Creating Price Indexes

Creating Price Indexes Chapter 10 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

Background • Thousands of price indicators are used to track prices of • Stocks • Bonds • Commodities • Foreign exchange rates • Mortgages • Options • Futures Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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. Stock Splits and DJIA Divisor • As an example, consider the hypothetical stocks 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. Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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. Chapter 10: Creating Price Indexes

S&P 500 Stocks Composite Index • Equation used to calculate S&P500 • Automatically adjusts for stock splits, etc. • Base period of 1941-1943 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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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) Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

One-Period Return for an Index • The percentage change in an index is • A rt < 0 represents price depreciation while a rt > 0 represents price appreciation • To calculate the total one-period rate of return adjust for cash dividends Chapter 10: Creating Price Indexes

One-Period Return for an Index • Example: The closing value for the S&P500 Index in 1994 was 459.27 while the closing value in 1995 was 615.93. Cash dividends for the S&P500 during 1995 were 15.93. Calculate the total return for the S&P500 during 1995. Chapter 10: Creating Price Indexes

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) Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

Geometric Mean Example • Example: Given the following asset prices, calculate the geometric mean of the annual returns 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. Chapter 10: Creating Price Indexes

Geometric Mean Example • However, if you calculate the arithmetic mean return, the result is positive: AMR: (50% + -33.33%)  2 = 8.335% • The geometric mean calculation, however, does reflect a zero percent rate of return GMR: (1.50 × 0.666)½ - 1 = 0 Chapter 10: Creating Price Indexes

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] = PT/P0 • Maximizing GMR is more interesting to money managers • Can be achieved by maximizing arithmetic mean and minimizing risk • GMR  AMR – 0.5VAR(r) Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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: Chapter 10: Creating Price Indexes

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

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 $110.25 • If the investment actually grew to $108, the multi-period historical returns should be averaged using GMR • ($108/100)1/2 –1 = 0.03923 = 3.923% Chapter 10: Creating Price Indexes

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. Chapter 10: Creating Price Indexes

Example:Compounding Returns over Multiple Periods • An investment earned the following returns over the last three years: GMR = (1.111)(0.978)(1.033)1/3 –1 = 1.12241/3 – 1 = 3.92% annual return. The total 3-year return is 12.24%. AMR = 11.1% + -2.2% + 3.3% = 12.2%  3 = 4.07% Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 • 1.0112 – 1 = 12.68% Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

Hyperinflation • Some countries have experienced extraordinarily high inflation • Brazil, Israel, Mexico • Disrupts a country’s capital markets Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

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 Chapter 10: Creating Price Indexes

Creating Price Indexes