The investment horizon problem: a possible resolution

Abstract

In the conventional model of investments, the optimal fractions in the risky assets do not depend on the time horizon. This is against empirical evidence, and against the typical recommendations of portfolio managers. We demonstrate that if the intertemporal coefficient of relative risk aversion is allowed to depend on time, or the age of the investor, the investment horizon problem can be resolved. Accordingly, the only standard assumption in applied economics/finance that we relax in order to obtain our conclusion, is the state and time separability of the intertemporal felicity index in the investor’s utility function. We include a random, idiosyncratic horizon of the consumer, which is essential when there is no intermediate consumption, we consider bequest, and also demonstrate that preferences aggregate.

Keywords:

• The life cycle model
• optimal investments
• the horizon problem
• directional derivatives
• Kuhn–Tucker
• the Saddle Point Theorem
• time consistency

AMS Subject Classifications:

• G11
• D91
• 91G10

Acknowledgements

I want to thank an anonymous referee who contributed to the end result. Any remaining errors are my responsibility.

Notes

No potential conflict of interest was reported by the author.

1 Formally the probability space is enlarged to accommodate this life-time distribution, which is really unproblematic here, because of independence.

2 A reader not familiar with this extension, may just as well as well imagine the horizon as fixed and equal to in the first part of the paper.

3 The same phenomenon occurs with recursive utility, but for a different reason, e.g. Aase [3].

4 Hence it must be a solution to a system backward stochastic differential equations.

5 If one of the agents is close to risk neutral, this agent will dominate in the representation (52(52) ) provided his or her relative wealth is not too low, in which case the representative agent will be close to risk neutral as well.