Bond Demand in the Presence of Equities
The realism of the preceding analysis is limited by the fact that we have not allowed investors to hold equities. We now consider a scenario in which both bonds and equities are available to the investor. Table 8 reports optimal demands for equities and for 10-year indexed or nominal bonds by investors who are unconstrained (in panel A) or subject to borrowing and short-sales constraints (in panel B). For simplicity we do not allow investors to hold equities and both types of long-term bonds simultaneously.
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In a world with full indexation, the unconstrained demand for both long-term indexed bonds and equities is positive and often above 100%, implying that the investor optimally borrows to finance purchases of equities and indexed bonds. The portfolio share of indexed bonds exceeds that of equities, despite the higher Sharpe ratio of equities, because indexed bonds are much less risky than equities. As the coefficient of relative risk aversion increases, the demands for both long-term indexed bonds and equities fall, but the share of equities falls faster.
In the limit the infinitely risk-averse investor holds a portfolio equivalent to an indexed perpetuity as we have already discussed. When there are borrowing and short-sale constraints, investors with low risk aversion invest fully in equities as a way to maximize their risk and expected return without using leverage, while more risk-averse investors hold both indexed bonds and equities. Cash plays only a minor role and only in the portfolios of the most risk-averse investors, who are almost fully invested in indexed bonds.
In a world with no indexation, bonds play a much smaller role in optimal portfolios. Unconstrained investors with low risk aversion hold modest bond positions, but constrained investors hold only equities. As risk aversion increases, investors move into cash rather than long-term nominal bonds.
These findings are related to the “asset allocation puzzle” of Canner, Mankiw, and Weil (1997). Popular investment advisers often suggest that more conservative investors should have a higher ratio of long-term bonds to stocks in their portfolios. Canner, Mankiw, and Weil point out that this is inconsistent with the mutual fund theorem of static portfolio analysis, according to which risk aversion should affect only the ratio of cash to risky assets and not the relative weights on different risky assets.
Our analysis shows that static portfolio analysis can be seriously misleading when investment opportunities are time-varying and investors have long time horizons. The portfolio allocations to equities and indexed bonds in Table 8 are strikingly consistent with popular investment advice. Aggressive long-term investors should hold stocks, while conservative ones should hold long-term bonds and small amounts of cash. The explanation is that long-term bonds, and not cash, are the riskless asset for long-term investors.