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Creating Price Indexes

Creating Price Indexes. Chapter 10. 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

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Creating Price Indexes

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  1. Creating Price Indexes Chapter 10 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 10: Creating Price Indexes

  11. 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

  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 Chapter 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. Chapter 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 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

  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 Chapter 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 Chapter 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) Chapter 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 Chapter 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 Chapter 10: Creating Price Indexes

  20. 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

  21. 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

  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) Chapter 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 Chapter 10: Creating Price Indexes

  24. 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

  25. 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

  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] = 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

  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 Chapter 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: Chapter 10: Creating Price Indexes

  29. $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

  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 $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

  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. Chapter 10: Creating Price Indexes

  32. 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

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

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

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