Project 2

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Project 2

• Background Information and Analysis Guide

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Indexes

• Indexes are used to measure historical rates of
  return across several securities. Indexes can be
  used to
• measure the general performance of an economy
• serve as a benchmark for gauging the performance
  of a money manager (e.g., SP 500, SP 600,
  Russell 1000, Russell 2000, etc.)
• serve as a guide for passively managed mutual
  funds
• estimate the risk measures such as beta
• serve as underlying securities in various
  derivative securities

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Indexes

• The effectiveness of an index depends on
• Which securities are included in the index and
  how many
• How the index decides the changes to its
  components
• How the index is adjusted for changes in the
  securities
• Which method is used to calculate the index

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Indexes

• Indexes are generally constructed as
• It?Qit?Pit
• Where Qit is the quantity of security i used to
  construct the index at time t, Pit is the price
  of the security at time t.
• There are three types of indices price-weighted,
  value-weighted and equally weighted indices.

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Price-Weighted Index

• The quantity of each security is the same for all
  component securities.
• The value of a price-weighted index is found by
  adding the prices of each security and dividing
  by a divisor.
• The divisor is the number that is adjusted
  periodically for stock dividends, stock splits,
  and other changes.
• Higher priced stocks have greater impacts on the
  index. Why?
• Dow Jones Industrial Average (DJIA) is a
  price-weighted average of 30 U.S. industrial
  stocks

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Price-Weighted Index

• Example
• Adjustment of divisor for stock splits Suppose
  DJIA is composed of three stocks

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Value-Weighted Index

• Based on the total market value of each component
  security rather than just the price of each
  share.
• Stock splits do not affect the value of the
  index.
• The greater the market capitalization of a stock,
  the larger its influence.
• Examples of value-weighted index
• SP 500 index, NASDAQ Composite Index, NASDAQ
  100.

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Value-Weighted Index

• Based on the total market value of each component
  security rather than just the price of each
  share.
• Stock splits do not affect the value of the
  index.
• The greater the market capitalization of a stock,
  the larger its influence.
• Examples of value-weighted index
• SP 500 index, NASDAQ Composite Index, NASDAQ
  100.

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Equally Weighted Index

• Calculated by giving each security the same
  weight regardless of its price or market
  capitalization. An equal dollar amount is
  invested in each security in the index.
• The equally weighted index is computed as

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An Example

• Assume that we are interested in constructing a
  portfolio with two stocks ABC and XYZ. The
  information is given in the following table. If
  our initial wealth is 1 million, how do we
  construct a portfolio with value weight, price
  weight, and equal weight? How would our wealth
  change in each period? For convenience, lets
  assume fractional shares are allowed.

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An Example

• Initial wealth allocation
• Price weight
• Value weight
• Equal Weight

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An Example

• Wealth allocation at time 1
• Price weight
• Value weight
• Equal Weight
• Do we need to adjust our portfolio holdings under
  each case?
• Work out allocation at time 2

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Importance of Indexes

• Popular Indexes
• DJIA
• SP 500
• Russell 1000
• Russell 2000
• MSCI

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Size of Indexing
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Addition and Deletion Effects for Indexes

• Additions
• What to expect? Why?
• Deletions
• What to expect? Why?
• Empirical Evidence
• Chen, Noronha, and Singal (2004, Journal of
  Finance)

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Implication for Index Fund Investors

• NY Times (July 4, 2004)
• Chen, Noronha and Singal (2005, WP)
• Compare losses to two popular indexes
• Investors in funds indexed to SP 500 lose up to
  0.10 per year
• Investors in funds indexed to the Russell 2000
  lose up to 1.84 per year
• Total loss Up to 5 billion
• Note Losses in the mutual fund scandal were
  estimated between 0.5 and 3 billion.

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Project 2

• Construction of price-weighted average of five
  stocks. (Divisor is five)
• Return calculation
• Rt(Pt/Pt-1)-1
• Note you should have 7 time-series of returns, 5
  for stocks, one for DJIA, one for mini-DJIA
• If you prefer, you can place the returns in
  another worksheet.
• Mean, standard deviation, covariance matrix
• Mean Which Excel function to use?
• standard deviation Excel function
• Note Consider stacking the mean and standard
  deviations, so that your results look like this

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Project 2

• Covariance
• Use excel built-in function COVAR to find
  pair-wise covariances, and construct the
  covariance matrix
• select data analysis under tools menu.
• Select covariance from data analysis

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Project 2

• Covariance
• The input box for covariance should look like the
  chart
• Input range should be cell references for the
  entire region of returns 60 months of return X
  5 stocks. It is also desirable to put labels in
  the first row, so that it will be easier to see
  what each return series is.
• Output range where you want you output to go.
  Remember that you have five stocks, plus the
  headings. Therefore you need 6 rows and 6
  columns of space for the output.

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Project 2

• Covariance
• Output from covariance analysis with Excel add-in
  will only have half of the matrix, you need to
  fill in the missing values using the property
  that the covariance matrix is symmetry against
  its diagonal.

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Project 2

• Minimum-Variance Frontier Construction
• To construct a minimum-variance frontier, you
  need to graph the relation between the mean and
  standard deviation of the minimum-variance
  portfolios.
• With mean and covariance matrix identified in
  step 3, you can use matrix operation to find the
  mean and standard deviation for portfolios
  invested in these five stocks. You will need to
  put formulas for the mean and standard deviation
  (colored area in the Chart). I suggest that you
  put portfolio weights in rows.

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Project 2

• Minimum-Variance Construction
• Identify the means (from the lowest possible to
  the high possible returns, about 15 different
  values should be enough) that you intend to
  achieve as you target. For each target, find the
  minimum variance portfolio using solver. (What
  should be your target cell? What are your
  constraints?)

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Project 2

• Optimal portfolio with same risk as mini-DJIA
• Formulas similar to those in previous question.
  You only need to change the inputs to the solver.
• Out-of-sample validation
• We want how the optimal portfolio we constructed
  in step 5 performs in the future (month 61-72).
  Remember, in step 5, we have only used return
  data from month 1 through month 60.
• first identify the returns to our optimal
  portfolio in each month from month 61 through 72
• Find the average and standard deviation of these
  returns
• Compare with mean and standard deviation of DJIA
  and mini-DJIA over the same period

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Project 2

• Have fun!