Your Personal Geometric Frontier

I used to think there was a perfect investment that was best for everyone. A part of me still hopes that’s true, and I use this vision as a proverbial pot of gold to keep driving my investment research forward. But even if I reach the end of the rainbow, I now believe each investor should still use a perfect portfolio differently.

This is because everyone has their own personal geometric frontier.

Geometric Frontier

I spent last summer working through the concept of the geometric frontier, and how the peak provides the highest compound growth rate.

Near the end of the series, I discussed how understanding the shape of the entire curve is probably more important than just understanding the peak. An investor’s specific requirements and criteria can make investing at the peak sub-optimal.

In doing so, I gave the following example:

As an example, drawdowns can potentially cause real problems for retired investors. Therefore a meaningful reduction in potential drawdown for a small reduction in returns may make sense for a retiree. It will reduce the chance they run out of money if the markets goes south for a bit (often called sequence risk). Pension funds can also face similar challenges…

…Interestingly, the opposite is also arguably true. Steady income into an investment portfolio, can lead to a justified acceptance of larger drawdowns and increased “risk” in order to receive extra return.

In the prior post, you got a glimpse of how withdrawals effect investing growth rate and can make the lower return lower volatility strategy perform better.

This post will show this concept in more depth through our typical style…. with coin flips.

The Base Geometric Frontier

Here’s the coin flip game.

Heads: up 27%
Tails: down 13%
Arithmetic Return: 7%
Standard Deviation: 20%
Geometric Return: 5.1%

This coinflip is meant to be similar to an annual stock market return.

This frontier is a parabola since it’s mixing a portfolio and cash (were not mixing multiple assets), so it looks a bit different that the one above, but it’s still a geometric frontier (think back to the first post on balancing a portfolio).

On the chart 100% means betting everything you have on the coin flip. Less than 100% means to hold some cash back each flip. More than 100% means to take out a loan and invest more than you have on the bet (the interest rate is zero in this example).

You can see the peak is at 200% invested, or betting 2X what you have. This is a full Kelly bet.¹

Now we’re going to add a withdrawal or saving contribution to twist the frontier.

Withdrawals

Let’s start with withdrawals. Instead of compounding wealth forever unchecked, we will take away an amount after each flip.

We set the foundation for this in the last post: mixing multiplication and addition does strange things to compound growth.

Ok, so lets subtract 10% of the current wealth from the total each round after the coin flip. As an example, start with $1,000. Heads means the wealth goes to $1,270 after the flip, and then you subtract $100 for the withdrawal to end up at $1,170). Tails leads to $830 after the flip and subtract $100 to $730 after the withdrawal. Then repeat next round, with the amount to be subtracted being the 10% of the new wealth.

For this game, the subtracted amount isn’t fixed long term, which is probably a better model for real life. It is fixed short term and won’t respond to the next round’s random outcome, but will change later. Think of the withdrawal as: I have rent to pay, food to buy, a pre-planned a vacation with the family, car payment, etc., all set and planned for the year. Longer term costs can adjust later, but no matter what the market does now, spending is fixed for the near term.

With withdrawals you get this shape frontier. The peak is now less the 2X, at a recommended leverage of 180% of investable wealth.

If you read the prior post, this shouldn’t necessarily surprise you. The withdrawals amplify volatility drag, which therefore makes the optimally sized bet smaller (if you haven’t read the prior post, it will help in your understanding of of why this happens).

Contributions

And what about the opposite? What if we add 10% of the current wealth to the portfolio at the end of each round? You get this curve:

The peak compound growth rate is now more than 2X at about 220% of investable wealth. It’s the opposite effect of the withdrawals. Savings dampen the effect of volatility drag, meaning the investor can invest at a larger size and take on more risk. Interestingly, this investor would be investing “past the peak” of the coin flip, although not past their own personal peak.

You can see there is a clear link between risk, savings rates, and the peak of these curves.

Why is the Curve Moving?

So what’s happening here? Well, the withdrawals and the contributions are twisting the geometric frontier.

Withdrawals

Withdrawals make any loss behave like a deeper drawdown. Remember, going down 50% requires you to go up 100% to return to even. So the withdrawals exaggerate the pain from a compound growth perspective by “deepening” losses further than you would have experienced without any withdrawal.

A withdrawal amplifies the impact of a market downturn on the compound growth rate. This is why deep drawdowns are so painful for retires. And it’s one of the reasons why, when viewed from the average compound growth rate, the retiree should invest more conservatively than the true Kelly maximum.

Contributions

Contributions work the opposite. A loss doesn’t hurt as much because cash comes back into the portfolio from an outside source (not investing returns). So a portfolio that goes down down say 50%, but then gets 20% added back from saved wages doesn’t experience the same compounding pain as a portfolio without any new savings.³

So a younger investor who is adding a large amount to their portfolio each year could logically invest more aggressively than the true Kelly peak. They won’t feel market downturns the same way as they are bringing in dry powder to the battle each round.

Big Picture

Here is what the withdrawals/contributions landscape of the coin flip game looks like in the bigger picture. Along the top row, we have the withdrawal or contribution percentage. Along the left side we have leverage ratio, and in the chart we have the geometric return of the coin flip game for each combination. Red cells are low returns, green high. The red numbers are the peak for each withdrawal/contribution rate.

The three charts shown earlier are simply three columns on this chart plotted to show their curve.

Notice how as the withdrawals increase, the location of the peak goes down steadily. As contributions increase the optimal leverage increases.

More withdrawals call for more conservative investing, investing before the pure Kelly peak. Contributions calls for more aggressive investing, beyond the true Kelly peak of the coin flip.

The equation for finding the personal optimal leverage for each column is fairly simple:

Personal Optimal Leverage = Base Optimal Leverage Rate X (1 + Contribution Rate)

So for the 10% withdrawal rate example, you get:

180% = 200% x (1+ -10%)

Real World

Ok, so let’s take this concept back to the real world. This is just a coin flip game, with a simplified method to determine withdrawals and contribution amounts. Obviously the real world is more complicated than this.⁴

But this game gets to the spirit of how investing with contributions and withdrawals effect an investment. It shows that withdrawals and contributions mathematically alter the amount of risk an investor should take.

Withdrawls, call for less risk, contributions can allow for more risk.

The instinct we often have is that a younger investor should take on more risk because they are saving and that a retired investor who is spending should take on less risk isn’t just intuition. It’s supported by actual math.

So when you’re thinking about your own investment goals, keep this in mind.

Are you a young 20 something who is saving a large amount of their paycheck and you just have this sense you should be investing more aggressively than most? There’s a reason you may feel that way.

Did you just retire and you think maybe you should shift to a more conservative investment approach because you’re no longer adding to your investments each month? You’re gut is probably right.

Your Personal Geometric Frontier

We started years ago on our journey looking at simple portfolio sizing in how to balance a portfolio . This post mirrors that post, but now acknowledges that optimal leverage and rebalance ratio’s are not just a function of the investment properties, but also a function of the investor properties.⁵

Each investment has its own optimal leverage. But this is only true in isolation for the investment, and ignores how an investor plans to use the investment. It ignores the investor’s unique circumstances and goals. These individual situations mean we each have our own unique mix of assets which maximize our expected growth rate.

The specifics of our life mean each of us has our own personal geometric frontier.

Now the ultimate investing challenge is trying to find and understand this frontier, but hopefully you now have a better idea where to begin looking.

Footnotes:

1- For the coin flip the peak is 0.5/.13-0.5/.27= 1.994

3-In a way, the incoming funds make the portfolio behave less like one heading towards the geometric return, and one which behaves more like the arithmetic return.

4-One of the “problems” with this game is that it is very static. Each round is the exact same expected withdrawal or contribution. Real life is different, and very skewed in that we often don’t have unexpected contributions, but unexpected withdrawals for an emergency are common. So think about what would happen to this game if each round we had a random 10% chance of taking 30% withdrawal? Directionally, I think you may already see how that twists the personal geometric frontier. I’ll tackle this answer in future post.

This is one of the reasons very few invest in the S&P 500 at 2X even though that is roughly what has been optimal for the last 100 years to maximize compound growth. We all think we may need some of that wealth soon for an emergency.

5-I’ll tackle the personalized frontier with two assets in a future post.

6- I really tried to keep the term risk aversion out of this post to make its appeal broader. But if your are aware of my views on that subject, then you should see that perspective resting under the surface of this post. On paper, a retiree betting on the coin flip only up to 1.92 X leverage is more risk averse than a saver betting on the coinflip and levering up to 2.08 X. But really they aren’t any different. The are both maximizing their average growth rate. There isn’t any risk aversion in this example. If you study economics then you may see that this concept can explain why Kahneman and Tversky found that people are inconsistent in their risk preferences in some of his test.