When higher moments matter

“Full-scale optimization relies on sophisticated search algorithms to identify the optimal portfolio given any set of return distributions and based on any description of investor preferences" (Adler and Kritzman 2007).”

Rather than using summary statistics such as mean, variance, and correlation, full-scale optimization utilizes the full sample of returns based on plausible utility functions. Mean-variance optimization assumes either that returns are normally distributed or that investors have quadratic utility. However, asset returns are not exactly normally distributed in practice, and investors are rarely as averse to upside deviations as downside or prefer less wealth to more wealth.

While both approaches to optimization suffer from estimation error, mean variance optimization also incurs approximation error (Adler and Kritzman 2007). While full-scale optimization returns the in-sample optimal portfolio, parametric optimization returns an approximate in-sample optimal portfolio.

Historical returns are used to generate data for full-scale optimization. However, to incorporate an investor’s views regarding expected returns, we adjust the means of the data accordingly. We assume the historical experience for risk since we want to preserve and account for higher moments.

We include three common expected utility functions for Full-Scale optimization:

The most common utility function used is the power utility function. Power utility functions assume a preference for upside deviations and have positive slopes, which reflects a preference for increasing wealth. These utility functions assume that investors prefer to preserve the same percentage allocation to risky assets as their wealth changes.

The power utility function is similar to mean-variance (MV) optimization as it approximates the log-wealth utility by using a quadratic function. Power utility functions are always upward sloping in contrast to the quadratic function which has a turning point.

An investor might want to maintain a minimum standard of living, or need to maintain a wealth so as not to breach an agreement on a loan. A kinked utility function describes this preference, where it changes abruptly at a specified wealth or return level. This utility function penalizes solutions that breach a given threshold. It is defined by a log-wealth function above the threshold and by a steeper linear function below the threshold return.

If an investor must choose between certain gain and an uncertain outcome with a higher expected value he will often choose certain gain. If it is a choice between a certain loss and an uncertain outcome with a lower expected value he will often choose the uncertain outcome. This behavior is captured by an S-shaped value function (Kahnemann and Tversky 1979), where investors are risk-seeking below a certain threshold, and risk-averse above it.

The following video describes what is Full-Scale Optimization in the Windham Portfolio Advisor