© 2002 South-Western Publishing 1

Chapter 9

Stock Index Futures

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Outline

Introduction Stock indexes and their futures contracts Uses of stock index futures Hedging with stock index futures

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Introduction

The fastest growing segment of the futures market is in financial futures

– In 1972, physical commodities comprised over 95 percent of all futures volume

– Today, physical commodities amount to only one-third of total futures volume

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Stock Indexes and Their Futures Contracts

Stock indexes Stock index futures contracts The S&P 500 stock index futures contract Pricing of stock index futures

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Stock Indexes

Introduction Capitalization-weighted indexes

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Introduction

The S&P 500 index represents about 90% of all U.S. stock index futures trading

– First published in 1917– Currently one of the Commerce Department’s

leading indicators

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Capitalization-Weighted Indexes

The S&P 500 index is capitalization-weighted

– 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

8

Capitalization-Weighted Indexes (cont’d)

Assume only three firms are in an index

Assume the initial divisor is arbitrarily set at 2,700,000

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Capitalization-Weighted Indexes (cont’d)

Day 1

Index = 270,000,000/2,700,000 = 100.00

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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Capitalization-Weighted Indexes (cont’d)

Day 2

Index = 271,000,000/2,700,000 = 100.37

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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Capitalization-Weighted Indexes (cont’d)

Day 3 – B splits two for one

Index = 262,000,000/2,700,000 = 97.04

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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Stock Index Futures Contracts

As with other futures, a stock index future is a promise to:

– buy or sell – standardized units – of a specific index – at a fixed price – at a predetermined future date

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Stock Index Futures Contracts (cont’d)

Stock index futures are similar in every respect to a traditional agricultural contract except for the matter of delivery

– Index futures settle in cash rather than by delivery of the underlying asset

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The S&P 500 Stock Index Futures Contract

There is no actual delivery mechanism at expiration of an S&P 500 futures contract

– You actually deliver the dollar difference between the original trade price and the final price of the index at contract termination

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Pricing of Stock Index Futures

Elements affecting the price of a futures contract

Determining the fair value of a futures contract

Synthetic index portfolios Basic convergence

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Elements Affecting the Price of A Futures Contract

The S&P 500 futures value depends on four elements:

– 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

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Elements Affecting the Price of A Futures Contract (cont’d)

S&P 500 Stock Index

Futures

SPX Index

T-bill Rate Time until Settlement

SPX Dividend Yield

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Elements Affecting the Price of A Futures Contract (cont’d)

Stocks pay dividends, while futures do not pay dividends

– Shows up as a price differential in the futures price/underlying asset relationship

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Elements Affecting the Price of A Futures Contract (cont’d)

Stocks do not accrue interest

Posting margin for futures results in interest

– Shows up as a price differential in the futures price/underlying asset relationship

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Determining the Fair Value of A Futures Contract

The futures price should equal the index plus a differential based on the short-term interest rate minus the dividend yield:

TDRSeF )(

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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:

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

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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:

30.509,143.484,1 )365/121)(0110.0607(.

)(

e

SeF TDR

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Synthetic Index Portfolios

Large institutional investors can replicate a well-diversified portfolio of common stock by holding– 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

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Synthetic Index Portfolios (cont’d)

The futures approach has the following advantages over the purchase of individual stocks:

– Transaction costs will be much lower on the futures contracts

– The portfolio will be much easier to follow and manage

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Basic Convergence

As time passes, the difference between the cash index and the futures price will narrow

– At the end of the futures contract, the futures price will equal the index (basic convergence)

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Uses of Stock Index Futures

Speculation Spreading Arbitrage Anticipation of stock purchase or sale Hedging

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Speculation

Each one-point movement in the S&P 500 index translates to $250

– A person who is bullish could obtain substantial leverage by buying S&P contracts

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Spreading

Spreads using index futures can be used to speculate with reduced risk

– 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

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Arbitrage

Sometimes the market price of a futures contract temporarily deviates from the price predicted by pricing theory

– 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

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Anticipation of Stock Purchase or Sale

Futures contracts can be used to lock in a price in anticipation of a stock purchase or sale

– 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

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Hedging

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

– S&P futures are used by most large commercial banks and by many pension funds and foundations to hedge

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Hedging With Stock Index Futures

Systematic and unsystematic risk The hedge ratio Hedging in retrospect Adjusting market risk

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Systematic and Unsystematic Risk

Systematic factors are those that influence the stock market as a whole

– E.g., interest rates, economic indicators, political climate, etc.

– Systematic risk or market risk

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Systematic and Unsystematic Risk (cont’d)

Unsystematic factors are unique to a specific company or industry

– E.g., earnings reports, technological developments, labor negotiations, etc.

– Unsystematic risk

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Systematic and Unsystematic Risk (cont’d)

Proper portfolio diversification can virtually eliminate unsystematic risk

The market assumes that you have been smart enough to reduce risk through diversification– Beta measures the relative riskiness of a

portfolio compared to a benchmark portfolio like the S&P 500

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Systematic and Unsystematic Risk (cont’d)

Portfolio Variance

Number of Securities

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The Hedge Ratio

Introduction The market falls The market rises The market is unchanged

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Introduction

To construct a proper hedge, you must realize that portfolios are of– 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

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Introduction (cont’d)

To determine the hedge ratio, you need:

– The value of the chosen futures contract– The dollar value of the portfolio to be hedged– The beta of the portfolio

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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.

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.

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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:

contracts 17896.1779.0250$20.517,1

000,000,75$

betacontract futures P&S theof ueDollar val

portfolio theof ueDollar valHR

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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

A hedge will usually not be perfect because:

– 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

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Adjusting Market Risk

Futures can be used to adjust the level of market risk in a portfolio:

$250level futures

valueportfoliocontracts # currentdesired

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