Indexes and Funds

Indexes

Indexes are weighted averages of asset characteristics.

  • For example, it might be a weighted average of stock prices, stock returns, or bond yields.

Dow Jones

The Dow Jones Industrial Average (DJIA) is the oldest U.S. index, dating to 1896.

  • Since 1926 it has included 30 large stocks.
  • Originally a simple average of the prices.
  • Percentage change in the Dow was originally the return on a portfolio consisting of one share invested in each of the stocks in the index.

Dow Jones

  • The DJIA is a price-weighted average: the amount of money invested in each asset of the portfolio is proportional to the share price.
  • Due to splits and changes in the composition of the index, the DJIA is no longer a simple weighted average of prices.

Price-Weighted Indexes

Consider a price-weighted index of two stocks, \(X\) and \(Y\) .

  • The price of \(X\) is originally $25 and increases to $30.
  • The price of \(Y\) is originally $100 and decreases to $90.

Then

  • Initial index value = \(\frac{\$25+\$100}{2} = \$62.5\).
  • Final index value = \(\frac{\$30+\$90}{2} = \$60\).
  • Percentage change = \(\frac{-\$2.5}{\$62.5} = -0.04 = -4\%\).

Price-Weighted Indexes

Note that price-weighted indexes give higher priced stocks more weight.

  • The percentage change in stock \(X\) is:
\[\frac{\$30 - \$25}{\$25} = 0.2 = 20\%\]
  • The percentage change in stock \(Y\) is:
\[\frac{\$90 - \$100}{\$100} = -0.1 = -10\%\]

Price-Weighted Indexes

  • The overall percentage change in the index is
\[\% \text{change in index} = \frac{p^0_X}{p^0_X+p^0_Y} \Delta_X + \frac{p^0_Y}{p^0_X+p^0_Y} \Delta_Y\]
\[= 0.2 * 0.2 + 0.8 * (-0.1) = -0.04.\]
  • \(p^0_i\) is the initial price of stock \(i\).
  • \(\Delta_i\) is the percentage change in the price of stock \(i\).

Splits and Price-Weighted Averages

Suppose that stock \(Y\) split, causing its price to fall to $50.

  • This would cause a large fall in the value of the index, unless an adjustment is made to the divisor.
  • That is, the index value before the split is
\[\frac{\$25 + \$100}{2} = \$62.5.\]
  • The post-split divisor, \(d\), should be the value such that
\[\frac{\$25 + \$50}{d} = \$62.5.\]

Splits and Price-Weighted Averages

  • Hence, \(d\) falls from 2 to 1.2.
  • Notice that since the split causes the price of \(Y\) to fall, it’s relative weight in the portfolio will fall.
  • Movements in the price of \(Y\) will have a smaller impact on the index.

Standard and Poor’s Composite 500

The S&P 500 stock index has two advantages over the Dow:

  • It is comprised of 500 large stocks, and hence is more broadly based and a better indicator of the market as a whole.
  • It is a value-weighted, rather than price-weighted, index.
  • The market value or market capitalization of a firm is simply its total value on the market: price per share times the number of shares outstanding.
  • A value-weighted index weights each stock in the index according to its market cap.

Value-Weighted Indexes

If stock \(X\) currently has 20 shares trading in the market and stock \(Y\) only has 1 share, the market caps for \(X\) and \(Y\) are

\[\begin{split}MC^0_X & = 20 * \$25 = \$500\end{split}\]
\[\begin{split}MC^0_Y & = 1*\$100=\$100.\end{split}\]
  • A value-weighted index of the two stocks would give \(X\) five times the weight as \(Y\).
  • Compare to the price-weighted index which gives \(Y\) four times the weight.
  • Initially, the total stock on the market is equal to $500 + $100 = $600.

Value-Weighted Indexes

After the price changes, market caps become

\[\begin{split}MC^1_X & = 20 * \$30 = \$600\end{split}\]
\[\begin{split}MC^1_Y & = 1*\$90=\$90.\end{split}\]
  • The total value of stock is now $690.
  • If the initial value of the value weighted index was $100, after the price changes it would be \(\$100*\frac{\$690}{\$600} = \$115\).

Value-Weighted Indexes

  • In this case the value of the index rises since it gives a relatively higher weight to \(X\).
\[\begin{split}\% \text{change in index} & = \frac{MC^0_X}{MC^0_X + MC^0_Y} \Delta_X + \frac{MC^0_Y}{MC^0_X + MC^0_Y} \Delta_Y\end{split}\]
\[= \frac{5}{6} * 0.2 + \frac{1}{6} * (-0.1) = 0.15.\]

Equally-Weighted Indexes

One of the advantages of price-weighted and value-weighted indexes is that they correspond to buy-and-hold portfolio strategies:

  • A price-weighted index is equivalent to buying and holding one share (or an equal number of shares) of each stock in the index.
  • A value-weighted index is equivalent to buying and holding each share of the index in proportion to its market cap.

Equally-Weighted Indexes (Cont.)

In contrast, one could form an equally-weighted index, where all stocks receive the exact same weight.

  • This does not correspond to a buy-and-hold strategy.
  • Consider starting with with equal amounts of money invested in stocks \(X\) and \(Y\).
  • If the price of \(X\) increases by 20% and the price of \(Y\) falls by 10%, the dollar amount invested in each stock is no longer equal.
  • To keep the investment equally weighted, you would have to sell some shares of \(X\) and buy shares of \(Y\).

Other Indexes

There are a wide number of published indexes:

  • Sub-indexes of the S&P 500 and others above.

To hold these as part of a portfolio one could