Lifetime Active Portfolio Selection for Investments and Consumption – A Bull Bear Market Cycle Based Probabilistic Approach

Abstract

Beyond the single period Modern Portfolio Theory (Markowitz, 1955), the seminal work by Robert C. Merton (1969) solved elegantly a multi-period (finite time horizon) continuous portfolio optimization problem under the random walk market assumption. Due to the mathematical challenges, there has been little further theoretical breakthrough in nearly half century. Instead, popular financial planning and investment practice such as time diversification and target-date/glide-path based funds, deviate from the Merton/Samuelson’s teachings of long term static allocation. From a more realistic view of capital market and financial planning economics, what is the right asset allocation policy for lifetime investments and consumption?

In this study, I replace the simplified stock market assumption in the classical Merton model with a Markov chain Bull/Bear market regime switching formulation (Wonham, 1965). The bull or bear market probability is calculated based on the price history of a broad market index such as the S&P 500 Composite.

To maximize the total utility of discounted consumptions and bequest wealth value at the end of lifetime planning horizon, I derive a quantitative portfolio selection rule. It turns out that dynamic weight to risky asset is approximated solely as a quadratic function of the conditional bull market probability. From a market timing perspective, the active probabilistic rule has both trend following and mean reversion mechanisms at different stages of the anticipated market cycle. For practical purposes, I demonstrate the analytical approach in two sets of back-tests under different investment/consumption preferences.

First, for a pure investor with zero consumption requirements, the objective reduces to maximizing terminal wealth. I found the optimal dynamic investment leverage or exposure to a risky stock market index depends not only on investor risk preference and the index’s average attributes of market return premium and volatility, but also the “strength” of the bull/bear market cycles, and the point-in-time market regime probability itself.

Second, random walk or bull/bear market characteristics assume no impact on how an investor values the utility of his/her bequest wealth relative to lifetime consumption. In this case, the probabilistic portfolio selection is shown to out-perform easily in total utilities either the buy-and-hold static allocation or a typical linearly de-risking target-date glide path.

By integrating the cyclical capital market view into a multi-period portfolio optimization framework, the current probabilistic formulation is ground breaking, despite the need for further refinement and analysis. With a fast changing landscape in global markets, the current approach of market cycle based dynamic allocation and active management is particularly important. It has the potential to seriously impact the theory and practice of investment management and financial planning, for the long term.

Keywords: lifetime portfolio selection, investment and consumption, multi-period continuous portfolio optimization, bull/bear market cycle, Wonham filter, conditional bull market probability, trend-following, mean reversion, back test, target date fund, glide path

JEL Classification: C44, C61, D81, G11