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RESEARCHERS at Yale appear to have solved a big problem for mutual fund rating systems. In doing so, the research team may have also found a better way to pick winning stock funds.
--- The New York Times

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Indices

Professor Matthew Spiegel
February 16, 2005

If you click on either the Relative Returns graph on the home page or the Daily Returns link in the navigation pane you will see that we compare the return on the recommended Kalman filter portfolio with the Russell 3000. Now, you may be wondering what the Russell 3000 is and why we use it as our basis of comparison. However, before answering those two questions we need to understand what an index is and how to select one as a benchmark. This note examines how an index is created and tracked over time, a later note will look at how to select and use one as a benchmark.

There exist many different indices published by many different institutions. Yahoo! alone lists 46 on its web site at http://finance.yahoo.com/indices. Each index is nothing more than a portfolio of securities whose value is tracked over time. These portfolios cover a wide range of interests; from government bonds, to large capitalization stocks, to international currencies. This variety exists because different institutions and individuals need them for different purposes. For example, a multinational firm may want to know how the dollar is doing relative to other currencies, while a bank might want to know what the current yield is on a government bond coming due in five years.

Value Weighted Indices

While there are many different indices there are only three commonly used methods for creating one. The first (and generally most sensible and by far the most popular) is to "value weight" the components. To see how this works consider the soon to be famous Alpha VW Index that contains exactly two stocks. Our stocks will go by the rather clever names of A and B. Company A is worth $1 billion and company B is worth $3 billion. They thus have a total market value of $4 billion. Since A makes up ¼ of the aggregate value of the firms in the index it gets a weight of ¼. Similarly company B gets a weight of ¾. Typically, indices start with a value of $100 on day 0. (However, this is completely arbitrary and you can make it any number you wish. After all the initial "investment" is just pretend money, not real.) Because A makes up ¼ of the index this implies that the index begins with an investment of $25 (¼ of $100) in A, and $75 in B (since B is the other ¾ of the index). To keep the arithmetic as simple as possible assume both A and B start with a price per share of $5. This means the index owns 5 shares of A and 15 shares of B.

The next day suppose the price of A goes down to $4 and B up to $6. What happens now? The investment in A went from $25 to $20 (5×4), while the investment in B went from $75 to $90 (15×6). So the Alpha VW Index has gone from a value of $100 to $110. Not bad a 10% return in one day! However, this leads to an important question; since the price of each stock has changed does that mean we need to buy or sell shares of A and B to maintain the appropriate weights? The total value of company A has gone from $1 billion to ($1 billion)×(4/5) or $0.8 billion. Similarly company B has gone from $3 billion to ($3 billion)×(6/5) or $3.6 billion. Their total market value is now $4.4 billion. If the index is going to value weight each firm that must mean company A should now comprise 0.8÷4.4 or 18.2% of the index's value. Right now the index has a total value $110 of which $20 comes from A. That means A has a weight of 20/110 or 18.2%! It's a miracle! The weights have automatically adjusted with the values! There is no need to either buy or sell shares of either company A or B. In fact, this is true of any value weighted index. The weights automatically adjust with the value of each firm so as to make trading completely unnecessary. We will soon see why this is so important.

Equally Weighted Indices

Now that we have conquered the value weighted index the next step is to see how an equally weighted index works. Going back to our example, the Alpha EW Index will divide its funds equally between stocks A and B. Thus, on day 0 the index will hold $50 of A and $50 of B, or 10 shares of each. Now consider what happens on day 1. The 10 shares of A are now worth $40 (10×$4), while the 10 shares of B have a value of $60 (10×$6). This means the total value of the Alpha EW Index remains at $100. But look at what has happened to the weights. Even though the index has not changed the weights have. A is now 40% of the index's value while B is now 60%. Since this is an equally weighted index that means we need to buy $10 of A and sell $10 worth of B. The implicit buying and selling of shares by an index creates a wide array of problems. Worse yet, these problems all cause the reported index to overstate the returns an investor might hope to earn by duplicating the index's portfolio.

When an index buys or sells securities to rebalance it holdings it does so at the "end of day closing" prices which represent the last price at which a security traded that day. While the index can buy or sell these phantom shares at no cost real world investors cannot. For real shares buying and selling costs real money. That means if you attempt to duplicate the index's holdings you will find that your returns fall below those reported by the index itself. The difference can be substantial depending on the number of shares that must be traded each day and the commissions and other fees you need to pay to carry out the trades. The fact that nobody can hope to duplicate the returns in an equally weighted index makes it rather useless as a comparison tool. An index should tell you how well a particular investment strategy would have done had you followed it. Unfortunately, if an index is equally weighted there is no way to actually implement its strategy and that limits or even eliminates its usefulness as a financial tool.

Beyond the transactions cost issue outlined above, indices that require constant trading run up against another rather subtle (but potentially more serious) problem. The end of the day prices used to calculate the index's value are not what real investors see. Financial markets are maintained by liquidity providers that stand ready to buy and sell a security on demand. These individuals are compensated by the "bid-ask spread." This spread represents the difference between the price liquidity providers will pay to acquire and sell shares. The terms bid and ask come from the liquidity supplier's perspective. Thus, the bid is what he or she is will to pay to buy shares and the ask the price at which he or she will sell shares. (A moments thought should make clear that the bid is always below the ask.) Now return to the Alpha EW index. On day 1 the price of A fell and the price of B rose. What does that imply about the last transaction of the day? Since A fell in price it is likely that the last transaction involved a sale from the public to a liquidity provider. If so, that transaction occurred at the bid (the low price). But, notice the index had to buy shares of A and when it did so the assumption was that the price would equal the price from the last transaction. But, nobody was actually able to buy shares at that price! The final price was at the bid and if you had tried to buy shares you would have had to do so at the ask. For firm B's shares the same problem arises. Because B went up in value on day 1 the last transaction was at the ask. But, to duplicate the Alpha EW portfolio you had to sell shares of B and had you done so you would have sold at the bid (a lower price). Thus, the bid-ask spread introduces another bias in the reported returns from an equally weighted index; a bias that again causes the index to overstate what an investor might hope to earn by duplicating its holdings. Overall then, if you follow the returns from an equally weighted index you are probably looking at returns higher than you can ever obtain yourself were you to attempt to duplicate the index with real money.

Price Weighted Indices

There is one more index construction method that is worth reviewing but only because it is used to construct the Dow Jones Industrial Average (DJIA). The DJIA is a "price weighted" index. This method is so silly that if it were not for this index's long history it is doubtful anybody would take it seriously. To construct the DJIA you take a fixed number of shares from each of the thirty firms in the index and hold them. So the good news is that you can duplicate this index without trading on a daily basis. The bad news is the weights make absolutely no sense. Consider a third firm C that like B is worth $3 billion on day 0. Imagine B has 100 million shares outstanding and that C has 200 million shares outstanding. The price of each share of B must be worth $30 ($3 billion divided by 100 million) while the price of a C share equals $15 ($3 billion divided by 200 million). Thus, while these two firms have identical values the DJIA will give firm B twice as much weight in the index as firm C. If you think this makes any sense just note that the number of outstanding shares a firm has is completely arbitrary! Firm B can get the DJIA to cut its weight within the index by 50% by simply doing a two for one stock split (thereby doubling the number of shares it has outstanding and cutting its share price in half). Alternatively, C can increase its weight in the index by doing a so called "reverse stock split" in which it reduces the number of outstanding shares it has. Given all of this why follow the DJIA at all? Well for one your local news cast probably reports its value every night and for another you probably have some feel for what a "80 point" change means about the markets that day. However, if you can wean yourself from these two advantages you would get a much better understanding of the day's market moves by following a broader value weighted index.

Some value weighted indices:
S&P 500: Follows 500 of America's largest companies. This index covers about two-thirds of the stock market's total value.
Russell 2000: Follows 2000 of America's smallest companies. Despite the vast number of firms in this index it covers less than one-third of the stock market's total value.
Russell 3000: A broad index that includes almost all of the stock market's total value. This is the benchmark used by Alpha Investment Opportunities.
Wilshire 5000: Similar to the Russell 3000 but covers even more small company stocks. The returns from the Wilshire 5000 and the Russell 3000 tend to be nearly identical. The only reason Alpha Investment Opportunities does not use the Wilshire 5000 is that it is somewhat easier for us to obtain the Russell 3000's value on a daily basis.

Equally weighted index:
Value Line Arithmetic Index

Price weighted index:
Dow Jones Industrial Average

Alpha Investment Opportunities