ECONOMICS The Equity Premium Puzzle: A Model for Its Behavior Timo thy P. La velle In the United States, there exist independent markets for risk free government bond securities and risky equity securities. A rational/sophisticated investor will build a portfolio out of these two components in such a way as to optimize his portfolio in terms of the ratio of riskiness (or, more specifically, idiosyncratic variability) and expected return (Black and Litterman 1992). Once this equilibrium condition is met (an investor-specific optimal allocation between risky and risk free securities) the rules of arbitrage pricing should keep the return spread between these two securities at an equilibrium level (Brennan, Schwartz and Lagnado 1997). From this notion of an arbitrage pricing model, one can look at historical returns in the United States and appreciate that the aggregate return spread between the market index for equity securities and government debt instruments should be characterized by a function of expected returns, risk, and investor preference for risk. This premium has been roughly 6% over the long run (Kocherlakota 1996). Given the observed level of risk for each asset class, one can then derive the implied risk preference profile of the investing community in aggregate. This stream of analysis has been applied to what has come to be known as the equity premium puzzle. While the fact that a premium exists is not a puzzle to even the most naïve of market observers, it is the magnitude of this premium that is interesting and perplexing. No theory to date has been able to explain it fully, and thus, the focus of this article will be to demystify its enigmatic behavior using a dynamical systems approach. As stated previously, the premium return of equity securities over government debt securities is inevitable due to the greater systematic and idiosyncratic risk of equity returns in the market. The problem, however, is determining whether this premium is appropriate given the level of risk and investors’ risk aversion, or if it is, in fact, extreme. Siegel and Thaler use a utility preference model to describe the level of risk aversion implied by the long run 6% equity premium (Siegel and Thaler 1997). The following anecdote demonstrates this level of risk aversion. Given a fair bet with 50% probability of winning and exactly doubling one’s entire wealth and a 50% probability of losing and exactly halving one’s entire wealth, an individual with the implied level of risk aversion derived from the equity premium would be willing to pay 49% of his or her entire wealth to avoid the gamble (Siegel and Thaler 1997). Thus, if one has wealth of one million dollars and is given a 50% chance to either be worth two million dollars or 500 thousand dollars, that individual would rather pay 490 thousand dollars to avoid this bet. This person would rather sacrifice 490 thousand dollars with 100% certainty than have a 50%

The Equity Premium Puzzle- A Model for Its Behavior

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