A price-weighted index is the arithmetic average of current security prices. As such, movements in the series are influenced by the differeni: prices of the index components.
Computationally, a price-weighted index adds together the market price of each stock in the index and then divides this total by the number of stocks in the index. The returns on a price-weighted index could be matched by purchasing an equal number of shares ofeach stock represented in the index. Since the index is price-weighted, a percentage change in a high-priced stock will have a relatively greater effect on the index than the same percentage change in a low priced stock. Also, due to the price weighting, the denominator must be adjusted for stock splits and other changes in the index portfolio to maintain the continuity of the series,
sum of stock prices
price-weighted index= - - - " - - - - number of stocks in index adjusted for splits
The two major price-weighted indexes are the Dow Jones Industrial Average (DJIA) and the Nikkei Dow Jones Stock Average.
The DJIA is a price-weighted index that uses 30 stocks. Criticisms of the DJIA are:
• Limited number of stocks in the index.
• Downward bias in the computation of the index.
• Large size of the companies included in the index.
The Nikkei Dow is the arithmetic average of the prices of 225 stocks that trade in the first section of the Tokyo Stock Exchange.Itis calculated the same way as the DJIA.
The Nikkei Dow represents only 15% of the first section stocks.
A market value-weighted index is calculated by summing the total value (current stock price times the number of shares outstanding) of all the stocks in the index. This sum is then divided by a similar sum calculated during the selected base period. The ratio is then multiplied by the index's base value (typically 100).A value-weighted index assumes you make a proportionate market value investment in each company in the index. The
major problem with a value-weighted index is that firms with greater market capitalization have a greater impact on the index than do firms with lower market capitalization.
I [(priceroday) (number of shares outstanding) ]
----.=-=---='_=;_x base vear index value
I [(pricebase year )(number of shares ourstJ.nding)] .
Study Session 13 Cross-Reference to CFA Institute Assigned Reading#53 - Security-Market Indexes The following is a listing of the major market value-weighted indexes:
• Standard & Poor's 500 (S&P 500) Index Composite measures 500 firms.
• New York Stock Exchange Index considers all NYSE stocks in one of five value- weighted indexes: industrial, utility, transportation, financial, and the composite index.
• Other U.S. value-weighted series include the NASDAQ series, the AMEX Market Value Index, the Dow Jones Equity Market Index, the Wilshire 5000 Equity Index, and the Russell Index.
• International value-weighted indexes include the Morgan Stanley Capital
International Indexes, the Dow Jones World Stock Index, and the Salomon-Russell World Equity Index.
• Non-U.S. value-weighted national indexes include the Financial Times Actuaries Share Index, which represents stocks on the London Stock Exchange, and the Tokyo Srock Exchange Price Index, which represents stocks listed on the first section of the Tokyo Stock Exchange.
An unweighted index places an equal weight on the returns of all index stocks, regardless of their prices or market values. A$20 stock is just as important as a $4,000 stock, and a small-size company is just as important as a large-size company. The procedure used to compute an unweighted index value assumes that the index portfoLio makes and maintains an equaL doLLar investment in each stock in the index. In effect, you are working with percentage price changes.
The change in value of an unweighted index may be calculated using two methods:
I. Arithmetic mean: change in the average index value =LXi, where Xi =the
n
return on each srock from time =t to time =t + I.
2. Geometric mean: change in the average index value=~Xl xX2x ... xXn-1, where Xi =(l + HPR) =Price r+l for srock i.
Pricer
The use of the geometric mean rather than the arithmetic mean will result in a lower index value. Recall that the geometric mean of returns is always less than the arithmetic mean, unless all returns are equal.
•
•
•
The Value Line (VL) Composite Average is an equal-weighted index where VL's 1,695 stock returns are averaged using the geometric mean.
The Financial Times Ordinary Share Index is a geometric average of30 major stocks on the London Srock Exchange.
Most academic studies are conducted using arithmetically averaged equal-weighted indexes.
Smdy Session 13
Cross-Reference to CFA Institute Assigned Reading#53 - Security-Market Indexes Source and Direction of Bias
Price-weighting bias. Once a price-weighted index is established, the denominator must be adjusted to reflect stock splits and changes in the sample over time. After a stock split, the denominator is adjusted downward, so the index is the same before and after the split. This places a downward bias on the index because large successful firms tend tosplit their stocks more often than low growth stocks and will lose weight within the index simply by splitting their stock.
Value-weighting bias. The major problem with a value-weighted index is that firms with greater market capitalization have a greater impact on the index than do firms with lower market capitalization. Thus, iflarge market capitalization gro\'(th firms have exceptionally high rerurns, much of the S&P 500 Index return could be attributable to only a few firms.
Unweighted (i.e., equal-weighted) bias. As noted earlier, the use of the geometric mean rather than the arithmetic mean causes a downward bias in the index. The geometric average will always be lower than the arithmetic average unless all stocks have equal-percentage price changes.
Computing Price-Weighted, Market-Weighted, and Unweighted Indexes Example: Price-weighted index
Given the information for the three stocks presented inthefollowingfigure, calculateaprice-weighted and value-weighted index return overa I-nlOnth period.
Index Firm Data
As ofDecember31,2006 As ofJanuary 31, 2007
Number of Total Number of Total
Share Shares Market Share Shares Market
Price Outstanding Value Price Outstanding Value
(000's) (000's) (000's) (000's)
Stock X $10 3,000 $30,000 $20 3,000 $60,000
StockY $20 1,000 $20,000 $15 1,000 $15,000
StockZ $60 500 $30,000 $40 500 $20,000
Total $90 4,500 $80,000 $75 4,500 $95,000
Answer:
The price-weighted index is [(10 + 20 +60) / 3] = 30 as of December31 and [(20 + 15+ 40) / 3] =25 as of January31.Hence, the price-weighted I-month percentage return is:
-25 -1 =-16.7%
30
Study Session 13 Cross-Reference to CFA Institute Assigned Reading #53 - Security-Market Indexes
Value-weighted indexes normally use a beginning (base year) index value of 100. The total market values of the index portfolio on December 31 and January 3I are $80 million and $95 million, respectively. So the index value at the end of January is:
. d I .current total market value of index Stocks currentIn ex va ue= - - - -
base year total market value of index stocks x base year index value
current index value= $95 million x 100=118.75
$80 million
Thus, the value-weighted indexpetcentagereturn is:
(118.75/100) - 1=18.75%
. .
.: :: . ....•...-'...•..•...
Let's.l,ogkat anexampleofprice~\¥eighting t<tshqwhow these twO indexes are calculated
Answer:
100+10+1.==37 3 .
ãIfSto~kãã.Ad()ulMsãfhã::pã)t:Â~t() ...$2()(), the~inde)j~Y~J.J}e
. . .
lOP+10+1
Study Session 13
Cross-Reference to CFA Institute Assigned Reading #53 - Security-Market Indexes If Sto.ckC doubles inpric;:eto$]., the index vah;e
.~,
-;-t~:
lisFQS~~doubles...•ir ãã.prise~R~2QQ,Jh~ã.incieXgb~~"f~:'
";~Q?~OOOX$200+.1,Ooo,q?R><$1?+~O,OQO,?O?:'~lX100"12.5...'. .•...'::"::'ãã2'-',;-C~' .:-,;', '.'$'4"'0" ,-"}.,,;,,_ã,,VU<'-~-"ã:",::,'::,_-iã-:'>--'."_."",ããoãoãã()'•Nil..ã0ãããã '.. . '. ,...••.• , ' •••.• •.•,,' _ .-<.
~:,.'..-.:;:~:'.', :, ' .".'.:. .-:':~,--
Ifs~¥kCdoubles in price tQ$2,
- , - ' " _ ..
Example: Unweighted (equal-weighted) index
Calculate both the arithmetic and geometric unweighted iridexvalues for the three stocks described below, assuming an initial index value of 131.
Unweighted Index Data
Stock Initial Price Current Price Price Change
A $12.00 $15.00 +25%
B $52.00 $48.00 -7.7%
C $38.00 $45.00 +18.4%
Study Session 13 Cross-Reference toCFA Institute Assigned Reading #53 - Security-Market Indexes