Numerical Solution of the Static Portfolio Problem for Power Utility Investors

May 2012

This paper introduces a simple and reliable numerical method for a classic problem in portfolio choice: finding the single-period portfolio that maximizes the utility of an investor with power utility, the standard family of preferences with constant relative risk aversion. The problem has no closed-form solution, so it is usually solved numerically. The paper shows that this is more treacherous than it appears. For common choices of risk aversion, the objective function is discontinuous and riddled with false local optima, and a generic optimizer can converge to any one of them. What makes this dangerous is that the resulting portfolio weights look perfectly reasonable; the only symptom is that the portfolio would have gone bankrupt, producing a negative value, in at least one historical period. The paper’s contribution is a small modification to the objective function that removes these false optima, so that a standard search reliably finds the unique economically valid solution. It is written for a technical audience of researchers and quantitative practitioners who estimate optimal portfolios from data.

What This Paper Examines

• Why numerically maximizing power utility over observed returns is prone to false solutions.
• How discontinuities in the objective produce a large number of spurious local optima.
• Why those false optima are hard to detect and economically meaningless.
• How the problem worsens as the number of assets grows.
• A modification to the objective that makes the problem globally well-behaved.

Key Findings

• The power-utility portfolio problem is riddled with false local optima. For common, odd-integer choices of risk aversion, including the important log-utility case, the objective is discontinuous, and a generic optimizer can settle at any of many spurious solutions.
• Those false optima correspond to bankruptcy and are easy to miss. At all but one optimum, the portfolio produces a negative value in at least one period. The weights look plausible, so the failure is invisible unless one checks the path of wealth directly.
• The problem gets worse with more assets. As the number of assets and the length of the return history grow, the number of false local optima can rise into the thousands, making naive optimization increasingly unreliable.
• A quadratic extension near zero wealth makes the objective globally concave. Smoothly pasting a quadratic piece onto the utility function just above zero wealth gives it a continuous, monotonic gradient, so a standard Newton-Raphson search converges to the unique optimum.
• The fix enforces the no-bankruptcy constraint without distorting the answer. The modification leaves the objective unchanged in the economically relevant region, so it delivers the correct nonnegative-wealth optimum rather than an artifact of the numerical method.

The Authors

This paper is part of the long lineage of quantitative research that Versor’s founders began earlier in their careers and continue to build on at the firm today.

Ludger Hentschel, Founding Partner, Investment Advisor

Ludger Hentschel joined Versor Investments as a Founding Partner and is based in New York. Ludger has over 20 years of experience in quantitative research and investing.

Versor’s founders co-authored this research with John B. Long Jr., a colleague at the firm where this work was originally conducted.

Disclaimer: Past performance is not necessarily indicative of future results. Not an offer to sell or a solicitation of any type with respect to any securities or financial products.

Methodology: The paper analyzes the first-order conditions of the static power-utility maximization problem and shows analytically why the objective function has asymptotes and discontinuities, and how these generate a proliferation of false local optima whose second-order conditions are nonetheless satisfied. It illustrates the severity of the problem using long-run US equity data, including single-asset and multi-asset examples.

It then proposes the remedy: smoothly grafting a quadratic extension onto the power-utility function at a wealth level just above zero, which penalizes bankruptcy in a transparent way and renders the augmented objective globally concave. The paper relates this to classical penalty-method optimization, and demonstrates that standard Newton-Raphson searches then locate the unique optimum quickly across the same examples.