Kelly Investing is About Slope

I left you hanging a few weeks ago, and it’s time to pick the thread back up. In my prior post I explained why factors of safeties are important in engineering and equated the factor of safety to using partial Kelly in investing. But I didn’t explain how to use partial Kelly.

What I’m about to discuss is not the normal description of partial Kelly (half Kelly as an example). Partial Kelly, even full Kelly, is fundamentally about the slope of the geometric frontier.

In a prior post, I pointed out that the top of the geometric curve, the pure Kelly portfolio, is flat. Its slope is 0. What I didn’t point out is the slope at the far left edge of the geometric frontier equals the maximum possible Sharpe ratio.

The slope of the geometric frontier starts at the Sharpe ratio and falls slowly until reaching zero at the full Kelly point — the maximum geometric return.

Easiest to Understand Through Leverage

I’ve specifically stayed away from discussing leverage with Geometric Balancing. This has been very much on purpose both due to leverage aversion and other reasons I’ll explain later. This choice causes a problem now though.

The unlevered geometric frontier at first glance doesn’t fit traditional definitions of partial Kelly. A portfolio like half Kelly is often described as investing at half the allocation of the peak.

Holding half the full Kelly allocation, or picking the portfolio with half the volatility, doesn’t make any sense. In this chart from the geometric frontier post, the peak is 60%/40% stocks bonds. It’s silly to say half Kelly equates to investing at 30% stocks 20% bonds and 50% cash. This point isn’t even on the frontier.

However, Kelly is usually presented as the presence (or absence) of leverage (cash). In order to understand why the slope provides the key to unlocking partially Kelly, it’s easier to think about Kelly with leverage. Doing so makes makes this problem go away. So we start there and then return to the unlevered problem.

My first post on Kelly did introduce the concept of levering a portfolio, so let’s return to that example.

Here you can see the parabola of the portfolio’s geometric return as a function of the portfolio composition. In this case the chart is levering up or down an asset with an 8% geometric return. The “% of the asset” on the X-axis has a direct relationship to the volatility of the portfolio. From a scale perspective they are interchangeable, and I will flip it in the next chart.

The geometric frontier here starts at 0% on the left, rises up, and peaks at 125% of the portfolio (borrowing 25% cash). It is a smooth curve. Importantly, there isn’t a bend in the frontier like there is in our unlevered geometric frontier (at the tangency point).

What is Partial Kelly Really?

I’m going to talk through “half Kelly” here to start, but this works for all partial Kelly portfolios. “Half” Kelly typically entails investing with half the asset size a full Kelly portfolio requires. In this case, the peak is at 125% of the risky asset (uses 25% leverage). So half of 125% is 62.5%, leaving 37.5% cash (100% – 62.5%). Half Kelly cuts the volatility in half, while only reducing the return by about 2/3rds of a percent.

Seems like a decent trade off.

Half Kelly is often explained this way, so people think of it as investing with half the recommend position, or investing at half the recommended volatility.

The problem is, these metrics are not what Kelly is about. They are secondary. Kelly is about the slope of the geometric frontier.

The Slope is the Key

Kelly isn’t about volatility levels. It isn’t even really about position size. It’s about the risk return trade off between volatility and the geometric return. The slope of the curve. At a full Kelly peak, the trade off ceases to exist. The slope is flat.

Full Kelly is a portfolio section criteria which says to invest with more and more risk until additional risk provides no further benefit to the geometric return.

Remember, when using leverage the geometric frontier is a smooth parabola. Its starting slope equals the Sharpe ratio and moves downward to a final slope of zero at full Kelly. Therefore a half Kelly portfolio invests at a slope along the curve equivalent to half the Sharpe ratio.¹

It’s the point where the geometric frontier has flattened out halfway between the Sharpe ratio on the left, the flat top on the right.

Therefore it’s a criterion to allow the risk/return trade off to degrade halfway to zero. This is a very powerful concept. One that can be applied to all different geometric frontier shapes, not just smooth ones.

Unlevered Geometric Frontier

Let’s run through some examples, continuing to use half Kelly as the safety factor (equivalent to an engineering factor of safety of 2) for demonstration purposes.²

Take the geometric frontier, and invest at the point where the initial slope has been cut in half and:

Much of the time, a half Kelly portfolio will pick a spot shortly after the tangency portfolio (example 1). This is because the curve bends the most when cash falls out and the curve begins to move along the Markowitz frontier. Therefore the slope is most likely to cross the threshold right above the tangency portfolio.

But it doesn’t have to. If the curve bends softly through the tangency point, the half Kelly point will move much closer to the peak (example 2), especially when the unlevered curve doesn’t come close to turning over with an actual top. This risk return trade off warrants taking on more risk.

To the point that when the investing world is tame (volatility is low), half Kelly can even pick the portfolio on the the far right end of the geometric frontier (example 3). This happens because the slope at the the end of the curve (100% stocks in this example) is still quite steep. The risk return trade off at the end of the curve is still worth taking.

And on the flipside, when volatility is high across the board, and correlations don’t provide enough help, the portfolio moves into a defensive posture picking a portfolio left of the tangency portfolio, employing cash (example 4). The concept therefore reacts from very aggressive to very defensive depending on the risk/return opportunity the market provides.

Maybe you think half Kelly is too conservative. Maybe 3/4ths Kelly is the right factor of safety (engineering factor of safety of 1.33) for the assets in your portfolio. Then you would invest further out on the curve at the point the slope has decreased by 75%.

Dynamic Portfolio Selection

This method produces spectacular results because it’s dynamic to the market and the current risk/return trade off it’s offering. Many people prefer (and classical theory calls for) investing at the tangency portfolio, as it provides the best risk/ return trade off on the typical (arithmetic) efficient frontier. But as you see on the geometric frontier, the point isn’t as important.

Sometimes the tangency point is extremely conservative (example 2). If you’re not using leverage this point can be needlessly safe. At other times the tangency portfolio is still very volatile and risky (example 4). It can be way out near, or past, the peak of the curve (think March 2020). And in those cases, the tangency portfolio should be cut with cash to get the portfolio back safely on the correct side of the peak.

A static portfolio can’t respond like this. But a Geometrically Balanced one can.

Solves Two Problems With One Stone

What I really like about this concept is it can be used to solve two problems with investing.³ One, it works very well as the “factor of safety” in portfolio construction. But it also does a fine job addressing the question of selecting a point along the geometric curve to meet specific individuals invest criteria since the concept stops at a desired relative risk/return trade off.

Kelly is About Risk/Return Slope

So please remember, when you read about partial Kelly investing, its foundation comes from monitoring the risk/return slope of the geometric frontier.

Full Kelly moves along the curve until the slope is flat.

Partially Kelly pushes up the curve until the slope provides the desired risk/return.

We’ve discussed how errors make using a factor of safety important. Therefore use partial Kelly to select a risk/return point along the curve that meets your own personal criteria, and ensures errors don’t sink your portfolio.

In later posts I’ll show you how to implement this framework.

1-You may see similarities with volatility targeting here. There are similarities, but it’s not the same. It’s not targeting vol, it’s targeting a risk/return trade off.

2- I want to be clear I’m only using half Kelly as an example and not endorsing it as the best choice. It may be the right choice for some people, but I use it here as it seems to be the example people grasp the quickest.

3-I actually use a two step process to address these two issues separately. It’s slightly different than what I’ve described here, but would be much more complicated to explain. The concepts are fundamentally the same and I don’t believe my more complicated methods improve the portfolio much over this one.