**TOPICS:** Portfolio theory, manager selection, performance measurement

Economics (and, by extension, finance) is arguably one of the most mathematical fields of research. About 62% of Nobel laureates in economics have a finite Erdos number—a higher percentage than in chemistry or medicine. Also, the average Erdos number of Nobel laureates in economics is 4.11 (median 4), which is very close to the 3.36 average (median 3) of Fields medalists, compared to the 5.63 average (median 5) of Nobel laureates in physics (see Exhibit 1). But as Focardi and Fabozzi [2016] argued in a recent editorial, economists’ choice of math may be inadequate to model the complexity of social institutions.

It is easy for financial practitioners to relate to these authors’ arguments. Outperforming investment firms such as Renaissance Technologies, Two Sigma, D.E. Shaw, TGS, Winton, and so on, are led by eminent scientists and mathematicians without a finance background (Lux [2000]). In fact, those firms tend to discriminate against candidates with a finance or economics background (Manzoor [2013]). Even traditional “macro” hedge funds, formerly staffed with economists, are filling an increasing proportion of their portfolio management jobs with STEM (science, technology, engineering, and math) graduates (Childs and Wigglesworth [2016]). They have come to realize that math works—just not the math taught in financial programs.

In a constructive spirit, I offer some advice on how students can increase their chances of having a successful career in 21st century finance. Practitioners seeking to enhance their skillset may also draw some ideas.

**GRAPH THEORY, TOPOLOGY, DISCRETE MATH**

Econometric models are employed every day with blind fervor, without much challenge to their sense or effectiveness. And although multiple-testing has been denounced as outright unethical by the main professional body of statisticians (American Statistical Society [1999]), the Econometric Society remains shamefully silent on the subject. The consequences of using overly simplistic and outdated statistical methods are palpable: Professionals often complain that betas and factor loadings need to be “recalibrated” all the time. Cointegration relations are notoriously unstable (i.e., useless). Millions of regressions can be performed per minute, with automated “factor selection” routines that lead to discoveries such as GDP being predicted by the size of the male organ (Westling [2011]). A researcher dared theorize that “heightened self-esteem” may justify this finding… good grief! Beware of economists bearing “factors.”

Mathematicians learn beyond linear algebra and calculus, and so to them the love affair between economics and econometrics seems unhealthy. *Graph theory* is better suited to capture the complexity inherent to financial systems. The reason is, unlike the geometry that underlies all econometrics, graph theory inherits the notion of “hierarchy” that pervades *topological* objects. Hierarchical relations are key to understanding complex phenomena, such as contagion within a system. And although much has been written about the crisis of 2008, most financial researchers seem oblivious to this powerful tool at their disposal. In general, *discrete math* applications tend to be robust, because the structure imposes a countable (even if infinite) number of feasible solutions.

**INFORMATION THEORY, SIGNAL PROCESSING**

Much of financial theory deals with asymmetric information. A case in point is market microstructure, in which several models relate the size of the bid–ask spread to the probability that market makers may be providing liquidity to an informed trader. Unfortunately, researchers often ruin this promising idea by smashing it with their econometric sledgehammer. It would have made more sense to derive the informational content of trading sequences via *information theory*. There is some evidence that such an approach may have succeeded. Several mathematicians have made fortunes applying information theory and *signal processing* (Poundstone [2005], Patterson and Strasburg [2010]). But economists’ aversion to these tools dates back to the nasty dispute between Paul Samuelson and the proponents of Kelly’s criterion (Samuelson [1979]). It is time to move on, be practical, and use whatever works.

**MACHINE LEARNING**

In a remarkable study, De Miguel et al. [2009] showed that the out-of-sample performance of the 1/N or “naïve allocation” beats the entire set of commonly used portfolio optimization techniques. Think about it: Ignoring all risk factors delivers better results than taking them into consideration! The reason is that inverting covariance matrices often leads to errors of such magnitude that they entirely offset the benefits of diversification. This is not a good omen for linear finance, factor investing, and smart-beta funds.

Portfolio optimization is not the only possible way of building a portfolio. Several modern *unsupervised*
*machine learning methods* study the structure of financial “big data” and determine clusters of instruments that should share allocations (for a recent example, see López de Prado [2016]). So paradoxically, even though Markowitz-style allocation is optimal in sample, it loses out of sample to alternative methods that are suboptimal in sample.

**PARALLEL PROCESSING, QUANTUM SUPERCOMPUTING**

Financial economists have traditionally shown a strong preference for closed-form solutions. Like many pure mathematicians, they would have little use for a supercomputer, much less a quantum computer. Consider the National Hurricane Center: Weather systems are trivial compared to modern markets (after all, hurricanes do not adapt their behavior in a Lucasian manner), and yet weather centers around the world use petaflops of computing power every day.* Algorithm complexity*, *supercomputing*, and o*perations research* are three important courses missing in most finance syllabi. *Quantum computers* can solve NP-complete financial problems, such as the optimal trading trajectory problems (Rosenberg et al. [2016]).

**EXPERIMENTAL TECHNIQUES**

It is not possible to build a scientific discipline without experimentation. Financial academics have attempted to overcome the experimentation obstacle with an ingenious artifact: sophisticated historical simulations, or backtesting. But historical simulation is not out-of-sample experimentation, and most published backtests are tainted with overfitting and selection bias (Bailey et al. [2014]; Bailey and López de Prado [2014]). Investment firms are the closest things we have to financial laboratories. Students need to learn *mathematics by experimentation* in particular (Borwein and Bailey [2008]), and *experimental techniques* in general, so they can apply those on their jobs.

**A SOLUTION TO THE RESEARCH CRISIS**

Would you board a Boeing 787 knowing that your pilot has never flown a real plane—and all his or her experience comes from a (rather stylized) flight simulator of a Boeing 247? Investors demand a track record from portfolio managers, not a list of published (biased) backtests. This brings us to one solution to the multiple-testing crisis that has invalidated most results in empirical finance (Harvey et al. [2016]). Financial academics could commit a portion of their salaries (say 10%) to the validation of their articles. Those funds would be entrusted to the university’s endowment management company, which would invest them according to their published theory. This out-of-sample experiment would leave an undeniable, certifiable, and auditable trail, similar to the decades-long track record of the quant investors listed at the beginning of this article. Candidates for the Nobel Prize in physics have accumulated a wealth of experimental data corroborating their theories. In finance, that experimental evidence is called a *track record*. A simple brokerage account would provide some credibility that the theories taught to students are worth the rampant tuition fees.

**IDEOLOGUES MAKE POOR INVESTORS**

Some of my colleagues wonder why so many mathematicians become ultrawealthy or billionaire investors, while the number of billionaire finance academics is about zero (Horst [2016]). Very few mathematicians are managing funds, so they are scoring much higher success rates than their finance counterparts. One possible explanation is that mathematicians are not ideologues. Leading financial academic conferences often bear some resemblance to ecumenical councils, where one faction exposes another’s perceived heresy.

We will never know which financial theories are correct because finance is not an experimental science. Financial journals almost never retract a published article. Selection bias makes most claimed financial discoveries likely wrong (Harvey et al. [2016]). This calls for a moratorium on financial indoctrination. Professors do a disservice to their students when they present backtested theories as scientific “truth.” Students should be trained on abstract mathematical techniques to model complex systems, rather than dogmatic views of how markets should function in an axiomatic world. The hiring manager is the one who decides which theory will be tested out of sample, with real money at risk. Graduates need to know the tools required to execute that experiment rigorously, and ingrained prejudices only make that task harder.

University programs are designed by career academics with little or no industry experience. Some programs have adjunct level, nontenure track, professors dubbed “of practice.” Unfortunately, these professors do not run departments or design the syllabi. This state of affairs inevitably leads to students being trained to succeed in academia—not in the industry. If finance departments care about their students, they must rely much more on researchers who understand what it takes to succeed in the real world.

## ENDNOTE

The statements made in this communication are strictly those of the authors and do not represent the views of Guggenheim Partners or its affiliates. No investment advice or particular course of action is recommended.

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