Stock Indices

ETF investing has become one of the primary vehicles for many individual investors, since it offers a low-cost, low-maintenance approach to investing, with results that match the overall market. The reason for this is that most ETF strategies revolve around investing in market-index ETFs. A great reference for this type of investing can be found at the Canadian Couch Potato blog, or at the original couch potato site at MoneySense Magazine.

When I started off investing on my own, I purchased Canadian Market Index Stocks such as XIC and XIU. At a high level, I understood that an index based ETF basically held the holdings of the index it represented; for example, XIC holds all of the equities in the S&P/TSX Capped Composite Index. But what is an index anyways?

In its broadest sense, an index provides an overview of the performance of the underlying securities, either through a price-weighted or value-weighted average. What constitutes inclusion in that group of securities depends on what the index is trying to achieve. Standard & Poors is one of the primary index publishers, and they have several such as the S&P/TSX 60, S&P/TSX Capped Composite, S&P/TSX SmallCap, etc. Each of these indices is designed to be representative of one dimension of the overall market. The 60 acts as a subset of the S&P/TSX Composite, but caps the total number of equities at 60. The SmallCap attempts to provide an overall barometer on the performance of small cap stocks on the TSX, etc.

Excluding the focus of the index, the other primary factor to take into account is whether or not it is price-weighted or value weighted.

(For the discussions below, a worksheet is available to play with, which can be found here.)

Price-Weighted Indices

The most popular price-weighted index is the Dow Jones Industrial Average (DJIA). There are a number of articles on the Internet which speak to the history of the DJIA, and my interest is in what it means to be price-weighted.

Essentially, a price weighted index is exactly what it sounds like: an index whose value is more influenced by firms which have a higher price. The value of a price-weighted index is the sum of the prices of firms in that index, divided by some divisor. The divisor is the trickier part of the equation.

DJIA=\frac{\sum_{i=1}^{30} P_{i}}{divisor}

Where Pi is the price of firm i. When the DJIA was initially started, the divisor was 30, because there were 30 firms in the average. However, over time the divisor has had to be adjusted due to stock (reverse-)splits. I.e., when a stock splits, the total number of shares goes up, but the price of the stock goes down. However, the before and after values of the DJIA should be the same. With a little algebra, we can calculate the divisor after a (reverse-)split as:

divisor_{new}=\frac{(\sum_{i=1}^{30}P'_{i})\times divisor_{old}}{\sum_{i=1}^{30}P_{i}}

In the above, Pi are the prices before the (reverse-)split, and P’i are the prices after the split.

Over time, the divisor for the DJIA has become incredibly small, and the divisor as of 2010/07/02 was 0.132129493 (from 0.132319125). The reason for this is that, using the formula for divisornew above, over the years successive splits have made the divisor smaller and smaller.

The other thing to consider with a price weighted index is the weighting of the price itself, and how it affects the overall value of the index. That is, an equal percentage change for a stock will mean more (or less) if the stock price was large (small) relative to the index to begin with. For example, both IBM and Bank of America are stocks in the DJIA. However, they are incredibly different in terms of pricing. At one point during the week of October 29, 2012, IBM was trading at $193.27 a share, and Bank of America was trading at $9.12 a share. But why does this matter? It matters because a 5% shift in either stock will have a different overall effect on the index as a whole. A 5% shift in IBM results in a 77.289 point shift in the DJIA, but an identical shift of 5% for Bank of America only shifts the DJIA 3.6471 points!

This means that one should pay particular attention to what it means when news reports say that the Dow has moved – the context of the movement has to be taken into account to ensure that the movement isn’t being skewed by a heavy hitter such as IBM.

Value-Weighted Indices

On the other end of the spectrum are value-weighted indices.

Looking at the sample spreadsheet, when we enter values for the split multiple, the sum of the market caps does not change. This is because (reverse-)splits automatically account for changes in the number of shares outstanding, and share price. This simplifies computation of the index because there is no denominator which must be continuously adjusted.

The other thing to notice is that changes in share price in a value weighted index are scaled appropriately to the weight of the firm in the index, measured by market cap. This means that the bigger the firm by market cap, the more weight it has. This also demonstrates that movements of the share price will be properly reflected in the index, and that the percentage move of a given security in the index will be properly reflected in the change of the index itself. But why does this matter?

Consider two hypothetical stocks, each with a market cap of $1,000,000,000. Further, assume that stock #1 has a share price of $5.23, and stock #2 has share price of $198.53. This is summarized below:

Stock Share Price Shares Outstanding Market Cap
Stock 1 $5.23 191,204,588.91 $1,000,000,000
Stock 2 $198.53 5,037,022.11 $1,000,000,000

As discussed above for the DJIA, if both stocks change by 5%, this will not have an identical change on the stock. However, based on market cap alone, a movement of 5% in either stock will have the same effect on the inded. For this reason, a value-weighted index is often a better indicator of the performance of the cross-section of the market that the index is monitoring.

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