Reflecting At The Milestones: Implementing the Partial Kelly Strategy

We’ve taken a long journey together through the world of Geometric Balancing, traveling through some deep concepts together:

We’ve learned about the geometric return and why it’s all that matters.

We’ve acknowledged that the investing returns look mostly random.

We’ve explored why rebalancing improves returns by moving them from the geometric to the higher arithmetic return.

We’ve journeyed through how to balance a simple portfolio to maximize geometric returns.

We’ve evaluated which asset classes best fit inside this framework.

We’ve determined to treat leverage with caution, as it can cause untold damage to a portfolio when overused.

We’ve examined how much data you need to “understand and uncover” the true returns, standard deviations, and correlations.

We’ve built a simple framework for estimating returns, standard deviations, and correlations for stocks and bonds.

We’ve expanded that framework of maximizing geometric returns to include multiple assets and then cash.

We’ve watched as the concepts were implemented live, and saw how they navigated the Coronavirus Crash.

We’ve re-examined our inputs and realized they are full of errors. These errors can have serious consequences.

We’ve walked through the concept of the Geometric Frontier and how the Kelly Maximum is simply the peak of this frontier.

We’ve considered how input errors create a need for an investing factor of safety.

We’ve realized each person may have unique circumstances warranting different investing points.

And finally, we’ve constructed a concept of partial Kelly which uses the slope of the geometric curve to determine our own individual investment portfolio.

Quite a voyage. For many, these ideas were brand new. They may have stressed what you knew and assumed about investing. They may have stretched what you thought was possible with investing. We’ve grown through this trip.¹ Our understanding of investment markets is very different now than it was before this odyssey began.

And now it’s time to take this knowledge and return back to the full investing world, mark a milestone, and bring it all together. It’s time to combine everything above into an ultimate, usable, unified portfolio selection technique.

The Two Asset Partial Kelly Portfolio

As I did in a prior post, I’ve created a Google Sheet to educate how the entire framework unifies into a portfolio selection tool. It builds off the post, “optimum portfolio: two assets and cash“. In this case the sheet uses the S&P 500, long-term US government bonds, and cash.

This sheet is built and shared for educational purposes . It’s not a recommendation. I learn best by seeing a technique in action, and I’m sure some of you are the same way. The actual sheet is shown down below, but I’m going to work through the sections first.³

Dashboard

The first section is the Dashboard. It shows the geometric frontier, the selected portfolio on that frontier, and the data for two other key points on the frontier. This is where the input for your partial Kelly value is located as well.

The partial Kelly value is the key input that depends on you, the investor, personally. A value of 1 would be maximum risk and return, matching this portfolio. A value of 0.5 equates to half Kelly. A value of 0 will be 100% cash and no risk. This example uses 0.6 since the inputs below can be raw and error prone.⁴

Return, Standard Deviation, and Correlation Inputs

The second section pulls the inputs from live data found on the web. This section is nearly identical to the same section found in the prior post on two assets and Cash. It’s taking the prior standard deviation and correlation of the assets over a certain timeframe (20 days in this example) and the returns from the bond yield and the stock earnings yield plus half their typical variance.⁵

Geometric Markowitz Bullet

Next the calculations for The Geometric Markowitz Bullet. Portfolios of each possible Stock/Bond mix (step of 2%) are built and calculated with the equations discussed in the Geometric Frontier post:

Portfolio Arithmetic Return = Weight₁ x Return₁ + Weight₂ x Return₂

Portfolio St. Dev. = Sqrt[ (Weight₁ x St.Dev.₁)² + (Weight₂ x St.Dev.)² + 2 x Correlation x St.Dev.₁ x St.Dev.₂ x Weight₁ x Weight₂]

Geometric Return = Arithmetic Return– Standard Deviation² / 2

The Sharpe Ratio is also calculated in the left most column for use in the next section with the formula:

Sharpe Ratio = Arithmetic Return / Standard Deviation

Geometric Frontier

This section creates the geometric frontier. The geometric frontier begins at 100% cash at the bottom left and then grows towards 0% cash at the tangency portfolio. The tangency portfolio is the portfolio with the highest possible Sharpe ratio on the Markowitz Bullet. It is shown in orange.

At the tangency portfolio, the unlevered geometric frontier then bends and follows the geometric Markowitz bullet until reaching the Geometric Maximum, also know as the Kelly point. This is the last portfolio on the table with a calculated Geometric Return.

After developing the geometric frontier, the sheet finds the partial Kelly portfolio. At the partial Kelly point, the slope of the geometric frontier has degraded a certain percentage toward zero (read this post for the full details). So in this example, the original slope equals the maximum Sharpe ratio at 0.0983. A 0.6 partial Kelly portfolio means the slope of the frontier falls to

(1 – 0.6) x 0.0983 = 0.03930

or 60% of the way to zero. This portfolio is highlighted in red on the right.

All Together

For educational purposes, here is the full Google Sheet and a link to the file so you can see the formulas used. The sheet is dynamic, and should fully adapt to changes in inputs. As before, I’m the only one who has quality checked it, so if you see an error please let me know.

How Does This Backtest?

If we follow the precedent of the the prior back tests of the “401k post” and “two assets and cash” of these educational portfolios, and rebalance every month at the start of the month, a 0.6 partial Kelly portfolio produces the following results:

Look at that. This is actually quite close to the full Geometric Balancing with three assets, rebalanced frequently!

I want to caution you though, monthly rebalancing numbers can be sensitive to the date of rebalancing. There is some rebalancing timing luck coming from doing it at the start of the month which probably is only luck. It’s more realistic to view the returns slightly lower, leading to a Sharpe ratio in the the upper 0.7’s.

And even then, things can be tricky as I kept that back test through 2018 to match the others. If I include 2020, rebalancing at the start of the month produces the largest drawdown in March ’20 at 24%. So this back test is a bit rosy.

I’ll dig deeper into this and other backtests in later posts to show some interesting characteristics, but wanted to provide a comparison now to the prior numbers.⁶

Let’s Not Forget What Really Matters

As we take a moment to reflect at this mile marker, let’s not forget why we invest. Clearly, I like investing. You probably do too if you’ve traveled this far. But we’re not investing for the sake of investing.

We invest for what it will provide us in real life. Maybe you imagine the ability to retire earlier than planned. Maybe you imagine ability to retire with peace of mind. Maybe you imagine a higher standard of living. Maybe you imagine a larger house, or a vacation home. Maybe you imagine sending your children to the best colleges debt-free. Maybe you imagine the ability to give to charities. Maybe you imagine the ability to cooperate and help others.

No matter why you invest, we all imagine the ability to build a better life.

It’s going to take time. It takes hard work. However if you’ve journeyed this far, I believe you can reach your goals. Let’s keep traveling down that road.

Thanks again for reading the blog.

1-I’m very much included in this growth. I’ve had all these ideas, but having to actually put them into words in a cohesive explanation has very much solidified and molded them. It’s forced me to run into all sorts of tangents. Mentally, this entire philosophy was coherent and defendable.

3-I want to be clear: this isn’t the formulas for the strategy running at the top of the blog. That one is built around similar framework. It will be mostly similar. But in multiple posts, I’ve pointed out places where the strategy running at the top of the blog is different–where I would consider it improved.

4-Is this the correct value to use for partial Kelly? Not necessarily. I’m not really sure what the correct factor of safety would be for these inputs. But it is true these inputs will be quite off sometimes. Additionally, the partial Kelly number should include your own personal criteria, so there isn’t a universal answer for everyone. I’ll discuss in later posts ways to think about this number.

5-I’ve said this before, but these inputs for return, standard deviation, and correlation are quite raw. They are full of errors. They absolutely can be improved. They serve as a great starting point, and work very well as an educational tool to teach the portfolio construction concepts, but to really build a great strategy, they must be improved.

6-Readers are going to ask about a sheet like this with three assets. Maybe I’ll walk through that one too in a later post, but it’s a lot harder to build, explain, and understand. Two assets are much clearer because the geometric frontier above the tangency point is a smooth transition from one asset to another. With three assets that same transition is more complicated. Conceptually though, you can see through a two asset portfolio, how a portfolio of any size would work.