A theoretical investigation of randomized asset allocation strategies

Abstract

The traditional paradigm of Markowitz-Sharpe diversification stipulates the partition of wealth among the universe of all available investments. For an investor with constant relative risk aversion (CRRA) preferences, the optimal constantly rebalanced allocation is invariant in the dimension of time. In this paper the implications of reshuffling an investor's entire wealth among asset classes according to the stochastic outcome of a Bernoulli (zero or one) random variable is examined. In other words, at any point in time the investor is in only one, albeit random, asset class. The Bernoulli random variables can be constructed so that the investor obtains the exact same level of expected wealth as the ‘constantly rebalanced’ strategy. Over time, by the law of large numbers, this portfolio becomes randomly diversified. Technically, the probability density function (pdf) of the ‘constantly rebalanced’ strategy in continuous time is derived using a new proof that does not require Ito's lemma. The pdf of the randomized Bernoulli strategy (RBS) in continuous time is then derived and contrasted with the pdf arising from ‘constantly rebalanced’ diversification. It is shown that the two pdfs have the same probabilistic functioned form, namely, the log normal distribution, albeit with different parameter values. Although both strategies share the same expected value, the variance and skewness of the Bernoulli strategy is greater than its continuously rebalanced counterpart. Investors with mean-variance or CRRA utility will avoid randomization. However, those with a partially convex utility function or a preference for skewness are likely to select this strategy. As a by-product, an analytic expression is provided for the market timing penalty of a strategic asset allocator whose decisions are based on pure noise. Also provided is an application to the pricing of a second generation exotic option where the payoff function depends on the stochastic combination of two underlying assets.