### Impact of Minimum and Preferred Return Settings on Optimal Position Size

We welcome our very own Customer Relations Manager, Dana Lambert, for this guest post:

During the on-boarding phase and in meetings with clients, we often like to point out how minimum and preferred return settings influence optimal position size (OPS), as well as the right framework to think about these settings.

As a starting point, Alpha Theory was designed to derive OPS using a set of linear relationships between OPS and risk-adjusted return (RAR) and – if used – liquidity, beta and other thresholds. The way to think about this is straightforward enough, starting with the RAR bounds. If the parameters are set to 0% and 10% for minimum and maximum position sizes, respectively, and 0% and 50% for minimum and preferred returns, then the following will be the case.

(1) For a stock with a 50% or greater risk-adjusted return, all else equal, Alpha Theory will recommend a 10% position size.

(2) For shares with a 0% return, all else equal, the application will suggest a 0% position size.

(3) At a 25% return, the midpoint of the risk-adjusted return range, Alpha Theory will show a 5% position recommendation, the midpoint of the position size range.

Following is a graphical depiction of the linear relationship with OPS on the y axis and RAR on the x axis:

The ‘all else equal’ implies that no other portfolio-level constraints or Checklist items are in use. In short, Alpha Theory employs a linear scale to determine the optimal position weighting as well as for liquidity thresholds, beta, and confidence.

(For example, assuming the same minimum and maximum position size thresholds, and $10m and $20m average daily volume bounds for minimum and preferred liquidity: any stock with below $10m in liquidity will see a 0% position recommendation, a $20m or greater liquidity name will see no cut to OPS from the liquidity parameters, and a stock with $15m in ADV will see a 5% position suggestion – *assuming the 50% or greater return threshold is already met*. Further, stocks with a beta of 2.0 will see half the optimal position size recommendation versus stocks with a beta of 1.0 while betas of 0.5 will result in a doubling of OPS. Finally, for each 10% degradation in confidence, OPS will also fall 10%.)

What this implies is that by establishing a range for both RAR and position size, we’re establishing the degree of sensitivity for this linear relationship, or the slope of the line in our diagram above. In the example above, each 10% increment in RAR leads to a 2% jump in OPS. Of course, the next logical question is how one goes about *determining* minimum and preferred RAR, which involves more consideration for most versus setting the minimum and maximum *position size* (which most portfolio managers have long ago determined without substantial deliberation).

In setting the minimum RAR, we consider it best practice to use the 0% bound. The reason is simple, in that as long as there is some positive return available in a stock, we do not want Alpha Theory to suggest a zero weighting in that name. The reality is that as expected return approaches zero (say, 3% then 2% then 1% return left), the OPS will be suggesting some nominal position anyway, so risk will be minimized.

In establishing the *preferred* RAR level, one place to begin is to look across all of one’s stock-by-stock expected returns and note what the average or even just predominant trend is in RAR levels. If one’s entire book shows itself to offer no single stock with greater than say, a 20% probability-weighted return, then a 50% preferred return threshold is almost certainly too high. Another way to think about preferred return is to determine the 95% confidence interval of the RAR range. Find a preferred return that is close to but not ultimately the highest RAR in the book.

On the other hand, too low a preferred RAR may indeed make position size recommendations too sensitive to changes in expected return. So if one has the original 0% and 10% position limits from our first example, but RAR bounds are changed to 0% and 20% (rather than 0% and 50%), then only a 4% change in expected return will be required to change the position recommendation by 2%, a much higher ratio than the original 10% RAR change needed (2%/10% or 1:5 in the original example versus 2%/4% in the current example or 1:2). Clearly a 4% change in expected returns occurs much more frequently and easily than a 10% change, so the swing in OPS will be much more pronounced when a lower return delta exists. For this reason, we find ourselves recommending to many clients who have too low a preferred return threshold – and as a result who are seeing major and/or unsettling OPS changes – that they widen their RAR range, and in effect lessen the optimal position size sensitivity.

We always want to allow for personal preference and one’s own intuition to dictate the ‘business rules’ in Alpha Theory, and indeed there is not a ‘perfect’ level for most portfolio constraints. But we also want users to understand fully the implications of their choices, especially for critical fund-level parameters that influence the OPS framework. The relationship between RAR and position size is the starting point and key construct for the optimal position size calculation, and this post should serve to highlight how to consider one’s choices for these thresholds.