P
F
r
y
t=0
Borrow $100 for one year at 2%
Buy one asset in the spot market
Sell a forward contract for delivery of asset
Total Cash Flow
100
-100
0
0
+P
-P
t=1
Receive cash flow
Deliver the asset against the forward contract
Repay loan, including interest
Total Cash Flow
In equilibrium this will equal 0, not 8.
3
107
-102
8
+yP
+F
-P(1+r)
F-P(1+r-y)
F=P+P(r-y)
F520 – Futures
12
Assume the forward price is $92.
Is this a fair price?
F520 – Futures
13
Assume the forward price is $92.
Is this a fair price?
Cash-and-Carry Arbitrage Transaction
Prices for Analysis
Spot price of asset
Forward price of asset
Financing Rate
Cash flow yield
100
92
2%
3%
P
F
r
y
t=0 Borrow $100 for one year at 2%
Buy asset in the spot market
Sell a forward contract for delivery of asset
Total Cash Flow
100
-100
0
0
+P
-P
t=1 Receive cash flow on asset
3
+yP
Deliver share of asset against the forward contract 92
+F
Repay loan, including interest
-102
-P(1+r)
Total Cash Flow
-7 F-P(1+r-y)
14
In equilibrium this will equal 0, not -7.
F=P+P(r-y)
What is the fair price?
• F = P + P(r - y)
• F = $100 + $100(0.02 - 0.03) = $99
 P is the current spot price
 r is the finance rate, risk-free rate
 y is the cash yield, such as dividends
F520 – Futures
15
Using a Forward Price of 99, we obtain
the fair price.
Cash-and-Carry Stock Arbitrage Transaction
Prices for Analysis
Spot price of asset
Forward price of asset
Financing Rate
Cash flow yield
100
99
2%
3%
P
F
r
y
t=0 Sell the asset and receive
Invest the proceeds in an interest bearing account
Buy a forward contract for delivery of asset
Total Cash Flow
100
-100
0
0
+P
-P
t=1 Pay cash flow to person that we sold asset to
Buy asset at the forward price
Interest earned plus original principal
Total Cash Flow
In equilibrium this will equal 0.
F520 – Futures
-3
-yP
-99
-F
102
+P(1+r)
0 -F+P(1+r-y)
F=P+P(r-y)
16
F520 – Futures
17
Besides interest, what factors should be
considered in our carrying costs?
• Financing/Investment costs may differ (rb and rl)
Range of Prices
Range of
 F = P + P(rb - y)
Prices
 F = P + P(rl - y)
• Costs as a percent are deducted from y or added to r.
Most common for commodities
 Transaction costs
Costs as a percent are deducted
 Storage cost
from y or added to r. Most
common for commodities
 Insurance costs
 Transportation costs
F520 – Cha Futures
18
What market imperfections cause our arbitrage
pricing model to break down and provide us a range
of prices that we should consider?
• Interim cash flows are not considered (not important for
commodities, but are for dividend paying stocks)
• Differences between the borrowing and lending rate
• Proceeds from short selling may not be available
• Deliverable asset and settlement date may be unknown
(contract may be contingent or allow for a range of dates or
more than one type of asset is allowed for delivery)
• Deliverable asset may be a basket of securities which may
be difficult to track
• Differences in the tax treatment of securities and
forwards/futures
F520 – Futures
19
Futures Contracts
• Like forwards, a financial futures contract is an agreement
to buy or sell an asset at a specified time at a specified
price
• But Futures are:
– “Standardized” contracts
– Traded on an organized exchange which “guarantees”
performance
– “Marked” to market daily
– Supported by a “margin account” (performance bond)
– Only rarely do buyers and sellers of futures contracts accept or
make delivery. Most futures contracts are “offset” by taking
an opposite position.
F520 – Futures
20
Forward
0
1
2
Forward price agreed
between buyer / seller
Time of delivery / payment
agreed between buyer / seller
(3 months)
Futures
0
3 Months
Buyer pays forward
price / seller delivers
commodity
Marking to Market Daily
1
Futures price agreed
between buyer / seller
Time of delivery / payment
agreed between buyer / seller
(3 months)
2
3 Months
Buyer pays futures
price / seller delivers
commodity
F520 – Futures
21
Payoff of Future ($)
Payoff of Future ($)
Net Payoff of a Futures Contract
Net Payoff
Long Position in Futures Contract
0
Security Price
X–futures price
Short Position in Futures Contract
Net Payoff
0
Xfutures price
22
Security Price
Exchange vs. Over-the-Counter Markets
• Exchange Market is a market with a centralized exchange
and trading floor. Each futures exchange has a
clearinghouse which guarantees contract performance to
both parties
– Major products sold on Exchanges
• Options
• Futures
• Over-the-Counter market is a market with out a centralized
exchange or trading floor
– Major products sold Over-the-Counter
• Options
• Forwards
• Swaps
23
Advantages of Over-the-Counter markets
(relative to Exchanges)
• Exchange products may lack flexibility in the variety of
instruments available and the horizon over which they
trade.
• Exchanges are regulated by the government. While
providing some benefits, it also restricts the kinds of
trading that can be conducted.
• Complying with exchange rules and regulations may
increase the cost of trading on exchanges
F520 – Futures
24
Disadvantages of Over-the-Counter
Markets (relative to Exchanges)
• Credit risk -- both parties must trust each other to
complete the contract as promised.
• Difficulty of finding a trading partner -- It may be difficult
to find a counter party willing to buy (sell) a commodity at
a specific day in the future
• Difficulty of fulfilling an obligation without actually
completing delivery
F520 – Futures
25
Why Hedges are Imperfect
• Amount -- Since exchange traded derivatives have specified contract
amounts, if the amount that we wish to hedge is not exactly equal to
some multiple of the contract size, we must choose to either underhedge or over-hedge.
• Delivery Date -- If the date of our future transaction does not
perfectly match with the derivative delivery date, we must choose a
contract with a longer delivery date and close it (Future and spot
rates converge as they get closer to maturity, therefore the
correlation between the future and spot rate is not equal to 1.) out
early or an earlier delivery date and remain unhedged for a period of
time.
• Basis Risk – When price movements on the futures contract and the
underlying asset are not perfectly correlated, an imperfect hedge
results, even though they are the same asset. This risk can be
estimated by obtaining the correlation between the two prices from
past price movements.
26
Why Hedges are Imperfect (cont.)
• Cross-hedging Risk -- When we try to hedge an asset with a
similar, but not identical asset. For example, hedge platinum
with a gold futures contract. While historically highly
correlated, there is the risk that these past price patterns may not
continue.
Maturity of Contract Security -- For interest rate derivatives, if
the item we are hedging does not have the same maturity of the
item that we are using to hedge the instrument, we are exposed
to risk. For short-term financial contracts we estimate the
number of contracts needed as
$ amt. of security duration of asset.
# of Contracts = ------------------ X -----------------------$ amt. of contract dur of derivative sec
• Transaction Costs -- Commissions and margin requirements
take up cash and make it difficult to get a perfect hedge.
27
Hedging Illustration
• Page 175 of text
Assume that a gold mining company expects to sell 1,000
ounces of gold one week from now and that the
management of a jewelry company plans to purchase 1,000
ounces of gold one week from now. The managers of both
the gold mining company and the jewelry company want to
lock in today’s price – that is, they both want to eliminate
the price risk associated with gold one week from now.
The cash price for gold is currently $352.40 per ounce.
The cash price is also called the spot price. The futures
price for gold is currently $397.80 per ounce. Each futures
contract is for 100 ounces.
F520 – Futures
28
Devise the appropriate hedge
F520 – Futures
29
TABLE 10-2: How well did your hedge fair? Assume the
following prices in one week – cash $304.20 and futures $349.60.
Information at time of hedge (t=0)
Cash Price
$ 352.40
Futures Price
$ 397.80
Number of ounces to hedge
1000
Number of ounces per futures contract
100
Number of futures contracts used
10 calc
Information at end of hedge (t=1 week)
Cash Price
$ 304.20
Futures Price
$ 349.60
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 1000 oz. @ spot
1,000 * $ 352.40 $ 352,400
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $
F $
Basis $
352.40
397.80
(45.40)
At time hedge is Lifted
Value of 1000 oz. @ spot
1,000 * $ 304.20 $ 304,200
Buy 10 futures contracts
+10*100* $349.60 $ 349,600
S $
F $
Basis $
304.20
349.60
(45.40)
Summary
Gain (Loss) Cash Mkt. $ (48,200)
Net Position
Gain (Loss) Futures
$0
$ 48,200
30
TABLE 10-3: How well did your hedge fair? Assume the
following prices in one week – cash $392.50 and futures $437.90.
Information at time of hedge (t=0)
Cash Price
$ 352.40
Futures Price
$ 397.80
Number of ounces to hedge
1000
Number of ounces per futures contract
100
Number of futures contracts used
10 calc
Information at end of hedge (t=1 week)
Cash Price
$ 392.50
Futures Price
$ 437.90
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 1000 oz. @ spot
1,000 * $ 352.40 $ 352,400
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $
F $
Basis $
352.40
397.80
(45.40)
At time hedge is Lifted
Value of 1000 oz. @ spot
1,000 * $ 392.50 $ 392,500
Buy 10 futures contracts
+10*100* $437.90 $ 437,900
S $
F $
Basis $
392.50
437.90
(45.40)
Summary
Gain (Loss) Cash Mkt. $ 40,100
Net Position
Gain (Loss) Futures
$0
$ (40,100)
31
TABLE 10-4 (Basis Widens): How well did your hedge fair? Assume
the following prices in one week – cash $304.20 and futures $385.80.
Information at time of hedge (t=0)
Cash Price
$ 352.40
Futures Price
$ 397.80
Number of ounces to hedge
1000
Number of ounces per futures contract
100
Number of futures contracts used
10 calc
Information at end of hedge (t=1 week)
Cash Price
$ 304.20
Futures Price
$ 385.80
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 1000 oz. @ spot
1,000 * $ 352.40 $ 352,400
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $
F $
Basis $
352.40
397.80
(45.40)
At time hedge is Lifted
Value of 1000 oz. @ spot
1,000 * $ 304.20 $ 304,200
Buy 10 futures contracts
+10*100* $385.80 $ 385,800
S $
F $
Basis $
304.20
385.80
(81.60)
Summary
Gain (Loss) Cash Mkt. $ (48,200)
Net Position
Gain (Loss) Futures
($36,200)
$ 12,000
32
TABLE 10-5 (Basis Widens): How well did your hedge fair? Assume
the following prices in one week – cash $392.50 and futures $474.10.
Information at time of hedge (t=0)
Cash Price
$ 352.40
Futures Price
$ 397.80
Number of ounces to hedge
1000
Number of ounces per futures contract
100
Number of futures contracts used
10 calc
Information at end of hedge (t=1 week)
Cash Price
$ 392.50
Futures Price
$ 474.10
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 1000 oz. @ spot
1,000 * $ 352.40 $ 352,400
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $
F $
Basis $
352.40
397.80
(45.40)
At time hedge is Lifted
Value of 1000 oz. @ spot
1,000 * $ 392.50 $ 392,500
Buy 10 futures contracts
+10*100* $474.10 $ 474,100
S $
F $
Basis $
392.50
474.10
(81.60)
Summary
Gain (Loss) Cash Mkt. $ 40,100
Net Position
Gain (Loss) Futures
($36,200)
$ (76,300)
33
What would the hedge look like
if we entered a forward contract?
Assume a forward price of $370 per ounce
F520 – Futures
34
Cross-Hedging Illustration
• Page 178 of text
Suppose that a mining company on a far-away planet plans
to sell 2,500 ounces of kryptonite one week from now and
that a jewelry company plans to purchase the same amount
of kryptonite in one week. Both parties want to hedge
against price risk. However, kryptonite futures contracts
are not currently traded. Both parties believe that there is a
close relationship between the price of kryptonite and the
price of gold. Specifically, both parties believe that the
cash price of kryptonite will remain at 40% of the cash
price of gold. The cash price of kryptonite is currently
$140.96 per ounce and the cash price of gold is currently
$352.40 per ounce. The futures price of gold is currently
$397.80 per ounce.
F520 – Futures
35
Devise the appropriate cross-hedge
F520 – Futures
36
TABLE 10-6 (Basis&Ratio Stable): How well did your hedge fair?
Information at time of hedge (t=0)
Information at end of hedge (t=1 week)
Cash Price Kryptonite
$ 140.96 Cash Price Kryptonite
$ 121.68
Cash Price Gold
$ 352.40 Cash Price Gold
$ 304.20
Futures Price Gold
$ 397.80 Futures Price
$ 349.60
Number of ounces Kryptonite hedged
2500
Number of ounces per futures contract
100
Ratio (t=0)
Ratio (t=1)
Number of futures contracts used
10 calc
40.00%
40.00%
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 2500 oz. @ spot
2,500 * $ 140.96 $ 352,400
At time hedge is Lifted
Value of 2500 oz. @ spot
2,500 * $ 121.68 $ 304,200
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $ 352.40
F $ 397.80
Basis $ (45.40)
Buy 10 futures contracts
+10*100* $349.60 $ 349,600
S $ 304.20
F $ 349.60
Basis $ (45.40)
Summary
Gain (Loss) Cash Mkt. $ (48,200)
Gain (Loss) Futures
Net Position
$0
$ 48,200
37
TABLE 10-7 (Basis Stable, Ratio Changes):
How well did your hedge fair?
Information at time of hedge (t=0)
Information at end of hedge (t=1 week)
Cash Price Kryptonite
$ 140.96 Cash Price Kryptonite
$ 112.00
Cash Price Gold
$ 352.40 Cash Price Gold
$ 304.20
Futures Price Gold
$ 397.80 Futures Price
$ 349.60
Number of ounces Kryptonite hedged
2500
Number of ounces per futures contract
100
Ratio (t=0)
Ratio (t=1)
Number of futures contracts used
10 calc
40.00%
36.82%
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 2500 oz. @ spot
2,500 * $ 140.96 $ 352,400
At time hedge is Lifted
Value of 2500 oz. @ spot
2,500 * $ 112.00 $ 280,000
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $ 352.40
F $ 397.80
Basis $ (45.40)
Buy 10 futures contracts
+10*100* $349.60 $ 349,600
S $ 304.20
F $ 349.60
Basis $ (45.40)
Summary
Gain (Loss) Cash Mkt. $ (72,400)
Gain (Loss) Futures
Net Position ($24,200)
$ 48,200
38
TABLE 10-8 (Basis Stable, Ratio Changes):
How well did your hedge fair?
Information at time of hedge (t=0)
Information at end of hedge (t=1 week)
Cash Price Kryptonite
$ 140.96 Cash Price Kryptonite
$ 121.68
Cash Price Gold
$ 352.40 Cash Price Gold
$ 392.50
Futures Price Gold
$ 397.80 Futures Price
$ 437.90
Number of ounces Kryptonite hedged
2500
Number of ounces per futures contract
100
Ratio (t=0)
Ratio (t=1)
Number of futures contracts used
10 calc
40.00%
31.00%
Short (Sell) Hedge by Gold Mining Company
Cash Market
At time hedge is placed
Value of 1000 oz. @ spot
2,500 * $ 140.96 $ 352,400
At time hedge is Lifted
Value of 1000 oz. @ spot
2,500 * $ 121.68 $ 304,200
Futures Market
Basis
Sell 10 futures contracts
10*100* $397.80 $ 397,800
S $ 352.40
F $ 397.80
Basis $ (45.40)
Buy 10 futures contracts
+10*100* $437.90 $ 437,900
S $ 392.50
F $ 437.90
Basis $ (45.40)
Summary
Gain (Loss) Cash Mkt. $ (48,200)
Gain (Loss) Futures
Net Position ($88,300)
$ (40,100)
39
Copper Cash and Forward Prices
LME Official Prices (US$/tonne) for 13 September 2013
COPPER
Cash Buyer
Cash Seller &
Settlement
3-months Buyer
3-months Seller
15-months Buyer
15-months Seller
7,028.00
7,028.50
7,059.50
7,060.00
7,235.00
7,245.00
2,204.60
$3.19
lbs per metric tonne
price per pound
$3.19
$3.20
$3.20
$3.28
$3.29
http://www.lme.com/metals/non-ferrous/copper/
F520 – Futures
40
Copper Futures, price per pound, 25,000 pounds per contract
Daily Settlements for Copper Future Futures (FINAL) - Trade Date: 09/13/2013
Month
SEP 13
OCT 13
NOV 13
DEC 13
JAN 14
FEB 14
MAR 14
APR 14
MAY 14
JUN 14
JLY 14
AUG 14
SEP 14
Open
3.2010
3.2060
3.2015
3.2000
3.2030
3.2090
3.2350
3.2115
3.2255
3.2200
3.2405
High
|
3.2300
3.2255
3.2200
3.2270
3.2100
3.2120
3.2355
3.2115
3.2255
3.2200
3.2505B
3.2475
Low
3.1950
3.1905
3.1910
3.1905
3.1980
3.2020
3.2040
3.2115
3.2150
3.2200
3.2405
Last
3.2100
3.2040
3.2070
3.2040
-
Change
-.0055
-.0060
-.0065
-.0065
-.0065
-.0060
-.0070
-.0070
-.0065
-.0065
-.0065
-.0065
-.0065
Settle
3.2070
3.2030
3.2035
3.2035
3.2075
3.2110
3.2150
3.2190
3.2230
3.2275
3.2315
3.2360
3.2405
Prior Day
Estimated
Open
Volume
Interest
597
2,244
366
2,086
138
1,577
39,396 106,197
24
1,619
30
1,108
3,979
25,423
4
553
238
2,739
4
620
27
1,628
1
649
6
1,195
http://www.cmegroup.com/trading/metals/base/copper_quotes_settlements_futures.html
F520 – Futures
41
Product Calendar for Copper Futures
First
Holding
Last
Holding
First
Position
Last
Position
First Notice
Last Notice
First
Delivery
Last
Delivery
Contract
Month
Product
Code
First Trade
Last Trade
Settlement
SEP 2013
HGU13
09/20/2010
09/26/2013
09/26/2013
-
08/29/2013
09/27/2013
08/30/2013
09/27/2013
09/03/2013
09/30/2013
OCT 2013
HGV13
10/31/2011
10/29/2013
10/29/2013
-
09/27/2013
10/30/2013
09/30/2013
10/30/2013
10/01/2013
10/31/2013
NOV 2013
HGX13
11/30/2011
11/26/2013
11/26/2013
-
10/30/2013
11/27/2013
10/31/2013
11/27/2013
11/01/2013
11/29/2013
DEC 2013
HGZ13
09/20/2010
12/27/2013
12/27/2013
-
11/27/2013
12/30/2013
11/29/2013
12/30/2013
12/02/2013
12/31/2013
F520 – Futures
42
What is the estimated forward price?
• Dividend: 0.00% on a commodity
• Rate: 0.00% to 0.30% for 3 month to 1 year rate
http://finance.yahoo.com/bonds
• F = P + P(r - y)
• F = 3.19 + 3.19 (r - 0.00)*(month/12)
Rate = 0.00%
Forward Months
3.19
1
3.19
2
3.19
3
3.19
4
3.19
5
3.19
6
Rate = 0.30%
Forward Months Forward Months
3.19
7
3.19
1
3.19
8
3.19
2
3.19
9
3.19
3
3.19
10
3.19
4
3.19
11
3.19
5
3.19
12
3.19
6
F520 – Futures
Forward Months
3.20
7
3.20
8
3.20
9
3.20
10
3.20
11
3.20
12
43
Using Futures to Replicate the Returns of a
Portfolio
• Futures Price is for delivery in 1 year
• F = P + P(r - y)
 P is the current spot price –
Assume S&P500 is 1,000
and I buy 1000 contracts (my total exposure is $1 million)
 r is the finance rate, risk-free rate of 4%
 y is the cash yield, such as dividends of 2%
• Futures Price (based on fair value formula)
F = $1,000 + $1,000(0.04 - 0.02)
= $1,020
F520 – Futures
44
S&P500 Stocks
• Assume I buy the stocks representing the S&P500 Index
and hold it to its delivery date in 1 year. What is my dollar
return if in 1 year the spot price is $1100? Note that the
spot price and futures price should converge on the
delivery date since both have 2 days to settlement.
(P1 - P0 + D)
($1.1m-$1m+$1m*.02) = $120,000
F520 – Futures
45
S&P500 Futures
• How many Futures contracts do I need
Information from earlier
– Current spot price – on S&P500 is $1,000
– Current futures price _ S&P futures is $1,020
an emini S&P futures controls 50 S&P indexes
a S&P futures controls 250 S&P indexes
– The total exposure I want is $1 million
• The calculation for the number of contracts is similar to a
cross-hedge.
# futures = Market value of portfolio / Cash Value of 1 future
# futures = $1 mil / $1000*50 using the emini
# futures = 20 contracts
F520 - Futures
46
•
•
•
Assume I buy 20 Futures contract and hold them till the delivery date in 1 year.
What is my return if on the date of delivery the futures price is $1100?
(P1 - P0)
($1100-$1020)*50*20 = $80,000
How can the return on a futures replicate the return on holding the S&P 500 contract?
What were the cash outflows on the futures contract at t=0?
Nothing!
So what should I do with the money? So far I have nothing invested in the futures
contract. Remember a margin simply means I have funds accessible to my broker in
a liquid account (assume a money market earning the risk-free return)
Invest the full $1 million in the risk-free rate (this is far more than I need for the
margin), earning 4% in our example.
Assume I have the same $1 million that I assumed for the spot purchase.
$1 mil *.04 = $40,000 return on investment in risk-free rate
(This underestimates my return since I did not take into account that the futures price
is increasing and marking to market will add to the amount of funds in my investment
account that earns the risk-free rate.)
•
So what is my return on the Futures in the S&P500 and the risk free portfolio?
$40,000 + $80,000 = $120,000 on my original $1 million of investment or 12%
Note, both made $120,000 on my investment. Both required $1 million as the initial
investment. Therefore, the futures portfolio can replicate the expected return in the
underlying asset.
47
Adjusting for the Hedge Ratio
• What would I do if the portfolio had a beta of 1.2. This is greater risk
than the S&P500 contract. How could I mimic the expected return on
this portfolio?
# futures = ($1 mil / $1000*50 using the emini)*1.2
# futures = 20*1.2 = 24 contracts
• Our expected dollar return
(P1 - P0)
($1100-$1020)*50*24 = $96,000 from the futures
$1 mil *.04 = $40,000 from the risk free return
$40,000 + $96,000 = $136,000
• Since this is a cross-hedge (portfolio beta of 1.2 and S&P500 futures
beta of 1.0) the hedge will not be perfect. This is an approximate
number of contracts.
F520 – Futures
48
S&P 500 (^GSPC)
-SNP
1,687.99 4.57(0.27%)
Sep 13, 2013
http://finance.yahoo.com/q?s=^GSPC&ql=1
Daily Settlements for E-mini S&P 500 Future Futures (FINAL)
Month
Open
High
Low
Last
Change
Trade Date: 09/13/2013
Settle
Prior Day
Estimated
Open
Volume
Interest
SEP 13
1684.75
1690.50
1680.50
1689.50
+3.50
1688.50
756,480 2,261,091
DEC 13
1678.25
1683.75
1674.00
1682.75
+3.75
1682.00 1,162,403
MAR 14
1669.00
1676.75B 1668.00A 1676.50A
+3.75
1675.50
105
2,953
JUN 14
1665.00
1668.75B
SEP 14
-
837,087
1661.50
1668.75B
+3.75
1669.25
8
441
1658.75B 1656.75A
1658.75B
+3.75
1662.25
0
6
Total
1,918,996 3,101,578
http://www.cmegroup.com/trading/equity-index/us-index/e-mini-sandp500_quotes_settlements_futures.html
F520 – Futures
49
What is the estimated forward price?
• Dividend: 2.04%
http://finance.yahoo.com/q?s=SPY&ql=0
• Rate: 0.00%
http://finance.yahoo.com/bonds
• F = P + P(r - y)
• F = 1,687.99 + 1,687.99 (0.00 - 0.0204)*(month/12)
Rate = 0.00%
Forward Months
1,685.12
1
1,682.25
2
1,679.38
3
1,676.51
4
1,673.64
5
1,670.77
6
Rate = 0.30%
Forward Months Forward Months
1,667.90
7
1,685.54
1
1,665.03
8
1,683.09
2
1,662.16
9
1,680.65
3
1,659.29
10
1,678.20
4
1,656.42
11
1,675.75
5
1,653.56
12
1,673.30
6
F520 – Futures
Forward Months
1,670.86
7
1,668.41
8
1,665.96
9
1,663.51
10
1,661.07
11
1,658.62
12
50
Resources for Calculations
Category:
Fund Family:
Net Assets:
Yield:
Fund Inception Date:
Legal Type:
Large Blend
SPDR State Street Global Advisors
135.73B
2.04%
Jan 22, 1993
Exchange Traded Fund
http://finance.yahoo.com/q/pr?s=SPY+Profile
US Treasury Bonds Rates http://finance.yahoo.com/bonds
Maturity
Yield
Yesterday
Last Week
3 Month
0.00
0.00
0.00
6 Month
0.01
0.01
0.04
2 Year
0.43
0.44
0.45
3 Year
0.86
0.88
0.88
5 Year
1.69
1.71
1.75
10 Year
2.88
2.90
2.93
30 Year
3.83
3.85
3.86
F520 – Futures
Last Month
0.02
0.05
0.32
0.67
1.47
2.71
3.75
51