Few people balance a portfolio properly. Most have no idea even where to start. In this post, I’m going to show you the proper method to balance a portfolio for the long term. To do this we will analyze the simplest portfolio possible: a risky asset (i.e, stocks) and a risk free asset (i.e cash or treasury bills). Is it possible that cash can do more than just reduce risk, and actually help the portfolio grow faster?
My original epiphany about the market centered around the ideas of the geometric average and its absolute importance to long term investing. Problem was, I didn’t understand how to construct a portfolio around this idea. I’d never seen an article outlining the math or the methods behind balancing a portfolio geometrically. So I was going to have to figure out the math myself. Through my search, I ultimately uncovered the foundation behind the Kelly Criterion.
The Portfolio Building Blocks
Let’s start with the simplest portfolio imaginable. A risky asset (i.e. S&P500 index) and a risk free asset (i.e. cash or treasury bills). How would you combine these two together to maximize long term returns?
Let’s say the risk free rate is 7% and you expect a 10% return for the index, with a 20% standard deviation. We start here:
Many people believe that cash will never improve the returns of the index over the long run. It certainly looks like you would much rather be 100% in stocks. And if you only cared about the next “event” (i.e. day), you would be correct. But over the long term you need cash to improve your returns.
Remember, the geometric return is the only thing that matters in the long term. And here are the geometric returns of these assets:
Now they look much closer to each other. Cash has the same geometric and arithmetic returns because it doesn’t have any risk or volatility (within reason). The risky asset’s arithmetic return is reduced by :
standard deviation 2 / 2
20%2 / 2 = 2%
Resulting in an 8% geometric return. It’s still higher than the return on cash. So you may be tempted to believe full index investing is still superior. But lets try mixing them.
What happens When We Mix the Assets
What are the properties of a portfolio of 50% cash and 50% index? The arithmetic return is fairly straightforward:
50% x 7% + 50% x 10% = 8.5%.
The standard deviation of the portfolio is a bit more complicated on the surface:
√ (50%2 x 02 + 50% 2 x 20%2 ) = ?
But because the risk free asset doesn’t fluctuate, it simplifies down to:
50% x 20% = 10%
Leading to the geometric return of this portfolio:
Arithmetic Return – Standard Deviation 2 / 2 = Geometric Return.
8.5% – 10%2 / 2 = 8%.
How about that, A 50% / 50% portfolio of a risky asset and cash produces the exact same long term geometric return, with half the volatility as the risky asset. Clearly the mixed portfolio is a superior investment to the pure risky asset.
The Optimal Mix.
Interestingly, the optimal mixture of two assets is dependent on the standard deviation and can be easily understood visually in this chart:
The bottom the mixing range is the arithmetic average return minus the variance (standard deviation2 ). The geometric return rests right in the middle of the range. In our example, the risk free rate sits 25% inside the range1, indicating the optimal portfolio is 25% cash and 75% risky assets.
The following chart shows the expected return of all possible mixed portfolios (each dot shows a change in portfolio construction by 5%) 3:
It’s clear, there is an obvious unique maximum, and cash absolutely does increase a portfolio’s long term returns.
What if the Risk Free Rate Lies Outside the Mixing Range?
So we now understand how to mix assets when the risk free rate falls within the mixing range. But what if it’s outside the range as shown below? Lets evaluate the same scenario when the risk free asset return is 5%.
In this case, the proper portfolio leverages the risky asset, and borrows the risk free asset. Here we would prefer to create a portfolio of 125% of the risky asset and borrow 25% from cash 4 . The expected return of the range of portfolios is shown in following chart:
Ok what if the risk free rate is above the mixing range at an 11% return?
Here the correct move “shorts” the risky asset by 25% and places 125% of the assets into cash 5.
Now i believe this situation is very, very rare in the real world, but it is still useful to understand the nature of the portfolio.
Key Insights.
We could discuss and evaluate many properties of this simple example (and we will). But for now, I want to highlight two key takeaways:
- Be Careful with Leverage. Reviewing the leverage chart, It’s obvious how quickly the returns tail away after the peak. Too much leverage truly will ruin your long term returns. Applying leverage without concern for these concepts will ultimately lead to ruin. This is where the the Kelly Criterion becomes absolutely critical, in sizing your position with leverage or cash.
- Every point on these curves has the same Sharpe Ratio. The Sharpe Ratio is a ratio of the risk adjusted return of a portfolio to its volatility, and is a very common ratio used in investing. However, these charts prove its flaw when used in isolation. The Sharpe Ratio provides no information about the expected geometric return. Clearly, it’s possible to have a great Sharpe Ratio that is actually bound to lose money over time if the sizing is incorrect.
The Foundation of Geometric Balancing
In a nutshell, I’ve just explained how Geometric Balancing works. With two assets, you simply develop the mixing range, and then figure out the percentage of cash to add (or leverage) to the portfolio. This way you end up with a portfolio tuned for long term returns. The math gets much more complicated when you bring in multiple assets, and with the introduction of correlation between those assets 6. Additionally I add another level of complication by making these variables dynamic. However, even the foundational concepts presented here are amazingly powerful.
Next, I want to pivot a little bit and use this understanding of optimal portfolio construction to describe the actual workings of the market itself. In doing so, we will re-create the efficient market theory correctly, solving many of the unexplained issues economists have with efficient markets. But that revelation will have to wait for the another post.
Part 2, Building portfolios with two risky assets here.
1: 25% = (7% – 6%) / (10% – 6%)
3: The full equation of the geometric return of a cash/asset mix is as follows:
{(Percent asset)*(asset return) + (percent cash) x (cash return)} – ½ x {(percent asset) x (st.dev asset)}2
4: -25% = (5% – 6%) / (10% – 6%)
5: 125% = (11% – 6%) / (10% – 6%)
6: As some will be quick to point out, without good estimates of the standard deviation, returns, and correlation of the assets, the entire process just produces garbage, which is why I put in all sorts of checks and limits into the strategy. And I’ve worked to develop quality estimates.
>> 6: As some will be quick to point out, without good estimates of the standard deviation, returns, and correlation of the assets, the entire process just produces garbage, which is why I put in all sorts of checks and limits into the strategy. And I’ve worked to develop quality estimates. <<
This tiny footnote needs to be front and centre.
You need to tell us about your 'checks and limits'.
You need to tell us how you estimate SD, returns & correlations – over what periods, otherwise as you quite rightly mention – it's garbage.
Really love this blog and the concepts presented however, I do have to agree with Anonymous that this whole method relies upon accurate forecasts of average returns for each risky asset, and the covariance matrix. Without that, your optimal portfolio could end being way off, and even if you use half Kelly could end up using too much leverage.
Matt could you share any further information on this?
Just to let you know the link at the end of the article is formatted incorrectly.
Thanks.
Your blog is interesting but lacks details on how to test and implement some of these ideas. Is it done by purpose?
Partially yes. Science has shown you remember things better if you have to work for them. I’d rather you remember these ideas than just take a formula without understanding why it works. There’s enough in my most recent post to implement this strategy though if you piece it together.
That said Jim seems to be giving it a good go of implementing the concepts. https://www.bogleheads.org/forum/viewtopic.php?f=10&t=303649
Hi, thanks for graphs, but what is the meaning in ‘The bottom the mixing range’? Ok, It is the arithmetic average return minus the variance (standard deviation2 (btw there are no 2nd footnote)).
But it’s not minimum for mixing. In case of mixing 100% of asset1 and 0% of asset, then geometric return is 7%, not 6% as it shows on graph.
The mixing range is the range over which it would be useful to hold both cash and the asset together.
Not sure which chart you are saying should be 7% geo return.
I skipped footnote #2 because I had so many “squareds” in the post i didn’t want to confuse someone.
Bit late, but how did you get 6% = bottom of mixing range ? shouldnt be 7% ??
10% – 20%^2 = %
The answer to that equation is 6% but I’m not understanding what the meaning of (avg return – Variance) is.
Isn’t the bottom of the mixing range 7% because you’d hold 100% cash and 0% risky asset?
Pls explain what % of cash and risky asset you would hold to get a 6% geo return. Thanks!
To quote from your post: “And if you only cared about the next “event” (i.e. day), you would be correct. But over the long term you need cash to improve your returns.”
What if my next ‘event’ is a year from now and I don’t care about what happens in the meantime? If I set my ‘event’ to equal my investment horizon is all of it necessary?
Lets say you buy a bond with a 5% payout a year from now. The bond price will fluctuate in value over that period. The daily geometric return will be lower than arithmetic return if we decide to focus on the daily price changes during that period. However if you decide to focus only on the bond redemption date the geometric return over that period will be equal to arithmetic (5%).
To ask a direct question, can you beat that 5% return by daily balancing the bond with cash by optimizing the geometric return as described in your blog?
If the return of cash is close enough to 5% so that it falls within the mixing range, then theoretically yes.
If we’re 100% cash, the mixing range suggests we would earn 6%, but the risk free rate is 7%. How does this make sense?
…or is the mixing the range the range of likely options give the average? Why did you use the arithmetic mean minus Standard deviation squared?
In the example the mixing range is 10% to 6%. The risk free rate sits 25% into this at 7%, so 25% cash, 75% risky assets maximize the geometric return.
I think the confusion here is where you are getting a 6% geo return from? If you just hold 100% cash you’ll get 7%. What allocation of cash and risky asset would get you 6% geo return?
It’s also not clear what is the meaning of the equation: avg return – variance
It IS clear that the geo return = avg return – var/2
I agree with this.
Why are you using avg return – variance to caculate the bottom of the mixing range and using avg return – var/2 in other places?
One is the mixing range. The other is the geometric return.
Hi, thanks for the article, it’s great! I have some doubts:
1. What risk-free asset offers a 7% return? Is this blog only directed to the United States? In Spain, for example, getting 1% is already an achievement and logically it does not beat inflation …
2.Can geometric performance be applied on a monthly scale? It would be more convenient to rotate between funds for example every end of the month and not every day. It is clear that if you operate with a USA Broker that does not charge commissions and there is abundant liquidity, the fork or slippage will not be a problem.
I don’t think any risk free rate of 7% exists today, but it has in the past. I just used it as an example.
Yes, bit does work monthly. But the drawdowns will be worse and more timing dependent on when in the month you rebalance.
https://breakingthemarket.com/optimum-portfolio-two-assets-and-cash/
A 7% may have existed in the past but then a higher expected return for the market would have been appropriate.
It’s quite misleading to mention the S&P 500 in the example given those returns.
Using more reasonable long-term numbers for the risk-free asset and equity market pair one would put us in the “Risk Free Rate Lies Outside the Mixing Range” situation.
The concept is easier to grasp when you start with the risk free rate within the mixing range, and it certainly has been inside the mixing range at times in the past. I agree it’s usually outside the mixing range for the S&P 500.
The formula of: geometric_mean = arithmetic_mean – variance / 2 is just an approximate formula, even for normal distributions. The whole mixing range concept relies on this approximation.
So I ran some simulations, using 7% return on cash and drawing from a normal distribution with mean of 10% return, standard deviation 20%. I found that the optimal allocation to the risky asset was actually higher than 75%, maybe even as high as 80% (however, it didn’t make much of a difference for the expected return). I than repeated the same experiments, but instead of using a normal distribution, I used a log-normal distribution (again with mean of 10% return and standard deviation 20%). This time I found the optimal allocation to the risky asset to be 100% or higher!
I think this implies a more careful look at the distributions is warranted, and I think instead of using approximate formulas, finding approximate maximum using monte carlo simulations would be preferable.
I haven’t read this yet, but have been told a few years ago Markowitz wrote a book where he compared that formula against 6 other methods. They labeled the formula HL after Henry Latane, who took up Kelly’s work after he died. Markowitz found this formula to be the most accurate method for real investments of all he tested.
https://twitter.com/goldstein_aa/status/1323641396801187844?s=20
I find very different results with the lognormal test when I run it.
That’s an interesting comparison of different methods, thanks for linking!
So it seems that given the real-world distribution of returns, the HL formula works quite well. But it seems to me that if we had a good approximation of the distribution of returns for the various assets, it’d be better to forgo the approximate formula and just find the best allocation using empirical methods, especially since for combining many assets the formula may become intractable anyway.
> I find very different results with the lognormal test when I run it.
Ah, I found an issue with my lognormal distribution results. I’m now seeing that maybe 80%-85% risky asset is the optimal for it.
Hi, this is a great article. What rolling window do you use for the standard deviation and arithmetic return?