Pascal Traccucci, Luc Dumontier, Guillaume Garchery and Benjamin Jacot present an extended reverse stress test (ERST) triptych approach with three variables: level of plausibility, level of loss and scenario. Any two of these variables, can be derived, provided the third is given as input. A new version of the Levenberg-Marquardt optimisation algorithm is introduced to derive the ERST in certain complex cases.

**Introduction: the case of ARP portfolios**

Academic theory has been mined to support the development of investment solutions containing an ever-increasing number of factors. Over the last decade, academics and practitioners have shown traditional asset classes offer limited diversification, especially in market downturns. In response, they have delved into modern portfolio theory (MPT) to identify the microeconomic factors that are the backbone of alternative risk premia (ARP) solutions. The ARP 1.0 approach combines 10–15 different long/short portfolios capturing standard investment styles such as value, carry, momentum, low risk and liquidity across a broad range of traditional asset classes. For further diversification, the ARP 2.0 approach combines up to 30 strategies by including investment banking-style premia likely to use instruments with quadratic profiles.

Many risk management frameworks cannot properly account for nonlinear profiles and assess the risk of loss associated with combining an unusually high number of strategies. Specifically, historical value-at-risk is an instantaneous risk indicator and does not correspond to a clearly identified scenario; hence the need for complimentary stress tests. To build a stresstesting tool, the dataset must be simplified, and historical or predefined scenarios are used without quantifying their plausibility. Thus, parametric VAR imposes dependence on a model to benefit from an analysis framework in the form of a VAR and a sensitivity of this VAR to all the parameters of the model. This requires several numerical problems to be addressed, especially in case of quadratic profit and loss (P&L). This article presents an innovative approach: the extended reverse stress test (ERST), following on from the work of Breuer et al (2009) and Mouy et al (2017). This approach is able, with low technical costs,1 to deliver two of three parameters, provided the third is given as input. The three parameters are scenario, level of plausibility and level of loss (see figure 1). The result is a more meaningful risk measure and one that corresponds to a clearly identified scenario.

In what follows, S is defined as a scenario. It is a vector with length n, which equals the number of risk factors to which the portfolio is exposed. In addition, the covariance matrix of the risk factors will be denoted by…