# Active and Passive Portfolio Management with Latent Factors

## TL;DR

This paper develops a convex analysis-based method for optimal portfolio selection that combines active and passive strategies in a latent factor market model, incorporating filtering for hidden states and demonstrating improved performance through backtests.

## Contribution

It introduces a closed-form solution for portfolio optimization with latent Markovian factors, including a filtering approach for unobservable states, and validates it with historical backtests.

## Key findings

- Optimal portfolio strategy is the posterior average of state-specific strategies.
- The solution is unique and derived in closed form.
- Backtests show improved portfolio performance.

## Abstract

We address a portfolio selection problem that combines active (outperformance) and passive (tracking) objectives using techniques from convex analysis. We assume a general semimartingale market model where the assets' growth rate processes are driven by a latent factor. Using techniques from convex analysis we obtain a closed-form solution for the optimal portfolio and provide a theorem establishing its uniqueness. The motivation for incorporating latent factors is to achieve improved growth rate estimation, an otherwise notoriously difficult task. To this end, we focus on a model where growth rates are driven by an unobservable Markov chain. The solution in this case requires a filtering step to obtain posterior probabilities for the state of the Markov chain from asset price information, which are subsequently used to find the optimal allocation. We show the optimal strategy is the posterior average of the optimal strategies the investor would have held in each state assuming the Markov chain remains in that state. Finally, we implement a number of historical backtests to demonstrate the performance of the optimal portfolio.