Roughly speaking, a tensor is an array (generalization of a matrix) of numbers that transform according to certain rules when the array’s coordinates change. Fixed-income returns across countries can be seen as residing on tensor-like multidimensional data structures. Hence a tensor-valued approach allows identifying common factors behind international yield curves in the same way as principal components analysis identifies key factors behind a local yield curve. Estimated risk factors can be decomposed into two parallel risk domains, the maturity domain, and the country domain. This achieves a significant reduction in the number of parameters required to fully describe the international investment universe.

The below are quotes from the paper and some other articles (links below quote). Headings and text in brackets have been added.

The post ties in with the SRSV summary on information efficiency.

### What is a tensor?

“A Tensor is a mathematical object similar to, but more general than, a vector and often represented by an array of components that describe functions relevant to coordinates of a space. Put simply, a Tensor is an array of numbers that transform according to certain rules under a change of coordinates.” [DeepAI]

“An nth-rank tensor in m-dimensional space is a mathematical object that has n indices and m^n components and obeys certain transformation rules. Each index of a tensor ranges over the number of dimensions of space…Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.” [MathWorld]

“A tensor is often thought of as a generalized matrix. That is, it could be a 1-D matrix (a vector is actually such a tensor), a 3-D matrix (something like a cube of numbers), even a 0-D matrix (a single number), or a higher dimensional structure that is harder to visualize. The dimension of the tensor is called its rank. But this description misses the most important property of a tensor! A tensor is a mathematical entity that lives in a structure and interacts with other mathematical entities. If one transforms the other entities in the structure in a regular way, then the tensor must obey a related transformation rule…This ‘dynamical’ property of a tensor is the key that distinguishes it from a mere matrix. It’s a team player whose numerical values shift around along with those of its teammates when a transformation is introduced that affects all of them. Any rank-2 tensor can be represented as a matrix, but not every matrix is really a rank-2 tensor.” [Medium]

### What do tensors have to do with global fixed income markets?

“Global fixed income returns span across multiple maturities and economies, that is, they naturally reside on multidimensional data structures referred to as tensors.”

“Market participants have long recognized the importance of identifying the common factors that affect the returns of securities within asset classes. In such a task it is critical to distinguish the common risks that have general impact on the returns of most securities from the idiosyncratic risks that influence securities individually.”

“A significant portion of the fixed income literature has been devoted to the technique of principal component analysis (PCA) to provide a parsimonious interpretation to the dynamics of the term structure of interest rates. Empirical results suggest that three latent factors, referred to as level, slope and curvature, are required to almost fully reflect the behaviour of the entire term structure…The degree of robustness in these findings has made PCA a fundamental building block for characterizing single-economy interest rate curves.”

“However, the growing interconnectedness of the international markets presents a major challenge for risk management of fixed-income securities, owing to the high correlation between interest rates across maturities and countries…Trades are typically hedged by offsetting their domestic-curve principal components. This leaves such strategies unprotected to cross-country risk arising from global macroeconomic events… For this reason, a parsimonious model to describe the co-variation of interest rates at the relevant maturities and in the relevant countries appears necessary… investors to adequately identify and manage risk.”

It is only through tensor analysis that we have the opportunity to develop sophisticated models capturing the interactions between the entirety of interest rate curves. This motivates the development of multilinear techniques, which have eventually found its place in many real-world applications where tensors naturally reside…Accordingly, we have developed a framework that employs the structure-aware multilinear algebra to rigorously model the risk factors shared by an international universe of fixed income returns.”

### What does the tensor model tell about global risk factors?

“We introduce a tensor-valued approach to model the global risks shared by multiple interest rate curves…The estimated risk factors can be analytically decomposed into two parallel domains of risk: (i) maturity-domain factors which are shared by all countries; and (ii) country-domain factors which are shared by all maturities.”

“By virtue of the multilinear approach (as opposed to the current ‘flat-view’ multivariate ones), the proposed approach has been shown to decompose the overall multivariate covariance structure of international asset returns into maturity-domain covariance and country-domain covariance. In this way, the proposed analysis: (i) achieves a significant reduction in the number of parameters required to fully describe the international investment universe; and (ii) offers a physically interpretable setting for estimating and identifying global risk factors.”

“An empirical analysis confirms the existence of common global risk factors shared by eight developed economies. The resulting maturity-domain and country-domain factors are shown to provide compact and physically meaningful insight into the global macroeconomic environment.”

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Ralph Sueppel is founder and director of SRSV, a project dedicated to socially responsible macro trading strategies. He has worked in economics and finance for over 25 years for investment banks, the European Central Bank and leading hedge funds. At present, he is head of research and quantitative strategies at Macrosynergy Partners.