Why Choosing the Right Objective Function Matters: How to Achieve Maximum Information Ratio with Portfolio Optimization

In this article we explore the complex hurdles that active portfolio managers encounter and how portfolio optimization is a crucial component of their daily routines.

Portfolio optimization stands as a core component within an active manager's strategic framework to consistently outperform benchmarks. This is a complex process aiming to balance risk, returns, and compliance with diverse constraints. These constraints cover a range of practical conditions, including sector delineations, asset allocation thresholds, and limits on position sizes. This process is truly multifaceted, grappling not only with a variety of restrictions, but also selecting between various objectives.

The objective could be to minimize risk, maximize return, or enhance risk-adjusted returns, typically using utility-based optimization or Information Ratio (IR) approach.

Utility-based optimization, rooted in the Markowitz framework, emphasizes risk-return trade-offs. This method employs a utility function, typically defined as the expected return of the portfolio minus a risk-aversion coefficient multiplied by variance. This can be mathematically expressed as:

Where E[Rp] is the expected return of the portfolio, Var[Rp] is the variance (or risk) of the portfolio returns, and λ is the coefficient of risk aversion (the higher the λ, the more risk averse the investor).

On the other hand, the Information Ratio, another measure of risk-adjusted return, focuses on maximizing the ratio of active return to tracking error. The Information Ratio (IR) is defined as follows:

Where E[Rp - Rb] is the expected active return—that is, the difference between the portfolio return Rp—and the benchmark return Rb, σ(Rp - Rb) is the standard deviation of the active return, also known as tracking error.

One critical aspect of portfolio optimization involves limiting the number of assets in a portfolio. While it might seem counterintuitive to restrict potential investments, managing too many assets can dilute a portfolio's effectiveness.

This might lead to higher costs, reduced diversification benefits, and a deviation from the investment strategy. However, optimizing portfolios, especially for max IR, becomes more challenging when there are cardinality constraints. (e.g., constraints on the number of assets).

Many optimization tools lack support for solving maximum IR problems with cardinality constraints. As a workaround, active managers sometimes replace the IR with utility function defined as a difference between active return and tracking error.

While this simplifies the problem, it's a paradox—they seek to improve IR while optimizing based on another metric, which could affect their strategy, since those two optimization problems are not equivalent and those could result in different allocations as we shall see below.

To highlight the difference between Max Utility and Max IR optimization, we will define both optimizations within the FactSet Portfolio Optimizer (FPO). Following this, we'll conduct various analyses within the FactSet Programmatic Environment. For these analyses, we'll utilize the available FPO, Portfolio Analysis (PA) modules, and a comprehensive range of Python-based visualization tools.

Consider an optimization setup utilizing the Russell 1000 as a benchmark and an investment universe of identical stocks. We will introduce the following constraints, all of which aim to emulate the practical limitations encountered in portfolio management: sector and asset relative weights must fall within -3% to 3%, a maximum of 150 assets, a minimum of 20 assets, and a minimum weight threshold of 0.05%.

The alpha definition is derived from one of the Quant Factor Library analyst sentiment factors: QFL_PTGT_EST_REV(0,75D). That sentiment factor is based on Price Target Estimate Revisions with 75-day consensus window.

In Figure 1 below, we illustrate the solutions for Max IR optimization (represented as a blue point) and Max Utility optimization (represented as a red point), alongside the entire Efficient Frontier. The Efficient Frontier is the result of the Min Tracking Error optimization with a lower limit on the active return, considering all other weight and cardinality constraints as defined previously.

Figure 1: Efficient Frontier

It can be observed that both solutions are situated on the Efficient Frontier as anticipated, however, they significantly differ from each other. For instance, the Max IR portfolio achieves an Information Ratio (IR) of 0.575, while the Max Utility portfolio only reaches an IR of 0.442.

Let's delve into deeper analysis of the two portfolios, examining sector and risk factor exposures.

As depicted in Figure 2, the sector exposure of both Max IR and Max Utility portfolios displays significant variations in their active sector weights. Max Utility seems to have a stronger preference for the Communication Services, has higher weightings in Real Estate, and is more underweight in Information Technology compared to the Max IR portfolio.

A marked difference is observed in the Materials sector, which is considerably more favored by Max IR. In consumer sectors, Max IR is more weighted to Consumer Staples, in contrast to Max Utility, which is more weighted to Consumer Discretionary.

In terms of Healthcare, Energy, and Industrials sectors, both portfolios are underweighted to varying degrees, with Max IR being more underweight in Energy and less so in Healthcare.

To wrap it up, distinct sector preferences between Max IR and Max Utility clearly highlight the differences between their individual investment strategies.

Figure 2: Sector Exposure

Next, we compare the risk factor exposures of the two portfolios (Figure 3). The Max IR and Max Utility portfolios both show different exposures to various factors in comparison to the benchmark.

Looking at the two portfolios, Max IR and Max Utility, we see some key differences in their investment approaches across several risk factors such as Dividend Yield, Momentum, Liquidity, Size, and Volatility.

Specifically, Max IR tends to place less emphasis on Dividend Yield, indicating a less conservative focus on immediate income generation from dividends. It has a stronger preference for Momentum, showing it favors stocks with more upward price trends. In terms of Liquidity, Max IR leans into investments with more Liquidity.

On the other hand, Max Utility shows a different picture. It has a more noticeable tilt towards larger companies. However, it has a slightly negative exposure to Liquidity.

Both portfolios, along with their benchmark, showcase an aversion to volatility. That reflects a more conservative stance toward market uncertainties.

Figure 3: Risk Factor Exposure

Again, we conclude that in relation to their exposure to risk factors, the two portfolios display significant differences. That reaffirms our thesis: These are two different optimization problems that, in most instances, lead to significantly different optimal portfolios—despite their objectives seeming similar in terms that both risk and return coefficients in both cases are one.

In conclusion, choosing the appropriate objective function is of utmost importance in portfolio optimization. Access to sophisticated solutions that offer such functionality is invaluable for active managers navigating the intricacies of investment decisions in today's dynamic financial landscape.

This blog post is for informational purposes only. The information contained in this blog post is not legal, tax, or investment advice. FactSet does not endorse or recommend any investments and assumes no liability for any consequence relating directly or indirectly to any action or inaction taken based on the information contained in this article.