Slides by: Pamela L. Hall, Western Washington University Francis & IbbotsonChapter 10: Creating Price Indexes1 Creating Price Indexes Chapter 10

Francis & IbbotsonChapter 10: Creating Price Indexes 2 Background  Price indexes –Designed to summarize price behavior of market goods The Consumer Price Index (CPI) is used to measure the monthly rate of inflation (or deflation) –Contains 300 heterogeneous goods The S&P500 Composite Index –Contains value of 500 U.S. stocks

Francis & IbbotsonChapter 10: Creating Price Indexes 3 Background  Thousands of price indicators are used to track prices of –Stocks –Bonds –Commodities –Foreign exchange rates –Mortgages –Options –Futures

Francis & IbbotsonChapter 10: Creating Price Indexes 4 Constructing a Stock Market Indicator  Stock market average –Weighted or unweighted average stock price  Index number –Number void of units of measure –Designed to avoid distortions –A base period is selected as starting point  Price indicators used as a –Standard of comparison or benchmark For example, compare mutual fund’s performance to S&P500

Francis & IbbotsonChapter 10: Creating Price Indexes 5 Principles for Constructing an Indicator  Sample size –Should be large enough to statistically represent population of interest Small samples are subject to larger sampling errors Large sample is unnecessary if elements within sample are highly positively correlated  Representativeness –All securities in sample should have characteristics of interest

Francis & IbbotsonChapter 10: Creating Price Indexes 6 Principles for Constructing an Indicator  Weighting –May be proportional to total market value of outstanding shares –Equal weighting reflects equal likelihood of selecting security used in indicator Appropriate if investors select stocks irrationally or randomly Critics argue that small companies receive too large a weight –Equal weighting ignores fact that large companies provide more investment opportunities

Francis & IbbotsonChapter 10: Creating Price Indexes 7 Principles for Constructing an Indicator  Convenient units –Facilitate answering questions Index numbers are more convenient than average numbers DJIA is an ‘inconvenient unit’ –A ‘point’ from 1940s is not the same as a point today –Few investors understand what a DJIA ‘point’ means –Inflation can distort the numbers –Unit of measurement needs to be free from distortion

Francis & IbbotsonChapter 10: Creating Price Indexes 8 Principles for Constructing an Indicator  Uniform definition –The way the price index is computed should never change  Economical –Computational costs need to be considered Are shrinking due to computerized nature, but argues for small samples  Timeliness –Ideally would reflect changes immediately  Descriptive title –Title should not be misleading

Francis & IbbotsonChapter 10: Creating Price Indexes 9 Contrasting Two Well-Known Stock Market Indicators  Dow-Jones Industrial Average (DJIA) –Begun in 1884 with 11 stocks –Average has contained 30 stocks since 1928 –Only large, successful firms are in the average

Francis & IbbotsonChapter 10: Creating Price Indexes 10 Dow-Jones Industrial Average  Misleading name –Only large firms are in the average –New firms are not included –Some firms may be more utility than industrial firms  DJIA Divisor –In 1928 the prices of the 30 stocks were summed and divided by 30 However, stock splits and stocks dividends impact the divisor

Francis & IbbotsonChapter 10: Creating Price Indexes 11 Stock Splits and DJIA Divisor  As an example, consider the hypothetical stocks StockPrice X$50 Y$10 Total$60 Average60/2 = 30 StockPrice X$25 Y$10 Total$35 Average35/2 = 17.5 If Stock X undergoes a 2 for 1 stock split The stock split changed the price per share, but the stockholder’s wealth has remained the same—each stockholder in X has twice as many shares as before. If the divisor remains at 2, the average will drop, even though the aggregate market value of X remains the same. The divisor value must drop to reflect the stock split.

Francis & IbbotsonChapter 10: Creating Price Indexes 12 Dow-Jones Industrial Average  Points –DJIA is price-weighted More weight is given to higher priced stocks –Each point represents a few pennies of stock price Converting each point to a stock price is inconvenient

Francis & IbbotsonChapter 10: Creating Price Indexes 13 S&P 500 Stocks Composite Index  First developed in 1923 –Contained 233 stocks –Has been at the 500 stock level since 1957  Uses a market weighting scheme –Each security’s weight is based on the total market value of the firm Corresponds to the investment opportunities that exist in U.S.

Francis & IbbotsonChapter 10: Creating Price Indexes 14 S&P 500 Stocks Composite Index  Equation used to calculate S&P500  Automatically adjusts for stock splits, etc.  Base period of with a base index value of 10  Index components change slightly each year  500 stocks in index are about 17% of the stocks listed on NYSE  But aggregate market value is > 50% of aggregate market value of all stocks listed on NYSE & AMEX

Francis & IbbotsonChapter 10: Creating Price Indexes 15 S&P 500 Stocks Composite Index  S&P500 is more representative of U.S. common stock investing than DJIA  S&P500 Index is slightly less timely than DJIA –Some of the component stocks are not as actively traded as the 30 stocks in DJIA

Francis & IbbotsonChapter 10: Creating Price Indexes 16 Maintaining A Price Index  Stock market indicators require frequent revisions –Adjustments must be made for stock dividends and stock splits –Changing the number of stocks in sample –Substituting new stocks for ones that have become unsuitable

Francis & IbbotsonChapter 10: Creating Price Indexes 17 Maintaining the DJIA  Changes due to stock splits and dividends have already been demonstrated  DJIA contains only 30 stocks—this has not changed since 1928  Over the years many substitutions have been made –In 1999 the first technology stocks were added Microsoft and Intel (also were first OTC stocks to be included in average)

Francis & IbbotsonChapter 10: Creating Price Indexes 18 Maintaining the S&P500  S&P500 calculation automatically adjusts for stock dividends and splits  Sample size is adequate –500 stocks since 1957  Stocks are added or deleted due to –Listings or delistings –Mergers & acquisitions –Bankruptcy  Changes to index are not as important as changes to DJIA due to small weight of each individual stock

Francis & IbbotsonChapter 10: Creating Price Indexes 19 DJIA vs. S&P500  High positive correlation exists between the two stock market indicators, despite their differences –Due to large amount of undiversifiable risk in the U.S. equity markets

Francis & IbbotsonChapter 10: Creating Price Indexes 20 One-Period Return for an Index  The percentage change in an index is  A r t 0 represents price appreciation  To calculate the total one-period rate of return adjust for cash dividends

Francis & IbbotsonChapter 10: Creating Price Indexes 21 One-Period Return for an Index  Example: The closing value for the S&P500 Index in 1994 was while the closing value in 1995 was Cash dividends for the S&P500 during 1995 were Calculate the total return for the S&P500 during 1995.

Francis & IbbotsonChapter 10: Creating Price Indexes 22 Statistics  Four investment statistics are commonly calculated –Expected return (Chapter 7) –Arithmetic average return (Chapter 2) –Variance (or Standard Deviation) (Chapter 2) –Geometric mean return (Chapter 2)

Francis & IbbotsonChapter 10: Creating Price Indexes 23 Geometric Mean  The geometric mean (GMR) differs from the arithmetic mean (AMR) in that the geometric mean –Considers the compounding of rates of return –GMR usually less than AMR  GMR will equal AMR when there is no risk

Francis & IbbotsonChapter 10: Creating Price Indexes 24 Geometric Mean Example  Example: Given the following asset prices, calculate the geometric mean of the annual returns YearPrice Begin Price End 2001$40$ $60$40 If you bought the asset for $40 at the beginning of 2001 and you sold it for $40 at the end of 2002, you have not earned a positive rate of return over the 2 years.

Francis & IbbotsonChapter 10: Creating Price Indexes 25 Geometric Mean Example  However, if you calculate the arithmetic mean return, the result is positive: YearReturnReturn Relative % %0.666 AMR: (50% %)  2 = 8.335%  The geometric mean calculation, however, does reflect a zero percent rate of return GMR: (1.50 × 0.666) ½ - 1 = 0

Francis & IbbotsonChapter 10: Creating Price Indexes 26 Different Investment Goals  Maximizing an investor’s terminal wealth is the same as maximizing the investor’s geometric mean return over the planning horizon –(1+GMR) T = [Terminal Wealth/Beginning Wealth] = P T /P 0 –Maximizing GMR is more interesting to money managers Can be achieved by maximizing arithmetic mean and minimizing risk –GMR  AMR – 0.5VAR(r)

Francis & IbbotsonChapter 10: Creating Price Indexes 27 Contrasting AMR and GMR  GMR should be used for –Measuring historical returns that are compounded over multiple time periods  AMR should be used for –Future-oriented analysis where the use of expected values is appropriate

Francis & IbbotsonChapter 10: Creating Price Indexes 28 Example: GMR vs AMR  An investment costs $100 and it is equally likely to –Lose 10% or –Earn 20%  The probability distribution of such an investment is: OutcomeProbabilityRate of ReturnProduct Up50%+20%10% Down50%-10%-5% Total100%E(r) = 5%

Francis & IbbotsonChapter 10: Creating Price Indexes 29 Example: GMR vs AMR  If we held the investment for 2 years, the following outcomes exist: T=0 T=1T=2 $100 $120 (50%) $90 (50%) $81 (25%) $108 (50%) $144 (25%) The expected terminal value is $110.25, or $100  (1.05) 2. $110.25

Francis & IbbotsonChapter 10: Creating Price Indexes 30 Example: GMR vs AMR  Expectations about the future should use the E(r) –If $100 is compounded at 5% annually for two years, the expected terminal value is $  If the investment actually grew to $108, the multi-period historical returns should be averaged using GMR –($108/100) 1/2 –1 = = 3.923%

Francis & IbbotsonChapter 10: Creating Price Indexes 31 Compounding Returns over Multiple Periods  Various periodic price relatives can be compounded to obtain a new rate of return for the entire period –3 monthly returns can be compounded to determine 1 quarterly return –12 monthly returns can be compounded to determine 1 annual return, etc.

Francis & IbbotsonChapter 10: Creating Price Indexes 32 Example: Compounding Returns over Multiple Periods  An investment earned the following returns over the last three years: YearReturn 111.1% 2-2.2% 33.3% GMR = (1.111)(0.978)(1.033) 1/3 –1 = /3 – 1 = 3.92% annual return. The total 3-year return is 12.24%. AMR = 11.1% % + 3.3% = 12.2%  3 = 4.07%

Francis & IbbotsonChapter 10: Creating Price Indexes 33 Consumers Price Index (CPI)  Each month the U.S. Government’s Bureau of Labor Statistics computes the CPI  300 goods and services are included in the market basket  Represents food, clothing, housing, medical, etc., that the typical U.S. urban consumer would purchase  Many cost-of-living allowances (COLAs) are based on CPI  Most countries’ CPIs inflate almost every month  Causes purchasing power risk  Unfortunately, many people pay no attention to inflation

Francis & IbbotsonChapter 10: Creating Price Indexes 34 Purchasing Power Risk  Some investments pay a fixed dollar amount that do not rise in tandem with inflation –Coupon and principal payments on bonds  These investors will experience a loss in purchasing power over time

Francis & IbbotsonChapter 10: Creating Price Indexes 35 Measuring Inflation  Inflation is measured as the percentage change in CPI  If the value of a basket of goods rises from $200 to $202 in one month, inflation for the month is: –(202 – 200)/200 = 1%  This monthly inflation rate can be annualized – – 1 = 12.68%

Francis & IbbotsonChapter 10: Creating Price Indexes 36 Nominal Returns Exceed Real Returns  Nominal rate of returns –Advertised money rate of return –Not inflation-adjusted  Real rate of return –Removes inflation from the nominal return

Francis & IbbotsonChapter 10: Creating Price Indexes 37 A Handy Approximation  The previous equation can be modified:  However, the (Inflation  Real) value is usually quite small, so the following approximation is often used  Nominal = Real + Inflation or  Real = Nominal - Inflation

Francis & IbbotsonChapter 10: Creating Price Indexes 38 Empirical Research  Studies have been conducted analyzing the impact of inflation on securities’ returns –Find that both nominal and real returns of common stocks are negatively correlated with rate of inflation –Only real estate provided investors with a complete hedge against actual and unanticipated rates of inflation –T-bills and T-bonds were complete hedges against actual inflation –Real historical bond and bill returns may be zero or negative after considering taxes, management costs and inflation –Common stocks sometimes yield negative real returns

Francis & IbbotsonChapter 10: Creating Price Indexes 39 Hyperinflation  Some countries have experienced extraordinarily high inflation –Brazil, Israel, Mexico  Disrupts a country’s capital markets

Francis & IbbotsonChapter 10: Creating Price Indexes 40 The Bottom Line  When creating a price index the sample should be sufficiently large & representative of the population of interest  Price index should be consistently defined and stated in convenient units  Difficulties arise when dealing with stock splits, stock dividends, mergers and bankruptcies  While the DJIA has some deficiencies, it is still highly correlated with S&P500

Francis & IbbotsonChapter 10: Creating Price Indexes 41 The Bottom Line  When calculating average rates of change can use either AMR or GMR  AMR is most popular but when computed over multiple time periods leads to errors  GMR is appropriate for compounding returns over time

Francis & IbbotsonChapter 10: Creating Price Indexes 42 The Bottom Line  Governments around the world compute CPI to measure country’s inflation rate  Most countries experience inflation although some experience hyperinflation  Inflation results in purchasing power risk  It is important to understand the difference between nominal rates of return and real rates of return