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Part-04: Stock Market Indices
1
Introduction
Standard question asked on a daily basis by:
Investors
Traders
Market Analysts
“What Happened to the Market Today?”
2
Introduction (cont…)
The term “market” refers to the Stock Market.
3
Introduction
Why focus on the market as a whole?
The market consists of a multitude of assets.
The performance of each asset cannot be realistically tracked at the same time to draw meaningful conclusions.
Thus there is a tendency to focus on a summary measure of the market’s performance.
4
Types of Indices
5
Stock Indices
An Index Number is a summary measure or a representative measure of the market’s performance.
Its value is a barometer of changes in the overall performance of the market. 6
Types of Indices
There are 3 different ways of
Types of Indices
computing stock market indices. Price Value Equally Weighted Weighted Weighted Indices Indices Indices 7
Price Weighted Indices Specify the number of stocks constituting the index Add up all the latest prices of all the component stocks Divide the price aggregate by a number known as the Divisor 8
Formula
A price weighted index consists of N stocks
The day of computation =t
The value of the index = It
Price of stock i on day t = Pit
9
Hypothetical Example
Take an index consisting of 5 stocks
Assume that we are standing on the base date.
The Base Date is the date on which the index is being computed for the first time.
On the Base Date, the Divisor can be set equal to any value.
A logical value for the Divisor is the number of stocks in the index.
10
Prices of Constituent Stocks STOCK
PRICE
ACC
907
Bombay Dyeing
81
Colgate Palmolive
211
Escorts
68
Hindustan Lever
732
TOTAL
1999 11
Initial Index Value 1999 It = ______ = 399.80 5
12
The Following Day
Assume that at the end of the next day the stock prices are: STOCK ACC Bombay Dyeing Colgate Palmolive Escorts Hindustan Lever TOTAL
PRICE 925 90 225 75 750 2065 13
New Index Level 2065 It+1 = ______ = 413 5
14
Conclusion
The market has moved up since the index level has risen.
Conclusion is valid
Every component stock has moved up in value.
15
Changing the Divisor
16
Stock Split - A Different Scenario
Assume that at the end of the first day, ACC announces a 3:1 split.
An investor will receive 3 new shares for every share that he is holding.
17
Second Day’s Prices - Post Split STOCK
PRICE
ACC
308
Bombay Dyeing
90
Colgate Palmolive
225
Escorts
75
Hindustan Lever
750
TOTAL
1448 18
Comparison of Prices With & Without the Split STOCK ACC Bombay Dyeing Colgate Palmolive Escorts Hindustan Lever TOTAL
PRICE - If no split 925
PRICE - With split 308
90
90
225
225
75
75
750
750
2065
1448 19
Comparison of Prices (Cont…)
If we compare the two sets of prices we see that:
All the stocks have the same value except for ACC
20
Calculation of Index After the Split using the Existing Divisor 1448 It+1 = ______ = 289.60 5
21
Erroneous Conclusion
Since the index value on the base date was 399.80, the index has fallen.
This is an erroneous conclusion since: Every Stock Has Risen In Value As Compared To The Base Date 22
Reason For Wrong Deduction
The decline is purely due to the stock split
due to which ACC has only 1/3rd of its value
How de we account for the split?
Obviously we have to change the divisor 23
Adjusting the Divisor
At the end of the day on which ACC undergoes a 3:1 split
Only ACC’s price will be impacted.
It will come down to 1/3rd of the existing value.
There will be no change in the prices of the other stocks.
24
Theoretical Post-Split Prices STOCK
PRICE
ACC
302.33
Bombay Dyeing
81
Colgate Palmolive
211
Escorts
68
Hindustan Lever
732
TOTAL
1394.33 25
The New Divisor
The new divisor X should be such that the pre- & post-split index value is the same.
If we denote the adjusted divisor by X: (1394.33 / X) = 399.80
The new divisor is 3.4876 26
The New Divisor (Cont…)
The validity of the new divisor can be checked by recomputing the index level on the next day. 1448 It+1 = ______ = 415.1852 3.4876
27
The New Divisor - Conclusion
After making the necessary adjustments to the divisor, we would conclude that the market has risen in value
Which is the correct conclusion in this case
28
Situations Warranting Re-calculation of Divisor
We need to adjust a divisor under the following situations:
A stock undergoes a split
A stock undergoes a reverse split
A company declares a stock dividend
A company is deleted from the index and replaced with another. 29
Handling Stock Dividends
From a mathematical standpoint, a stock split and a stock dividend are identical
20% stock dividend 1 new share will be issued for every 5 existing shares
This is exactly analogous to a 6:5 split.
Procedure for adjusting the divisor will be identical to that for a stock split
30
Changing the Composition of the Index
Assume that Escorts, which has a price of 75, is deleted at the end of the 2nd day It is replaced with Ranbaxy which has a price of 120. STOCK PRICE ACC 308 Bombay Dyeing 90 Colgate Palmolive 225 Ranbaxy 120 Hindustan Lever 750 TOTAL 1493
31
Changes in Composition and Divisor Adjustment
The new divisor X, should be such that: 1493 It+1 = ______ = 415.1852 X
The value of X turns out to be 3.5960 32
Price weighted indices
33
Examples of Price Weighted Indices
The Dow Jones Industrial Average commonly known as the DOW and abbreviated as DJIA
It consists of 30 Blue Chip companies
The Nikkei Index in Japan is also a price weighted index
It consists of 225 stocks 34
The Importance of Price
High priced stocks carry more weight in the case of price weighted indices than low priced stocks.
35
Illustration of Relative Importance
Case A: Consider a 20% increase in the price of ACC
Case B: Consider a 20% increase in the price of Colgate, which is priced lower
36
Illustration of Relative Importance STOCK
Day 1
Day 2: Case A
Day 2: Case B
ACC
900
1080
900
Bombay Dyeing Colgate
90
90
90
200
200
240
Palmolive Escorts
80
80
80
Hindustan Lever TOTAL
700
700
700
1970
2150
2010 37
Illustration (Cont…)
Assume that the divisor is 5
Index value on Day 1 = 394
Case A: Index value on Day 2 = 430
Increase of 9.14%
Case B: Index value on Day 2 = 430
Increase of 2.03% 38
Value weighted indices
39
Value Weighted Indices
Unlike price weighted indices, these indices use Market capitalization, not just price
Market capitalization is defined as: price * no. of shares outstanding 40
Formula
41
Explanation of Symbols
Pib ≡ Price of stock i on the base date Qib ≡ Number of shares outstanding of stock i on the base date Pit ≡ Price of stock i on date t Qit ≡ No. of shares outstanding of stock i on day t M ≡ No. of companies constituting the index on the base date N ≡ No. of companies constituting the index on day t
Note: M need not equal N
42
Starting Index Value
We have assigned the index a value of 100 on the base date.
This is common but is arbitrary
In practice, one can assign any value.
43
The Divisor
Once again a Divisor has been used
The treatment of the divisor in this case is slightly different
On the base date, the divisor is always assigned a value of 1.0 44
Hypothetical Example
Take the same 5 stocks
We will focus on their market capitalizations, not just their prices.
45
Market Capitalizations on the Base Date STOCK
No. of Shares (Q) 1,000,000
Market Cap.
ACC
Price (P) 907
Bombay Dyeing
81
500,000
40,500,000
Colgate Palmolive Escorts
211
700,000
147,700,000
68
200,000
13,600,000
Hindustan Lever
732
1,500,000
1,098,000,00 0 46
907,000,000
Total Market Capitalization
ΣPiQi = 2,206,800,000
47
Market Capitalizations on the Next Day STOCK ACC Bombay Dyeing Colgate Palmolive Escorts Hindustan Lever
Price (P) 925
No. of Market Cap. Shares (Q) 1,000,000 925,000,000
90
500,000
45,000,000
225
700,000
157,500,000
75
200,000
15,000,000
750
1,500,000
1,125,000,00 0
48
Total Market Capitalization & New Index Level
ΣPiQi = 2,267,500,000
New Index Level is: 2,267,500,000
It+1 = ___________ x 100 = 102.7506 2,206,800,000
49
Conclusion
We will conclude that the market has risen in value
This is a valid inference because every component stock has gone up in value as compared to the base date. 50
The Divisor
The Divisor need not be adjusted for
Stock Splits
Reverse Splits
Stock Dividends
51
Rationale – a theoretical standpoint
The below will not have any impact on the market capitalization of the stock
Stock splits
Reverse splits
Stock dividends
The decrease/increase in the stock price will be exactly offset by an increase/decrease in the no. of shares 52 outstanding
Example HLL
HLL post a 3:1 split
Market price
750
250
No. of shares outstanding
1,500,000
4,500,000
Market capitalization
1,125,000,000 1,125,000,00 0 53
Change in Composition and the Divisor
The divisor will have to be adjusted if there is a change in the composition.
Assume that Escorts is replaced by Ranbaxy. Escorts
Ranbaxy
Price
75
120
No. of shares outstanding
200,000
100,000 54
Post-Change Market Cap. STOCK ACC Bombay Dyeing Colgate Palmolive Ranbaxy Hindustan Lever
Price (P) 925
No. of Shares (Q) 1,000,000
Market Cap. 925,000,000
90
500,000
45,000,000
225
700,000
157,500,000
120
100,000
12,000,000
750
1,500,000
1,125,000,0 00 55
New Divisor
The new market capitalization is 2,264,500,000
The new divisor should be such that: 1 2,264,500,000 _ x ___________ x 100 = 102.7506 X 2,206,800,000
X is therefore equal to 0.9987 56
Relative Importance of Stocks
In a value weighted index, the importance of a stock would depend on its market value
A given percentage change in the value of a large cap firm, will have a greater impact on the index, than a similar percentage change in the value of a 57
Examples of Value Weighted Indices
Standard & Poor’s S&P 500
It consists of 500 stocks.
Nasdaq 100 index
In India - the Sensex and the Nifty
The Sensex consists of 30 stocks.
The Nifty is based on 50 stocks.
58
Equally weighted indices
59
Equally Weighted Indices
The value of an equally weighted index consisting of N stocks is given by
60
Prices & Returns is equal to (1 + rit) where rit is the rate of return on stock i between day t-1 and day t
61
The Formula in Terms of Returns
It
=
62
Explanation of Formula is the arithmetic average of the returns on all the component stocks Thus Value of the index = (Index level on the previous day) * (average return on all the stocks)
63
Tracking portfolois
64
Mimicking Portfolios
A mimicking portfolio also known as a Tracking Portfolio is formed to replicate the behaviour of the index
“Passive investors” are sometimes quite content to acquire portfolios of stocks that mirror the movements in an index
65
Replication Techniques
The method for forming a tracking portfolio depends on the nature of the index
66
To track… Price weighted index The investor
Equally weighted index The investor
Value weighted index
The investor needs to needs to needs to invest invest a fraction of purchase an an equal his wealth in each equal number amount in every component stock, of shares of stock that that is equal to the every stock constitutes the ratio of its market that index capitalization divided constitutes the by the total market index capitalization of all the assets in the 67 index
Portfolio Rebalancing
68
Portfolio Rebalancing
Once a tracking portfolio is formed it will not continue to track the index automatically forever
There are circumstances under which the index ought to be rebalanced if it is to maintain its property of being a tracking portfolio.
The circumstances under which the portfolio has to be rebalanced depend on the nature 69 of the index.
Events that Warrant Rebalancing Price weighted index
Equally weighted index
Value weighted index
One of the constituent stocks in the index: Undergoes a split Undergoes a reverse split Or pays a stock dividend If there is a change
Has to be rebalanced every day, unless none of the component stocks changes in price - because a price change in even 1 stock is adequate to ensure that the portfolio is no longer equally weighted.
If there is a change in the compositio n of the index
in the composition of the index
70
Equally Weighted Tracking Portfolio
71
An Equally Weighted Tracking Portfolio
Assume that We have Rs. 500,000 We wish to form an equally weighted portfolio consisting of 4 four stocks. STOCK
PRICE
Alfa Laval
50
Atlas Copco
100
Sandvik
40
Sulzer
125 72
Equally Weighted…(Cont…)
We have Rs. 500,000
We need to invest Rs. 125,000 in each stock
We will have to buy:
2,500 shares of Alfa Laval
1,250 shares of Atlas Copco
3,125 shares of Sandvik
1000 shares of Sulzer 73
Equally Weighted …(Cont…)
If we assume that the index level is 100 our portfolio will be worth 5000 times the index Assume that the prices on the next day are as follows. STOCK Alfa Laval Atlas Copco Sandvik Sulzer
PRICE 40 125 50 100 74
Equally Weighted …(Cont…)
The total portfolio value is Rs. 512,500 of which:
100,000 is in Alfa Laval
156,250 is in Atlas Copco
156,250 is in Sandvik
100,000 is in Sulzer
75
Equally Weighted …(Cont…)
The percentage of each stock in the portfolio STOCK
PERCENTAGE
Alfa Laval
19.51%
Atlas Copco
30.49%
Sandvik
30.49%
Sulzer
19.51% 76
Equally Weighted… (Cont…)
The portfolio is no longer equally weighted. The total portfolio value is 512,500 We need to invest 128,125 in each stock
To reset the weights to 0.25 each, we will have to: Sell 225 shares of Atlas Sell part of our holdings in Atlas Copco Copco & Sandvik Sell 562.5 shares of Sandvik Invest the proceeds in Buy 703.125 shares of Alfa Laval Alfa Laval & Sulzer Buy 281.25 shares of Sulzer
77
Rebalancing at Zero Cost
If we assume that there are no transactions costs, we can rebalance at no cost.
Inflow from rebalancing is: (225 x 125) + (562.50 x 50) = $56,250
Outflow from rebalancing is: (703.125 x 125) + (281.25 x 100) = $56,250
78
The New Index Level
It+1 = It x (1 + = 100 x (1 +
) )
= (-0.20 + 0.25 + 0.25 - 0.20) _____________________ 4 = 0.025 It+1 = 102.50 79
Mimicking Property
The portfolio value is: 512,500 = (102.50 x 5,000)
Thus the portfolio continues to mimic the index
80
Price Weighted Tracking Portfolio
81
A Price Weighted Tracking Portfolio
Assume that we wish to form a price weighted tracking portfolio by buying 1000 shares STOCK Alfa Laval Atlas Copco Sandvik Sulzer
PRICE 50 100 40 125 82
Price Weighted…(Cont…)
Assume that the divisor is 4.0, the index level will be:
It =
50 + 100 + 40 + 125 ________________ = 78.75 4.0
83
Price Weighted…(Cont…)
The value of our tracking portfolio will be: {1000 x (50 + 100 + 40 + 125)} = 315,000 = 4000 x 78.75
Thus the portfolio will be worth 4000
84
Stock Split
Assume that Atlas Copco undergoes a 2:1 split The post split theoretical value of the stock will be 50 The new divisor should be such that: 50 + 50 + 40 + 125 _______________ = 78.75 ⇒ X = 3.3651 X 85
Rebalancing
In order to ensure that our portfolio continues to mimic the index, we need to rebalance in such a way that the portfolio value remains unchanged.
86
Rebalancing (Cont…)
The portfolio value = 4000 x 78.75 = 1000 X 4 x 78.75
If the portfolio value is to remain unchanged: 1000 x 4 x 78.75 = N x 3.3651 x 78.75 N = no. of shares of each stock that is to be held after rebalancing 87
Rebalancing (Cont…)
Therefore: 1000 x 4 N = ______ = 1188.672 3.3651
88
Rebalancing (Cont…)
Assume that fractional shares can be bought & sold
We need to buy
188.672 shares of Alfa Laval, Sandvik, and Sulzer,
We need to sell
811.328 shares of Atlas Copco
Because we would have 2000 shares after the split and we need only 1188.672 shares89
Rebalancing (Cont…)
Inflow: 811.328 x 50 = 40,566.40
Outflow: 188.672 x (50 + 40 + 125) = 40,564.48
Once again if we ignore transactions costs we can rebalance at zero net cost 90
Value Weighted Tracking Portfolio
91
A Value Weighted Tracking Portfolio
Consider a value weighted portfolio consisting of the following 4 stocks.
STOCK
Price (P)
MRF J.K. Tyres Apollo Tyres Vikrant
20 40 50
No. of Shares (Q) 100,000 50,000 100,000
10
100,000
Tyres
Market Cap. 2,000,00 0 2,000,00 0 5,000,00 0 1,000,00 0
92
Value Weighted…(Cont…)
The total market capitalization is 10,000,000
Assume:
Base period market capitalization = 16,000,000
Divisor = 1
Index level = 62.50
93
Value Weighted… (Cont…)
Consider a person with a capital of 200,000
In order for his portfolio to mimic the index, it must have:
2,000,000 ________ x 200,000 = 40,000 in MRF 10,000,000 94
Value Weighted…(Cont…)
2,000,000 _________ x 200,000 = 40,000 in J.K. Tyres 10,000,000 5,000,000 _________ x 200,000 = 100,000 in Apollo Tyres 10,000,000 1,000,000 ________ x 200,000 = 20,000 in Vikrant Tyres 95
Value Weighted… (Cont…)
Thus he needs to buy
2000 shares of MRF
1000 shares of JK Tyres
2000 shares of Apollo Tyres
2000 shares of Vikrant Tyres
The total value of the portfolio is 3200 times the index level of 62.50 96
Value Weighted…(Cont…)
Assume that Vikrant Tyres is replaced by CEAT
Price = 35
No. of shares outstanding = 100,000
The total market capitalization of the four companies after the change will be 12,500,000 97
The Divisor
The divisor must be changed in such a way that the index level remains unchanged. 1 x 12,500,000 x 100 __ _________ = 62.50 X 16,000,000 ⇒ X = 1.25
98
Value Weighted…(Cont…)
For the portfolio to remain value weighted, the investor must have:
2,000,000 _________ x 200,000 = 32,000 in MRF 12,500,000
2,000,000 x 200,000 = 32,000 in J.K. Tyres ________ 12,500,000
99
Value Weighted…(Cont…)
5,000,000 x 200,000 = 80,000 in Apollo Tyres ________ 12,500,000
3,500,000 x 200,000 = 56,000 in CEAT ________ 12,500,000
100
Value Weighted…(Cont…) The investor requires
The investor needs to
1600 shares of MRF
Sell 400 shares of MRF
800 shares of J.K. Tyres
Sell 200 shares of J.K. Tyres
1600 shares of Apollo
Sell 400 shares of Apollo
Tyres
Tyres Sell 2000 shares of Vikrant Tyres
1600 shares of CEAT
Buy 1600 shares of CEAT
101
Value Weighted…(Cont…)
Inflow = 400x20 + 200x40 + 400x50 + 2000x10 = 56,000
Outflow = 1600x35 = 56,000
Once again, if we ignore transactions costs, we can rebalance at zero cost.
102
Base period capitalization
103
Changing the Base Period Capitalization
We have just seen one way of handling a change in the composition of a “value weighted index”
Adjust the Divisor
104
Base Period…(Cont…) Adjust the Divisor
Keep the Divisor fixed at 1.0
This is the approach used for the S&P500
In India we follow this procedure for the Sensex and the Nifty
The base period capitalization remains fixed in such cases and is never changed
The base period capitalization is changed
105
Base Period…(Cont…)
In our illustration when the market capitalization changed from 10,000,000 to 12,500,000 we changed the divisor to 1.25 to keep the index level fixed at 62.50.
In India we would have changed the base period capitalization instead.
12,500,000 _________ x 100 = 62.50 ⇒ X = 20,000,000 X
106
Base Period…(Cont…)
For subsequent computations of the index we would use the changed base period capitalization, unless there were to be another change in the index composition. 107
1
Introduction
Standard question asked on a daily basis by:
Investors
Traders
Market Analysts
“What Happened to the Market Today?”
2
Introduction (cont…)
The term “market” refers to the Stock Market.
3
Introduction
Why focus on the market as a whole?
The market consists of a multitude of assets.
The performance of each asset cannot be realistically tracked at the same time to draw meaningful conclusions.
Thus there is a tendency to focus on a summary measure of the market’s performance.
4
Types of Indices
5
Stock Indices
An Index Number is a summary measure or a representative measure of the market’s performance.
Its value is a barometer of changes in the overall performance of the market. 6
Types of Indices
There are 3 different ways of
Types of Indices
computing stock market indices. Price Value Equally Weighted Weighted Weighted Indices Indices Indices 7
Price Weighted Indices Specify the number of stocks constituting the index Add up all the latest prices of all the component stocks Divide the price aggregate by a number known as the Divisor 8
Formula
A price weighted index consists of N stocks
The day of computation =t
The value of the index = It
Price of stock i on day t = Pit
9
Hypothetical Example
Take an index consisting of 5 stocks
Assume that we are standing on the base date.
The Base Date is the date on which the index is being computed for the first time.
On the Base Date, the Divisor can be set equal to any value.
A logical value for the Divisor is the number of stocks in the index.
10
Prices of Constituent Stocks STOCK
PRICE
ACC
907
Bombay Dyeing
81
Colgate Palmolive
211
Escorts
68
Hindustan Lever
732
TOTAL
1999 11
Initial Index Value 1999 It = ______ = 399.80 5
12
The Following Day
Assume that at the end of the next day the stock prices are: STOCK ACC Bombay Dyeing Colgate Palmolive Escorts Hindustan Lever TOTAL
PRICE 925 90 225 75 750 2065 13
New Index Level 2065 It+1 = ______ = 413 5
14
Conclusion
The market has moved up since the index level has risen.
Conclusion is valid
Every component stock has moved up in value.
15
Changing the Divisor
16
Stock Split - A Different Scenario
Assume that at the end of the first day, ACC announces a 3:1 split.
An investor will receive 3 new shares for every share that he is holding.
17
Second Day’s Prices - Post Split STOCK
PRICE
ACC
308
Bombay Dyeing
90
Colgate Palmolive
225
Escorts
75
Hindustan Lever
750
TOTAL
1448 18
Comparison of Prices With & Without the Split STOCK ACC Bombay Dyeing Colgate Palmolive Escorts Hindustan Lever TOTAL
PRICE - If no split 925
PRICE - With split 308
90
90
225
225
75
75
750
750
2065
1448 19
Comparison of Prices (Cont…)
If we compare the two sets of prices we see that:
All the stocks have the same value except for ACC
20
Calculation of Index After the Split using the Existing Divisor 1448 It+1 = ______ = 289.60 5
21
Erroneous Conclusion
Since the index value on the base date was 399.80, the index has fallen.
This is an erroneous conclusion since: Every Stock Has Risen In Value As Compared To The Base Date 22
Reason For Wrong Deduction
The decline is purely due to the stock split
due to which ACC has only 1/3rd of its value
How de we account for the split?
Obviously we have to change the divisor 23
Adjusting the Divisor
At the end of the day on which ACC undergoes a 3:1 split
Only ACC’s price will be impacted.
It will come down to 1/3rd of the existing value.
There will be no change in the prices of the other stocks.
24
Theoretical Post-Split Prices STOCK
PRICE
ACC
302.33
Bombay Dyeing
81
Colgate Palmolive
211
Escorts
68
Hindustan Lever
732
TOTAL
1394.33 25
The New Divisor
The new divisor X should be such that the pre- & post-split index value is the same.
If we denote the adjusted divisor by X: (1394.33 / X) = 399.80
The new divisor is 3.4876 26
The New Divisor (Cont…)
The validity of the new divisor can be checked by recomputing the index level on the next day. 1448 It+1 = ______ = 415.1852 3.4876
27
The New Divisor - Conclusion
After making the necessary adjustments to the divisor, we would conclude that the market has risen in value
Which is the correct conclusion in this case
28
Situations Warranting Re-calculation of Divisor
We need to adjust a divisor under the following situations:
A stock undergoes a split
A stock undergoes a reverse split
A company declares a stock dividend
A company is deleted from the index and replaced with another. 29
Handling Stock Dividends
From a mathematical standpoint, a stock split and a stock dividend are identical
20% stock dividend 1 new share will be issued for every 5 existing shares
This is exactly analogous to a 6:5 split.
Procedure for adjusting the divisor will be identical to that for a stock split
30
Changing the Composition of the Index
Assume that Escorts, which has a price of 75, is deleted at the end of the 2nd day It is replaced with Ranbaxy which has a price of 120. STOCK PRICE ACC 308 Bombay Dyeing 90 Colgate Palmolive 225 Ranbaxy 120 Hindustan Lever 750 TOTAL 1493
31
Changes in Composition and Divisor Adjustment
The new divisor X, should be such that: 1493 It+1 = ______ = 415.1852 X
The value of X turns out to be 3.5960 32
Price weighted indices
33
Examples of Price Weighted Indices
The Dow Jones Industrial Average commonly known as the DOW and abbreviated as DJIA
It consists of 30 Blue Chip companies
The Nikkei Index in Japan is also a price weighted index
It consists of 225 stocks 34
The Importance of Price
High priced stocks carry more weight in the case of price weighted indices than low priced stocks.
35
Illustration of Relative Importance
Case A: Consider a 20% increase in the price of ACC
Case B: Consider a 20% increase in the price of Colgate, which is priced lower
36
Illustration of Relative Importance STOCK
Day 1
Day 2: Case A
Day 2: Case B
ACC
900
1080
900
Bombay Dyeing Colgate
90
90
90
200
200
240
Palmolive Escorts
80
80
80
Hindustan Lever TOTAL
700
700
700
1970
2150
2010 37
Illustration (Cont…)
Assume that the divisor is 5
Index value on Day 1 = 394
Case A: Index value on Day 2 = 430
Increase of 9.14%
Case B: Index value on Day 2 = 430
Increase of 2.03% 38
Value weighted indices
39
Value Weighted Indices
Unlike price weighted indices, these indices use Market capitalization, not just price
Market capitalization is defined as: price * no. of shares outstanding 40
Formula
41
Explanation of Symbols
Pib ≡ Price of stock i on the base date Qib ≡ Number of shares outstanding of stock i on the base date Pit ≡ Price of stock i on date t Qit ≡ No. of shares outstanding of stock i on day t M ≡ No. of companies constituting the index on the base date N ≡ No. of companies constituting the index on day t
Note: M need not equal N
42
Starting Index Value
We have assigned the index a value of 100 on the base date.
This is common but is arbitrary
In practice, one can assign any value.
43
The Divisor
Once again a Divisor has been used
The treatment of the divisor in this case is slightly different
On the base date, the divisor is always assigned a value of 1.0 44
Hypothetical Example
Take the same 5 stocks
We will focus on their market capitalizations, not just their prices.
45
Market Capitalizations on the Base Date STOCK
No. of Shares (Q) 1,000,000
Market Cap.
ACC
Price (P) 907
Bombay Dyeing
81
500,000
40,500,000
Colgate Palmolive Escorts
211
700,000
147,700,000
68
200,000
13,600,000
Hindustan Lever
732
1,500,000
1,098,000,00 0 46
907,000,000
Total Market Capitalization
ΣPiQi = 2,206,800,000
47
Market Capitalizations on the Next Day STOCK ACC Bombay Dyeing Colgate Palmolive Escorts Hindustan Lever
Price (P) 925
No. of Market Cap. Shares (Q) 1,000,000 925,000,000
90
500,000
45,000,000
225
700,000
157,500,000
75
200,000
15,000,000
750
1,500,000
1,125,000,00 0
48
Total Market Capitalization & New Index Level
ΣPiQi = 2,267,500,000
New Index Level is: 2,267,500,000
It+1 = ___________ x 100 = 102.7506 2,206,800,000
49
Conclusion
We will conclude that the market has risen in value
This is a valid inference because every component stock has gone up in value as compared to the base date. 50
The Divisor
The Divisor need not be adjusted for
Stock Splits
Reverse Splits
Stock Dividends
51
Rationale – a theoretical standpoint
The below will not have any impact on the market capitalization of the stock
Stock splits
Reverse splits
Stock dividends
The decrease/increase in the stock price will be exactly offset by an increase/decrease in the no. of shares 52 outstanding
Example HLL
HLL post a 3:1 split
Market price
750
250
No. of shares outstanding
1,500,000
4,500,000
Market capitalization
1,125,000,000 1,125,000,00 0 53
Change in Composition and the Divisor
The divisor will have to be adjusted if there is a change in the composition.
Assume that Escorts is replaced by Ranbaxy. Escorts
Ranbaxy
Price
75
120
No. of shares outstanding
200,000
100,000 54
Post-Change Market Cap. STOCK ACC Bombay Dyeing Colgate Palmolive Ranbaxy Hindustan Lever
Price (P) 925
No. of Shares (Q) 1,000,000
Market Cap. 925,000,000
90
500,000
45,000,000
225
700,000
157,500,000
120
100,000
12,000,000
750
1,500,000
1,125,000,0 00 55
New Divisor
The new market capitalization is 2,264,500,000
The new divisor should be such that: 1 2,264,500,000 _ x ___________ x 100 = 102.7506 X 2,206,800,000
X is therefore equal to 0.9987 56
Relative Importance of Stocks
In a value weighted index, the importance of a stock would depend on its market value
A given percentage change in the value of a large cap firm, will have a greater impact on the index, than a similar percentage change in the value of a 57
Examples of Value Weighted Indices
Standard & Poor’s S&P 500
It consists of 500 stocks.
Nasdaq 100 index
In India - the Sensex and the Nifty
The Sensex consists of 30 stocks.
The Nifty is based on 50 stocks.
58
Equally weighted indices
59
Equally Weighted Indices
The value of an equally weighted index consisting of N stocks is given by
60
Prices & Returns is equal to (1 + rit) where rit is the rate of return on stock i between day t-1 and day t
61
The Formula in Terms of Returns
It
=
62
Explanation of Formula is the arithmetic average of the returns on all the component stocks Thus Value of the index = (Index level on the previous day) * (average return on all the stocks)
63
Tracking portfolois
64
Mimicking Portfolios
A mimicking portfolio also known as a Tracking Portfolio is formed to replicate the behaviour of the index
“Passive investors” are sometimes quite content to acquire portfolios of stocks that mirror the movements in an index
65
Replication Techniques
The method for forming a tracking portfolio depends on the nature of the index
66
To track… Price weighted index The investor
Equally weighted index The investor
Value weighted index
The investor needs to needs to needs to invest invest a fraction of purchase an an equal his wealth in each equal number amount in every component stock, of shares of stock that that is equal to the every stock constitutes the ratio of its market that index capitalization divided constitutes the by the total market index capitalization of all the assets in the 67 index
Portfolio Rebalancing
68
Portfolio Rebalancing
Once a tracking portfolio is formed it will not continue to track the index automatically forever
There are circumstances under which the index ought to be rebalanced if it is to maintain its property of being a tracking portfolio.
The circumstances under which the portfolio has to be rebalanced depend on the nature 69 of the index.
Events that Warrant Rebalancing Price weighted index
Equally weighted index
Value weighted index
One of the constituent stocks in the index: Undergoes a split Undergoes a reverse split Or pays a stock dividend If there is a change
Has to be rebalanced every day, unless none of the component stocks changes in price - because a price change in even 1 stock is adequate to ensure that the portfolio is no longer equally weighted.
If there is a change in the compositio n of the index
in the composition of the index
70
Equally Weighted Tracking Portfolio
71
An Equally Weighted Tracking Portfolio
Assume that We have Rs. 500,000 We wish to form an equally weighted portfolio consisting of 4 four stocks. STOCK
PRICE
Alfa Laval
50
Atlas Copco
100
Sandvik
40
Sulzer
125 72
Equally Weighted…(Cont…)
We have Rs. 500,000
We need to invest Rs. 125,000 in each stock
We will have to buy:
2,500 shares of Alfa Laval
1,250 shares of Atlas Copco
3,125 shares of Sandvik
1000 shares of Sulzer 73
Equally Weighted …(Cont…)
If we assume that the index level is 100 our portfolio will be worth 5000 times the index Assume that the prices on the next day are as follows. STOCK Alfa Laval Atlas Copco Sandvik Sulzer
PRICE 40 125 50 100 74
Equally Weighted …(Cont…)
The total portfolio value is Rs. 512,500 of which:
100,000 is in Alfa Laval
156,250 is in Atlas Copco
156,250 is in Sandvik
100,000 is in Sulzer
75
Equally Weighted …(Cont…)
The percentage of each stock in the portfolio STOCK
PERCENTAGE
Alfa Laval
19.51%
Atlas Copco
30.49%
Sandvik
30.49%
Sulzer
19.51% 76
Equally Weighted… (Cont…)
The portfolio is no longer equally weighted. The total portfolio value is 512,500 We need to invest 128,125 in each stock
To reset the weights to 0.25 each, we will have to: Sell 225 shares of Atlas Sell part of our holdings in Atlas Copco Copco & Sandvik Sell 562.5 shares of Sandvik Invest the proceeds in Buy 703.125 shares of Alfa Laval Alfa Laval & Sulzer Buy 281.25 shares of Sulzer
77
Rebalancing at Zero Cost
If we assume that there are no transactions costs, we can rebalance at no cost.
Inflow from rebalancing is: (225 x 125) + (562.50 x 50) = $56,250
Outflow from rebalancing is: (703.125 x 125) + (281.25 x 100) = $56,250
78
The New Index Level
It+1 = It x (1 + = 100 x (1 +
) )
= (-0.20 + 0.25 + 0.25 - 0.20) _____________________ 4 = 0.025 It+1 = 102.50 79
Mimicking Property
The portfolio value is: 512,500 = (102.50 x 5,000)
Thus the portfolio continues to mimic the index
80
Price Weighted Tracking Portfolio
81
A Price Weighted Tracking Portfolio
Assume that we wish to form a price weighted tracking portfolio by buying 1000 shares STOCK Alfa Laval Atlas Copco Sandvik Sulzer
PRICE 50 100 40 125 82
Price Weighted…(Cont…)
Assume that the divisor is 4.0, the index level will be:
It =
50 + 100 + 40 + 125 ________________ = 78.75 4.0
83
Price Weighted…(Cont…)
The value of our tracking portfolio will be: {1000 x (50 + 100 + 40 + 125)} = 315,000 = 4000 x 78.75
Thus the portfolio will be worth 4000
84
Stock Split
Assume that Atlas Copco undergoes a 2:1 split The post split theoretical value of the stock will be 50 The new divisor should be such that: 50 + 50 + 40 + 125 _______________ = 78.75 ⇒ X = 3.3651 X 85
Rebalancing
In order to ensure that our portfolio continues to mimic the index, we need to rebalance in such a way that the portfolio value remains unchanged.
86
Rebalancing (Cont…)
The portfolio value = 4000 x 78.75 = 1000 X 4 x 78.75
If the portfolio value is to remain unchanged: 1000 x 4 x 78.75 = N x 3.3651 x 78.75 N = no. of shares of each stock that is to be held after rebalancing 87
Rebalancing (Cont…)
Therefore: 1000 x 4 N = ______ = 1188.672 3.3651
88
Rebalancing (Cont…)
Assume that fractional shares can be bought & sold
We need to buy
188.672 shares of Alfa Laval, Sandvik, and Sulzer,
We need to sell
811.328 shares of Atlas Copco
Because we would have 2000 shares after the split and we need only 1188.672 shares89
Rebalancing (Cont…)
Inflow: 811.328 x 50 = 40,566.40
Outflow: 188.672 x (50 + 40 + 125) = 40,564.48
Once again if we ignore transactions costs we can rebalance at zero net cost 90
Value Weighted Tracking Portfolio
91
A Value Weighted Tracking Portfolio
Consider a value weighted portfolio consisting of the following 4 stocks.
STOCK
Price (P)
MRF J.K. Tyres Apollo Tyres Vikrant
20 40 50
No. of Shares (Q) 100,000 50,000 100,000
10
100,000
Tyres
Market Cap. 2,000,00 0 2,000,00 0 5,000,00 0 1,000,00 0
92
Value Weighted…(Cont…)
The total market capitalization is 10,000,000
Assume:
Base period market capitalization = 16,000,000
Divisor = 1
Index level = 62.50
93
Value Weighted… (Cont…)
Consider a person with a capital of 200,000
In order for his portfolio to mimic the index, it must have:
2,000,000 ________ x 200,000 = 40,000 in MRF 10,000,000 94
Value Weighted…(Cont…)
2,000,000 _________ x 200,000 = 40,000 in J.K. Tyres 10,000,000 5,000,000 _________ x 200,000 = 100,000 in Apollo Tyres 10,000,000 1,000,000 ________ x 200,000 = 20,000 in Vikrant Tyres 95
Value Weighted… (Cont…)
Thus he needs to buy
2000 shares of MRF
1000 shares of JK Tyres
2000 shares of Apollo Tyres
2000 shares of Vikrant Tyres
The total value of the portfolio is 3200 times the index level of 62.50 96
Value Weighted…(Cont…)
Assume that Vikrant Tyres is replaced by CEAT
Price = 35
No. of shares outstanding = 100,000
The total market capitalization of the four companies after the change will be 12,500,000 97
The Divisor
The divisor must be changed in such a way that the index level remains unchanged. 1 x 12,500,000 x 100 __ _________ = 62.50 X 16,000,000 ⇒ X = 1.25
98
Value Weighted…(Cont…)
For the portfolio to remain value weighted, the investor must have:
2,000,000 _________ x 200,000 = 32,000 in MRF 12,500,000
2,000,000 x 200,000 = 32,000 in J.K. Tyres ________ 12,500,000
99
Value Weighted…(Cont…)
5,000,000 x 200,000 = 80,000 in Apollo Tyres ________ 12,500,000
3,500,000 x 200,000 = 56,000 in CEAT ________ 12,500,000
100
Value Weighted…(Cont…) The investor requires
The investor needs to
1600 shares of MRF
Sell 400 shares of MRF
800 shares of J.K. Tyres
Sell 200 shares of J.K. Tyres
1600 shares of Apollo
Sell 400 shares of Apollo
Tyres
Tyres Sell 2000 shares of Vikrant Tyres
1600 shares of CEAT
Buy 1600 shares of CEAT
101
Value Weighted…(Cont…)
Inflow = 400x20 + 200x40 + 400x50 + 2000x10 = 56,000
Outflow = 1600x35 = 56,000
Once again, if we ignore transactions costs, we can rebalance at zero cost.
102
Base period capitalization
103
Changing the Base Period Capitalization
We have just seen one way of handling a change in the composition of a “value weighted index”
Adjust the Divisor
104
Base Period…(Cont…) Adjust the Divisor
Keep the Divisor fixed at 1.0
This is the approach used for the S&P500
In India we follow this procedure for the Sensex and the Nifty
The base period capitalization remains fixed in such cases and is never changed
The base period capitalization is changed
105
Base Period…(Cont…)
In our illustration when the market capitalization changed from 10,000,000 to 12,500,000 we changed the divisor to 1.25 to keep the index level fixed at 62.50.
In India we would have changed the base period capitalization instead.
12,500,000 _________ x 100 = 62.50 ⇒ X = 20,000,000 X
106
Base Period…(Cont…)
For subsequent computations of the index we would use the changed base period capitalization, unless there were to be another change in the index composition. 107
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