Weighing methods in index construction

CFA level I / Equity Investments: Market Organization, Market Indices, and Market Efficiency / Security Market Indices / Weighing methods in index construction

Index weighting Index weighting decides how much of each security will be included in an index and has a direct bearing on the price movements of an index. Index weighting is done on the following four methods:

Price-weighted index, which assigns a weight based on the price of a security divided by the sum of prices of all the securities.

W_i = Pi/∑P_i , where P_i is the price of security, P₁, P₂…. P_n are the prices of all constituent securities.

Advantage: Simple to use

Disadvantage: A security with a high price will have more effect on the movement of the index. Stock split for security may cause changes in the weights of all constituent securities

Example: DJIA

Example 2

Consider an index comprising 3 securities with equal weight (say 1 each). Assume Divisor = 5 Security No. of shares in index (a) Initial Price (b) Value (a*b) Initial weight (%) Final Price A 1 10 10 16.67 11 B 1 20 20 33.33 19 C 1 30 30 50.00 31 Total 60 100 Index value 12 Solution: Initial weight for A = (1*10)/(1*10+1*20+1*30)= 10/60=16.67 percent. Similarly, weights calculated for security B and security C are 33.33 percent and 50.00 percent respectively. Security No. of shares in index (a) Final Price (b) Final Value Final weight (%) Price return (%) Final Price A 1 11 11 18.03 10.00 11 B 1 19 19 31.15 -5.00 19 C 1 31 31 50.82 3.33 31 Total 61 100 Index value 12.2 Initial index value = 12 Final index value = 12.2 Price return = (12.2-12)/12*100= 1.67 percent.

Note: If the dividend details are given total return can be calculated by adding the dividend paid to change in price.

Example 3

Now, suppose security B goes for the 2-for-1 split at the end of the final day. Calculate the impact of the split on the Divisor and the weight of securities. Solution: Security Final Price Pre-split weight (%) Price post split Post split weight (%) A 11 18.03 11 21.36 B 19 31.15 9.5 18.45 C 31 50.82 31 60.19 Total 61 100 51.5 100 Divisor 5 4.22 Index value 12.2 12.2 Note that the divisor value is derived by dividing the post-split total value (51.5) by the index value before the split (12.2) so that the final index value remains same (at pre-split level). The weight of each security is simply the weighted average of each security.

The equal-weighted index assigns weight on the basis of the number of securities. For example, if there are 10 securities in an index, each security will have 10 percent weight (1/10) in it. To construct an equal weighted portfolio, an equal value is assigned to all the constituent securities and a number of share in the index for each security is derived by dividing the value by share price of the security. See the example below:

Example 4

Compute the return of an equal-weighted index with the following data (assuming the divisor as 5): Security Initial Price (a) Value (b) Number of shares in index (b/a) Initial weight (%) Final Price A 10 100 10 33.33 11 B 20 100 5 33.33 19 C 30 100 3.33 33.33 31 Total 300 100 Index value 60 Solution: Security No. of shares in index Final Price (a) Value (b) Final weight (%) A 10 11 110.00 35.68 B 5 19 95.00 30.81 C 3.33 31 103.33 33.51 Total 308.33 100 Index value 61.67 The final weight of A = 110/308.33=35.68 percent. Index return = (61.67-60)/60 = 2.78 percent Or Index return = Average of returns of each constituent security = (10.00 -5.00 + 3.33)/3 = 2.78 percent.

Note the difference in index return for the price and equal weighted returns. Also, note that the weight of the securities whose prices have increased relatively more have also increased relative to other securities. So, to replicate the return of the equal-weighted index, we need to decrease the weight of those securities and increase the weight of the securities whose prices have come down in the next period.

The key advantage of an equal-weighted index is again simplicity.

Disadvantage: Under-representation of securities that contribute the highest value to the target market and over representation for securities with a lower value. Secondly, the index needs to re-balanced once the price of securities changes (example above). This rebalancing exercise costs money and erodes index return.

Example: Value Line Composite Average

Market capitalization weighted index assigns weight on the basis market capitalization of the security. Market capitalization is calculated by multiplying the total outstanding shares of a company by the share price. The weight of security is calculated by dividing its market capitalization by the sum of the market capital value of all the constituent securities. The market capitalization weighted index will have a momentum effect in the indices because the weight of the securities who have risen will have more weight in the indices.

Another version of this index methodology is the float-adjusted market capitalization method. Market float of a company is the number of shares available for trading to the public.

Advantage: The weight of securities reflect the value each security contributes to the target market.

Disadvantage: Securities whose prices have risen or fallen the most tend to have greater or lower weight in the index. An investor replicating this index is prone to the risk of having a greater weight on the overvalued securities and less weight on the undervalued securities.

Example: S&P500

Example 5

Compute the return of the following market-capitalized index with a divisor of 1,000: Security No. of shares outstanding (a) Initial Price (b) Market capitalization (a*b) Initial weight (%) Final Price A 20,000 10 200,000 34 11 B 4,000 20 80,000 14 19 C 10,000 30 300,000 52 31 Total 580,000 100 Index value 580 Solution: Security No. of shares outstanding (a) Final Price (b) Market capitalization (a*b) Final weight (%) A 20,000 11 220,000 36 B 4,000 19 76,000 13 C 10,000 31 310,000 51 Total 606,000 100 Index value 606 Index return = (606-580)/580 = 4.5 percent

Fundamental-weighted indexing tries to address the disadvantage of market capitalization method, which places too much emphasis on the price movement of a security. It uses a host of other variables like revenue, EPS growth, cash flow, book value, dividend and number of employees. These indices are more likely to have a value tilt. Also, fundamentally weighted indices generally will have a contrarian effect as the portfolio weights will shift away from securities that have increased in relative value and toward securities that have fallen in relative value during the rebalancing of the portfolio.

Check your concepts:

(46.3) The return of a float-adjusted market capitalization weighted index is most impacted by the return of which of the following securities? The index contains the following three securities:

Security	Market Capitalization	Float factor
A	$200 million	0.70
B	$150 million	0.90
C	$230 million	0.60

(a) Security A
(b) Security B
(c) Security C

(46.4) An index consists of three securities. The details about the securities are given in the following table:

Security	Price at the beginning of the period	Price at the end of the period	Total shares outstanding
A	$100	$150	10,000
B	$20	$20	40,000
C	$5	$1	300,000

Which of the following weighted index is most likely to provide the best return for the given index?

(a) Market capitalization weighted index
(b) Equal-weighted index
(c) Price-weighted index

(46.5) Which of the following weighted -index is most suitable for the securities that tend to follow mean reversion in their prices?

(a) Price-weighted index
(b) Equal-weighted index
(c) Market capitalization weighted index

Solutions:

(46.3) Correct Answer is A: For a float-adjusted market capitalization weighted index, the higher is the value of float-adjusted market capitalization of a security, higher will be its impact on the return of the overall index. Float-adjusted market capitalization for security A = $200*0.7 = $140 million. Similarly, the float-adjusted market capitalization for security B and security C are $135 million and $138 million respectively. So, the return of security A will have the most impact on the return of the overall index.

(46.4) Correct Answer is C: Only one security has increased in price. That security has the highest weight in the price-weighted index. So, the price-weighted index will have the highest return.

(46.5) Correct Answer is B: For the securities following the mean reversion in prices, the returns would be maximized if we decrease the weight of the securities that have gone up and increase the weight of the securities that have come down. That's what we do in the equal-weighted index. Hence, the equal-weighted index is most likely to have the highest return.

Exam Alert: This is the most important section of this chapter and candidates are likely to get a couple of questions from this area. Candidates should be comfortable with the calculation of return for each type of index and their disadvantages.