# Smart betas, return models and the tangency portfolio weights

## TL;DR

This paper derives formulas for optimal investment portfolios that maximize returns relative to risk, showing how they can match or differ from popular investment strategies.

## Contribution

The paper provides closed-form expressions for tangency portfolio weights under various return models, including a compound symmetric correlation matrix and CAPM.

## Key findings

- Tangency portfolio weights derived for a compound symmetric correlation matrix return model.
- Tangency portfolio weights for CAPM return model may differ significantly from market-weighted portfolios.
- Tangency portfolios can range from highly diversified to concentrated depending on the return model.

## Abstract

In this paper, we analytically derive closed-form expressions for the tangency portfolio weights: the fully invested portfolio that maximizes the expected return over the risk-free rate, relative to the volatility of the portfolio return. We explicitly derive this portfolio from a range of underlying return models and show examples where it coincides with different well-known smart beta products. Specifically, we find the closed-form expression for the tangency portfolio weights for a return model with compound symmetric correlation matrix. We also deduce the tangency portfolio weights for the CAPM return model and illustrate in a case study that the estimated tangency portfolio weights may distinctly deviate from the market value weighted portfolio. Furthermore, we show that depending on the return model, the tangency portfolio weights may take a diverse set of shapes; from very diversified to highly concentrated portfolios.