Position sizing is a core degree of freedom investors can use to generate outsized returns. In this article, we’ll focus on the two main approaches to quantifying skill set in this area. First, we introduce the equally weighted portfolio comparison, a chart that compares the returns of an active portfolio to those of a similar portfolio, where all positions are given an equal weight. Second, we introduce the position sizing scorecard where the return on invested capital is plot against each position’s weight.

Equal Weighted Comparison compares the returns of the actual portfolio with that of an equally weighted version (orange line in the chart below). An equally weighted portfolio is a portfolio in which each position has an equal value in relation to the portfolio committed to it.

For example, an equally weighted portfolio comprised of four stocks would imply that each position has 25% of the portfolio value committed. The success of a sizing strategy can be determined by noticing which portfolio has performed better (or has the largest returns). If the equally weighted portfolio has outperformed the active, then sizing has been unsuccessful and the portfolio returns would have improved if no active sizing had taken place, as shown in Figure 1.

This metric buckets positions based on their % size out of the total portfolio and compares them to the ROIC (return on invested capital) earned through these positions. Here, a positive correlation between position size and ROIC is desirable. In other words, positions that were sized up the most ought to have yielded the highest ROIC. This implies that the position sizing strategy has been successfully executed. An asset manager would be sizing positions such that as a position is sized up, that position earns a larger return. An example of this scenario can be seen in Figure 2. It is important to note that during the time interval this metric captures, positions can roam through different buckets as they are sized up or down. Typically, as positive momentum begins to build for a position, the manager will size up.

Additional insights that can be drawn from Figure 2 include the cost of research the manager pays on smaller positions. When entering a new position, the manager may choose to start small to determine if the position is worth allocating further capital. If these positions detract value from the portfolio, this can be considered as the cost of research. The manager can choose to size up these positions if they believe it will be profitable to do so, or exit the position if it is not worth allocating additional capital.

Conversely, the largest position size bucket has earned little ROIC, suggesting that by the time this position had roamed through all sizing buckets, the momentum had died.

Immediately, the equally weighted returns comparison and position sizing scorecard appear to be telling the same story. If position sizing has been executed successfully, the largest positions should be earning the largest return on invested capital, and this will outperform the equal weights return. Both metrics consider the performance of the stock and consider the size of each position.

However, you may have noticed that in the figures above, a contradiction seems to be occurring (note that these metrics refer to the same portfolio). We see that the equally weighted portfolio is continuously outperforming the active portfolio, while on the other hand, the position sizing scorecard indicates position sizing has been executed successfully. How is this possible?

One may assume that for the equally weighted portfolio to be outperforming the actual portfolio, the position sizing scorecard should show a negative correlation. The smallest positions are earning the largest returns, then once sized up to an equal share of the portfolio, the returns are increased. The opposite can be said for large positions which could be detracting value from the portfolio. These losses would shrink when the position is sized down to an equal share of the portfolio.

While it’s often the case that both charts tend to be leading to the same conclusion about a manager’s sizing skills, there are situations where this is not true. Certain portfolio configurations result in a situation where the scorecard has a clear positive correlation, but the equally weighted portfolio is continuously outperforming the active one. This can be hard to reason without understanding that there are combinations of portfolio configuration and market movements where this will occur.

The reason these situations can occur is down to the subtle differences between how ROIC and contribution are calculated. To be more precise, how PnL is aggregated in both metrics. There are a few ways of defining ROIC however, in this example, and as in most places throughout the Novus Platform, ROIC is defined as the PnL at the end of the day divided by the market value of the stock at the beginning of the day as seen in Equation (1). For other ways to define ROIC, please see this ROIC Novus University article.

When finding the ROIC over a time period for more than a single stock, this needs to be compounded (using FVSCHEDULE in Excel), further straying from how contribution is defined in equation (2).

The difference in how PnL is aggregated, alongside specific market movements, explains the apparent contradiction in the scenario above. In the next section, we will work through an example that illustrates which market configurations may lead to the scenario above. To anticipate the conclusions, you will see that if the smallest position has the largest market movements in the right direction before it is sized up, then one might find themselves with a positively correlated scorecard, but smaller returns than an equally weighted portfolio.

The following video contains a walk-through of an anomaly that can occur between Position Sizing and Equal Weighted Portfolio analysis.

It’s important to bear in mind that this video example is not the only configuration that causes this combination of an underperforming active portfolio with a positive sizing scorecard. It’s also important to remember that these metrics are providing insights from different angles of position sizing. The position sizing scorecard should not be considered alone to determine how successful a sizing strategy has performed. As we have seen, if a position is large enough, small market movements will still provide a large ROIC, and large movements on smaller sizes, will not result in a large ROIC. Thus, positions that are earning the largest return are not necessarily earning the highest ROIC.

On the other hand, only considering the equally weighted portfolio does not give much insight into individual positions. Generally, it is difficult to determine how smaller positions are performing when compared with larger ones, without the scorecard. Both metrics are designed to be used in tandem to paint a clear picture of how successful active sizing has performed.