The Law of One Price
Prices and Economic Incentives: Comparing Apples and Assets
We expect the same thing to sell for the same price. This is the Law of One Price. Why should this be true? Common sense dictates that if you could buy an apple for 25¢ and sell it for 50¢ across the street, everyone would want to buy apples where they are cheap and sell them where they are priced higher. Yet this price disparity will not last: As people take advantage, prices will adjust until apples of the same quality sell for the same price on both sides of the street. Furthermore, a basket of apples must be priced in light of the total cost of buying the fruit individually. Otherwise, people will make up their own baskets and sell them to take advantage of any mispricing.6 The arbitrage relationship between individual asset prices and overall portfolio values is explored later in this chapter.
The structure imposed on prices by economic incentives is the same in financial markets as in the apple market. Yet a different approach must be taken to determine what constitutes the "same thing" in financial markets. For example, securities are the "same" if they produce the same outcomes, which considers both their expected returns and risk. They should consequently sell for the same prices. Similarly, equivalent combinations of assets providing the same outcomes should sell for the same price. Thus, the criteria for equivalence among financial securities involve the comparability of expected returns and risk. If the same thing sells for different prices, the Law of One Price is violated, and the price disparity will be exploited through arbitrage. Thus, the Law of One Price imposes structure on asset prices through the discipline of the profit motive. Similarly, if stocks with the same risk have different expected returns, the Law of One Expected Return is violated.
Economic Foundations of the Law of One Price
The Law of One Price holds under reasonable assumptions concerning what investors like and dislike and how they behave in light of their preferences and constraints. Specifically, our analysis assumes the following:
More wealth is preferred to less. Wealth enhancement is a more comprehensive criterion than return or profit maximization. Wealth considers not only potential returns and profits but also constraints, such as risk.7
Investor choices should reflect the dominance of one investment over another. Given two alternative investments, investors prefer the one that performs at least as well as the other in all envisioned future outcomes and better in at least one potential future outcome.
An investment that generates the same return (outcome) in all envisioned potential future situations is riskless and therefore should earn the risk-free rate. Lack of variability in outcomes implies no risk. Thus, strategies that produce riskless returns but exceed the risk-free return on a common benchmark, such as U.S. Treasury bills, must involve mispriced investments.
Economic incentives ensure that two investments offering equivalent future outcomes should, and ultimately will, have equivalent prices (returns).
The proceeds of a short sale are available to the investor. This assumption is easiest to accept for large, institutional investors or traders who may be considered price-setters on the margin. Even if this assumption seems a bit fragile, market prices generally behave as if it holds well enough.8 The nature and significance of short sales are discussed more later in this chapter.
Systematic, persistent deviations from the Law of One Price should not occur in efficient financial markets.9 Deviations should be relatively rare or so small as to not be worth the transactions costs involved in exploiting them. Indeed, when arbitrage opportunities do appear, those traders with the lowest transactions costs are likely to be the only ones who can profitably exploit them. The Law of One Price is largelybut not completelysynonymous with equilibrium, which balances the forces of supply and demand.