The ETF-Index Pricing Relationship
By John J. Neumann

In the more than a decade since the first exchange-traded products were introduced, these securities have become progressively more popular investment vehicles among professional and individual investors alike. As of March 2008, there were more than 600 exchange-traded fund portfolios being actively traded and several hundred more in registration.

Not only has the number of ETF offerings increased, but the size of the individual ETF issues has grown. At year-end 2007, the S&P 500 index-tracking ETF (SPY, or the SPDR) had 25 times more shares outstanding than in 1998, and its average daily trading volume grew at an average annual rate of 40.5 percent over that time. DIA, or ''Diamonds,'' which was introduced in 1998 to track the portfolio of 30 stocks in the Dow Jones Industrial Average (''DJIA'') and is the focus of this article, has seen its daily average volume grow by 41 percent per year on average over that same period.

ETFs in general allow investors to gain from intraday moves of an index since they trade as individual company stocks do, with continuously updated bid/ask quotes. They support investors' pursuit of passive investment strategies with more direct control of tax events (and generally lower expense ratios) than mutual funds. Thus, if investors believe that large, blue-chip-type companies are going to have a good (bad) day on average, but are unable to pinpoint exactly which of the 30 stocks in the DJIA might experience the most significant price increases (declines), Diamonds now provide a vehicle with which they can try to profit from their expectation. This is not possible with a traditional index mutual fund, since these funds calculate their net asset value once a day, after the markets have closed, so buyers and sellers of shares do so at the same price. Additionally, unlike mutual funds, ETFs can be sold short to gain if the index is expected to decline.

If the individual stocks in the DJIA index are all subject to their own buying and selling pressures throughout the day, DIA, as a stand-alone security, must also be subject to buying and selling pressures. One can envision a piece of news being released which motivates investors to transact in the ETF— but not the individual companies—perhaps because the news is so fundamental a systematic news event, in modern portfolio theory terms, that it is expected to have an on average effect on ''the market.'' Conversely, and perhaps more clearly, news released about individual companies in the index might attract investor attention to these individual stocks more so than to the portfolio as a whole, especially during earnings reporting periods. What, then, is the mechanism that assures that the share price of an independent ETF security does not become disconnected from the underlying portfolio of stocks to which it is tied and consistently fulfills its objective of tracking the underlying index portfolio? In a word, arbitrage.

Arbitrage And The ETFs
Strictly speaking, an arbitrage is a mispricing between two (or more) assets or bundles of assets which are identical in some way and related via a well-defined pricing relationship. The mispricing may exist because one asset (or bundle) is overpriced for a period of time, because the other asset (or bundle) is underpriced for a period of time or because both assets (bundles) are mispriced. An arbitrage strategy attempts to lock in a risk-free profit by exploiting the mispricing with a virtually costless portfolio that calls for short positions in the overpriced asset(s) and long positions in the underpriced asset(s), the latter financed using the short proceeds. Implemented on a large scale, the net selling pressure of the short positions causes the price(s) of the overpriced asset(s) to decline, while the net buying pressure forces the price(s) of underpriced asset(s) to rise. The expectation is that when prices return to levels at which the well-defined relationship once again holds, this portfolio can be unwound to leave a net cash amount as the investor's arbitrage profit.

The Diamonds prospectus reads: ''[T]he Sponsor's aim in designing DIAMONDS was to provide investors with a security whose initial market value would approximate one-hundredth (1/100th) the value of the DJIA.'' Figure 1 reveals that from January 20, 1998, the day the Diamonds ETF started trading, until December 31, 2005, the ratio of the closing values of the Dow Jones Industrial Average to the per-share price of Diamonds ranged from 98.4526 to 102.4855. The mean ratio was 99.8978, with a standard deviation of 0.2204, indicating that roughly 95 percent of the time, this ratio has been between 99.457 and 100.3386. Interestingly, the standard deviation of this ratio was highest in the years immediately after the introduction of this price-weighted ETF, perhaps reflecting a learning curve among the financial markets.

Figure 1


Source: Values are taken at the close. Index values from Yahoo Finance and DIA prices from CRSP. Data from January 20, 1998 through December 31, 2005.

Once the ratio between these two ''baskets of securities'' was pegged at 100:1, a pricing relationship was established the violation of which presents opportunities for arbitrage trading. When a significant enough violation occurs, astute (or well-positioned) investors can swoop in, capture profits and force prices back in line. Thus, the price pressures unleashed by arbitrage activity help to maintain the integrity of the relationship between an ETF security and its underlying index portfolio. I note here that this activity originates in two possible ways. The first is the ability of well-capitalized, professional investors to take long positions in the underpriced security and short positions in the overpriced security, and then unwind those positions by interacting with the ETF-issuing trust to make exchanges of ETF shares and the underlying stocks (Mehta, 2002 and Venkatesh, 2002). This is the primary mechanism by which the ETF and underlying index remain closely tied together. The second sees individual arbitrageurs unwind the positions with offsetting sell and buy-to-cover transactions in the financial markets. I'll start with the latter. I use Diamonds in this demonstration because it is much easier to envision simultaneous buy or sell trades of 30 stocks than of 500 stocks (using the S&P 500 and SPY) or even 100 stocks (using the NASDAQ-100 Index and QQQQ). In addition, as a price-weighted index, the DJIA is calculated in a way that is distinct among indexes so that implementing an arbitrage strategy with the DJIA stocks and Diamonds calls for a straightforward rule of starting with one share of each component, and then scaling it up to the limit of available investable funds.

Example 1— DJIA:DIA Ratio Less Than 100:1
The examples use market data from April 11, 2006, when the closing price of Diamonds was $110.93 per share and the closing level of the DJIA was 11089.63, calculated by dividing the $1,385.44 sum of the prices of the 30 component stocks by the index divisor, then at 0.12493117. On this day, the DJIA:DIA ratio was 99.96962:1.

Suppose, however, that this ratio was observed to be 96.049:1 at some point during the day, significantly below the lower end of the range in which roughly 95 percent of the actual daily values have historically fallen. This would occur if the DJIA were at its proper level but the Diamonds price was $115.458 per share instead of $110.93. Or, it would be observed if Diamonds were properly priced at $110.93 but the DJIA was too low at 10,654.69. It could also happen with combinations of DJIA values above 10654.69 and Diamonds prices below $115.458. The critical insight is that it does not matter which of these scenarios has caused the ratio to fall below 100:1. With it on the low side of 100:1, the ETF is overvalued relative to the index, or the index is net undervalued relative to the ETF, and so a knowledgeable investor could exploit the mispricing by short-selling DIA and buying shares of the 30 index stocks.

Figure 2

Columns 2 and 3 of Figure 2 outline two distinct cases where the price of the ETF is above its correct $110.93 value and the DJIA index is below the 11089.63 level where it should be. Case 1 has the price of Diamonds at $113.49 per share with the DJIA at 10900.58. Case 2 sets the ETF price at $115 per share and the index at 11045.61. In both cases, the ratio of DJIA level to DIA price is 96.049:1. I present two scenarios to reinforce the idea that the arbitrage strategy and payoff is the same for any given value of the DJIA:DIA ratio, regardless of what specific prices and values the ETF and index take to produce that ratio.

Since the DJIA index effectively includes one share of each component stock, and the total cost of one share of each stock is the sum of the 30 stock prices, the investor needs $1,361.82 (=10900.58*0.12493117) to purchase one share of each index component as part of the arbitrage strategy to exploit the mispricing depicted in Case 1 and $1,379.94 (=11045.61* 0.12493117) to purchase the shares in Case 2. A short sale of 12 ETF shares raises the required funds in each case so that, as expected, the cash flows of the arbitrage portfolio net to virtually $0, ignoring commissions for the moment.

These two cases illustrate how the selling pressure on the ETF forces its price down while the buying pressure on the index component stocks forces their prices up, until both the ETF and DJIA reach levels where arbitrage exploitation is no longer profitable. Once that happens and the ratio resettles at 100:1, each portfolio can be unwound to lock in the identical $54.28 arbitrage profit. When the index returns to its correct value of 11089.63, the sum of the prices of the 30 component stocks is $1,385.44. This amount is received when the shares are sold, and these funds are used to cover the short positions in the Diamonds, which cost a total of $1,331.16 at the true per-share price of $110.93. This activity leaves a cash balance of $1,385.44 - $1,331.16 = $54.28, again ignoring commissions, with no positions in any securities or liabilities remaining. The initial 6 cents retained from establishing the portfolio brings the total arbitrage profit to $54.34. As shown in the ''Expectation'' row of Figure 2, this amount is exactly the difference between the respective posited prices of Diamonds and their correct price ($110.93) scaled by the 12 shares, plus the difference between the posited levels of the index and its 11089.63 true value scaled by the index divisor.

The $54.34 is the maximum profit to be reaped from arbitrage exploitation when the DJIA:DIA ratio is 96.049. If an investor unwinds the arbitrage portfolio early, prior to the ratio resettling at 100:1, smaller arbitrage profits are realized. The fourth and fifth columns of Figure 2 revisit Cases 1 and 2 with the change that the investor unwinds the arbitrage portfolio when the ratio has risen from 96.049:1 to 99:1 instead of all the way back to 100:1.

Example 2—Ratio Greater Than 100:1
An inflated DJIA:DIA ratio signals a situation where the ETF is undervalued relative to the index, or, alternatively, the stocks of the index are net overvalued relative to the ETF. Figure 3 summarizes two scenarios where the index:ETF ratio is well above its 100:1 peg at 104.058:1. These scenarios capture the realistic cases where both assets are mispriced in producing the 104.058:1 ratio. In Case 1, the DJIA is 11238.26 and higher than its actual April 11, 2006 value of 11089.63, while the price of Diamonds at $108 per share is lower than the $110.93 price on April 11. Case 2 depicts the DJIA at 11350.65 and the ETF price at $109.08. The arbitrage strategy to exploit such a severe deviation from the peg ratio in this direction is to short sell the 30 component stocks in the index and use those proceeds to buy the relatively undervalued shares of the ETF. Columns 2 and 3 of Figure 3 outline the trades and maximum net arbitrage profit that can be earned by unwinding the arbitrage portfolio when the ratio falls back to exactly 100:1, while columns 4 and 5 illustrate what happens if the unwinding occurs prematurely at a DJIA:DIA ratio of 101:1.

Figure 3

Professional Investors And Restrictions On Individuals
Each strategy, whether exploiting the high or low ratio mispricing, involves 31 trades to set up the portfolio: 30 buys or sells for the component stocks and one for the ETF―and 31 trades to unravel it. Therefore, transactions costs are key. For an individual investor using the oft-advertised price of $14.95/trade from an online broker, the total cost is $926.90. Thus, an investor needs to scale up the 96.049:1 position by at least 18 times to cover these costs and realize profits. That calls for short selling 216 shares of the ETF and buying 18 shares of each component stock to produce an arbitrage profit of 18*$54.34 = $978.12 - $926.90 = $51.22. The 104.058:1 position would likewise need to be scaled up 17 times to produce a net profit of 17*$56.66 = $963.22 - $926.90 = $36.32. Of course, with risk-free profits on the menu, an investor is motivated to scale it up to as much as he/she can afford!

Unfortunately, individual investors face obstacles in implementing these strategies. For one, they are generally required to keep short sale proceeds in their brokerage accounts and would not be able to use them to cover the required purchases. Further, with the average online brokerage account, it would be virtually impossible to execute all 31 trades simultaneously as required by this strategy. That is a must, as any delay or fracture in establishing the arbitrage positions puts the success of the strategy at great risk while other investors quickly squeeze out the arbitrage profits. Thus, professional traders are usually best-positioned to take advantage of these short-lived opportunities because of their full-time attention to the financial markets, available technology, collateral at their disposal and fewer restrictions on the use of short sale proceeds. It can be shown how a trader employs futures contracts in the DJIA index along with positions in the ETF to capitalize on opportunities when the index:ETF ratio deviates substantially from 100:1. When it is below 100:1, the portfolio consists of a short position in Diamonds and a long position in the index futures contract. Above 100:1, the strategy entails a long position in the ETF and a short index futures position. These cases would demonstrate that while a profit is certain to be captured, for a given DJIA:DIA ratio value, the amount of that profit depends on the extent of the index mispricing.

Authorized Participants (''APs'') have an additional option by which they can exploit an index:ETF mispricing by virtue of the fact that they have contracted with the ETF issuer and are able to ''create'' or ''redeem'' shares. So, when the DJIA:DIA ratio is sufficiently higher than 100:1, an AP might start with the steps I outlined earlier, but instead of selling DIA shares to purchase index components shares when the ratio resettles at 100:1, they swap their DIA shares with the ETF issuer in exchange for index component shares which are surrendered to cover the short positions. Excess components are the source of the arbitrage profit here. If the DJIA:DIA ratio is profitably below 100:1, an AP could begin with the steps I outlined earlier, but unwind the portfolio when prices adjust and the ratio returns to 100:1 by swapping index components shares to the ETF issuer (rather than selling them) in exchange for new DIA shares, use these to cover the short positions and sell leftover components shares to lock in the profit. I illustrate this case in the next section by returning to the two cases from Figure 2.

Authorized Participants When DJIA:DIA Ratio Is 96.049:1
The key distinction in transacting with the ETF issuer is that all exchanges must be done in 50,000-lot creation units. 21 May/June 2008 Thus, sufficient quantity of the component stock shares must be surrendered to receive 50,000 shares of Diamonds, or a minimum of 50,000 shares of Diamonds must be surrendered in exchanges for component shares.

When the price of the ETF is $113.49 ($115) per share, 50,000 shares have a value of $5.6745M ($5.75M). When the DJIA index is 10900.58 (11045.61), the sum of the prices of the 30 component stocks is $1361.82 ($1379.94). In both cases, short sales of 50,000 shares of the ETF on the secondary market are enough to finance the purchase of exactly 4,167 shares of each index component stock. Once the selling-and-buying pressure unleashed by these trades moves prices so that the ratio resettles at 100:1, 50,000 shares of Diamonds are worth $5,546,500 ($110.93 per share) and are equivalent to $5,546,500/$1,385.44 = 4,004 shares of each component. The AP at that point swaps 4,004 of his/her 4,167 shares of each component for 50,000 ETF shares, uses these shares to cover the original short positions and sells the remaining 4167 - 4004 = 163 shares of each component for 163*$1,385.44 = $225,826.72 (sum of the 30 prices at the reestablished index level of 11089.63 is $1,385.44). This amount represents the captured arbitrage profit, as shown below.

Conclusion
The fundamental lesson from these analyses is that it does not matter, in the first case, how the DJIA:DIA ratio came to be low at 96.049:1. Recognition that this signals a relative ETF overvaluation (or index component net undervaluation) calls for a strategy of short selling the ETF shares and using the proceeds to buy shares of each index component stock, which produces a maximum unscaled arbitrage profit of $54.34 for the individual investor and 163 excess component shares for the AP. In the second case, where the ratio is 104.058:1, how the ratio became inflated is irrelevant. An investor merely needs to recognize this as the signal of a relative ETF undervaluation (or index component net overvaluation) and act accordingly. The strategy to short sell shares of the index component stocks and buy the ETF produces a maximum unscaled arbitrage profit of $56.66 in the individual case. This is the essence of the risk-free arbitrage profit. The establishment of a (virtually) zero-cost portfolio results in the same, sure profit outcome when the portfolio positions are closed out, and the price pressures exerted by investors attempting to exploit even the slightest mispricings are what maintain the peg ratio between an ETF and its underlying index.

References
Bodie, Z., Kane, A., and Marcus, A., Investments (Boston, 2005), 6th Edition, Irwin.
Mehta, N. ''Getting Saved on Wall Street: ETFs: A New Religion is Ten Years Old,'' Traders Magazine, 15 (Dec. 1, 2002), 32-34.
PDR Services, LLC. Diamonds Trust, Series 1 Prospectus. (New York, February 2006), American Stock Exchange LLC.
Venkatesh, B. ''NAV discount and ETFs,'' Businessline (July 14, 2002).