# Multi-goal Optimization

Parametric optimization models in the WPA

## Parametric Utilities

Parametric optimization is a tool which investors can use to construct a portfolio that maximizes expected returns based on absolute and (or) relative risk. We have four optimization objectives that are specific to an investor’s concern with absolute or relative performance.
Mean-variance (MV) optimization is a portfolio construction technique that identifies combinations of assets that offer the highest expected return for a given level of risk. It assumes the investor is indifferent to a portfolio’s tracking error against a benchmark. This is the framework that Harry Markowitz introduced.
Mean-tracking error (MTE) optimization is a portfolio construction technique that identifies combinations of assets that offer the highest expected return for a given tracking error. It assumes that the investor is indifferent to a portfolio’s total volatility.
Mean-variance-tracking error (MVTE) optimization is a portfolio construction technique that maximizes expected return while minimizing variance and tracking error against a benchmark. It is useful to investors who are concerned with controlling both absolute and relative risk.

## Theoretical Foundations

The theoretical foundation of our analysis is based on portfolio theory, which was introduced in 1952 by Harry Markowitz. His innovation, which is sometimes called mean-variance optimization, requires estimates of expected returns, standard deviations, and correlations. With this information, we combine assets efficiently so that for a particular level of expected return the efficiently combined assets offer the lowest level of expected risk, usually measured as standard deviation or its squared value, variance. A continuum of these portfolios plotted in dimensions of expected return and standard deviation is called the efficient frontier. We identify portfolios along the efficient frontier by maximizing a measure of investor satisfaction defined by the following quantity
$U_{MV}(w) = \mu' w - \lambda w' \Sigma w$
Some investors also care about relative risk; that is, performance relative to a benchmark. Relative risk is measured as tracking error. Just as standard deviation measures dispersion around an average value, tracking error also measures dispersion, but instead around a benchmark’s returns. It is the standard deviation of relative returns. In this case we identify efficient portfolios by substituting tracking error aversion for risk aversion and tracking error for standard deviation, as shown
$U_{MTE}(w) = \mu' w - \lambda_{TE} (w-b)' \Sigma (w-b)$
In many situations, investors care about both absolute and relative performance. They typically deal with concern about relative performance by employing ad hoc constraints to mean-variance optimization in order to prevent the solutions from deviating too far from the benchmark. We address this dual focus more rigorously by augmenting the definition of investor satisfaction to include both measures of risk explicitly
$U_{MVTE}(w) = \mu' w - \lambda w' \Sigma w - \lambda_{TE} (w-b)' \Sigma (w-b)$
This approach produces an efficient surface in three dimensions - expected return, standard deviation, and tracking error. The efficient surface is bounded on the upper left by the traditional mean-variance efficient frontier. The right boundary of the efficient surface is the mean-tracking error efficient frontier. It comprises portfolios that offer the highest expected return for varying levels of tracking error. The lower boundary of the efficient surface represents combinations of the minimum risk portfolio and the benchmark portfolio.
The Efficient Surface
The multi-goal optimization approach typically yields an expected result that is superior to constrained mean-variance optimization.
• For a given expected return, it typically produces a portfolio with a lower standard deviation and less tracking error
• For a given standard deviation, it typically produces a portfolio with a higher expected return and less tracking error
• For a given tracking error, it typically produces a portfolio with a higher expected return and a lower standard deviation