Portfolio Choice with State-Dependent Adjustments: Analyzing Leveraged Positions Without Parametric Assumptions

Abstract

In this paper, we discuss a new method for solving the portfolio choice problem which models state-dependent quantity adjustment using boundary crossing events. This new method has several advantages: first, we can solve for the optimal portfolio weights without parametric assumptions, by deriving them directly from the data. Next, we can describe the full distribution of the portfolio's value, not just its moments. Finally, we can easily deal with important practical issues, such as transaction costs, leveraged positions and no-ruin conditions, or the cost of margin financing. In particular, the method allows us to analyze leveraged positions in discrete time under zero ruin probability. Analyzing historical stock data suggests that historically, the log-optimal portfolio was a not too extensive leveraged purchase of a diversified stock portfolio, therefore leveraging does not necessarily imply risk-seeking behavior. We also show that depending on how much weight we allocate to this diversified stock portfolio, the downside risk measured as 5% VAR of the portfolio's value may be decreasing or increasing over time. Consequently, an objective functions which incorporates the VAR of the portfolio's value or operate with VAR constrains result in horizon-dependent portfolio weights. We also present some evidence suggesting that the log-optimal portfolio weights are time-dependent. Finally, irrespective of what weight we chose for the diversified stock portfolio, it is log-optimal to reduce exposure to the stock market if the predicted volatility is high, and increase it in low volatility periods.