Why Maximizing Treynor or Jensen’s Alpha Can Concentrate a Portfolio

Summary

The document considers how to optimize a portfolio for the Treynor ratio or Jensen’s alpha, in contrast with maximizing the Sharpe ratio. Its central point is that maximizing either measure by itself can produce a degenerate solution: allocate all weight to the single asset with the highest value of the chosen measure. These ratios, as presented, do not reward diversification through a direct penalty for portfolio return variability.

The answer then gives a broader construction for a quantitative asset attribute. If portfolio exposure is defined as the weighted sum of asset-level attribute values, one can target a unit of exposure while minimizing portfolio variance. The resulting weights depend on the inverse covariance matrix and the attribute vector. This offers a way to incorporate risk and diversification constraints, but the document does not define a full optimization framework for Treynor or Jensen objectives, address practical constraints such as long-only weights, or compare empirical outcomes.

Key ideas

• Maximizing Treynor ratio or Jensen’s alpha alone can assign all capital to one asset.
• These measures do not directly reward diversification in the formulation discussed.
• A portfolio can target a unit exposure to an asset attribute while minimizing variance.
• The minimum-variance exposure portfolio uses the inverse covariance matrix and the attribute vector.

Tags

# Create optimal portfolio by Treynor and Jensens Alpha


# Create optimal portfolio by Treynor and Jensens Alpha












I would like to know which formula to use in order to optimize a portfolio based on highest Treynor and Jensens Alpha. I am aware that usually one optimize a portfolio by highest Sharpe ratio (the tangency portfolio) by following formula:

$$\textbf{w}_{tan} = \frac{1}{(\boldsymbol{\mu}^e)^\top\boldsymbol{\Sigma}^{-1}\textbf{1}}\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}^e. $$

Which formula can I use to maximize Treynor or Jensens Alpha, or do I need to create a maximization problem?

## Answer by Brian B (score 5)

https://quant.stackexchange.com/a/11366

This optimization is trivial

$$ w^{T,J}_i = \begin{cases} 1 \quad \text{if } i=\arg \max_i R^{T,J}(S_i) \\0 \quad \text{otherwise} \end{cases} $$

That is to say, when you optimize only one weight will be nonzero. That's because these ratios incorporate no notion of distributional width, and therefore do not reward diversification.

With no concentration penalty, they will simply put all their weight on the most "attractive" asset.

More generally, if you take any quantitative attribute $a$ of a stock, and apply it as a whole to the portfolio by summing it over the entire collection, you get exposure

$$ a_\bf{w}=\bf{w}^*\bf{a} $$

To achieve unit exposure to this attribute while minimizing variance, you optimize (using Lagrange multipliers), obtaining

$$ \bf{w}_a=\frac{\bf{\Sigma}^{-1}\bf{a}}{\bf{a}^*\bf{\Sigma}^{-1}\bf{a}} $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.