1. 2-1 Stock Market Indexes Importance/Uses People often use them to measure the “health” of stock market They provide the basis for some of the most popular mutual funds They provide the basis for some of the most popular exchange-traded funds (‘ETFs’) --- examples, SPY and QQQ They’re benchmarks for measuring portfolio manager’s performance They are used as the “underlying” (basis) for some of the most widely used derivatives such as futures and options They provide the basis for ongoing arbitrage opportunities “… everybody talks about them but few people understand them…” Dow Jones Industrial Average S&P 500 Index

2. 2-2 Stock Market Indexes (continued)  Domestic indexes within a country may track all stocks, a representative sample, or an industry.  Global indexes include stocks from many countries to help overcome differences. Question: How are the indexes calculated? Answer: There are 3 different weighting methods:  Price-weighted average (Dow Jones Industrial Average ‘DJIA’)  Market-value weighted index (S&P 500, Nasdaq 100, Wilshire 5000)  Equally weighted index (Value Line Index)

3. 2-3 3 Price-weighted index  Just add up the prices and adjust the total by a divisor  Biased towards high price stocks (i.e., higher priced stocks influence the index number more than lower priced stocks) Market-value weighted index  Use the market capitalizations of the companies  Biased toward large market value companies An equal weighted (unweighted)  Just use equal weighted average returns  The more numerous smaller companies are more important than with the other methods Stock Market Index Calculation Methods

4. 2-4 CFALA/USC CFA Review Level 1, SS-13 4 Price Weighted Example Dow Jones Industrial Average (DJIA):  Thirty large cap stocks.  High price stocks carry more weight than low price stocks.  High growth companies with stock splits lose relative importance, thus downward bias.  For stock splits, they adjust divisor downwards to compensate. (instead of 30, divisor is currently about 0.125 to adjust for past stock splits and composition changes.) DJIA Companies: 3M · American Express · AT&T · Boeing Co · Caterpillar · Chevron · Cisco Systems · The Coca-Cola Company · DuPont · ExxonMobil · General Electric · Goldman Sachs Group Inc – The Home Depot · Intel · IBM · Johnson & Johnson · JPMorgan Chase · McDonald's · Merck · Microsoft · Nike Inc Pfizer · Procter & Gamble · The Travelers Companies · United Technologies UnitedHealth Group Verizon Communications · Visa Inc Wal-Mart · The Walt Disney Company

5. 2-5 5 Market Value Weighted Most stock market indices are calculated this way. Uses market value weights: Price per share × Number of Shares Outstanding for each company (some use float instead of outstanding, publicly available shares). Changes in large cap stocks (high market value) are relatively more important. Automatically adjusts for stock splits and stock dividends by adjusting number of shares.

6. 2-6 6 Unweighted / Equal Weighted Cumulatives of the arithmetic average of the percentage changes in price for all stocks in index. Equivalent to investing the same $ amount in each stock, then rebalancing each period. An equal weighted (unweighted) index is biased towards the returns of smaller companies relative to a value weighted index because small companies are quite numerous.

7. 2-7 7 Index computation examples

8. 2-8 8 Index at time 0 Index at time 1

9. 2-9 CFALA/USC CFA Review Level 1, SS-13 9

10. 2-10 CFALA/USC CFA Review Level 1, SS-13 10 If Index was 100 on day 0, it would also be 100 on day 1 I 0 (1+%  ) = I 1

11. 2-11 CFALA/USC CFA Review Level 1, SS-13 11 Summary

12. 2-12 Another Example: Data to Construct Stock Indexes CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 450 0.1$450 x 0.1 = $45 1) Construct each of the 3 types of indices. 2) Examine how each index changes if C’s stock price increases 10% and GOOG’s price drops by 20%. 3) Examine the impact of a 2:1 stock split by GOOG. 4) Examine the implications of index type on mutual fund managers whose funds are supposed to replicate each index’s return.

13. 2-13 Stock Price Indices (continued) CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 450 0.1$450 x 0.1 = $45 1) Construct each of the 3 types of indices. Price-weighted average: index is one share of each company divided by divisor. Index value = (20 + 450) / 2 = 235. (Initially, let divisor = # of companies.) Market-value weighted index: companies are weighted by their market value. Index value = we can use any number initially. Let’s let index initially = 100. When we calc the index, we’ll add the current market caps of C and GOOG, multiply by 100 and then divide that total by $145 billion. Initially: ($100B+$45B)*(100/$145B) Equally weighted index: simply calculate the returns for each company and average. Index value = any number initially. For simplicity, let’s let index initially = 100.

14. 2-14 Stock Price Indices (continued) CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 450 0.1$450 x 0.1 = $45 2) Examine how each index changes if C’s stock price increases 10% and GOOG’s price drops by 20%. Price-weighted average: --- First, calculate the new stock prices: C’s new stock price = $20x(1+0.10) = $22. GOOG’s new stock price = $450x(1-.20) = $360. Index value now = (22 + 360) / 2 = 191. Change in index = (New value – Old value)/ Old value = (191-235)/235 = -0.187 or -18.7%. NOTE --- the higher priced stock has more impact on the index.

15. 2-15 Stock Price Indices (continued) CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 4500.1$450 x 0.1 = $45 2) Examine how each index changes if C’s stock price increases 10% and GOOG’s price drops by 20%. Market-value weighted index: Calculate the new market values: C’s new market value = $20x(1+0.10)x(5.0) = $110 billion. GOOG’s market value = $450x(1-.20) x(0.1) = $ 36 billion. Index value now = ($110B+$36B)*(100/$145B) = 100.69. Change in index = (100.69-100)/100 = 0.0069 or 0.69%. NOTE --- the company with the higher market value has more impact on the index.

16. 2-16 Stock Price Indices (continued) CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 4500.1$450 x 0.1 = $45 2) Examine how each index changes if C’s stock price increases 10% and GOOG’s price drops by 20%. Equally weighted index: Change in index = [(10% + (-20%)]/2 = -5%. NOTE --- the companies have equal impact on index.

17. 2-17 Stock Price Indices (continued) CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 450 0.1$450 x 0.1 = $45 3) Examine the impact of a 2:1 stock split by GOOG. Price-weighted average: In step 1, we calculated an index of 235. --- Calculate the new stock price for GOOG when it splits: GOOG’s new stock price = $450 / 2 = $225. --- Calculate the new divisor needed to keep index unchanged at 235: Index = 235 = (20 + 225) / (new divisor). A little algebra: new divisor = (20 + 225) / 235 = 1.042553 NOTE --- because we’ve had a stock split, we have to calculate new divisor that we will use from now on ---- until the next unusual event that makes us change it. Examples of events: stock splits, dividends>10%, and changes of companies.

18. 2-18 Stock Price Indices (continued) CompanyStock Price ($/share) Shares Outstanding (in Billions) Market Value ($ Billions) = Stock Price x Shares Out. Citigroup (C)$ 205.0 $20 x 5 = $100 Google (GOOG)$ 450 0.1$450 x 0.1 = $45 3) Examine the impact of a 2:1 stock split by GOOG. Market-value weighted index: No changes necessary – stock split has no impact on index. Equally weighted index: No changes necessary – stock split has no impact on index.

19. 2-19 Stock Price Indices (continued) 4) Now examine the implications of index type on mutual fund managers whose funds are supposed to replicate each index’s return. Suppose the fund manager is managing $145,000. Price-weighted average: - The manager initially has to invest in equal number of shares of each stock. - Let n = number of shares we want to solve for: (n)($20) +(n)($450) = $145,000 (n)($20 + $450) = $145,000 n = $145,000/ ($20 + $450) = 308.51 shares in each stock. We will not have to rebalance our holdings of each stock when prices change. Exception --- when stocks pay dividends --- we will have to reinvest in equal shares.

20. 2-20 Stock Price Indices (continued) 4) Now examine the implications of index type on mutual fund managers whose funds are supposed to replicate each index’s return. Suppose the fund manager is managing $145,000. Market –value weighted average: - The manager initially has to invest in proportion to market value. -- $ to invest in C = ($145,000)(100/145) = $100,000. So shares of C = $100,000/($20 per share) = 5,000 shares. -- $ to invest in GOOG = ($145,000)(45/145) = $45,000. So shares of GOOG = $45,000/($450 per share) = 100 shares. We will not have to rebalance our holdings of each stock when prices change. Exception --- when stocks pay dividends --- we will have to reinvest them in proportion to market values at that time.

21. 2-21 Stock Price Indices (continued) 4) Now examine the implications of index type on mutual fund managers whose funds are supposed to replicate each index’s return. Suppose the fund manager is managing $145,000. Equally weighted average: - The manager initially has to invest equal $ amounts in each stock. -- $ to invest in C = ($145,000/2) = $72,500. So shares of C = $72,500 /($20 per share) = 3,625 shares. -- $ to invest in GOOG = ($145,000/2) = $72,500. So shares of GOOG = $45,000/($450 per share) = 161.11 shares. We WILL have to rebalance our holdings of each stock when prices change. To convince yourself, look at the case where C increases by 10% and GOOG drops by 20%.