Optimum Portfolio: Two Assets and Cash

Time to get to some unfinished business. A few months ago I said I would shortly discuss adding cash to create a small geometrically optimized portfolio. I’ve taken too long to get this post out. So here is the process to maximize the geometric return of two assets and cash. Additionally I’ll show you how to obtain the required inputs in near real time.

Two Uncorrelated Assets and Cash

Lets start with an uncorrelated example, which is surprisingly straight forward once you’ve grasped the ideas in my prior posts (if you haven’t read that post I recommend you start there).

Lets imagine two assets and a risk free rate with the following properties:

• Risk Free rate: 2%
• Risky Asset #1: 3% Return, 20% Standard Deviation
• Risky Asset #2: 9% Return, 40% Standard Deviation
• The assets are uncorrelated

Then create the mixing ranges of the two assets as I outlined in this post. You get this chart:

Now calculate the optimum portfolios (as described here and here) for the three different potential two-way combinations:

• Cash to Asset #1
• Cash to Asset #2
• Asset #1 to Asset #2

And write these values in a table under the chart. No leverage for now, so anything less than 0 or greater than 100%, just put down 100% or 0%. For our example you get:

Now pick the smallest asset allocation in the three possible two-way portfolios. For our example you get this:

Finally, if these assets don’t add up to 100%, put the rest into cash:

And this portfolio provides the maximum geometric return for the two uncorrelated assets and cash. It’s that simple.

Let’s Add Correlation

Lets say the two assets have a negative correlation of 0.5. This adjusts the chart as follows (as explained in this post):

And that’s where we’re going to stop with a precise Kelly maximum portfolio for now. I’ve tried my hardest over the last five months to figure out a simple graphical way to explain the process with correlation, and I can’t find one. It gets too complicated and I don’t want to take away from the simplicity of what’s above.

Luckily, if you run through the same process outlined for uncorrelated assets from here (only applying the correlation to the mixture) you still get a very similar portfolio to the precise solution as long as the correlations are not near one.¹

If the risk free rate doesn’t overlap any of the individual asset mixing ranges, then this method is precise. This is usually the case with government bonds and a market index. Even if it does overlap some, the method is still fairly precise. It’s only way off when the risk free rate falls well within the individual asset mixing ranges, and even then it works better than ignoring correlation entirely.

A Backtest of This Framework

Let’s take this portfolio construction philosophy, and apply it to the S&P 500, the 30 Yr Treasury, and Cash. Calculate the predicted future returns, standard deviations and correlation with the methods identified here. Rebalance Monthly. You get this for 1978-2018:

It’s roughly the same returns as the portfolio without cash, but with a bit lower drawdown and slightly higher Sharpe ratio. As stated above, the strategy doesn’t call for cash often, but when it does it’s useful.

There’s nothing overly complicated here. It’s just two foundational investments. The inputs are simple:

• Bond yield
• Index PE Ratio
• Trailing 20 day returns to calculate simple standard deviation and correlation
• Risk Free Rate

Implemented with Live Data

Since I didn’t figure out a clean way to deal with the correlation issue, I decided to provide something else. I’ve taken this framework and put it into a google sheets document.

With the “googlefinance” command, google sheets lets you easily access stock data in near real time. It’s a wonderful tool to simply, quickly pull raw investment information without knowing much about coding. With the sheet below, you can see how to actually implement a simple two asset portfolio, balanced to maximize geometric return with current data.

I’ve shared the file for everyone to see the math. I’m the only one who has quality checked it, so if you see an error please let me know. The lookback can also be changed if you don’t like 20 days (you have to save it to your own account)

What Have You Done For Me Lately?

I showed the backtest through 2018 to match with the others on the site, but ran it through early April 2020. In 2019, the backtest indicated a 26.5% return.

This year hasn’t been as strong

Corey Hoffstein often writes about the impact of rebalancing timing luck. He’s very correct, it really matters to portfolio performance. The rebalance timing for this recent downturn was critical. The strategy called for 90% stocks and 10% bonds on the March 2nd rebalance (first day of the month). Not much protection for the oncoming storm. Eight days later the rebalanced portfolio would have been 58% cash with half the drawdown yet to come. Quite a difference.

I like frequent rebalancing because I believe it helps returns. I also like that frequent rebalancing responds to volatile markets and smooths out rebalancing timing luck.

If you rebalanced this signal daily, the total drawdown for the year would have fallen 13.6% peak to trough. Year to date it bottomed out down 9.2% before starting to slowly climb again. That’s pretty strong defense for a two asset strategy which positioned itself aggressively in 2019.

The Next Portfolio Construction Technique

Once again, this is a pretty raw strategy using the Geometric Balancing philosophy. There’s only two assets. The inputs are crude, and the portfolio is mathematically an approximation in certain environments. Yet not many strategies can match it’s results.

There are lots of portfolio construction techniques. 60/40. Risk Parity. Trend. Minimum Variance. Long-Short Factors, etc. In time geometric return optimization (of which Geometric Balancing is a finely tuned version) will join and possibly surpass all of them.

I invite you to take these ideas. Incorporate them into your own investing. Reap the benefits.

1-If they are near 1, and the risk free rate overlaps either asset’s mixing range, this method could be quite wrong and will over allocate to the lower returning asset vs. using cash.