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.
P

I=
;
i
D
i = 1,..., N.
EXAMPLES
STOCK SPLITS
1.
2.
(P1  P2  ... PN ) / D1  I1
1
(P1  P2  ... PN ) / D2  I1
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
CHANGE OF STOCKS IN THE INDEX
1.
2.
(P1  P2 (ABC)  ... PN ) / D1  I1
(P1  P2 ( XYZ)  ... PN ) / D2  I1
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
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.
(P1  P2  ... PN ) / D1  I1
(P1P2 ,...,PN PN+1 ) / D2 I1
1.
(30 + 50 + 40 + 60 + 20)/5 = 40
D=5
I = 40.
2.
(30 + 50 + 40 + 60 + 20 + 35)/D = 40
D = 5.875.
VALUE WEIGHTED INDEXES
S & P500, NIKKEI 250, VALUE
LINE
It 
N
N
P
ti ti
P
Bi Bi
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.
THE RATE OF RETURN ON THE INDEX
I t +1 I t
R It 

It
N
P
t +1i t +1i
VB

N P
ti ti
VB
N


P  N tiPti
t +1i t +1i
N P
ti ti
but, N t +1i  N ti . Thus,
N (P P )


N P
ti
N P
t +1i
ti ti
ti
;
ti ti
VB
Pt 1i  Pti
 N tiPti P
ti
R It 
,
 N tiPti
N PR


N P
ti ti
ti
. Rewrite this as :
ti ti
N ti Pti
 [
]R ti, or
 N tiPti
Vi
  R ti . Finally,
VI
R It   w ti R ti . Notice, again, that :
N tiPti
Vti
wi 

.
 N Bi PBi VBI
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.
R It   w ti R ti .
N tiPti
Vti
wi 

.
 N Bi PBi VBI
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.
R
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.
Proof: By definition, the portfolio’s β is:
COV(R P , R I )
βP 
.
VAR(R I )
Substituti ng for R P ; R P   w i R i ,
βP 
COV([  w i R i ], R I )
VAR(R I )
.
Recall that the covariance is
a linear operator, thus :
w iCOV(R i , R I )
βP  
, or :
VAR(R I )
 COV(R i , R I 
βP   wi 
 w iβi .

 VAR(R I ) 
This concludes the proof.
STOCK PORTFOLIO BETA
STOCK NAME
FEDERAL MOUGUL
MARTIN ARIETTA
IBM
US WEST
BAUSCH & LOMB
FIRST UNION
WALT DISNEY
DELTA AIRLINES
PRICE SHARES
18.875
73.500
50.875
43.625
54.250
47.750
44.500
52.875
9,000
8,000
3,500
5,400
10,500
14,400
12,500
16,600
VALUE
169,875
588,000
178,063
235,575
569,625
687,600
556,250
877,725
3,862,713
WEIGHT BETA
.044
.152
.046
.061
.147
.178
.144
.227
1.00
.80
.50
.70
1.1
1.1
1.4
1.2
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
A STOCK PORTFOLIO BETA
STOCK NAME
BENEFICIAL CORP.
CUMMINS ENGINES
GILLETTE
KMART
BOEING
W.R.GRACE
ELI LILLY
PARKER PEN
PRICE
SHARES
40.500
64.500
62.000
33.000
49.000
42.625
87.375
20.625
11,350
10,950
12,400
5,500
4,600
6,750
11,400
7,650
VALUE
WEIGHT BETA
459,675
706,275
768,800
181,500
225,400
287,719
996,075
157,781
3,783,225
.122
.187
.203
.048
.059
.076
.263
.042
.95
1.10
.85
1.15
1.15
1.00
.85
.75
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)
S&P500I
Daily Price
2 yrs (daily)
Monthly Price
3 (5) yrs
(Monthly)
Monthly Price
5 yrs (Monthly)
www.quote.bloomberg.com
Bridge Information Systems
www.bridge.com
Nasdaq Stock Exchange www.nasdaq.com
Media General Fin. Svcs. (MGFS)
www.mgfs.com
S&P500I
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
+ 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
(via brokerage subscription)
*Updating frequency.
S&P500I
Daily Price
5 yrs (Daily)
Sources of calculated Betas
And calculation inputs
Source
Index Data Horizon
Market Guide
S&P500I
Monthly Price
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)
5 yrs (Monthly)
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)
Quicken.Excite.com
1.23
MSN Money Central
1.20
DailyStock.com
1.21
Standard & Poors Compustat Svcs
S&P500I
S&P500I
S&P Personal Wealth
1.2287
S&P Company Report)
1.23
Monthly P ice
Monthly Price
3 (5) yrs
5 yrs (Monthly)
Charles Schwab Equity Report Card 1.20
S&P Stock Report
1.23
AArgus Company Report
1.12
Market Guide
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
S&P500I
Daily Price
5 yrs (Daily)
S&P500I
Monthly Price 5 yrs (Monthly)
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 OPTIONS
THE 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.
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, $m.
AT EXPIRATION
STRATEGY INITIAL
CASH FLOW
Hold the
portfolio
I1 < X
I1 ≥ X
-V0
V1
V1
Buy n puts
-n P($m)
n(X- I1)($m)
0
TOTAL
-V0 –n P($m) V1+n($m)(X- I1)
V1
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:
ER i  rF  βi (ER M  rF )
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 :
ER p  rF  β p (ER I  rF ).
Second, rewrite the CAPM result,
with actual returns:
R p  rF  β p (R I  rF ).
In a more refined way, using V and I
for the portfolio and index market
values, respectively:
V1 - V0  D P
I1 - I 0  D I
 rF  β p [
 rF ].
V0
I0
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.
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:
V1 - V0
I1 - I 0
 q P  rF  β p [
 q I  rF ]
V0
I0
Which may be rewritten as:
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ]
V0
I0
Notice that the ratio V1/ V0 indicates
the portfolio required protection
ratio.
FOR EXAMPLE:
V1
 .90,
V0
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:
V1
I1
 1  q P  rF  β p [
- 1  q I - rF ],
V0
I0
and solving for X:
V1
X
( ) * 1  q P  rF  β p [
- 1  q I - rF ].
V0
I0
The solution is:
I0 V1
X  [( ) * q p - (βp )q I  (1  rF )(βp - 1)].
βp V0
2.
The number of puts is:
V0
n  βp
.
($m)I 0
We rewrite the Profit/Loss table for the
protective put strategy:
AT EXPIRATION
STRATEGY INITIAL
CASH FLOW
Hold the
portfolio
I1 < X
I1 ≥ X
-V0
V1
V1
Buy n puts
-n P($m)
n(X- I1)($m)
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($m)(X- I1)
We can solve for V1 the equation:
V1
I1
 1  q P  rF  β p [ - 1  q I - rF ]
V0
I0
I1
V1  V0 (1  q P  rF  βp [ - 1  q I - rF ])
I0
V0
V1  βp I1
I0
 V0 1  rF  q p  βp (q I  1  rF ) 
From the profit/loss table, The floor level:
Floor level = V1+n($m)(X- I1),
Which can be rewritten as:
Floor level = V1+n($m)X – n($m)I1
Substituting for n:
V0
n  βp
.
($m)I 0
V0
Floor level  V1  β p
($m)X
($m)I 0
V0
 βp
($m)I 1
($m)I 0
V0
V0
Floor level  V1  β p X - β p I1.
I0
I0
But,
V0
V1  β p I1
I0
 V0 1  rF  q p  β p (q I  1  rF ) .
Thus, substitution of V1 into the equation
for the Floor Level, yields:
Floor level 
V0
β p X  V0 [β pq I  q p  (1  rF )(1  β p )].
I0
It is important to observe that the final
expression for the Floor Level is in
terms of known parameter values. That
is, management knows the minimum
portfolio value at time 1, at the time
the strategy is opened!!!
A SPECIAL CASE: NOTICE THAT IF
β = 1 AND IF THE DIVIDEND RATIOS
ARE EQUAL, qP =qI, THEN:
V1 - V0 I1 - I 0

,
V0
I0
I1
and : V1  V0 .
I0
Moreover,
V1
X  I 0 ( )*;
V0
and
V0
n
.
($m)I 0
In this case, the Floor level
 V0 (V1/V0 ) .
*
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 riskfree rate is 10%, while the portfolio and
the index annual dividend payout rates
are 5% and 6%, respectively. The data is
summarized below:
V1
V0  $25,000,000; ( )*  .9;
V0
I 0  1,250;
$m  $250;
The annual rates are :
rF  10%; q p  5%; q I  6%.
Finally, β  2.4.
Solution: Purchase
V0
n  βp
($m)I 0
$25,000,000
n  2.4
 192 puts.
($250)(1,2 50)
The exercise price of the puts is:
I 0 V1
X  [( ) * q p - (β p )q I  (1  rF )(β p - 1)].
β p V0
1,250
X
[.9  .025  (2.4).03  (1  .05)(2.4  1)
2.4
X  1,210.
Solution:
Purchase n = 192 six-months puts
with X= 1,210.
The Floor level is calculated as
follows:
Floor level 
V0
βp
X  V0 [β p q I  q p  (1  rF )(1  β p )].
I0
$25,000,000
 2.4
1,210
1,250
 $25,000,000[2.4(.03) - .025  (1  .95)(1 - 2.4)]
 $22,505,000.
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.
A SPECIAL CASE: Let us assume that
in the above example, βp = 1 and qP
=qI, THEN:
V1
X  I0 ( ) *
V0
 $25,000,000(.9)  $22,500,000.
V0
n
($m)I 0
$25,000,000

 80 puts and
$250(1250)
Floor level  V  V0 (.9)
*
1
*
 $25,000,000(.9)
 $22,500,000.