Creating Price Indexes

1
Creating Price Indexes

• Chapter 10

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 SP500 Composite Index
• Contains value of 500 U.S. stocks

3
Background

• Thousands of price indicators are used to track
  prices of
• Stocks
• Bonds
• Commodities
• Foreign exchange rates
• Mortgages
• Options
• Futures

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 funds performance to
  SP500

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

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

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

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

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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

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

11
Stock Splits and DJIA Divisor

• As an example, consider the hypothetical stocks

Stock Price
X 50
Y 10
Total 60
Average 60/2 30
Stock Price
X 25
Y 10
Total 35
Average 35/2 17.5
If Stock X undergoes a 2 for 1 stock split
The stock split changed the price per share, but
the stockholders wealth has remained the
sameeach stockholder in X has twice as many
shares as before.
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

13
SP 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 securitys weight is based on the total
  market value of the firm
• Corresponds to the investment opportunities that
  exist in U.S.

14
SP 500 Stocks Composite Index

• Equation used to calculate SP500

• 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 gt 50 of aggregate
  market value of all stocks listed on NYSE AMEX

15
SP 500 Stocks Composite Index

• SP500 is more representative of U.S. common
  stock investing than DJIA
• SP500 Index is slightly less timely than DJIA
• Some of the component stocks are not as actively
  traded as the 30 stocks in DJIA

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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

17
Maintaining the DJIA

• Changes due to stock splits and dividends have
  already been demonstrated
• DJIA contains only 30 stocksthis 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)

18
Maintaining the SP500

• SP500 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

19
DJIA vs. SP500

• 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

20
One-Period Return for an Index

• The percentage change in an index is

• A rt lt 0 represents price depreciation while a rt
  gt 0 represents price appreciation
• To calculate the total one-period rate of return
  adjust for cash dividends

21
One-Period Return for an Index

• Example The closing value for the SP500 Index
  in 1994 was 459.27 while the closing value in
  1995 was 615.93. Cash dividends for the SP500
  during 1995 were 15.93. Calculate the total
  return for the SP500 during 1995.

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)

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

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Geometric Mean Example

• Example Given the following asset prices,
  calculate the geometric mean of the annual returns

Year PriceBegin PriceEnd
2001 40 60
2002 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.
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Geometric Mean Example

• However, if you calculate the arithmetic mean
  return, the result is positive

Year Return Return Relative
2001 50 1.50
2002 -33.33 0.666
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
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Different Investment Goals

• Maximizing an investors terminal wealth is the
  same as maximizing the investors geometric mean
  return over the planning horizon
• (1GMR)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)

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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

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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

Outcome Probability Rate of Return Product
Up 50 20 10
Down 50 -10 -5
Total 100 E(r) 5
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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.
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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

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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.

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Example Compounding Returns over Multiple
Periods

• An investment earned the following returns over
  the last three years

Year Return
1 11.1
2 -2.2
3 3.3
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
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Consumers Price Index (CPI)

• Each month the U.S. Governments 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

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

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

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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

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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

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

39
Hyperinflation

• Some countries have experienced extraordinarily
  high inflation
• Brazil, Israel, Mexico
• Disrupts a countrys capital markets

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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 SP500

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

42
The Bottom Line

• Governments around the world compute CPI to
  measure countrys 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