Dimensions of Return

This post originally appeared on osam.com as part of a new push to accelerate the velocity of our firm’s research and tackle “big” questions in investing.

There are three universal dimensions of return that drive the performance of all strategies—regardless of investment style or asset class: consistency, magnitude, and conviction. These dimensions serve as levers that can increase or decrease performance of any strategy. They also provide context for why portfolios are constructed in the manner that they are. This piece will attempt to create a framework for evaluation and to identify which of the dimensions have a disproportionate influence on performance. In applying the framework to the Russell 1000® and 2000® Value, and the top and bottom large and small cap managers, I find that the dimensions provide insight as to which skills differentiate top and bottom professional managers.

Investing in any asset class, be it public equities or seed stage venture capital consists of two critical decisions: what to buy and sell (selection) and in what proportions (weighting). To understand the drivers of return, an investor must disaggregate the impacts of these selection and weighting decisions. Selection decisions can be evaluated through the dimensions of consistency and magnitude. Weighting decisions can be evaluated through the dimension of conviction.

Having studied markets for almost two decades, I have found the existing knowledge base to be abundant on investment selection and sparse on weighting, or portfolio construction. This piece breaks with existing literature, which conflates the effects of selection and weighting decisions in an overarching assessment of “skill”. Evaluating selection and weighting as distinct skills provides unique insight into what drives manager returns, and how active manager returns might be improved with no additional improvement in selection abilities. My hope is that this framework contributes to the portfolio construction literature as an alternative perspective to the theoretically beautiful, but over-utilized and impractical, modern portfolio theory

Before we can get into the practical application, bear with me as I build intuition for the framework. Caution: there is some math involved. If Greek letters evoke some inner anxiety, skip over the equations and focus on the concepts.

CONSISTENCY – HOW OFTEN POSITIONS WIN

Consistency measures the performance impact of how often winning investments are selected.

The return of a portfolio over any holding period is the weighted average of the underlying position returns and weights. If a position is held at a 1% weight and it appreciates 10% over the holding period, its contribution to return is 0.1% (1% X 10%). The sum of all individual contributions is the portfolio return, which is the weighted average of position returns:

To isolate consistency, we need to level the playing field across portfolio positions by neutralizing the investor’s expression of preference for one investment over another through position weights. This can be done by assuming that each position receives the same weight. When you assume positions have the same weight, you get the portfolio’s equal-weighted return, defined as:

The equal-weighted portfolio is the simplest expression of neutrality because its weighting scheme suggests that the expected probability of some investment outcome is the same for each position. Said another way, uncertainty as to which investments will win or lose is at its maximum. The outcome has nothing to do with manager skill beyond simply selecting the investments from a wider opportunity set. To understand the connection between expected outcomes and how positions are weighted, we can look to probability and information theory.

In attempting to predict the outcome of a fair coin toss—fair in the sense that each toss is 50% likely to be heads or tails—there exists no edge to betting on one outcome over the other, despite our behavioral biases to the contrary. The chart below illustrates the amount of uncertainty as the outcome of the toss becomes more certain, either a tails or heads outcome. Notice how uncertainty (measured on the vertical axis) falls as the probability of tossing heads (moving to the right on the horizontal axis) or tails (moving to the left) increases. As uncertainty decreases, an investor should revise his bets accordingly by increasing the wager—more on this later when we discuss the weighting component of skill. But for now, our equal-weighted portfolio is evaluated as if it were a series of fair coin tosses where each position either wins or loses.

We can break out the winning and losing positions from the equal-weighted portfolio return in equation (2) as follows:

After some manipulation, we can derive the average return of winners and losers as:

Knowing the number of winning and losing positions in a portfolio is useful because it allows you to calculate a batting average. The batting average quantifies how often a manager picks winners. It is a measure of breadth of wins across the portfolio and is the yard stick for consistency.

A batting average can be generated relative to any objective—a benchmark index, a fixed return, or just whether a positive return is produced. The batting average (]]>𝐵𝑝𝑅𝐶𝑅𝐶𝑅𝑀𝑅𝑒𝑅𝐶𝑅𝑀𝑅𝑒𝑅𝐶𝑅𝑀𝑅𝐾