An Asset Allocation Puzzle: Comment

Abstract

Should the proportion of risky assets in the risky part of an investor’s portfolio depend on the investor’s risk aversion? According to basic financial theory, in particular the mutual-fund separation theorem with a riskless asset, the answer is no. The theorem states that rational investors should divide their assets between a riskless asset and a risky mutual fund, the composition of which is the same for all investors. Risk aversion affects only the allocation between the riskless asset and the fund. However, Niko Canner et al. (1997), CMW hereafter, observed that popular investment advice does not conform to this theory. They reported the stocks, bonds, and cash allocations recommended by four advisors for conservative, moderate, and aggressive investors. As shown in Table 1, which is reproduced from CMW, the advisors recommend a bond/stock ratio that varies directly with risk aversion. For example, Fidelity recommends a bond/stock ratio of 1.50 for a “conservative” (more riskaverse) investor, a ratio of 1.00 for a “moderate” (less risk-averse) investor, and a ratio of 0.46 for an “aggressive” (still less risk-averse) investor. The inconsistency between such advice and the separation theorem is called an asset allocation puzzle by CMW. They attempted to solve the puzzle by relaxing key assumptions in the theory, but finally reached a negative conclusion: “Although we cannot rule out the possibility that popular advice is consistent with some model of rational behavior, we have so far been unable to find such a model” (p. 181). However, they suggested that consideration of intertemporal trading might help resolve the puzzle. In the present paper, we provide theoretical support for the popular advice. The two key insights are that the investor’s horizon may exceed the maturity of the cash asset and that the investor rebalances the portfolio as time passes. If the investor’s horizon exceeds the maturity of cash, which might be a money-market security with maturity of one to six months, then cash is not the riskless asset as is commonly assumed in the basic theory. In a theory allowing portfolio rebalancing, as opposed to a buy-and-hold framework, it is not unreasonable to assume that the investor can synthesize a riskless asset (a zero-coupon bond maturing at the horizon) using a bond fund and cash. Then bonds will be both in the (synthetic) riskless asset and in the risky mutual fund and we show that in this case the theoretical bond/stock ratio varies directly with risk aversion for any hyperbolic absolute risk aversion (HARA) investor. As an example of the type of results that a specific model can produce, we provide a continuous-time model with closed-form solutions, which produces theoretical bond/stock ratios similar to the popular advice. The present paper is organized as follows: in the next section, we analyze the popular advice in terms of the theory of mutual-fund separation of David Cass and Joseph E. Stiglitz (1970). We show that this theory is relevant both in static and dynamic frameworks and use it to analyze the popular advice in complete and incomplete markets. In Section II, we analyze the popular advice in the context of Robert C. Merton’s (1971) continuous-time statement of mutualfund separation and present an illustrative model in which a CRRA investor makes continuous-time portfolio decisions under interest rate and stock price uncertainty. In Section III, numerical results are compared with the popular advice. Section IV is a conclusion. * Bajeux-Besnainou: Department of Finance, School of Business and Public Management, George Washington University, 2023 G Street NW, Washington, DC 20052; Jordan: National Economic Research Associates, 1255 23rd Street NW, Washington, DC 20037; Portait: CNAM and ESSEC, Finance Chair CNAM, 2 Rue Conte, Paris, France. This research was supported by a grant from the Institute for Quantitative Investment Research. We thank two anonymous referees for their comments. 1 HARA functions include quadratic utility, which is one way of justifying mean-variance preferences, and constant relative risk aversion (CRRA) utility. Both quadratic and CRRA utility were considered in the CMW analysis.