Long-run total real returns of the stock market are approximately equal to long-run real earnings growth plus average dividend yield. However, earnings can be distributed to shareholders not only via dividends, but via buybacks and debt retirement. Thus the total returns minus earnings growth can be considered as implied dividend yield. This quantity must be stable in the long run. If the converse is true: this quantity is abnormally high in the last few years, then the market is overpriced. A measure of this bubble is (detrended) cumulative sum of differences. We regress next year's implied dividend yield upon this current bubble measure. We simulate future returns, starting from current market conditions. We reject the conventional wisdom that currently the market is overpriced. In our model the current market is undervalued and is likely to grow faster than historically.

We consider a multi-step algorithm for the computation of the historical expected shortfall such as defined by the Basel Minimum Capital Requirements for Market Risk. At each step of the algorithm, we use Monte Carlo simulations to reduce the number of historical scenarios that potentially belong to the set of worst scenarios. The number of simulations increases as the number of candidate scenarios is reduced and the distance between them diminishes. For the most naive scheme, we show that the L p-error of the estimator of the Expected Shortfall is bounded by a linear combination of the probabilities of inversion of favorable and unfavorable scenarios at each step, and of the last step Monte Carlo error associated to each scenario. By using concentration inequalities, we then show that, for sub-gamma pricing errors, the probabilities of inversion converge at an exponential rate in the number of simulated paths. We then propose an adaptative version in which the algorithm improves step by step its knowledge on the unknown parameters of interest: mean and variance of the Monte Carlo estimators of the different scenarios. Both schemes can be optimized by using dynamic programming algorithms that can be solved off-line. To our knowledge, these are the first non-asymptotic bounds for such estimators. Our hypotheses are weak enough to allow for the use of estimators for the different scenarios and steps based on the same random variables, which, in practice, reduces considerably the computational effort. First numerical tests are performed.

The objective of this paper is to develop a duality between a novel Martingale Entropy Optimal Transport problem (D) and an associated optimization problem (P). In (D) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by (Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018) but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al. The Problem (D) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive relaxation terms D, which may not have a divergence formulation. In Problem (P) the objective functional, associated via Fenchel coniugacy to the terms D, is not any more linear, as in OT or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a non linear subhedging value. Our results allow us to establish a novel nonlinear robust pricing-hedging duality in financial mathematics, which covers a wide range of known robust results in its generality.

This paper introduces a new functional optimization approach to portfolio optimization problems by treating the unknown weight vector as a function of past values instead of treating them as fixed unknown coefficients in the majority of studies. We first show that the optimal solution, in general, is not a constant function. We give the optimal conditions for a vector function to be the solution, and hence give the conditions for a plug-in solution (replacing the unknown mean and variance by certain estimates based on past values) to be optimal. After showing that the plug-in solutions are sub-optimal in general, we propose gradient-ascent algorithms to solve the functional optimization for mean-variance portfolio management with theorems for convergence provided. Simulations and empirical studies show that our approach can perform significantly better than the plug-in approach.

This paper presents novel evidence for the prevalence of deviations from rational behavior in human decision making - and for the corresponding causes and consequences. The analysis is based on move-by-move data from chess tournaments and an identification strategy that compares behavior of professional chess players to a rational behavioral benchmark that is constructed using modern chess engines. The evidence documents the existence of several distinct dimensions in which human players deviate from a rational benchmark. In particular, the results show deviations related to loss aversion, time pressure, fatigue, and cognitive limitations. The results also demonstrate that deviations do not necessarily lead to worse performance. Consistent with an important influence of intuition and experience, faster decisions are associated with more frequent deviations from the rational benchmark, yet they are also associated with better performance.

The concept of univariate Range Value-at-Risk, presented by Cont et al. (2010), is extended in the multidimensional setting. Traditional risk measures are not well suited when dealing with heavy-tail distributions and infinite tail expectations. The multivariate definitions of robust truncated tail expectations are provided to overcome this problem. Robustness and other properties as well as empirical estimators are derived. Closed-form expressions and special cases in the extreme value framework are also discussed. Numerical and graphical examples are provided to examine the accuracy of the empirical estimators.

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance.

In recent decades, the male-female wage gap has fallen, while the skilled-unskilled wage gap has risen in advanced countries. The rate of decline in the gender wage gap has tended to be greater for unskilled than skilled workers, while the rate of increase in the skill wage gap has tended to be greater for male than female workers. To account for these trends, we develop an aggregate production function extended to allow for gender-specific capital-skill complementarity, and estimate it using shift-hare instruments and cross-country panel data from OECD countries. We confirm that information and communication technology (ICT) equipment is not only more complementary to skilled than unskilled workers but also more complementary to female than male workers. Our results show that changes in gender and skill premia are the outcome of the race between progress in ICT and advances in female educational attainment and employment.

Feature selection in machine learning is subject to the intrinsic randomness of the feature selection algorithms (for example, random permutations during MDA). Stability of selected features with respect to such randomness is essential to the human interpretability of a machine learning algorithm. We proposes a rank based stability metric called instability index to compare the stabilities of three feature selection algorithms MDA, LIME, and SHAP as applied to random forests. Typically, features are selected by averaging many random iterations of a selection algorithm. Though we find that the variability of the selected features does decrease as the number of iterations increases, it does not go to zero, and the features selected by the three algorithms do not necessarily converge to the same set. We find LIME and SHAP to be more stable than MDA, and LIME is at least as stable as SHAP for the top ranked features. Hence overall LIME is best suited for human interpretability. However, the selected set of features from all three algorithms significantly improves various predictive metrics out of sample, and their predictive performances do not differ significantly. Experiments were conducted on synthetic datasets, two public benchmark datasets, and on proprietary data from an active investment strategy.

Interbank contagion can theoretically exacerbate losses in a financial system and lead to additional cascade defaults during downturn. In this paper we produce default analysis using both regression and neural network models to verify whether interbank contagion offers any predictive explanatory power on default events. We predict defaults of U.S. domiciled commercial banks in the first quarter of 2010 using data from the preceding four quarters. A number of established predictors (such as Tier 1 Capital Ratio and Return on Equity) are included alongside contagion to gauge if the latter adds significance. Based on this methodology, we conclude that interbank contagion is extremely explanatory in default prediction, often outperforming more established metrics, in both regression and neural network models. These findings have sizeable implications for the future use of interbank contagion as a variable of interest for stress testing, bank issued bond valuation and wider bank default prediction.