## The Ratio That Broke Investors’ Brains

If you are going to use some set of risk-reward measures (such as Sharpe Ratio) to judge investment performance, you need to keep your eyes on the prize: The overall performance of your portfolio, not that of the individual portfolio components. I was reminded of this the other day reading this article from Institutional Investor: “The Sharpe Ratio Broke Investors’ Brains” by Richard Wiggins. The article makes the case that analysts have become obsessed with the individual Sharpe Ratios of potential investees. As a result, they miss opportunities to improve the Sharpe Ratio of their portfolio. I thought I would demonstrate what they were getting at.

The prize: The overall performance of your portfolio, not that of the individual portfolio components.

This is the ninth in my Hedge Fund Hacks series.

## Teamwork Works

When I was a younger chap at high school in the UK, I played rugby. If you have ever played the game, you will know it can be quite … physical. In my particular age-group, we had a shortage of large players. Thanks to some great coaching, we learned that playing as a team more than made up for lack of physical size. By the time I and my cohorts made it to the First XV, we were able to put together a 20-1 season and a successful tour of British Columbia against teams that looked like giants to us! The point is, a great team will beat a bunch of disorganized raw talent any day.

The same is true with a portfolio of traders or trading systems.

It seems intuitive that combining high-performance programs will naturally lead to a high-performance portfolio. In fact, the interaction among the return streams that make up the portfolio is every bit as important to overall portfolio performance. In this article, I will explore the impact of adding high-performance programs to a portfolio versus adding programs that may have lower performance but play well together.

## A Simple Model

I am going to use the simplest model possible and measures of performance and component interaction that everyone is familiar with. This is not to endorse these measures of performance and interaction. The concepts extend to your preferred measures. I am then going to throw in a couple of real-world examples to show this is not just an academic exercise.

Our example looks like this:

- Portfolio: two assets.
- Performance Measure: Sharpe Ratio.
- Measure of Interaction: Correlation Coefficient.

## The Basic Markowitz Bullet

It is worth taking a quick moment to remind ourselves how the Markowitz Bullet works because I will be using it in an unfamiliar way to make my point later on.

Read my previous post for a detailed tour of the Markowitz Bullet.

Asset A has a Sharpe Ratio of 0.99 at a 0% risk-free rate. This odd-ish choice will make sense later. Asset B has a Sharpe Ratio 1/2 that of Asset A with 3/4 of the volatility. The two assets have a Pearson correlation coefficient of 0.

### The Efficient Frontier

The curve is made up of a dot for each combination of Assets A and B in increments of 1%. So the large dot at the upper right is 100% Asset A. The large dot lower right is 100% Asset B. Given an either-or choice between A and B, we would choose Asset A. We can do better, at least in terms of the Sharpe Ratio of our final portfolio, by substituting some exposure to Asset B in place of Asset A. While the replacement of Asset A by Asset B reduces the expected return of the portfolio, it also reduces the volatility of the portfolio by more, so the Sharpe Ratio (proportional to Return / Vol) improves. Up to a point. If we add too much Asset B, the portfolio Sharpe begins to fall again. The upper half of the curved line is called the Efficient Frontier because at any point on the line there is no portfolio that can be constructed that has both higher return AND lower volatility.

### The Optimal Portfolio

The portfolio with the maximum Sharpe Ratio is marked by the dot part-way along the curve. It is also known as the minimum mean-variance portfolio and is the optimal portfolio in this paradigm. The straight line on the chart passes through (0,0) because we are assuming the risk-free rate of return = 0% and the maximum Sharpe portfolio. This is tangent to the curve, and since Sharpe is proportional to return / vol, the slope of the line is proportional to the Sharpe Ratio. The line is also known as the Capital Asset Line. By employing leverage, we can move our portfolio anywhere (in theory) along the line.

## Optimal Portfolio vs. Varying Asset B Sharpe Ratio

So now let’s explore what happens as we change the Sharpe Ratio of Asset B.

First notice that in this case, the zero-Sharpe asset doesn’t even make it into an optimal portfolio – you would always be better off using Asset A alone in this case. As the Sharpe of the second asset increases, the Sharpe of the portfolio increases. This is what leads asset allocators to focus so much on Sharpe: if all my portfolio components have high Sharpe Ratios, my portfolio will have a high Sharpe.

## High Sharpe Ratio – High Correlation Assets

That’s all very well as far as it goes. In this example, we have a very low, and somewhat rare, correlation coefficient between the assets. Let’s take a look at what happens in a more realistic situation, where the correlation coefficient is much higher.

## Optimal Portfolio vs. Varying Asset Correlation

Correlation matters. A lot. Let’s take a look.

I hope by now I have made the point that correlation is just as important, if not more so than raw individual performance in terms of impact on the portfolio in this paradigm. The corollary being, if you focus too much on the search for high Sharpe Ratio programs to add to your portfolio, you risk missing programs with attractive correlation characteristics. In fact, if you search a database and filter for, say, Sharpe greater than x, you won’t even be aware of these programs.

## Optimal Portfolio vs. Asset B Sharpe Ratio AND Correlation

Now let’s pull all this together onto a single chart. I am going to use the x and y axes to indicate the Sharpe Ratio of Asset B (which I am going to vary) and the correlation between Assets A and B (which I am going to vary), respectively. I am going to indicate the Sharpe Ratio of the resulting portfolio (on the z-axis) using a colored contour system. Let’s take a look and then I will walk you through it.

### Exploring the Sharpe Surface

To help you get oriented on this chart, notice that the large purple area top-left indicates a portfolio Sharpe Ratio of just below 1 (0.99, in fact) which is a portfolio of 100% Asset A. In this region, the Sharpe of Asset B and the correlation between A and B is insufficient to induce any allocation to B in the optimal portfolio. There is a general increase in optimum portfolio Sharpe Ratio as we move towards the bottom right. This corresponds to increasing Sharpe of Asset B and decreasing correlation between A and B. Notice that the gradient of the surface (the rate of increase in portfolio Sharpe Ratio) increases quicker as correlation decreases than as the Sharpe Ratio of Asset B increases – you get more bang for your buck from attractive correlation characteristics than higher Sharpe Ratios.

### Some Landmarks

To act as landmarks, I have marked white dots where all the optimal portfolios from our preceding examples are located:

- The dot right in the middle at Sharpe = 0.5, Correlation = 0 is the optimal portfolio from the basic Markowitz Bullet.
- The 5 dots in a horizontal line (which includes the one above) are from the second chart showing constant correlation of 0, with varying Sharpe Ratio for Asset B.
- The 5 dots close together in a horizontal line at the top right are the high-Sharpe Asset B, high correlation chart.
- The 5 dots in a vertical line (which includes the first point above) are from the fourth chart showing constant Asset B Sharpe Ratio and varying correlation.

The temptation is to look for programs on the right-hand side of the chart by setting a minimum program performance measure during the screening process. This can be a mistake. Here’s the hack: Use correlation characteristics as part of your initial screen. You should look for programs that play well together – be prepared to sacrifice a little performance to get a good team player.

If you focus too much on the search for high Sharpe Ratio programs to add to your portfolio, you risk missing programs with attractive correlation characteristics.

There are literally thousands of programs in the top half of the chart. There will be barely any in the bottom right-hand corner – hens’ teeth, if you will. However, there are a good number in the bottom-left of the chart – the CTA space is home to most of them, as well as some vol. strategies that play both sides of the smile.

## Meanwhile, In the Real World

Lest you believe I am spinning you a tail about mythical programs inhabiting the lower half of my chart, I have a real-life example. Asset A in all my charts is none other than the S&P 500 Total Return Index (9 years and 9 months monthly ending October 2020 provided by Yahoo! Finance).

The red dot on the chart marks the location of 3D Capital’s 3D Defender Program. It currently has a Sharpe since inception of 0.46 and a correlation to the S&P500TR of -0.33. An aggressive allocation to 3D Defender in a hypothetical portfolio alongside the S&P had the potential to raise the expected Sharpe from 1 (the S&P alone) to 1.3 – not too shabby. (*Please note that 3D Defender was enhanced in November 2013 (7 years ago) which resulted in the program’s Sharpe Ratio nearly doubling while maintaining its negative correlation to the S&P 500. Past performance is not necessarily indicative of future results. 3D Defender involves risk of loss and may not achieve its objectives.)*

If you are interested in learning more, reach out to my good friend Tom O’Donnell – he will be happy to introduce you to their program. You can also visit their website: 3D Capital Management.

Finally, my alter-ego at Fides Asset Management constructs programs from traders with very low correlation to one another specifically to take advantage of this particular hedge fund hack. Reach out if you want to learn more.

Photo Credit: CaseOriginals on Pixabay

Hi! Thanks for the content, would it be possible to obtin your .py file for the Optimal Portfolio vs. Asset B Sharpe Ratio AND Correlation ?

It is for my personal curiosity!

Best,