investment viewpoints

Optimising predictive patterns for portfolios

Optimising predictive patterns for portfolios


As part of its on-going, quantitative research initiative, Lombard Odier Investment Managers invited Professor Semyon Malamud for an interactive session at the firm to meet our quantitative researchers from New York, London, Geneva and Hong Kong.

Professor Malamud is an Associate Professor at EPFL (École Polytechnique Fédérale de Lausanne), and a research fellow at the Centre for Economic Policy Research. He earned a PhD in mathematics from the Swiss Federal Institute of Technology, Zurich.

Professor Malamud presented a mathematical model that is designed to optimise portfolio construction. Following the presentation, the visiting academic discussed recent developments in the fields of trend-following and momentum research with the LOIM team.

Professor Malamud’s model aims to construct a generic portfolio with low risk and high returns, and find predictive relationships between signals and returns. The signals are built from historical data and can include momentum, carry, volatility or fundamental strategies. The returns belong to multiple asset classes such as bonds, currencies, commodities or stocks.

The model aims to construct a generic portfolio with low risk and high returns, and find predictive relationships between signals and returns.

In essence, the model attempts to optimise predictive relationships between models’ signals and future returns. Among other features, the model takes into account the rebalancing frequency and can have any number of different signals and returns streams. It can incorporate specific constraints, transaction costs, etc.

The ultimate aim is to build a portfolio with the following characteristics:

  • High expected returns.
  • High predictability – this involves finding signals containing significant and robust information about returns.
  • Low risk.
  • Robust (or free of so-called noise) – this means that the predictive relationship that is identified in the data must not be spurious and must be stable over time.
  • Looks for the ground state of the predictive relationship.

“I propose a mathematical methodology where return is predicted by a number of signals, not just risk. My method takes into account these five criteria (as above) and the general formula aims to detect multi-variate predictive patterns in data,” said Malamud. “It’s a natural way to identify weak vs strong signals, and it can be applicable to any data – stocks, futures, commodities, currencies or derivatives.”

“The aim is to build a portfolio that is the most resilient to the breakdown of the predictive relationships between signals. We want to find the ones that are the most robust.”


The spectral method to reduce noise

Malamud then moves beyond linear geometry, to Perron-Frobenius theory and historical physical probabilities, or what is deemed a spectral method in order to reduce noise. Perron-Frobenius imagines a discretized state space model – a state of the system follows a (high dimensional) Markov chain.

The objective is to find the non-linear ground state of the system for the largest Eigenvalues.1 It’s possible to construct a transition probability matrix that encodes information about the joint behaviour of signals and returns. But this matrix contains a lot of noise and Perron-Frobenius can be applied to extract the ground state.

Lastly, Malamud incorporates some of the theories developed by the US economist, Stephen Ross, in his 2011 paper entitled “The Recovery Theorem.2” The theorem takes multiple option prices and calculates, under certain technical conditions, the market consensus for the physical probabilities. Malamud applies this technique using historical underlying price data (instead of options) in order to find physical probabilities and thereby build optimal portfolios.

“I believe this Perron-Frobenius theory, this ability to identify predictive, but non-linear, predictive relationships from transition probabilities and prices – both physical and risk-neutral - is a very powerful tool that can be used to construct portfolios.” 

Finally, Malamud summarised his findings: “I use spectral analysis to find the ‘market ground state’: the most robust portfolio in terms of return predictability and risk. More generally, my approach allows me to decompose returns into a basis of ‘optimally predictable’ portfolios that is better suited for portfolio analysis than the more standard PCA (price component analysis)”.


Please find key terms in the glossary.



1 Eigenvalues and Eigenvectors are at the core of PCA analysis and are used to reduce noise in data.

2 The Recovery Theorem by Stephen A. Ross. NBER Working Paper No. 17323, August 2011. Please see:


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