# Michaud Optimization

*The Michaud efficient frontier is a generalization of the Markowitz mean-variance efficient frontier that includes investment uncertainty in investment information in defining portfolio optimality.*

Markowitz mean-variance optimization provides the classical definition of portfolio optimality. Markowitz efficient portfolios have maximum return for a given level of risk or, equivalently, minimum risk for a given level of return assuming perfect knowledge of risk and return distributions. However, Markowitz optimization has important limitations in practice. Without numerous constraints, the procedure is typically unstable and often leads to unintuitive and unmarketable portfolios. Moreover, since input assumptions are usually wrong, mean-variance portfolios are optimized to something other than the truth and can be improved upon by assuming a plausible region around estimates to contain the truth, rather than assuming that mean-variance inputs are identical to the truth.

Markowitz efficiency, while theoretically sound given its assumptions, is limited in practice by how information is used in modern computers. Markowitz optimization is built on the assumption that risk-return inputs are accurate to 16 decimal places. While such precision is essential for many scientific and engineering applications, it is absurd for finance. The mismatch between optimization precision and level of investment information certainty results in portfolios that have limited, if any, investment value.

Michaud optimization (Resampled Efficiency), invented and patented by Richard Michaud and Robert Michaud, addresses information uncertainty in risk-return estimates. The procedure produces multiple sets of statistically-equivalent risk-return estimates surrounding the original estimates. These estimates are then used to compute multiple efficient frontiers, each optimized for one of the many possible ways in which assets may perform relative to plausible variation in the inputs. The figure shows mean-variance plots of one such set of efficient frontiers developed from statistically equivalent risk-return estimates. Each alternative frontier is drastically different from the others, even though they are all statistically equivalent and based on the same risk-return estimates. As each frontier represents many portfolios, the variability exceeds what is immediately apparent in the figure. If two mean-variance investors both wanted high risk portfolios, one might invest in portfolio A, while the other might choose portfolio B, despite starting with statistically equivalent risk-return estimates.

*A similar figure appears in Efficient Asset Management by Richard Michaud and Robert Michaud, published in 2008 by Oxford University Press. This version of the figure is for illustrative purposes only.*

New Frontier develops its efficient frontier by averaging all of the frontiers developed from the statistically equivalent risk-return estimates. For example, the high risk portfolio on the Michaud Resampled Efficient Frontier is created by averaging the high risk point on all the efficient frontiers shown in the figure. Michaud optimization is an averaging process that combines all the alternative efficient frontiers into a new efficient frontier and set of optimized portfolios.