Game-Theoretic Foundations for Probability and Finance

16Equity Premium and CAPM

We now apply the results of Chapters 14 and 15 to four closely related topics in finance: the equity premium, the capital asset pricing model (CAPM), the Sharpe ratio, and the theoretical performance deficit. The first two topics are standard; the other two have been studied in the context of log‐optimal investment, discussed in Exercises 12.8 and 12.9. The usual treatments of these topics rely on exogenous probabilities and assumptions about investors' risk preferences, whereas the theory of this chapter uses only the game‐theoretic mathematics of the preceding chapters and relies for its application to actual markets only on the assumption that a given numeraire is efficient in our game‐theoretic sense.

Throughout this chapter, we consider a fixed market space with three positive continuous gales , , and . The mathematical results we obtain do not, of course, depend on any particular interpretation of these objects, but in the application we are discussing they are interpreted as follows:

• is a trader's bank account, perhaps a money market account, that pays ...