The Zeta Coefficient – A New Way to Forecast Returns Using Risk


Peter White


Machine Learning


July 29, 2019

Image Credit: Debby Wong /

"Risk comes from not knowing what you’re doing"
- Warren Buffet

As far as superhuman powers go, the ability to accurately forecast returns probably wouldn't be many people's first choice. It's not as sexy as super speed or laser eyes. But, imagine the edge you'd have as an investor if you could look at an opportunity and predict the returns with precision. You would essentially remove all of the risk from investing: you'd be playing the game on easy mode!

Risk and returns are inter-twined. If you can assess risk accurately, you can generally predict returns accurately as well. Risk is the secret ingredient to investing in the same way that love is the secret ingredient to cooking. So, risk assessment is one of the most complex parts of investing and underpins many of our decisions. But, how well do we really understand it?

Each of us develops an intuitive understanding of risk during the course of our life, but it's a very difficult thing to quantify. It’s like trying to put a number on fun. The concept is too abstract, too vague to measure. We all know that dinner with friends is more fun than getting punched in the face, but how can you show that mathematically? The same issues arise when trying to compare investment risks. There's no obvious way to objectively measure risk, so, we often have to make a judgement call. And, you don't need me to tell you that when it comes to judgement calls that people don't always make the right decisions.

So, while you understand risk as a concept, it's not inherently clear how to factor it into your investment decisions. It's arguably the most important thing, but you can't really put it in the recipe because it's not tangible. You want a meal made with love, but if you saw a recipe that called for one cup of love, you'd be a bit sceptical. So, how do you properly factor risk into your return predictions?

CAPM – An Old Friend

It seems as though everything under the sun has been used to predict returns at one time or another, but one of the most well-known methods is the capital asset pricing model. Most financial professionals are familiar with CAPM and its ingenious method of exploring the relationship between risk and reward. It's a beautiful model that won over people's hearts and even won a Nobel Prize in 1990.

CAPM is a beautiful theory, but, much like origami master Robert J. Lang, only works on paper. Beyond the laundry list of assumptions it makes, CAPM has been around for so long and is in use by so many people that it's virtually impossible to gain an edge with it. And, despite the shortcomings, CAPM is still widely used as a tool because it simply and elegantly predicts expected returns in the face of risk. However, there are few--if any--who would say that CAPM is a complete and perfect model.

ZCAPM – The New Kid in Town

If Jay-Z has taught us anything, it's that if you want something to seem cool, you add a Z to the name, and ZCAPM is no different. Developed earlier this year by Wei Liu, James W. Kolari, and Jianhua Z. Huang, the ZCAPM is like CAPM on steroids. CAPM compares a security's return to the return of the market. ZCAPM adds another important factor, which it calls Zeta. Much like Beta in CAPM is a measure of a security's volatility compared to market returns, Zeta analyzes a security’s volatility compared to the cross sectional return dispersion of the market. That is, it measures how much a stock moves compared to the standard deviation of all of the individual stock returns during a particular period.

The idea that return dispersion is a predictive factor is nothing new (the idea has been well documented), but using it in this application is a real leap forward. ZCAPM is ready to be applied to portfolio construction, and, although the tests are minimal so far, early results are quite promising. Long only portfolios showed similar volatility to the CRSP index, but with somewhat higher returns (between 1.06% and 1.10% to the CRSP’s 0.89%). Other numbers are similarly impressive, although the results have yet to be verified by others.

So, what's the catch? Firstly, it's really hard to say “ZCAPM” aloud, which I think will really hold it back. But, more importantly, the premise is new and untested. Lots of models with long only portfolios look promising under initial test conditions, but they fall apart under rigorous examination. Also, further research is needed into how ZCAPM applies to other asset classes, including bonds and real estate, as well as international stocks. However, while both the scope and the breadth of the zeta coefficient are limited, the results are such that it is worth taking the time to investigate it further.

Zeta and Machine Learning

ZCAPM is an interesting concept, but for you to really tap into the full power of the coefficient, it's better to incorporate it into a machine-learning model. There is so much unavoidable risk when it comes to investing that you want to do your best to remove any potential source of human error that might compound it. And, the best way to remove human error in decision-making is to remove the human. Machine-learning algorithms can allow us to spot patterns and trends that are either too subtle for humans to notice or buried under too much data. And, more importantly, algorithms are free of the cognitive biases that haunt our choices, leaving them free to potentially make better decisions.

That said, machine-learning models are only as good as the inputs you feed them. The more significant the inputs, the more accurate the model. Think of your model like a dog sled where the inputs are your noble huskies. The more good dogs you have, the better the result. The ZCAPM results seem to indicate that the zeta coefficient contains an important signal about future performance, and you can harness that in your predictions. That's the sign of a good husky.

Investing is as much about beating everyone else as it is about beating the market, so, you need to stay a step ahead. Doing what everyone else is doing is not enough; you have to actively seek out the next thing. Machine learning helps immensely in this task as it allows you to develop and test new factors and models with a level of efficiency and complexity that human beings can never replicate. If the right algorithm can develop Star Trek like translating abilities, imagine what it can do for your portfolio.

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