## Arbitrage Opportunities And Profits

An arbitrage opportunity arises when an investor can construct a zero investment portfolio that will yield a sure profit. To construct a zero investment portfolio one has to be able to sell short at least one asset and use the proceeds to purchase (go long on) one or more assets. Borrowing may be viewed as a short position in the risk-free asset. Clearly, any investor would like to take as large a position as possible in an arbitrage portfolio.

An obvious case of an arbitrage opportunity arises when the law of one price is violated. When an asset is trading at different prices in two markets (and the price differential exceeds transaction costs), a simultaneous trade in the two markets can produce a sure profit (the net price differential) without any investment. One simply sells short the asset in the high-priced market and buys it in the low-priced market. The net proceeds are positive, and there is no risk because the long and short positions offset each other.

In modern markets with electronic communications and instantaneous execution, arbitrage opportunities have become rare but not extinct. The same technology that enables the market to absorb new information quickly also enables fast operators to make large profits by trading huge volumes the instant an arbitrage opportunity appears. This is the essence of index arbitrage, to be discussed in Part VI and Chapter 21.

From the simple case of a violation of the law of one price, let us proceed to a less obvious (yet just as profitable) arbitrage opportunity. Imagine that four stocks are traded in an economy with only four distinct, possible scenarios. The rates of return of the four stocks for each inflation-interest rate scenario appear in Table 11.1. The current prices of the stocks and rate of return statistics are shown in Table 11.2.

Eyeballing the rate of return data, it is not obvious that an arbitrage opportunity exists. The expected returns, standard deviations, and correlations do not reveal any particular abnormality.

Consider, however, an equally weighted portfolio of the first three stocks (Apex, Bull, and Crush), and contrast its possible future rates of return with those of the fourth stock, Dreck. These returns are derived from Table 11.1 and summarized in Table 11.3, which

High Real Interest Rates |
Low Real Interest Rates | |||

High Inflation |
Low Inflation |
High Inflation |
Low Inflation | |

Probability: |
.25 |
.25 |
.25 |
.25 |

Stock | ||||

Apex (<4) |
-20 |
20 |
40 |
60 |

Bull (6) |
0 |
70 |
30 |
-20 |

Crush (Q |
90 |
-20 |
-10 |
70 |

Dreck (D) |
15 |
23 |
15 |
36 |

Table 11.2 Rate of Return Statistics

Current |
Return |
Standard |
Correlation Matrix | ||||

Stock |
Price |
Deviation (%) |
A |
a |
C |
D | |

A |
$10 |
25 |
29.58% |
1.00 |
-0.15 |
-0.29 |
0.68 |

B |
10 |
20 |
33.91 |
-0.15 |
1.00 |
-0.87 |
-0.38 |

C |
10 |
32.5 |
48.15 |
-0.29 |
-0.87 |
1.00 |
0.22 |

D |
10 |
22.25 |
8.58 |
0.68 |
-0.38 |
0.22 |
1.00 |

High Real Interest Rates |
Low Real Interest Rates | |||

High Inflation |
Low Inflation |
High Inflation |
Low Inflation | |

Equally weighted portfolio | ||||

(4, B, arid C) |
23.33 |
23.33 |
20.00 |
36.67 |

Dreck |
15.00 |
23.00 |
15.00 |
36.00 |

reveals that the equally weighted portfolio will outperform Dreck in all scenarios. The rate of return statistics of the two alternatives are

Mean |
Standard Deviation |
Correlation | |

Three-stock portfolio |
25.83 |
6.40 |
.94 |

Dreck |
22.25 |
8.58 |

Because the two investments are not perfectly correlated, there is no violation of the law of one price. Nevertheless, the equally weighted portfolio will fare better under any circumstances; thus any investor, no matter how risk averse, can take advantage of this perfect dominance. Investors will take a short position in Dreck and use the proceeds to purchase the equally weighted portfolio.1 Let us see how it would work.

Suppose we sell short 300,000 shares of Dreck and use the $3 million proceeds to buy 100,000 shares each of Apex, Bull, and Crush. The dollar profits in each of the four scenarios will be as follows:

Stock |
Dollar Investment |
High Real Interest Rates |
Low Real Interest Rates | ||

High Inflation |
Low Inflation |
High Inflation |
Low Inflation | ||

Apex |
$ 1,000,000 |
$-200,000 |
$ 200,000 |
$ 400,000 |
$ 600,000 |

Bull |
1,000,000 |
0 |
700,000 |
300,000 |
-200,000 |

Crush |
1,000,000 |
900,000 |
-200,000 |
-100,000 |
700,000 |

Dreck |
-3,000,000 |
—450,000 |
-690,000 |
—450,000 |
-1,080,000 |

Portfolio |
0 |
$ 250,000 |
$ 10,000 |
$ 150,000 |
$ 20,000 |

The first column verifies that the net investment is zero. Yet our portfolio yields a positive profit in any scenario. This is a money machine. Investors will want to take an infinite

### 1 Short selling is discussed in Chapter 3.

position in such a portfolio because larger positions entail no risk of losses, yet yield evergrowing profits. In principle, even a single investor would take such large positions that the market would react to the buying and selling pressure: The price of Dreck has to come down and/or the prices of Apex, Bull, and Crush have to go up. The arbitrage opportunity will then be eliminated.

CONCEPT CHECK ^ QUESTION 1

Suppose that Dreck's price starts falling without any change in its per-share dollar payoffs. How far must the price fall before arbitrage between Dreck and the equally weighted portfolio is no longer possible? (Hint: What happens to the amount of the equally weighted portfolio that can be purchased with the proceeds of the short sale as Dreck's price falls?)

The idea that market prices will move to rule out arbitrage opportunities is perhaps the most fundamental concept in capital market theory. Violation of this restriction would indicate the grossest form of market irrationality.

The critical property of a risk-free arbitrage portfolio is that any investor, regardless of risk aversion or wealth, will want to take an infinite position in it. Because those large positions will force prices up or down until the opportunity vanishes, we can derive restrictions on security prices that satisfy a "no-arbitrage" condition, that is, prices for which no arbitrage opportunities are left in the marketplace.

There is an important difference between arbitrage and risk-return dominance arguments in support of equilibrium price relationships. A dominance argument holds that when an equilibrium price relationship is violated, many investors will make portfolio changes. Individual investors will make limited changes, though, depending on their degree of risk aversion. Aggregation of these limited portfolio changes is required to create a large volume of buying and selling, which in turn restores equilibrium prices. By contrast, when arbitrage opportunities exist each investor wants to take as large a position as possible; hence it will not take many investors to bring about the price pressures necessary to restore equilibrium. Therefore, implications for prices derived from no-arbitrage arguments are stronger than implications derived from a risk-return dominance argument.

The CAPM is an example of a dominance argument, implying that all investors hold mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their portfolios toward the underpriced and away from the overpriced securities. Pressure on equilibrium prices results from many investors shifting their portfolios, each by a relatively small dollar amount. The assumption that a large number of investors are mean-variance sensitive is critical; in contrast, the implication of a no-arbitrage condition is that a few investors who identify an arbitrage opportunity will mobilize large dollar amounts and restore equilibrium.

Practitioners often use the terms "arbitrage" and "arbitrageurs" more loosely than our strict definition. "Arbitrageur" often refers to a professional searching for mispriced securities in specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free) arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it from pure arbitrage.

To leap ahead, in Part VI we will discuss "derivative" securities such as futures and options, whose market values are completely determined by prices of other securities. For example, the value of a call option on a stock is determined by the price of the stock. For such securities, strict arbitrage is a practical possibility, and the condition of no-arbitrage leads to exact pricing. In the case of stocks and other "primitive" securities whose values are not determined strictly by another asset or bundle of assets, no-arbitrage conditions must be obtained by appealing to diversification arguments.

324 PART III Equilibrium in Capital Markets

11.2

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