STOCK INDEX FUTURES

A STOCK INDEX IS A SINGLE NUMBER BASED ON

INFORMATION ASSOCIATED WITH A BASKET STOCK PRICES

AND QUANTITIES.

A STOCK INDEX IS SOME KIND OF AN AVERAGE OF THE PRICES AND THE QUANTITIES OF THE STOCKS THAT

ARE INCLUDED IN THE BASKET.

THE MOST USED INDEXES ARE

A SIMPLE PRICE AVERAGE

AND

A VALUE WEIGHTED AVERAGE.

STOCK INDEXES - THE CASH MARKET

A. AVERAGE PRICE INDEXES: DJIA, MMI:

N = The number of stocks in the index

D = Divisor

P = Stock market price

INITIALLY D = N AND THE INDEX IS SET AT A GIVEN LEVEL. TO ASSURE INDEX CONTINUITY, THE DIVISOR IS

CHANGED OVER TIME.

N.1,..., = i ;D

P = I i

EXAMPLES

STOCK SPLITS

1.

2.

1. (30 + 40 + 50 + 60 + 20) /5 = 40

I = 40 and D = 5.

2. (30 + 20 + 50 + 60 + 20)/D = 40

The index remains 40 and the new divisor is D = 4.5

(P P P D I1 2 N 1 1 ... ) /

(P P P D I1 2 N 2 1 1

2... ) /

CHANGE OF STOCKS IN THE INDEX

1.

2.

1. (30 + 20 + 40 + 60 + 50)/5 = 40

I = 40 and D = 5.

2. (30 + 120 + 40 + 60 + 50)/D = 40

The index remains 40 and the new divisor is D = 7.5

(P P ABC) P D I1 2 N 1 1 ( ... ) /

(P P XYZ) P D I1 2 N 2 1 ( ... ) /

STOCK #4 DISTRIBUTED 40% STOCK DIVIDEND

(30 + 120 + 40 + 60 + 50)/D = 40

D = 7.5. Next,

(30 + 120 + 40 + 36 + 50)/D = 40

The index remains 40 and the new divisor is D = 6.9

STOCK # 2 SPLIT 3 TO 1.

(30 + 40 + 40 + 36 + 50)/D = 40

The index remains 40 and the new divisor is D = 4.9

ADDITIONAL STOCKS

1.

2.

1.

(30 + 50 + 40 + 60 + 20)/5 = 40

D = 5 I = 40.

2.

(30 + 50 + 40 + 60 + 20 + 35)/D = 40

D = 5.875.

(P P P D I1 2 N 1 1 ... ) /

121+NN21 ID/)PP,...,P(P

VALUE WEIGHTED INDEXES

S & P500, NIKKEI 250, VALUE LINE

B = SOME BASIS TIME PERIOD

INITIALLY t = B THUS, THE INITIAL INDEX VALUE IS SOME ARBITRARILY CHOSEN VALUE: M.

For example, the S&P500 index base period was 1941-1943 and its initial value was set at M = 10. The NYSE index base period was Dec. 31, 1965 and its initial value was set at M = 50.

IN P

N Ptti ti

Bi Bi

THE RATE OF RETURN ON THE INDEX

titi

ti1i+tti

ti1i+t

titi

titi1i+t1i+t

titi

titi1i+t1i+t

t

t1+tIt

PN

)P(PN

Thus, .N N but,

;PN

PNPN

VB

PNVB

PN

VB

PN

I

II R

: thatagain, Notice, .Rw R

Finally,.RV

V

or ,R]PN

PN[

:as this Rewrite.PN

RPN

,PN

PPP

PN R

titiIt

tiI

i

tititi

titi

titi

tititi

titi

ti

ti1ittiti

It

.V

V

PN

PNw

BI

ti

BiBi

titii

Conclusion:

The return on a value weighted index in any period t, is the weighted average of the individual stock returns; the weights are the dollar

value of the stock as a proportion of the entire index

value.

.Rw R titiIt

.V

V

PN

PN w

BI

ti

BiBi

titii

THE BETA OF A PORTFOLIO

THEOREM:

A PORTFOLIO’S BETA IS THE WEIGHTED AVERAGE OF THE BETAS

OF THE STOCKS THAT COMPRISE THE PORTFOLIO. THE WEIGHTS ARE

THE DOLLAR VALUE WEIGHTS OF THE STOCKS IN THE PORTFOLIO.

In order to prove this theorem, assume that the index is a well

diversified portfolio, I.e., it represents the market

portfolio.

In the proof, P denotes the portfolio; I, denotes the index and I denotes the individual stock; i = 1, 2, …, N.

R

proof. theconcludes This

.β w )VAR(R

R,COV(Rwβ

:or ,)VAR(R

)R,COV(Rwβ

: thusoperator,linear a

is covariance that theRecall

.)VAR(R

)R,]RwCOV([ β

,Rw R;for R ngSubstituti

.)VAR(R

)R,COV(R β

iiI

IiiP

I

IiiP

I

IiiP

iiPP

I

IPP

Proof: By definition, the portfolio’s β is:

STOCK PORTFOLIO BETA

FEDERAL MOUGUL 18.875 9,000 169,875 .044 1.00MARTIN ARIETTA 73.500 8,000 588,000 .152 .80IBM 50.875 3,500 178,063 .046 .50US WEST 43.625 5,400 235,575 .061 .70BAUSCH & LOMB 54.250 10,500 569,625 .147 1.1FIRST UNION 47.750 14,400 687,600 .178 1.1WALT DISNEY 44.500 12,500 556,250 .144 1.4DELTA AIRLINES 52.875 16,600 877,725 .227 1.2

3,862,713

PORTFOLIO BETA: .044(1.00) + .152(.8) + .046(.5) + .061(.7)

+ .147(1.1) + .178(1.1) + .144(1.4)

+ .227(1.2) = 1.06

STOCK NAME PRICE SHARES VALUE WEIGHT BETA

BENEFICIAL CORP. 40.500 11,350 459,675 .122 .95CUMMINS ENGINES 64.500 10,950 706,275 .187 1.10GILLETTE 62.000 12,400 768,800 .203 .85KMART 33.000 5,500 181,500 .048 1.15BOEING 49.000 4,600 225,400 .059 1.15W.R.GRACE 42.625 6,750 287,719 .076 1.00ELI LILLY 87.375 11,400 996,075 .263 .85PARKER PEN 20.625 7,650 157,781 .042 .75

3,783,225

A STOCK PORTFOLIO BETA

STOCK NAME PRICE SHARES VALUE WEIGHT BETA

PORTFOLIO BETA: .122(.95) + .187(1.1) + .203(.85)

+ .048(1.15) + .059(1.15) + .076(1.0)

+ .263(.85) + .042(.75) = .95

Sources of calculated Betas

And calculation inputs

Source Index Data Horizon Value Line Investment Survey NYSECI Weekly Price 5 yrs(Monthly)*

Bloomberg S&P500I Weekly Price 2 yrs (Weekly)

www.quote.bloomberg.com

Bridge Information Systems S&P500I Daily Price 2 yrs (daily)

www.bridge.com

Nasdaq Stock Exchange www.nasdaq.com

Media General Fin. Svcs. (MGFS) S&P500I Monthly Price 3 (5) yrs www.mgfs.com (Monthly)

Quicken.Excite.com www.quicken.excite.com

MSN Money Central www.moneycentral.msn.com

DailyStock.com www.stocksheet.com

Standard & Poors Compustat Svcs S&P500I Monthly Price 5 yrs (Monthly)

+ Dividend

S&P Personal Wealth www.personalwealth.com

S&P Company Report (via brokerage)

Charles Schwab Equity Report Card

S&P Stock Report (via brokerage account)

Argus Company Report S&P500I Daily Price 5 yrs (Daily)

(via brokerage subscription)

*Updating frequency.

Sources of calculated Betas

And calculation inputs

Source Index Data Horizon Market Guide S&P500I Monthly Price 5 yrs (Monthly)

www.marketguide.com

Yahoo!Finance www.yahoo.marketguide.com

Motley Fool www.fool.com

WWorldly Investor www.worldlyinvestor.com

Individual Investro www.individualinvestor.com

Quote.com www.quote.com

Equity Digest (via brokerage account)

ProVestor Plus Company Report (via brokerage account)

First Call (via brokerage account)

Sources of calculated Betas

And calculation inputs

Example: ß(GE) 6/20/00

Source ß(GE) Index Data Horizon Value Line Investment Survey 1.25 NYSECI Weekly Price 5 yrs (Monthly)

Bloomberg 1.21 S&P500I Weekly Price 2 yrs (Weekly)

Bridge Information Systems 1.13 S&P500I Daily Price 2 yrs (daily)

Nasdaq Stock Exchange 1.14

Media General Fin. Svcs. (MGFS) S&P500I Monthly P ice3 (5) yrs Quicken.Excite.com 1.23

MSN Money Central 1.20

DailyStock.com 1.21

Standard & Poors Compustat Svcs S&P500I Monthly Price 5 yrs (Monthly)

S&P Personal Wealth 1.2287

S&P Company Report) 1.23

Charles Schwab Equity Report Card 1.20

S&P Stock Report 1.23

AArgus Company Report 1.12 S&P500I Daily Price 5 yrs (Daily)

Market Guide S&P500I Monthly Price 5 yrs (Monthly)

YYahoo!Finance 1.23

Motley Fool 1.23

WWorldly Investor 1.231

Individual Investor 1.22

Quote.com 1.23

Equity Digest 1.20

ProVestor Plus Company Report 1.20

First Call 1.20

STOCK INDEX OPTIONS

ONE CONTRACT VALUE =

(INDEX VALUE)($MULTIPLIER)

One contract = (I)($m)

ACCOUNTS ARE SETTLED BY CASH SETTLEMENT

STOCK INDEX OPTIONS

WSJ

STOCK INDEX OPTIONSTHE MAIN REASON FOR THE

DEVELOPMENT OF INDEX OPTIONS WAS TO ENABLE PORTFOLIO AND FUND MANAGERS TO HEDGE THEIR POSITIONS. ONE OF THE BEST STRATEGIES IN THIS CONTEXT IS THE

PROTECTIVE PUTS. THAT IS, IF THE MARKET VIEW IS THAT THE MARKET IS

GOING TO FALL IN THE OFFING, PURCHASE PUTS ON THE INDEX.

QUESTIONS: 1. WHAT EXERCISE PRICE WILL GUARANTY THE PROTECTION LEVEL REQUIRED BY THE MANAGER.?

2. HOW MANY PUTS TO BUY?

THE ANSWERS ARE NOT EASY BECAUSE THE UNDERLYING ASSET - THE INDEX - IS

NOT THE SAME AS THE PORTFOLIO WE ARE TRYING TO PROTECT. WE NEED TO USE

SOME RELATIONSHIP THAT RELATES THE THE INDEX VALUE TO THE PORTFOLIO

VALUE.

STRATEGY

INITIAL CASH FLOW

AT EXPIRATION

I1 < X I1 ≥ X

Hold the portfolio

Buy n puts

-V0

-n P($m)

V1

n(X- I1)($m)

V1

0

TOTAL -V0 –n P($m) V1+n($m)(X- I1)

V1

The protective put consists of holding the unaltered portfolio and purchasing n puts. The premium, the exercise price and the index are levels and must be multiplied by the $ multiplier, $m.

ONE SUCH RELATIONSHIP COMES FROM THE CAPITAL ASSET

PRICING MODEL WHICH STATES THAT FOR ANY SECURITY OR

PORTFOLIO, i:

the expected excess return on the security and the expected excess return on the market

portfolio are linearly related by their beta: )r(ERβrER FMiFi

THE INDEX TO BE USED IN THE STRATEGY, IS TAKEN TO BE A PROXY FOR THE MARKET

PORTFOLIO, M. FIRST, REWRITE THE ABOVE EQUATION FOR THE INDEX I AND ANY PORTFOLIO P :

).r(ERβrER FIpFp

].rI

D I - I[βr

V

D V - VF

0

I01pF

0

P01

Second, rewrite the CAPM result, with actual returns:

Notice that in this expression the returns on the portfolio and on the index are in terms of their initial values, indicated by V0, I0 , plus any cash flow, dividends in this case , minus their terminal values at time 1, indicated by V1 and I1.

).r(RβrR FIpFp In a more refined way, using V and I for the portfolio and index market values, respectively:

NEXT, USE THE RATIOS Dp/V0 AND DI/I0 AS THE PORTFOLIO’S DIVIDEND PAYOUT RATE, qP, AND THE INDEX’

DIVIDEND PAYOUT RATE, qI, DURING THE LIFE OF THE OPTIONS AND

REWRITE THE ABOVE EQUATION:

]rqI

I - I[βrq

V

V - VFI

0

01pFP

0

01

]r - q 1 - I

I[βr q1

V

VFI

0

1pFP

0

1

Which may be rewritten as:

Notice that the ratio V1/ V0

indicates the portfolio required protection ratio.

FOR EXAMPLE:

,90.V

V

0

1

indicates that the manager wants the end-of-period portfolio market value, V1, to be down no more than 90% of the initial portfolio market value, V0. We denote this desired level by (V1/ V0)*.

We are now ready to answer the two questions associated with the protective put strategy:

1. What is the appropriate exercise price, X?

2. How many puts to purchase?

1. The exercise price, X, is determined by substituting I1 = X and the portfolio required protection level, (V1/ V0)* into the equation:

].r - q 1 - I

X[βr q1*)

V

V( FI

0pFP

0

1

1)].-)(βr(1 )q(β - q*)V

V[(

β

I X pFIpp

0

1

p

0

.($m)I

Vβn

0

0p

The solution is:

2. The number of puts is:

],r - q 1 - I

I[βr q1

V

VFI

0

1pFP

0

1

and solving for X:

STRATEGY

INITIAL CASH FLOW

AT EXPIRATION

I1 < X I1 ≥ X

Hold the portfolio

Buy n puts

-V0

-n P($m)

V1

n(X- I1)($m)

V1

0

TOTAL -V0 - nP($m) V1+n($m)(X- I1)

V1

We are now ready to calculate the floor level of the portfolio:

V1+n($m)(X- I1)

We rewrite the Profit/Loss table for the protective put strategy:

We can solve for V1 the equation:

]r - q 1 - I

I[βr q1

V

VFI

0

1pFP

0

1

)r1(qβqr1V

II

VβV

])r - q 1 - I

I[βrq(1VV

FIppF0

10

0p1

FI0

1pFP01

From the profit/loss table, The floor level:

Floor level = V1+n($m)(X- I1),

Which can be rewritten as:

Floor level = V1+n($m)X – n($m)I1

Substituting for n:

.II

Vβ-X

I

VβVlevelFloor

($m)I($m)I

Vβ

($m)X($m)I

VβVlevelFloor

10

0p

0

0p1

10

0p

0

0p1

.)r1(qβqr1V

II

VβV

But,

FIppF0

10

0p1

.($m)I

Vβn

0

0p

Thus, substitution of V1 into the equation for the Floor Level, yields:

.)]β)(1r(1qq[βVXI

Vβ

levelFloor

pFpIp00

0p

It is important to observe that the final expression for the Floor Level is in terms of known parameter values. That is, management knows the minimum portfolio value at time 1, at the time the strategy is opened!!!

.)/V(VV

levelFloor thecase, In this

.($m)I

Vn and

)*;V

V(IX Moreover,

.I

IV V :and

,I

I - I

V

V - V

*010

0

0

0

10

0

101

0

01

0

01

A SPECIAL CASE: NOTICE THAT IF β = 1 AND IF THE DIVIDEND RATIOS ARE EQUAL, qP =qI, THEN:

EXAMPLE:

A portfolio manager expects the market to fall by 25% in the next six months. The current portfolio market value is $25M. The portfolio manager decides to require a 90% hedge of the current portfolio’s market value by purchasing 6-month puts on the S&P500 index. The portfolio’s beta with the S&P500 index is 2.4. The index stands at a level of 1,250 points and its dollar multiplier is $250. The annual risk-free rate is 10%, while the portfolio and the index annual dividend payout rates are 5% and 6%, respectively. The data is summarized below:

2.4.β Finally,

6%.q 5%;q 10%;r

:are rates annual The

$250;$m 1,250;I

.9;)*V

V( 0;$25,000,00V

IpF

0

0

10

Solution: Purchase

puts. 19250)($250)(1,2

0$25,000,002.4n

($m)I

Vβn

0

0p

The exercise price of the puts is:

1,210.X

1).05)(2.4(1(2.4).03.025[.92.4

1,250X

1)].-)(βr(1 )q(β - q*)V

V[(

β

I X pFIpp

0

1

p

0

Solution:

Purchase n = 192 six-months puts

with X= 1,210.

The Floor level is calculated as follows:

0.$22,505,00

2.4)]-.95)(1(1.025-0[2.4(.03)$25,000,00

210,1250,1

0$25,000,002.4

.)]β)(1r(1qq[βVXI

Vβ

levelFloor

pFpIp00

0p

Holding the portfolio and purchasing 192 protective puts on the S&P500 index, guarantee that the portfolio value, currently $25M, will not fall below $22,505,000 in six months. Moreover, If the S&P500 index remains above the puts’ exercise price of 1,210, the portfolio market value in six months will exceed the floor level of $22,505,000.

.000,500,22$

)9(.000,000,25$

)9(.VV levelFloor

and puts 80$250(1250)

000,000,25$

($m)I

Vn

.000,500,22$)9(.000,000,25$

*)V

V(IX

*0

*1

0

0

0

10

A SPECIAL CASE: Let us assume that in the above example, βp = 1 and qP =qI, THEN: