# Revisiting the Basics of Return and Risk in Equilibrium

17 Pages Posted: 13 Aug 1998

Date Written: March 1998

### Abstract

Consider the following two well-diversified portfolios: The first consists of 100 preferred stocks and the second of 100 regular equity stocks. Which of the two portfolios is riskier? Common sense would seem to suggest that the equity portfolio is more risky. In terms of traditional concepts it may well be that the equity portfolio has a larger "beta", but the major reason for the suggested answer rests arguably more on the idea that the distributional spread of the equity portfolio exceeds the one of preferred stocks. That is, the equity portfolio return distribution has greater upside potential as well as downside risk. Despite the intuitive appeal of this idea, a problem arises from a traditional perspective since the upside/downside concept makes no reference to how the returns' distributions relate to factors influencing the broader opportunity set (e.g., the return on the market portfolio). Hence, to capture the texture of the proposed answer in a formal analysis requires assumptions which to a significant degree eliminate equilibrium features embraced by CAPM and APT. This paper develops an equilibrium model that differs materially from CAPM/APT type of theories concerning risk and return. The focus is on excess returns, i.e., returns minus the (spot) risk-free return, and the setting specified implies the following approximation in equilibrium: A constant gamma exists such that the excess return of portfolio one has the same distribution as that of portfolio two scaled by gamma. In other words, the two distributions of excess returns are the same, except for their scales. One can think of gamma as a relative (inverse) risk index; it exceeds 1 if and only if portfolio one is less risky than portfolio two. Because this concept of risk does not require perfectly dependent returns, the model here has unique features which CAPM or APT settings obviously lack. One can describe the above equilibrium property more intuitively. Consider the possibility of combining borrowing or lending with an investment in either of the two portfolios. Suppose there exists some mix of borrowing and a long position in the preferred stock portfolio so that the implied leveraged portfolio return distribution replicates the distribution of the equity portfolio return. If this is true, then the two excess return distributions are equivalent aside from the two distributions' connecting scale factor. The notion of relative risk in this scenario makes sense intuitively since it relates to whether borrowing or lending is necessary to equate the two portfolios' distributions. The set-up is a simple one-period model with no significant restrictions on preferences, except that more-is-better-than-less. Issues surrounding the modeling of the joint distribution of returns are more delicate. In broad terms, the return specification satisfies a one-factor model. It differs from the prototype model in two respects. First, the (random) commonality factor is always positive (rather than mean zero as is usually the case). Second, the factor model has a random response-coefficient. This crucial feature implies that the returns' distributions cannot be perfectly dependent. An equilibrium will therefore derive from portfolio optimization rather than a pure no-arbitrage argument. Further, an assumption on the slope-coefficients' distribution works such that any weighted average of slope-efficients will approximately have the same type of distribution. It follows that any portfolio combined with borrowing/lending can approximately replicate any other portfolio's return distribution. One can also derive the desired equilibrium in an approximate sense.

**JEL Classification:** G12

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