Risk Decomposition

Unified Approach to Portfolio Optimization using the `Gain Probability Density Function’ and Applications ArXiv ID: 2512.11649 “View on arXiv” Authors: Jean-Patrick Mascomère, Jérémie Messud, Yagnik Chatterjee, Isabel Barros Garcia Abstract This article proposes a unified framework for portfolio optimization (PO), recognizing an object called the `gain probability density function (PDF)’ as the fundamental object of the problem from which any objective function could be derived. The gain PDF has the advantage of being 1-dimensional for any given portfolio and thus is easy to visualize and interpret. The framework allows us to naturally incorporate all existing approaches (Markowitz, CVaR-deviation, higher moments…) and represents an interesting basis to develop new approaches. It leads us to propose a method to directly match a target PDF defined by the portfolio manager, giving them maximal control on the PO problem and moving beyond approaches that focus only on expected return and risk. As an example, we develop an application involving a new objective function to control high profits, to be applied after a conventional PO (including expected return and risk criteria) and thus leading to sub-optimality w.r.t. the conventional objective function. We then propose a methodology to quantify a cost associated with this optimality deviation in a common budget unit, providing a meaningful information to portfolio managers. Numerical experiments considering portfolios with energy-producing assets illustrate our approach. The framework is flexible and can be applied to other sectors (financial assets, etc). ...

Trading with the Devil: Risk and Return in Foundation Model Strategies ArXiv ID: 2510.17165 “View on arXiv” Authors: Jinrui Zhang Abstract Foundation models - already transformative in domains such as natural language processing - are now starting to emerge for time-series tasks in finance. While these pretrained architectures promise versatile predictive signals, little is known about how they shape the risk profiles of the trading strategies built atop them, leaving practitioners reluctant to commit serious capital. In this paper, we propose an extension to the Capital Asset Pricing Model (CAPM) that disentangles the systematic risk introduced by a shared foundation model - potentially capable of generating alpha if the underlying model is genuinely predictive - from the idiosyncratic risk attributable to custom fine-tuning, which typically accrues no systematic premium. To enable a practical estimation of these separate risks, we align this decomposition with the concepts of uncertainty disentanglement, casting systematic risk as epistemic uncertainty (rooted in the pretrained model) and idiosyncratic risk as aleatory uncertainty (introduced during custom adaptations). Under the Aleatory Collapse Assumption, we illustrate how Monte Carlo dropout - among other methods in the uncertainty-quantization toolkit - can directly measure the epistemic risk, thereby mapping trading strategies to a more transparent risk-return plane. Our experiments show that isolating these distinct risk factors yields deeper insights into the performance limits of foundation-model-based strategies, their model degradation over time, and potential avenues for targeted refinements. Taken together, our results highlight both the promise and the pitfalls of deploying large pretrained models in competitive financial markets. ...

Market-Based Portfolio Variance ArXiv ID: 2504.07929 “View on arXiv” Authors: Unknown Abstract The variance measures the portfolio risks the investors are taking. The investor, who holds his portfolio and doesn’t trade his shares, at the current time can use the time series of the market trades that were made during the averaging interval with the securities of his portfolio and assess the current return, variance, and hence the current risks of his portfolio. We show how the time series of trades with the securities of the portfolio determine the time series of trades with the portfolio as a single market security. The time series of trades with the portfolio determine its return and variance in the same form as the time series of trades with securities determine their returns and variances. The description of any portfolio and any single market security is equal. The time series of the portfolio trades define the decomposition of the portfolio variance by its securities, which is a quadratic form in the variables of relative amounts invested into securities. Its coefficients themselves are quadratic forms in the variables of relative numbers of shares of its securities. If one assumes that the volumes of all consecutive deals with each security are constant, the decomposition of the portfolio variance coincides with Markowitz’s (1952) variance, which ignores the effects of random trade volumes. The use of the variance that accounts for the randomness of trade volumes could help majors like BlackRock, JP Morgan, and the U.S. Fed to adjust their models, like Aladdin and Azimov, to the reality of random markets. ...