Strategic Stock-Bond Allocation with Mean-Reverting Equity Returns
Summary
The report studies long-horizon allocation between stocks and bonds when equity returns follow a mean-reverting process. It asks how this feature changes the risk-return trade-off and whether some strategies can make the portfolio’s terminal value behave as though it has a lower bound under specified assumptions and horizons.
Rather than using the usual Hamilton-Jacobi-Bellman approach to optimize strategies that depend on market conditions or current wealth, the authors restrict attention to time-dependent allocations suited to strategic investors. They use calculus of variations to examine the broader family of extreme strategies, including those that are not necessarily optimal. The excerpt describes the research question and analytical approach, but provides no model parameters, empirical evidence, or quantitative results, so conclusions about practical performance cannot be drawn from it.
Key ideas
- The model considers strategic allocation between stocks and bonds when equity returns mean-revert.
- The analysis focuses on allocations that vary with time rather than market state or portfolio wealth.
- Calculus of variations is used to identify a family of extreme strategies.
- The report asks when the horizon portfolio value may have an effective lower bound under stated assumptions.
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Full text
# Strategic mean-variance investing under mean-reverting stock returns # Strategic mean-variance investing under mean-reverting stock returns In this report we derive the strategic (deterministic) allocation to bonds and stocks resulting in the optimal mean-variance trade-off on a given investment horizon. The underlying capital market features a mean-reverting process for equity returns, and the primary question of interest is how mean-reversion effects the optimal strategy and the resulting portfolio value at the horizon. In particular, we are interested in knowing under which assumptions and on which horizons, the risk-reward trade-off is so favourable that the value of the portfolio is effectively bounded from below on the horizon. In this case, we might think of the portfolio as providing a stochastic excess return on top of a "guarantee" (the lower bound). Deriving optimal strategies is a well-known discipline in mathematical finance. The modern approach is to derive and solve the Hamilton-Jacobi-Bellman (HJB) differential equation characterizing the strategy leading to highest expected utility, for given utility function. However, for two reasons we approach the problem differently in this work. First, we wish to find the optimal strategy depending on time only, i.e., we do not allow for dependencies on capital market state variables, nor the value of the portfolio itself. This constraint characterizes the strategic allocation of long-term investors. Second, to gain insights on the role of mean-reversion, we wish to identify the entire family of extremal strategies, not only the optimal strategies. To derive the strategies we employ methods from calculus of variations, rather than the usual HJB approach.
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