Demonstrating arbitrage using Diamonds and the Dow Jones Industrials index.
There are now no fewer than 450 exchange-traded fund portfolios (ETF, HOLDR, or iSHARES) at www.amex.com and listed on the NYSE, a menu of choices which facilitates investor pursuit of a variety of investment objectives. If an investor seeks to allocate a portion of a portfolio to broad equity or bond market indices, to domestic or global markets, in specific sectors or commodities, to follow growth or value-style strategies, or to one or more of the large-cap, small-cap, or mid-cap size segments of the market, there is likely an exchange-traded security available to meet that need.
Both the number of ETF offerings and size of the individual ETF issues have grown over the last decade. Exhibit 1 shows the outstanding shares of a sample of twelve ETF securities over 1998 to 2005, selected to include the most well-known ETFs but also to provide a snapshot of different sponsors or trustees (State Street Global Advisors for SPDRs, Barclays Global Investors for iSHARES, and Vanguard for VIPERs). At year-end 2005, Diamonds (DIA), which tracks the Dow Jones Industrial Average and is the subject of the body of this paper, had seven times as many shares outstanding as compared to 1999. The S & P 500 index-tracking ETF, SPiDeRs, had 17 times more shares than it did in 1998. QQQQ (split-adjusted) shares have grown from 121,400,000 to 501,950,000 since they were introduced in 1999. The iSHARES Russell 2000 security had 117,350,000 shares outstanding at the end of 2005, a rise from 4,500,000 when it was debuted in 2000. Other ETFs in Exhibit 1 reveal similar increases.
Exhibit 2 depicts aggregated average daily trading volume by year for this sample of twelve ETFs, and affirms the growth in popularity of these investment vehicles. Average daily volume in SPY rose by an annual average of 35.5% between 1998 and 2005. Diamonds volume increased by 40.8% on average per year over that time period, while trading volume in the QQQQ climbed by 37% on an average annual basis. Daily volume in the Energy SPDR increased by an average of 109.2% per year from 1999 to 2005, and by 72% annually in the streetTRACKS Wilshire REIT from 2001 to 2005. This is not so surprising given the performance of energy sector and REIT stocks over the recent past, and these statistics reveal that investors turned in no small way to the ETFs to participate in those market cycles.
Introduced in 1998, Diamonds is the ETF designed to track the portfolio of stocks in the Dow Jones Industrial Average (DJIA) index of 30 large, industry representative, long track-record companies. The individual stocks in the DJIA index are all subject to their own buying and selling pressures throughout the day, as company or macroeconomic news is digested by the markets and capitalized into the prices of these securities. Diamonds, 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 is expected to have an on-average effect on "the market." The most efficient way for the average investor to trade on this news, then, is to buy or sell "the market," as captured by the index upon which the ETF is based. Conversely, news might be released about individual companies in the index that attracts investor attention to these individual stocks, more so than to the portfolio as a whole, for example during earnings reporting periods. With these potentially disparate forces at work, it is natural for students to wonder what prevents a share of an independent ETF security from becoming too disconnected from the underlying portfolio of stocks to which it is tied so that it consistently fulfills its objective of tracking the underlying index. This article is an educational tool which answers that question within the context of a pedagogical treatment of arbitrage. In it, I describe the no-arbitrage pressures that enforce a relationship between an ETF and its underlying index portfolio, and demonstrate the trading strategies that could be implemented to exploit situations where the disconnect between the ETF and its index becomes too large. While this case uses the Diamonds ETF, a similar framework can be applied to other ETFs as well. I use Diamonds here because the underlying index includes only 30 stocks, and it is much easier to envision simultaneous buy or sell trades of 30 stocks than of the 500 stocks in the S & P 500 or the 100 stocks which constitute the index underlying QQQQ. In addition, as a price-weighted index, the DJIA is calculated in a way that is distinct. While most other indices are market-value-weighted, the DJIA achieves its broad market representativeness by effectively holding one share of each of the 30 companies in the index. Thus, implementing the arbitrage strategy with the DJIA stocks and Diamonds calls for a straightforward rule of starting with one share of each company and then scaling it up to the limit of available investable funds.
Since certain asset-pricing relationships in finance are based on a "no arbitrage" principle, and since students may receive employment upon graduation with hedge funds or trading desks that concentrate on the practice of arbitrage strategies, a fundamental understanding of the concept is desirable and serves both theoretical and practical objectives. A familiarity with ETF securities and their differences from index mutual funds that is fostered by the exercise is vitally important for students, given the expanding role of exchange-traded and index-linked products in the investment management and financial advisory industries. Indeed, if it is true that the innovation of ETFs "enriches the arbitrage opportunities surrounding an index" (Gastineau, 2004), and that financial innovations improve market efficiency (Ang and Cheng, 2005), then a tool such as this offers an important contribution in helping students understand how financial markets work. This article also adds to a sparse pedagogical literature on arbitrage, which includes Knoll (2005), who explains how applications of the no-arbitrage put-call parity relationship can be used to exploit differential tax treatments of economically equivalent financial positions; Wei (1997), who details triangular arbitrage; and articles on in-class simulations by Dubil (2004), Marshall (2004), and Alonzi, Lange, and Simkins (2000).
The paper begins by defining arbitrage and describing the relationship between Diamonds and the DJIA index. Following this, the arbitrage scenarios are established and strategies for exploiting them are detailed. The Conclusion discusses the impact of commissions, along with some pedagogical observations.
Arbitrage and the ETFs
Strictly speaking, an arbitrage is a mispricing between two (or more) assets or bundles of assets that 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. The strategy calls for short selling the overpriced asset and using the proceeds to buy the underpriced asset. The initial portfolio is virtually costless to the investor, with the expectation that when prices return to their proper levels and the well-defined pricing relationship is restored, the portfolio will be unwound, leaving a net cash amount as the investor's profit. Implemented on a large scale, the net selling pressure of the short positions causes the price(s) of the overpriced asset(s) to decline--so that the short position gains; and the net buying pressure forces the price(s) of underpriced asset(s) to rise--so that the long position gains. These gains are the product of a portfolio that costs virtually nothing ($0) to construct.
The key to isolating an arbitrage is the existence of a well-defined pricing relationship. In the finance textbooks, spot-futures parity and put-call parity are two such relationships whose violation can be shown to admit various profit opportunities. In that spirit, note this excerpt from the Diamonds prospectus: "The 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." Exhibit 3 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% 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, reflecting perhaps a learning curve among the financial markets.
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. Empirically, Exhibit 3 shows that this ratio has not deviated too significantly from its 100:1 objective over the eight years since the ETF security began trading. When it does, astute investors can swoop in, capture their profits, and force prices back in line to the 100:1 ratio. 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 large investors, specifically "authorized participants," to take long positions in the underpriced security and short positions in the overpriced security, and unwind those positions by transacting with the ETF-issuing trust (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 their positions with offsetting sell and buy-to-cover transactions in the secondary market. I focus on this one both because the pedagogical opportunities to draw out trading and financial markets dynamics are greater, and since students are better able to relate to the individual perspective.
Example 1: RATIO LESS THAN 100:1
The examples use market data from March 20, 2006, when the closing price of Diamonds was $112.69 per share and the closing level of the DJIA was 11274.53, calculated by dividing the $1408.54 sum of the prices of the 30 component stocks by the index divisor, then at 0.12493117. This highlights other pedagogical benefits of employing the DJIA index and ETF in this demonstration: it can be wrapped around a technical exposition of index calculations and the different index weighting schemes, followed by a broader discussion of the implications these schemes have for interpreting index moves. In fact, for the student to completely understand the steps to follow, a discussion of the unique DJIA calculation is probably warranted. Note that, on this day, the DJIA:DIA ratio was 100.04907:1.
Suppose, however, that this ratio was observed to be 96.0515x:1, significantly below the lower end of the range in which roughly 95% of the actual daily values have historically fallen. This would occur if the DJIA was at its proper level but the price of Diamonds was $117.38 per share instead of $112.69. Or, it would be observed if Diamonds were properly priced at $112.69 but the DJIA was too low at 10,824.05. Of course, it could also happen with combinations of DJIA values above 10,824.05 and Diamonds prices below $117.38. The critical insight for students in their understanding of arbitrage is that it does not matter which of these mispricings has caused the ratio to fall below 100:1. With the ratio on the low side of 100:1, the ETF is overvalued relative to the index, or the index is undervalued relative to the ETF, and so a knowledgeable investor possessing adequate capital would short sell Diamonds and buy the 30 stocks in the DJIA index in order to exploit the mispricing.
For purposes of clarity and to mimic the same building block approach as one might follow in the classroom, I discuss each of these scenarios in turn. First, I posit that the index is correct and only the ETF is mispriced. Second, I vary the index while positing that the ETF is priced fairly. Finally, I outline the most realistic example, in which both assets are presumed to be mispriced from their correct levels. Exhibit 4 dissects each of these scenarios.
Scenario 1: Diamonds are Overvalued
The first column of Exhibit 4 details the case where the DJIA index is presumed correct (held constant) but the ETF is overpriced at $117.38 when it should be $112.69. Since the DJIA index effectively includes one share of each stock, and the cost of one share of each stock is the sum of the 30 stock prices, the investor needs $1,408.54 to execute the purchases called for by the arbitrage strategy. To raise these funds, the investor short-sells 12 shares of the ETF to generate exactly $1408.54 at the posited price of $117.38:
Short sell Diamonds 12 shares*$117.38 = +$1408.56 Buy 1 share of each DJIA stock -$1408.54 $0.02
As expected, the net cost to the investor to set up this arbitrage portfolio is virtually $0, ignoring commissions for the moment. As the selling activity exerts downward pressure on the ETF price and the ratio returns to its pegged value of 100:1, an investor unwinds the arbitrage portfolio by buying 12 shares of the ETF to cover that short position, at a cost of $1,352.28. The single shares of each of the 30 companies in the index are sold for the same $1,408.54 that it cost to purchase them, since the index is unchanged. This produces a cash balance of $1,408.54 - $1,352.28 = $56.26 with no positions in any securities nor liabilities remaining. Adding the initial two cents surplus from establishing the arbitrage portfolio leaves a net arbitrage profit of $56.28, which is exactly the difference between the posited price of Diamonds ($117.38) and its correct price ($112.69), scaled by the 12 shares.
Scenario 2: DJIA Index is "Too Low"
The second column of Exhibit 4 accompanies the discussion of a scenario where the ETF is priced properly (held constant) but the index is too low--at 10824.05--because some of the component stocks are mispriced. If the DJIA is 10824.05, then the sum of the prices of its 30 component companies is $1352.26 (=10824.05*0.12493117). The arbitrage strategy to short sell the ETF and buy one share of each component can be executed once again at virtually no cost:
Short sell Diamonds 12 shares*$112.69 = +$1352.28 Buy 1 share of each DJIA stock -$1352.26 $0.02
In this case, the buying activity of investors who implement this strategy forces the prices of the mispriced components back to their proper levels, thereby moving the index back to its fair value of 11,274.53 and the DJIA:ETF ratio back near 100:1. At that point, it can be shown how the investor unwinds the arbitrage portfolio by selling the shares of the 30 component stocks for a total of $1,408.54 and uses those funds to cover the short position by buying back 12 ETF shares for $1,352.28. The difference, $1,408.54 - $1,352.28, is the $56.26 arbitrage profit, as it was in Scenario 1.
Scenario 3: Diamonds are Overvalued and DJIA is Low
Columns 3 and 4 of Exhibit 4 outline two cases where the price of the ETF is above its correct $112.69 value and the DJIA index is below the 11,274.53 level where it should be, because some of its component prices are too low. In both cases, the ratio of DJIA level to DIA price is 96.0515x:1, as it was in Scenarios 1 and 2.1 present two scenarios to reinforce the idea that the arbitrage strategy and payoff is the same regardless of what specific prices and values the ETF and index take so long as the ratio between them is sufficiently below 100:1.
At $113.51, short-selling twelve ETF shares produces $1,362.12; with the DJIA at 10,902.81, the sum of the prices of its 30 component companies is $1362.10 (=10902.81*0.12493117). The cash flows of the arbitrage portfolio net to virtually $0, as expected:
Short sell Diamonds 12 shares*$113.51 = +$1362.12 Buy 1 share of each DJIA stock -$1362.10 $0.02
Alternatively, with Diamonds priced at $115.57 and the DJIA at 11,100.68 (sum of 30 prices is $1,386.82), the arbitrage strategy would be:
Short sell Diamonds 12 shares*$115.57 = +$1386.84 Buy 1 share of each DJIA stock -$1386.82 $0.02
These two cases are the most realistic of those presented since they incorporate the financial market's intuition--which can be emphasized in the classroom--that 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, or nearly happens, and the ratio settles around 100:1, either portfolio can be unwound to lock in the same $56.26 arbitrage profit that was seen earlier:
ETF was $113.51 ETF was $115.57 Sell shares of +1408.54 +1408.54 component stocks Buy DIA to cover -12*112.69 = -1352.28 -12*112.69 =-1352.28 short position Arbitrage Profit $56.26 $56.26 Net $56.28 $56.28
Example 2: Ratio Greater Than 100:1
Exhibit 5 contains four scenarios where the index: ETF ratio is 104.0662:1, well above both the 100:1 peg relationship and the maximum of the daily historical ratios observed since 1998. Scenario 1 has the index held fixed at its correct level with Diamonds undervalued, while Scenario 2 assumes that the DIA exchange-traded fund is priced properly and the index is higher than its correct value. Scenarios 3 and 4, as before, capture more realistic cases, where both assets are mispriced, while exhibiting the 104.0662:1 ratio. All four represent what can be described as situations where the ETF is undervalued relative to the index, or, alternatively, the stocks of the index are net overvalued relative to the ETF. 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 ETF shares.
Scenario 1: Diamonds are Undervalued
In the first column of Exhibit 5, it is assumed that the index is at its correct level (held constant), but the ETF is undervalued at $108.34 when it should be $112.69. With the index at 11,274.53, the total prices of single shares of each stock in the index is $1,408.54, which the investor receives from the short sales. These proceeds finance the purchase of exactly 13 shares of the ETF, resulting in a portfolio which produces surplus funds of 12 cents (again ignoring commissions, for now):
Short sell 1 share of each DJIA stock +$1408.54 Buy Diamonds shares 13 shares*$108.34 = -$1408.42 $0.12
As multiple investors identify the mispricing and implement this strategy, buying pressure on the ETF moves its price back towards its fair value of $112.69, resettling the ratio between the index and the ETF to 100:1. Once this condition is reached, investors unwind the arbitrage portfolio to lock in the gain of the price rise in the ETF, by selling the Diamonds shares and using that money to buy one share of each stock in the index to cover those short positions. The sale of the ETF produces $1,464.97; the purchase of the stocks in the index costs $1,408.54, so the arbitrage investor retains $56.43 with no positions nor liabilities remaining plus the original $0.12 surplus, for a total profit of $56.55. As Exhibit 5 shows, this is exactly the difference between the correct price of the ETF and its assumed lower price in this example, scaled by the 13 shares.
Scenario 2: DJIA Index is "Too High"
The second column of Exhibit 5 details the case where the ETF is priced properly (held constant) but the index is too high--at 11,727.22--because some of the prices of the component stocks are elevated. At that level, the sum of the prices of the 30 component company stocks is $1,465.09 (=11,727.22* 0.12493117). The arbitrage portfolio to short sell one share each of the stocks in the index and use those proceeds to purchase 13 shares of the ETF is established while producing a cash inflow of $0.12:
Short sell 1 share of each DJIA stock +$1465.09 Buy Diamonds shares 13 shares*$112.69 = -$1464.97 $0.12
Investors who implement this strategy force the prices of the mispriced components back down to their proper levels, thereby moving the index back to its fair value of 11,274.53 and resettling the DJIA:DIA ratio near 100:1. The arbitrage portfolio is unwound by selling the 13 shares of the ETF for $1,464.97 and using those funds to buy-to-cover one share each of the 30 component stocks, for a total cost of $1,408.54. The difference, $1,464.97 - 1,408.54, is the $56.43 arbitrage profit (total $56.55), as it was in Scenario 1.
Scenario 3: Diamonds are Undervalued and DJIA is High
Columns 3 and 4 of Exhibit 5 present two cases where the ratio of the DJIA to the ETF price is 104.0662:1, caused by the ETF's price being below its correct $112.69 value and the DJIA index being above the 11,274.53 level where it should be, because some of its component prices are too high.
At 11,396.29, the sum of the 30 prices of the index component stocks is $1,423.75 (=11,396.29* 0.12493117). This amount is received from the short sales and used to buy 13 shares of the ETF, priced at $109.51 per share, for a total cost of $1,423.63. The self-financing arbitrage portfolio generates cash inflows of $0.12:
Short sell 1 share of each DJIA stock +$1423.75 Buy Diamonds 13 shares* $109.51 = -$1423.63 $0.12
In an alternative situation where Diamonds are priced at $111.01 per share and the DJIA is 11,552.39 (sum of 30 prices is $1,443.25), the arbitrage strategy would be:
Short sell 1 share of each DJIA stock +$1443.25 Buy Diamonds 13 shares* $111.01= -$1443.13 $0.12
These cases are the most realistic of those presented in this section since they incorporate the financial market's intuition--which can be emphasized in the classroom--that selling pressure on the 30 component stocks forces their prices (and thus the index) down, while buying pressure on Diamonds forces its price up, until both the ETF and DJIA reach levels where arbitrage exploitation is no longer profitable. Once that happens, or nearly happens, and the ratio settles around 100:1, either portfolio can be unwound to lock in the identical $56.55 total arbitrage profit that was modeled earlier:
ETF was $109.51 ETF was $111.01 Sell shares of Diamonds +13*112.69 = 1464.97 +13*112.69 = 1464.97 Buy 1 share of each -1408.54 -1408.54 component stock Arbitrage Profit $56.43 $56.43 Net $56.55 $56.55
Discussion and Conclusion
The fundamental lesson from the first case is that it does not matter how the DJIA:DIA (index:ETF) ratio came to be low at 96.0515x:1. Recognition that this signals a relative ETF overvaluation (or index component net undervaluation) calls for a simple strategy of short selling the ETF shares and using the proceeds to buy one share of each index component stock, which produces a maximum arbitrage profit of $56.28. Similarly, in the second case, where the ratio was 104.0662:1, how the ratio became inflated is irrelevant. An investor merely needs to recognize this as a signal of a relative ETF undervaluation (or index component net overvaluation) and act accordingly. The strategy of short selling shares of the index component stocks and buying the ETF produces a maximum arbitrage profit of $56.55. 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 if the portfolio is unwound once the ratio returns to its 100:1 peg value. If an investor unwinds the arbitrage portfolio early, prior to the ratio re-settling at 100:1, smaller arbitrage profits are reaped.
The effect of individual investor commissions on these strategies is not difficult to explain to students, especially with the familiarity they may have with on-line brokers who advertise frequently. 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 additional for the ETF--and 31 trades to unravel it. At a $10 per trade commission rate, the total cost is $620. Thus, to cover these costs and realize profits, an investor needs to scale up the 96.0515x:1 position by at least 12 times and the 104.0662:1 position by at least 11 times. In the first case, that calls for short selling 144 shares of the ETF and buying 12 shares of each component stock to produce an arbitrage profit of 12*$56.28 = $675.36 - $620 = $55.36. In the second, short sales of 11 shares of each component stock finances a purchase of 143 shares of Diamonds to generate an eventual arbitrage profit of 11*$56.55 = $622.05 - $620 = $2.05. Of course, with risk-free profits on the menu, the investor is motivated to scale it up to as much as he/she can afford!
This last statement can be used to segue the class into a discussion of the practicality of these opportunities for the average investor, and the structural differences between what professional and individual investors can do. Despite the illustration that the portfolios in the preceding examples actually generate small cash inflows at their formation, in addition to the back-end profit, I inform students that, generally speaking, individuals are required to keep short sale proceeds in their brokerage accounts and would not be able to use them to cover the purchases required by these strategies. Not only do they have to pay for the buy trades (even though the securities are sold within the T+3 settlement period), but depending on the size and status of their accounts, any short positions require a margin deposit as well. Further, with the typical on-line 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, as other investors quickly squeeze out the arbitrage profits and push prices back to proper levels. This raises the topic of programmed trades to elaborate on how professionals are usually the ones who take advantage of these shortlived opportunities because of available technology, full-time attention to the financial markets, collateral at their disposal, and fewer restrictions on the use of short sale proceeds.
In summary, this analysis of the relationship between an ETF and its underlying index portfolio of stocks can be used as the centerpiece to drive coverage in an investments class of a number of financial assets, financial markets dynamics and structural features, and finance concepts. It introduces ETFs to students, which can be contrasted to index mutual funds while supporting discussions about indices and their different weighting schemes. Stock trades, short positions, margin accounts, even limit and stop orders can be defined and explored. Structural issues such as short sale restrictions and margin posting come into play. And, of course, the important concept of asset mispricings, of the trading strategies which can be constructed to exploit them and the resulting market forces which they unleash, can be illuminated for students very graphically within this contextual, real-world example.
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John J. Neumann, The Peter J. Tobin College of Business, St. John's University
Exhibit 1: Shares Outstanding (in 000's) of selected exchange-traded products, 1998-2005.
Data is taken from CRSP Daily files and is split-adjusted.
ETF Name Ticker 1998 ETF 1999 2000 2001 S & P 500 SPDRs SPY 27,109 27,109 149,422 149,422 Diamonds Trust, Series 1 DIA 2,550 9,700 9,700 28,402 NASDAQ 100 Trust Shares QQQQ 121,400 131,350 587,000 iShares Russell 1000 Index IWB 3,700 4,700 iShares Russell 2000 Index IWM 4,500 42,600 iShares S & P SmallCap 600 IJR 1,200 12,750 Index Energy Select Sector SPDR XLE 9,100 2,100 10,150 Technology Select Sector XLK 20,000 25,700 52,250 SPDR streetTRACKS Wilshire REIT RWR 450 Vanguard Total Stock VTI 10,083 Market VIPERs Vanguard Mid Cap VIPERs VO Vanguard Small Cap VIPERs VB ETF Name 2002 2003 2004 2005 S & P 500 SPDRs 418,241 395,856 461,947 471,080 Diamonds Trust, Series 1 56,555 64,560 74,974 71,289 NASDAQ 100 Trust Shares 701,000 634,300 555,750 501,950 iShares Russell 1000 Index 14,800 31,200 29,100 35,600 iShares Russell 2000 Index 55,800 81,400 117,000 117,350 iShares S & P SmallCap 600 41,250 44,400 76,650 66,700 Index Energy Select Sector SPDR 12,650 22,450 44,503 49,656 Technology Select Sector 64,250 53,801 58,503 73,653 SPDR streetTRACKS Wilshire REIT 2,250 5,550 8,553 11,955 Vanguard Total Stock 16,407 23,566 36,031 44,614 Market VIPERs Vanguard Mid Cap VIPERs 1,005 16,264 Vanguard Small Cap VIPERs 3,310 4,448
Exhibit 2: Average Daily Trading Volume of Exchange-Traded Funds (ETFs).
The graph shows the sum of the average daily trading volume of the sample of ETFs in Exhibit 1 from 1998 to 2005. Trading volume is taken from the CRSP Daily files and is adjusted for splits.
Exhibit 3: Historical Ratios between DJIA 30 Index and Diamonds market price.
Summary Statistics on the daily ratio of the level of the DJIA 30 Index to the per share price of the DIAmonds ETF (both values at close). Index values are from finance. yahoo.com and Diamonds prices are taken from CRSP.
Average [sigma] Maximum Minimum 1998-2005 99.8978 0.2204 102.4855 98.4526 2005 100.0102 0.1197 100.4722 99.6885 2004 99.8373 0.1559 100.2977 99.3565 2003 99.7889 0.1676 100.3701 99.3318 2002 99.8430 0.2176 100.4599 99.1101 2001 99.8923 0.2087 100.5997 99.4663 2000 99.9356 0.3036 102.4855 98.4526 1999 99.9591 0.2104 100.5599 99.3561 1998 99.9165 0.2438 100.7562 98.9492
Exhibit 4: Four scenarios where the ratio of the Dow Jones Industrials Index to the market price of the Diamonds ETF, DJIA:DIA, is 96.0515x:1 (the peg ratio is 100:1). At this ratio, Diamonds are overpriced relative to the index, or the stocks of the index are underpriced relative to the ETF.
Diamonds shares are DJIA index is too low at overpriced at $117.38 10824.05 Portfolio Short sell 12 DIA shares and use these funds to buy 1 share each of the 30 component stocks Market Change Diamonds price DJIA returns to 11274.53, corrects by falling to bringing the DJIA:DIA ratio $112.69 from $117.38 back to 100:1. Sum of 30 prices is now $1408.54 Unwinding Sell the index component shares for a total of $1408.54. Buy-to-cover 12 shares of Diamonds at $112.69; cost: $1352.28. Arbitrage Profit 1408.54 - 1352.28 = $56.26 (net $56.28, plus the initial $.02 surplus) Expectation (117.38 - 112.69)*12 = (11274.53 - 10824.05)* $56.28 12493117 = $56.28 Diamonds too high at $113.51 and DJIA too Diamonds too high at $115.57 low at 10902.81 and DJIA too low at 11100.68 Portfolio Short sell 12 DIA shares and use these funds to buy 1 share each of the 30 component stocks Market Change DJIA rises to 11274.53. DJIA rises to 11274.53. Sum Sum of 30 prices is of 30 prices is $1408.54. $1408.54. Diamonds falls to $112.69 Diamonds falls to from $115.57 $112.69 from $113.51 Unwinding Sell the index component shares for a total of $1408.54. Buy-to-cover 12 shares of Diamonds at $112.69; cost: $1352.28. Arbitrage Profit 1408.54 - 1352.28 = $56.26 (net $56.28, plus the initial $.02 surplus) Expectation (11274.53 - 10902.81)*. (11274.53 - 11100.68)* 12493117 + (113.51 - 12493117 + (115.57 - 112.69)* 112.69)* 12 = $56.28 12 = $56.28
Exhibit 5: Four scenarios where the ratio of the Dow Jones Industrials Index to the market price of the Diamonds ETF, DJIA:DIA, is 104.0662:1 (the peg ratio is 100:1). At this ratio, Diamonds are underpriced relative to the index, or the stocks of the index are overpriced relative to the ETF.
Diamonds are underpriced at DJIA index is too high at $108.34 11727.22 Portfolio Short sell 1 share each of the 30 index component stocks, and buy 13 shares of Diamonds Market Change DIA price corrects by DJIA returns to 11274.53 rising to $112.69 from from 11727.22 and brings $108.34 DJIA:DIA ratio back to 100:1. Sum of 30 prices becomes $1408.54 Unwinding Sell the Diamonds shares for $112.69*13=$1464.97; use these funds to buy the index components for $1408.54 to cover the short. Arbitrage Profit $1464.97 - $1408.54 = $56.43 (net $56.55 with the initial $.12 surplus) Expectation (112.69 - 108.34)*13 = (11727.22 - 11274.53) $56.55 *.12493117 = $56.55 Diamonds too low at $109.51 and DJIA too Diamonds too low at $111.01 high at 11396.29 and DJIA too high at 11552.39 Portfolio Short sell 1 share each of the 30 index component stocks, and buy 13 shares of Diamonds Market Change DJIA falls to 11274.53, DJIA falls to 11274.53. Sum where the sum of 30 of 30 prices returns to component prices is $1408.54. Price of Diamonds $1408.54. shares rises from $111.01 to Diamonds share price $112.69 rises from $109.51 to $112.69 Unwinding Sell the Diamonds shares for $112.69*13=$1464.97; use these funds to buy the index components for $1408.54 to cover the short. Arbitrage Profit $1464.97 - $1408.54 = $56.43 (net $56.55 with the initial $.12 surplus) Expectation (11396.29 - 11274.53) (11552.39 - 11274.53)*. *.12493117 + (112.69 - 12493117 + (112.69 - 109.51)*13 = $56.55 111.01)*13 = $56.55
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|Author:||Neumann, John J.|
|Publication:||Review of Business|
|Date:||Sep 22, 2007|
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