In portfolio management, **optimization** is a mathematical process of assigning the proportions of various asset classes or investment styles to be held in a portfolio, in a way as to construct the most efficient portfolio, given the expected rate of returns, expected return dispersion, and some other measures of risk. Portfolio optimization takes place in two stages:

- Optimizing weights of the assets classes to be held in the portfolio
- Optimizing weights of assets within each asset class

**Approaches to Portfolio Optimization**

There is a variety of approaches and mathematical tools used in optimization, such as mean-variance approach, the resampled efficient frontier, the Black-Litterman Approach, Monte Carlo simulation, and mean-variance surplus optimization. Some investment advisers also rely on experience and rules of thumb in making asset allocation recommendations.

The **mean-variance approach** is a classic quantitative approach to strategic asset allocation. In determining a strategic asset allocation, an investor can plot efficient portfolios graphically on the **efficient frontier**, and choose an opportunity set from these efficient portfolios consistent with that investor’s risk tolerance. Efficient frontier is part of the **minimum-variance frontier** (MVF). In this approach, the quality of inputs is critical, as the recommended asset allocations are highly sensitive to small changes in inputs and estimation error.

A **resampled efficient portfolio**, suggested by Michaud (1989, 1998) and other researchers, is based on a simulation exercise using **mean-variance optimization** (MVO) and a set range of historical returns. Michaud defines it by the average weights on each asset class for simulated efficient portfolios with a particular return rank. The resampled efficient portfolios represent the **resampled efficient frontier [1]**.

The **Black-Litterman approach** was developed in 1990 by Fischer Black and Robert Litterman. This approach seeks to deal with the problem of estimation error in the MVO. The two versions are **unconstrained Black-Litterman model **(UBL) and **Black-Litterman model **(BL). In the UBL, the investor takes the weights of asset classes in a global benchmark as a starting point, and then adjusts the weights to reflect his or her views on the expected returns of asset classes according to a Bayesian procedure that considers the strength of his or her beliefs. The BL, by contrast, reverse engineers the expected returns implicit in a diversified market portfolio and combines them with the investor’s own views on expected returns in a systematic way.

As a computer-based technique, the **Monte Carlo simulation** has gained its popularity in many areas of investments. In establishing strategic asset allocation, Monte Carlo simulation involves calculating the outcomes resulting in a particular strategic asset allocation under random scenarios for investment returns, standard deviation, correlation, inflation, and other relevant variables. The major advantage of this application in active portfolio management is that it provides information about the range of possible results from a given asset allocation over the course of the investor’s time horizon, and the probability that each result will occur.

In cases where a liability will need to be funded when it comes due, asset allocation must address the risk characteristics of the liabilities in addition to those of assets. **Mean-variance surplus optimization** extends traditional MVO to incorporate liabilities. **Net worth** or **surplus** is the single variable that summarizes the difference between the market value of assets and liabilities in this regard. This approach is most frequently applied in managing projected benefit obligation (PBO).

**The Utilization of Optimizers**

In the practice of portfolio management, investors and advisers utilize and heavily rely on portfolio optimizers, which they believe can determine the viability of a potential asset mix, test for the validity of input assumptions, analyze capital needs, and manage investor expectations.

Portfolio optimizers use historical correlations and standard deviations as departure points for developing estimates. Once investors have established the inputs for each asset class, they can then calculate and derive the asset allocations for the portfolios on the efficient frontier.

A few popular optimization programs, including MATLAB Financial Toolbox, provides a suite of mathematical modeling and statistical analysis for performing asset allocation and risk assessment. The Financial Toolbox was designed to have basic features, such as mean-variance and CVaR-based object-oriented portfolio optimization, cash flow analysis, time-series modeling, basic SIA-compliant fixed-income security analysis, Black-Scholes, binomial option pricing, regression and estimation with missing data, GARCH estimation, and simulation.

FactSet Research Systems uses Barra Portfolio Optimizer to generate a list of necessary trades to construct an optimal portfolio. Some more advanced features of this optimizer include: 1) it sets limits on weights or turnover to ensure that the optimal portfolio complies with fund mandates; 2) it integrates security ranks or forecasted returns from alpha models.

Dr. Richard Michaud and Robert Michaud, Managing Directors at New Frontier Advisors, LLC, have developed an approach based on their resampled efficient portfolio. It uses a Monte Carlo simulation to create a variety of sets of alternative optimization inputs that are able to address some of the uncertainty inherent in the input estimates.

Similarly, the Markov Process International (MPI)’s flagship product, MPI Allocator, also seeks to address the “re-sampling.” It is another quick tool for asset allocation and optimization at both asset-class-level and fund-level construction. MPI Allocator was built with Calibrated Frontiers for re-sampling. Another unique feature is that it allows multiple models to choose from MVO, BL, downside optimization, and mean-benchmark tracking optimization.

**Would Optimizers Work For Investors?**

Some portfolio management professionals and investors, including me at the beginning stage of my career, place far too much faith in the mathematics of portfolio optimization techniques, and those recommendations that result from applying optimization techniques to the portfolios. In fact, investors would benefit considerably from realizing that there is a substantial gap between portfolio theories and real-world practice. Since the foundation for most optimizers is the MPT, which suffers from a series of flaws, portfolio optimizers have inherent limitations:

- Asset classes are not well defined, nor are they stable. Portfolio optimizers are sensitive to changes in expected return, standard deviation, and correlation inputs between asset classes.
- Although it is commonly thought that there are universally accepted benchmarks for each investment style, no consensus exists.
- Investors certainly care about expected return and the risk/return tradeoff, but in reality, they also care about other factors that the optimizers fail to take into account.
- It is highly doubtful that anyone can consistently accurately forecast the previous factors, and therefore, optimizers are incapable of providing forward-looking information.
- In some cases not all rational investors would want to own a portfolio that lies on the efficient frontier.
- Financial market is part of a dynamic economic system that has many complicate feedback processes; multiple extreme events occur more often than the theory suggests that they should be.
- In terms of private assets, many investors have a very strong preference for liquidity premium, so they demand compensation in the form of a substantially higher-than-expected return when making an illiquid investment.

[1] Resampled efficiency is a U.S. patent-protected procedure with worldwide patents pending. New Frontier Advisors, LLC, is the exclusive worldwide licensee.