# Covariance Matrix of Trends and Risk Premia in Portfolio Allocation

Optimal portfolio allocation is a hard problem to solve in finance. Determining the best way to allocate assets and strategies across different investment vehicles in order to achieve the best possible risk-adjusted return is a complex task that has been studied by mathematicians and financial experts for many years. Despite all of the research that has been done, there is still no definitive answer to this question. There are a number of methods available for finding an optimal portfolio, and each has its own strengths and weaknesses. Reference  introduced a new method for allocating trend following strategies that utilized not only the covariance matrix of returns but also the covariance matrix of trends and risk premia.

We derive a theoretical setting to yield implementable solutions of the allocation problem of trend following portfolios. The main formula of the paper describes the optimal portfolio as depending on the covariance matrix of returns, the covariance matrix of trends and the risk premia.

We implement the formula to gauge the performance of five well established portfolios (Agnostic Risk Parity, Markowitz, Equally Weighted, Risk Parity and Trend on Risk Parity), using daily data from futures markets of 24 stock indexes, 14 bonds indexes and 9 FX, from 1985 to 2020.

Our main empirical finding is the optimal combination of the three best portfolios produces a Sharpe ratio of 1:37, with their respective optimal weights of 19.5% (ARP), 51% (RP), and 30% (ToRP) which combines both traditional and alternative approach. Consistent with related recent literature, we confirm that RP portfolio, which is a proxy of the traditional and well diversified portfolio is a important driver of performance. Furthermore, we show that the combination between ARP and ToRP is the best solution in term of Sharpe ratio for the trend following approach and the alternative benchmark as they tend to minimize the correlation among assets.

In the context of a portfolio optimization problem, the article solved for the optimal allocation amongst a set of trend following strategies. It utilized the covariance matrix of returns, trends, and risk premia in its optimization algorithm. The allocation scheme combined both traditional and alternative approaches, offering a better Sharpe ratio than each of the previous methods individually. We note, however, that it still depends on historical data and has not been tested for robustness.

References

 Sébastien Valeyre, Optimal trend following portfolios, (2021), arXiv:2201.06635