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Basic DescriptionOn a basic level, nothing of the same value should ever be sold for two different prices, and thus there should be no arbitrage. However, this is not a perfect financial world in which we live. There are discrepancies that allow arbitrage to persist. For instance, arbitrage can be achieved through differing exchange rates in multiple countries or through differences in stock prices in different markets. These are two examples of one of the two types of arbitrage, pure arbitrage. Pure arbitrage is rare but highly valued in today’s financial world. On the other hand, relative value arbitrage is much more common and is the basis for many hedge funds and other trading businesses.
Pure arbitrage is defined as, “Generating riskless profit today by statically or dynamically matching current and future obligations to exactly offset each other, inclusive of incurring known financial costs.”  The example of differing exchange rates in different countries is a good example of pure arbitrage. If the exchange rate in the US is 10 dollars for 10 euros and the exchange rate in France is 10 euros for 700 rupees and the exchange rate in India is 700 rupees for 9 dollars, one can make a profit by converting dollars to rupees to euros back to dollars.
An example of pure arbitrage can be found in the stock markets. Any discrepancy between a stock price in two different markets is an opportunity for arbitrage. If a stock is selling higher on the New York Exchange than the London Exchange, one could purchase the stock in the London market for a lower price and sell it in the New York Exchange for a profit. Other examples of pure arbitrage are very rare in the current market. This can be attributed to the high speed nature of financial transactions and interactions with the Law of One Price, which states that the same item cannot sell for two different prices at the same time.  When such a discrepancy occurs, it is instantly taken advantage of to turn a profit. These actions immediately start to enforce the Law of One Price. Therefore, as transactions take advantage of the discrepancy it begins to stabilize, effectively eliminating the opportunity for pure arbitrage. In the example of differing exchange rates, the Law of One Price would cause the values to begin stabilize as more between currencies.
Relative value arbitrage is defined as, “Generating profit today by statically or dynamically matching current and future obligations to nearly offset each other, net of incurring closely estimable financing costs.”  Note that while pure arbitrage is riskless profit, relative value arbitrage regulates risk to estimable levels. The idea of relative value arbitrage rests on the substitution of risk. In practice, one must start with a broadly defined hedge with certain known risks. To control these risks, one must find and substitute a comparable risk that is, preferably, exactly opposite for both risks. Take US bonds, for example. Suppose one purchases $50 million in 30 year US bonds while at the same time selling short $51 million of a 26 year bond. The risks in this instance are the interest rates. The investor does not know how much profit he or she will be making off either of the bonds. However, the investor does know that the pair of investments has related interest rates. As the rate of one goes down, the other goes up and vice versa. Therefore, as one investment loses money, the other will gain it and none of the investment will be lost. Risk is still present, as the interest rates are not concretely linked and could differ slightly. However, the differences will be small and still allow for a profit.
Since the investor will have some operating costs, the profits made through the investments must be greater that the operating costs. Through this elimination of primary risk, relative value arbitrage is achieved. The basis of hedge funds and also the primary tool of many large financial firms is relative value arbitrage. They follow the same basic pattern, balancing risks in stocks and bonds to maximize profit. In order to generate substantial profit, these firms rely on a large number of different investments that must be constantly reevaluated to maximize the profits yielded. Relative value arbitrage is vital in today’s financial markets.
A third, less common type of arbitrage is tax arbitrage and is used in non-profit hospitals. As nonprofits, these hospitals are exempt from federal income taxes and are allowed to issue tax-exempt debt. Using these benefits, nonprofit hospitals can achieve another type of arbitrage, tax arbitrage. Tax arbitrage is defined as, “The use of proceeds from lower-cost tax-exempt bonds to finance the purchase of higher-yield securities.”  According to the Internal Revenue and Treasury regulations, tax arbitrage is highly illegal and much regulated. Despite these facts, nonprofit hospitals can evade the law through a simple technicality. The definition of tax arbitrage states that profits from tax-exempt bonds cannot be directly used to purchase higher-yield securities. Since profits must be directly used to purchase securities, nonprofit hospitals can use profits from tax-exempt bonds as long as the profits are used indirectly.
For example, consider a hospital with plans to build a new research wing. The hospital has assets worth $20 million in two funds: $5 million in the building and upkeep fund, $5 million in real estate, and $10 million in the “Just-in-case-we-get-sued” fund. The new wing is projected to cost $5 million. The hospital could liquidate some of its assets from its building fund to finance the building project. However, they can also use tax-exempt debt. This tax-exempt debt will have a small interest rate, because the people who purchase the debt are willing to accept smaller returns on their money because the profits are tax-exempt and thus are comparable to the profits from taxed profits from higher interest rates. Now the hospital has tax-exempt debt worth $5 million and is expected to pay $1 million in interest on that debt for a total cost of $6 million for the new wing. Since the hospital didn’t liquidate $5 million for the building project, it decides to invest with funds from its “Just-in-case-we-get-sued” fund. They invest $5 million in high-yield yield securities. Since the hospital is nonprofit, all profits from this investment are tax free. The hospital’s investments gain interest in the amount of $1.2 million during the time of the building project. In the end, the hospital has paid $1 million in interest but earned $1.2 million in interest as well. The hospital’s total assets (including the new $5 million wing) are now $20.2 million: $10.2 million in the “Just-in-case-we-get-sued” fund and $10 million in the real estate fund. Tax arbitrage has been achieved!
According to the federal government, in 2002 there were 1,276 nonprofit hospitals with tax-exempt debt. Of these, approximately 248 were earning returns from tax arbitrage. If tax arbitrage became illegal, the federal government estimates it can make $504 million more per year from nonprofit hospitals investments.  Approximately 64% of tax-exempt debt issued by nonprofit hospitals is believed to be making profits under tax arbitrage. 
Nonprofit hospitals are not the only places where tax arbitrage is practiced. Tax arbitrage also exists in colleges and universities. Such institutions of higher learning also enjoy the benefits of being able to use tax-exempt debt and are exempt from federal income taxes. The process of tax arbitrage for colleges and universities is much the same as it is for nonprofit hospitals, relying on the use of tax-exempt debt and investment in high-yield securities. The main difference between institutions of higher learning and nonprofit hospitals is the magnitude of tax arbitrage. Nearly 100% of tax-exempt debt issued by colleges and universities is believed to make profits under tax arbitrage. In addition to this, colleges and universities generally have larger assets to use for tax arbitrage. The $15 billion of tax-exempt debt used by nonprofit hospitals pales in comparison to the $290 billion tax-exempt debt used by colleges and universities.  The federal government estimates that it loses $5.5 billion in income from allowing colleges and universities to purchase tax exempts bonds. 
Why It's InterestingThe concept of arbitrage can be seen all across the financial world. While pure value arbitrage doesn’t exist, relative value arbitrage is used by many different investment institutions to make profits. Tax-arbitrage is cunningly used by colleges, universities, and nonprofit hospitals to generate profits. Arbitrage is used to determine the purchase prices of options through the Black-Scholes formula. All in all, arbitrage can be found. It may be a difficult search, but it can be found. So go out and search for it! Ways for making free money with little risk are out there just waiting to be claimed!
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1. Financial Engineering and Arbitrage in the Financial Markets by Robert Dubil
2. "Arbitrage." from Five-minute Mathematics by Ehrhard Behrends
3. Nonprofit Hospitals and Tax Arbitrage by Dennis Zimmerman and Kurt Seiburt
4. Tax Arbitrage by Colleges and Universities by Kristy Piccinini
Future Directions for this Page
Arbitrage Pricing Theorem
Black-Scholes Formula Explanation
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