### Parametric Optimization

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.

### Efficient Frontier

The efficient frontier is a set of optimal portfolios plotted in dimensions of expected return and standard deviation. The efficient frontier offers the highest expected return for a given level of risk, or the lowest risk for a given expected return.

### Efficient Portfolios for Multiple Objectives

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:

**Expected Return – Risk Aversion x Standard Deviation²**

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:

**Expected Return – Tracking Error Aversion x Tracking Error²**

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:

**Expected Return – Risk Aversion x Standard Deviation²
– Tracking Aversion x Tracking Error²**

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.

This 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

### Full-Scale Optimization

“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 data accordingly.

We include three common expected utility functions – the Power Utility, Kinked Utility, and the S-Shaped Utility functions.

We scale the historical data to conform to specified return expectations. We assume the risk estimate is the historical experience as it is not possible to adjust the second moment without affecting the higher order moments of the dataset.

**Optimization**

We maximize these equations with enumerated search procedures as there is no closed form solution to the above equations. We use a global search method that generates a population of candidate solutions to an optimization problem and evolves iteratively towards better solutions. The fitness of the candidate solution at each iterative step is evaluated and is mutated or recombined to form a new population. The new population is used in the next iteration of the genetic search. There is the possibility in using a global search that we don’t arrive at the global maximum. In addition, because of the randomness in searching, it is possible for different solutions given different searches.

**Related Articles**

- Adler, T. and Kritzman, M. , Mean-Variance versus Full-Scale Optimisation: In and Out of Sample, Journal of Asset Management, Vol. 7, 5, 302–311, 2007
- Cremers, J., Kritzman, M., Page, S., Optimal Hedge Fund Allocations: Do Higher Moments Matter?, Revere Street Working Papers, 272-13, 2004