Chapter 9
Stock Index Futures
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© 2004 South-Western Publishing
Outline
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Introduction
Stock indexes and their futures contracts
Uses of stock index futures
Hedging with stock index futures
Introduction
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The fastest growing segment of the futures
market is in financial futures
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In 1972, physical commodities comprised over
95 percent of all futures volume
Today, physical commodities amount to only
one-third of total futures volume
Stock Indexes and Their
Futures Contracts
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Stock indexes
Stock index futures contracts
The S&P 500 stock index futures contract
Pricing of stock index futures
Basis convergence
Stock Indexes
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Introduction
Capitalization-weighted indexes
Introduction
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The S&P 500 index represents about 90% of
all U.S. stock index futures trading
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First published in 1917
Currently one of the Commerce Department’s
leading indicators
Capitalization-Weighted Indexes
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The S&P 500 index is capitalizationweighted
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Each of the 500 share prices in the index is
multiplied by the number of outstanding shares
in that particular firm
Standard and Poor’s calculates the index by
adding these figures and dividing by the index
divisor
Capitalization-Weighted Indexes
(cont’d)
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Assume only three firms are in an index
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Assume the initial divisor is arbitrarily set
at 2,700,000
Capitalization-Weighted Indexes
(cont’d)
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Stock
Shares Out
Closing Price
Shares x Price
A
1,000,000
$10
10,000,000
B
5,000,000
$22
110,000,000
C
10,000,000
$15
150,000,000
Total
270,000,000
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Day 1
Index = 270,000,000/2,700,000 = 100.00
Capitalization-Weighted Indexes
(cont’d)
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Stock
Shares Out
Closing Price
Shares x Price
A
1,000,000
$11
11,000,000
B
5,000,000
$20
100,000,000
C
10,000,000
$16
160,000,000
Total
271,000,000
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Day 2
Index = 271,000,000/2,700,000 = 100.37
Capitalization-Weighted Indexes
(cont’d)
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Stock
Shares Out
Closing Price
Shares x Price
A
1,000,000
$12
12,000,000
B
10,000,000
$11
110,000,000
C
10,000,000
$14
140,000,000
Total
262,000,000
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Day 3 – B splits two for one
Index = 262,000,000/2,700,000 = 97.04
Stock Index Futures Contracts
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As with other futures, a stock index future
is a promise to:
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Buy or sell
Standardized units
Of a specific index
At a fixed price
At a predetermined future date
Stock Index Futures Contracts
(cont’d)
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Stock index futures are similar in every
respect to a traditional agricultural contract
except for the matter of delivery
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Index futures settle in cash rather than by
delivery of the underlying asset
The S&P 500 Stock Index
Futures Contract
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There is no actual delivery mechanism at
expiration of an S&P 500 futures contract
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You actually deliver the dollar difference
between the original trade price and the final
price of the index at contract termination
Pricing of Stock Index Futures
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Elements affecting the price of a futures
contract
Determining the fair value of a futures
contract
Synthetic index portfolios
Elements Affecting the Price of
A Futures Contract
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The S&P 500 futures value depends on four
elements:
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The level of the spot index
The dividend yield on the 500 stock in the index
The current level of interest rates
The time until final contract cash settlement
Elements Affecting the Price of
A Futures Contract (cont’d)
SPX Dividend Yield
SPX Index
S&P 500
Stock Index
Futures
T-bill Rate
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Time until Settlement
Elements Affecting the Price of
A Futures Contract (cont’d)
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Stocks pay dividends, while futures do not
pay dividends
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Shows up as a price differential in the futures
price/underlying asset relationship
Elements Affecting the Price of
A Futures Contract (cont’d)
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Stocks do not accrue interest
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Posting margin for futures results in
interest
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Shows up as a price differential in the futures
price/underlying asset relationship
Determining the Fair Value of A
Futures Contract
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The futures price should equal the index
plus a differential based on the short-term
interest rate minus the dividend yield:
F  Se
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( R  D )T
Determining the Fair Value of A
Futures Contract (cont’d)
Calculating the Fair Value of A Futures
Contract Example
Assume the following information for an S&P 500
futures contract:
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Current level of the cash index (S) = 1,484.43
T-bill yield ® = 6.07%
S&P 500 dividend yield (D) = 1.10%
Days until December settlement (T) = 121 = 0.33 years
Determining the Fair Value of A
Futures Contract (cont’d)
Calculating the Fair Value of A Futures
Contract Example
The fair value of the S&P 500 futures contract is:
F  Se( R  D )T
 1,484.43e
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(.0607.0110)(121/ 365)
 1,509.30
Synthetic Index Portfolios
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Large institutional investors can replicate a
well-diversified portfolio of common stock
by holding
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A long position in the stock index futures
contract and
Satisfying the margin requirement with T-bills
The resulting portfolio is a synthetic index
portfolio
Synthetic Index Portfolios
(cont’d)
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The futures approach has the following
advantages over the purchase of individual
stocks:
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Transaction costs will be much lower on the
futures contracts
The portfolio will be much easier to follow and
manage
Basic Convergence
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As time passes, the difference between the
cash index and the futures price will narrow
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At the end of the futures contract, the futures
price will equal the index (basic convergence)
Uses of Stock Index Futures
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Speculation
Spreading
Arbitrage
Anticipation of stock purchase or sale
Hedging
Speculation
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Each one-point movement in the S&P 500
index translates to $250
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A person who is bullish could obtain substantial
leverage by buying S&P contracts
Spreading
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Spreads using index futures can be used to
speculate with reduced risk
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E.g., a speculator believing the Nasdaq will
outperform the Dow Jones could employ an
intermarket spread by buying Nasdaq 100
futures and selling DJIA futures
Arbitrage
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Sometimes the market price of a futures
contract temporarily deviates from the
price predicted by pricing theory
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An arbitrageur could short the futures contracts
and buy stock if the price deviates upward
An arbitrageur could short the stock and buy
futures contracts if the price deviates downward
Anticipation of Stock Purchase
or Sale
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Futures contracts can be used to lock in a
price in anticipation of a stock purchase or
sale
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E.g., a portfolio manager might want to get out
of the market, but for tax reasons does not want
to sell securities until the new year
Hedging
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The primary purpose of S&P futures is to
facilitate risk transfer from one who bears
undesired risk to someone else willing to
bear the risk
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S&P futures are used by most large commercial
banks and by many pension funds and
foundations to hedge
Hedging With Stock Index
Futures
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Systematic and unsystematic risk
The need to hedge
The hedge ratio
Hedging in retrospect
Adjusting market risk
Systematic and Unsystematic
Risk
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Systematic factors are those that influence
the stock market as a whole
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E.g., interest rates, economic indicators,
political climate, etc.
Systematic risk or market risk
Systematic and Unsystematic
Risk (cont’d)
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Unsystematic factors are unique to a
specific company or industry
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E.g., earnings reports, technological
developments, labor negotiations, etc.
Unsystematic risk
Systematic and Unsystematic
Risk (cont’d)
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Proper portfolio diversification can virtually
eliminate unsystematic risk
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The market assumes that you have been
smart enough to reduce risk through
diversification
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Beta measures the relative riskiness of a
portfolio compared to a benchmark portfolio
like the S&P 500
Systematic and Unsystematic
Risk (cont’d)
Portfolio Variance
Number of Securities
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The Need to Hedge
Using Futures Contracts to Hedge Portfolios
You are the manager of a $100 million equity
portfolio. You are bullish in the long term, but
anticipate a temporary market decline.
How can you use futures contracts to hedge your
stock portfolio?
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The Need to Hedge (cont’d)
Using Futures Contracts to Hedge Portfolios
(cont’d)
If you are long stock, you should be short futures.
You need to calculate the number of contracts
necessary to counteract likely changes in the
portfolio value.
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The Hedge Ratio
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Introduction
The market falls
The market rises
The market is unchanged
Introduction
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To construct a proper hedge, you must
realize that portfolios are of
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Different sizes
Different risk levels
The hedge ratio incorporates the relative
value of the stock and futures, and
accounts for the relative riskiness of the
two portfolios
Introduction (cont’d)
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To determine the hedge ratio, you need:
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The value of the chosen futures contract
The dollar value of the portfolio to be hedged
The beta of the portfolio
Introduction (cont’d)
Determining the Factors for A Hedge
Suppose the manager of a $75 million stock
portfolio (with a beta of 0.9 and a dividend yield of
1.0%) wants to hedge using the December S&P 500
futures.
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On the previous day, the S&P 500 closed at
1,484.43, and the DEC 00 S&P 500 futures closed
at 1,517.20.
Introduction (cont’d)
Determining the Factors for A Hedge (cont’d)
The value of the futures contract is:
$250 x 1,517.20 = $379,300
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Introduction (cont’d)
Determining the Factors for A Hedge (cont’d)
The hedge ratio is:
Dollar val ue of the portfolio
HR 
 beta
Dollar val ue of the S & P futures contract
$75,000,000
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 0.9  177.96  178 contracts
1,517.20  $250
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The Market Falls
Using the Hedge in A Falling Market
Assume the S&P 500 index falls 5%, from 1,484.43 to 1,410.20
after three months.
Given beta, the portfolio should have fallen by 5.0% x 0.9 =
4.5%, which translates to $3,375,000. However, you receive
dividends of 1% x .333 x $75,000,000 = $250,000.
If you sold 178 contracts short at 1,517.20, your account will
benefit by (1,517.20 – 1,410.20) x $250 x 178 = $4,761,500.
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The Market Falls (cont’d)
Using the Hedge in A Falling Market (cont’d)
The combined positions (stock, dividends, and
futures contracts) result in a gain of $1,636,500.
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The Market Rises
Using the Hedge in A Rising Market
Assume the S&P 500 index rises from 1,484.43 to 1,558.70
after three months.
Given beta, the portfolio should have advanced by 5.0% x 0.9
= 4.5%, which translates to $3,375,000. You still receive
dividends of 1% x .333 x $75,000,000 = $250,000.
If you sold 178 contracts short at 1,517.20, your account will
lose (1,517.20 – 1,558.70) x $250 x 178 = $1,846,750.
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The Market Rises (cont’d)
Using the Hedge in A Rising Market (cont’d)
The combined positions (stock, dividends, and
futures contracts) result in a gain of $1,778,250.
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The Market is Unchanged
Using the Hedge in An Unchanged Market
Assume the S&P 500 index remains at 1,484.43 after three
months.
There is no gain on the stock portfolio. However, you still
receive dividends of 1% x .333 x $75,000,000 = $250,000.
If you sold 178 contracts short at 1,517.20, your account will
benefit by (1,517.20 – 1,484.50) x $250 x 178 = $1,455,150.
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The Market is Unchanged
(cont’d)
Using the Hedge in An Unchanged Market
(cont’d)
The combined positions (stock, dividends, and
futures contracts) result in a gain of $1,705,150.
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Hedging in Retrospect
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A hedge will usually not be perfect
because:
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It is not possible to hedge exactly
Stock portfolios seldom behave exactly as their
beta suggests
The futures price does not move in lockstep with
the underlying index (basis risk)
The dividends on the S&P 500 index do not
occur uniformly over time
Adjusting Market Risk
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Futures can be used to adjust the level of
market risk in a portfolio:
Portfolio value   βdesired  βcurrent 
# contracts 
Futures level  $250
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Adjusting Market Risk (cont’d)
Determining the Number of Contracts Needed to
Increase Market Exposure
Suppose the manager of a $75 million stock portfolio with a
beta of 0.9 would like to increase market exposure by
increasing beta to 1.5. Yesterday, DEC 00 S&P 500 futures
closed at 1517.20
How can the manager use futures to accomplish this goal,
assuming the composition of the stock portfolio remains
unchanged?
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Adjusting Market Risk (cont’d)
Determining the Number of Contracts Needed to
Increase Market Exposure (cont’d)
The manager should go long futures and hold them with the
stock portfolio. Specifically, he should purchase 119 S&P 500
futures contracts:
$75 million  (1.50  0.90)
# contracts 
 119
1517.20  $250
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