Uncovering the Mathematics behind the World’s most Profitable Hedge Fund

A Continual Learner
9 min readJan 8, 2022

Renaissance Technologies (RenTec), led by Jim Simons, is an American hedge fund specializing in systematic trading using quantitative models derived from mathematical and statistical analysis.

RenTec currently manages over $10 Billion for Medallion Fund, but what is most impressive, is that this Fund has always made profit, year after year. From 1990 to 2018, RenTec posted a return of 66%/yr on average, even with 5% management fee and 44% performance fee. RenTec’s Medallion Fund has generated $100B+ in cumulative profits.

The secret? Be right 50.75% of the time, 100% of the time. Just as casinos handle so many daily bets that they only need to profit from a bit more than half of those wagers, RenTec only needed to be right 50.75% of the time, all the time.

What differentiated RenTec against other hedge funds was the trading system, governed by specific, preset rules aimed at removing emotions and irrationality by analyzing risk, and hedge investment positions.

Jim Simon’s Medallion Fund has had enormous returns from a diversified portfolio generating relatively little volatility and correlation to the overall market. This is how he did it.


James H. Simons graduated from Massachusetts Institute of Technology with a Bachelor’s in Mathematics at age 20. And at age 23, he graduated from the University of California, Berkeley with a Ph.D. in Mathematics. At age 26, he joined the Institute for Defense Analyses, an elite research organization part of the National Security Agency. The IDA taught Simons how to develop mathematical models to discern and interpret patterns in seemingly meaningless data.

After 4 years at IDA, Jim Simons chaired the Mathematics Department at Stony Brook University. In 1974, Simons made a discovery related to quantifying shapes in curved, three-dimensional spaces. He showed his work to Shiing-Shen Chern, who realized the insight could be extended to all dimensions. Chern and Simons published “Characteristic Forms and Geometric Invariants,” a paper that introduced Chern-Simons invariants — an invariant is a property that remains unchanged, even while undergoing particular kinds of transformations — which proved useful in various areas of mathematics.

Chern-Simons theory had applications to a range of areas in physics, including condensed matter, string theory, and supergravity. It even became crucial to methods used by Microsoft and others in their attempts to develop quantum computers capable of solving problems vexing modern computers, such as drug development and artificial intelligence. In 1976, they were awarded Oswald Veblen Prize for Geometry, the highest award for geometry.

While at Stony Brook University, he started trading commodities, which is a marketplace for buying, selling, and trading raw materials or primary products. His company, Monemetrics, made profit but the volatility of the market and emotional swings of day-trading were a gut-wrenching experience for Simons. He set out to establish a Hedge Fund, Renaissance Technologies, that incorporated complex mathematics to find signals that predicted price movements. In order to find signals, he needed data.

There is no Data like more Data

In the beginning, the data consisted of annual and quarterly earnings reports, records of stock trades by corporate executives, government reports, and economic predictions and papers. Then they began collecting every trade order, including those that hadn’t been completed. They also checked for commodity exchanges, futures tables, and archives of the Wall Street Journal and other newspapers. including the British pound, Swiss franc, Eurodollars, and commodities including wheat, corn, and sugar, dating back as far back as the 1800s. They also collected weather patterns, social media feeds, and satellite imagery, even going as far as collecting astrological data.

The data was used to detect unusual and idiosyncratic spikes, dips, or gaps in their collection of prices. By identifying comparable trading situations and tracking what subsequently happened to prices, they could develop a sophisticated and accurate forecasting model capable of detecting overlooked patterns and anomalies.

The result? The Medallion Fund would employ a single, monolithic trading model so it could draw on the firm’s vast trove of pricing data, detecting correlations, opportunities, and other signals across various asset classes. Though this meant employees had full access to the system’s code, as staffers could walk out the door, join a rival, and tap Renaissance’s secret, Jim Simons believed a single trading model could help solve the scalability of the portfolio.

Just as astronomers set up powerful machines to continuously scan the galaxy for unusual phenomena, the Medallion Fund’s model would scan the market and generate predictions for various commodity prices based on complex patterns, clusters, and correlations — an early form of machine learning.

The model adapted to its own trading system by giving a degree of probability to every guess and updating its best estimates as it received new information. The model was able to analyze large amounts of messy, complicated pricing data to predict future prices, discerning true signals from random market fluctuations. The model factored in risk, cost, impact, and market structure, and optimized buying in and out throughout the hour in unpredictable ways, preserving its trading signal.

It wasn’t immediately obvious why some of the new trading signals worked, but as long as they had p-values, or probability values, under 0.01 — meaning they appeared statistically significant, with a low probability of being statistical mirages — they were added to the system.

Renaissance Technologies concluded that there are reliable mathematical relationships between all these data. Applying data science, the researchers achieved a better sense of when various factors were relevant, how they interrelated, and the frequency with which they influenced shares. Below are the strategies and formulas implemented for the Medallion Fund’s model.

Reversion-to-the-Mean Strategy

Reversion-to-the-mean strategy is retracing a condition back to its long-run average state. Clusters of stocks will return to their historic norms, reverting to its understood state or secular trend. For example, buy futures contracts if they opened at unusually low prices compared with their previous closing price, and sell if prices began the day much higher than their previous close. Constructing a portfolio of these investments figured to dampen the fund’s volatility, giving it a high Sharpe ratio.

Hidden Markov Chain Strategy

Renaissance Technologies first incorporated Baum-Welch algorithm analyzing Hidden Markov Chain, which are sequences of events in which the probability of what happens next depends only on the current state, not past events. In a Markov chain, it is impossible to predict future steps with certainty, but future steps can be predicted with some degree of accuracy if one relies on a capable model.

A hidden Markov Chain is one in which the chain of events is governed by unknown, underlying parameters or variables. One sees the results of the chain but not the “states” that help explain the progression of the chain. Some investors liken financial markets, speech recognition patterns, and other complex chains of events to hidden Markov models.

The Baum-Welch algorithm provided a way to estimate probabilities and parameters within these complex sequences with little more information than the output of the processes. Baum’s algorithm, which allows a computer to teach itself states and probabilities, is seen as one of the twentieth century’s notable advances in machine learning, paving the way for breakthroughs affecting the lives of millions in fields from genomics to weather prediction.

Interestingly, Baum-Welch Algorithm is utilized for Speech-recognition as the computers are fed data of recorded speech and written text to develop a probabilistic, statistical model. Language could be modeled like a game of chance. At any point in a sentence, there exists a certain probability of what might come next, which can be estimated based on past, common usage. The algorithm was able to create a model capable of digesting uncertain jumbles of information and generating reliable guesses about what might come next.

The same principle was used to predict how the market would react. That is why Renaissance Technologies hired employees from IBM’s Thomas Watson Research Centre that focused on speech-recognition and incorporated the mathematics into the Renaissance’s new Fund, Renaissance International Equity Fund.

Pairs Strategy

Pairs strategy involves matching a long position with a short position in two stocks with a high correlation. Renaissance Technologies uncovered a series of combination effects, such as the propensity of pairs of investments — such as gold and silver, or heating oil and crude oil — to move in the same direction at certain times in the trading day compared with others.

Statistical Arbitrage Strategy

As a more advanced pairs strategy, Statistical Arbitrage, or stat arb strategy utilizes hundreds of different securities and determines the correlations. Time frame of Statistical Arbitrage can be as low as few seconds but up to multiple days. There are two phases: “scoring” provides a ranking to each available stock according to investment desirability, and “risk reduction” combines desirable stocks into a specifically-designed portfolio aiming to lower risk. The fund will open both a long position and short position simultaneously to take advantage of inefficient pricing in correlated securities. Then wait until prices diverge beyond a certain threshold, then short “winner” and buy “loser.” Finally, reverse positions once they converge.

Eliminating Slippage

Slippage refers to the difference between the expected price of a trade and the price at which the trade is executed. This normally transpires during high periods of volatility as well as periods whereby orders cannot be matched at desired prices. Renaissance discovered their trades had transaction costs, or slippage, for their trades and began writing a computer program to track how far their trades strayed from the ideal state, in which trading costs barely weighed on the fund’s performance. The firm could run a simulator that subtracted these trading costs from the prices they had received, instantly isolating how much they were missing out.

Brownian Model

2-dimensional random walk of a silver atom on an Ag(111) surface

The Brownian Motion is the random motion of particles suspended in a medium. This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of a plant immersed in water. In 1905, Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules. The Brownian Motion models for the financial market, as these assets have continuous prices evolving continuously in time. Renaissance Technologies used the concept of Brownian Motion to model random behavior of assets and commodities that evolves over time.


Leverage magnification

Leverage results from using borrowed capital as a funding source when investing to generate returns on risk capital. RenTec’s Medallion Fund typically levers 12.5x (and can get up to 20x). Without leverage, its returns are comparable to S&P 500. The firm makes 150k-300k small trades a day and holding periods are very short, usually around for 2 days.

Success of Renaissance Technologies

Simons came from a different world and enjoyed a unique perspective. He was accustomed to scrutinizing large data sets and detecting order where others saw randomness. But the biggest factor to his success and to his Hedge Fund was his employees: 250 staffers and over sixty PhDs, including experts in artificial intelligence, quantum physicists, computational linguists, statisticians, and number theorists. Scientists and mathematicians are trained to dig below the surface of the chaotic, natural world to search for unexpected simplicity, structure, and even beauty. Simons’s culture focused on insisting everyone at the firm actively share their work with each other. Though Jim Simons retired in 2009 to pursue philanthropic calling, the success of Renaissance Technologies continues to this day.



A Continual Learner

With passion for technology, military, and economics, In Bok Lee has created A Continual Learner with the purpose of sharing knowledge.