In Artificial Intelligence, interpreting the results of a Machine Learning technique often termed as a black box is a difficult task. A counterfactual explanation of a particular "black box" attempts to find the smallest change to the input values that modifies the prediction to a particular output, other than the original one. In this work we formulate the problem of finding a counterfactual explanation as an optimization problem. We propose a new "sparsity algorithm" which solves the optimization problem, while also maximizing the sparsity of the counterfactual explanation. We apply the sparsity algorithm to provide a simple suggestion to publicly traded companies in order to improve their credit ratings. We validate the sparsity algorithm with a synthetically generated dataset and we further apply it to quarterly financial statements from companies in financial, healthcare and IT sectors of the US market. We provide evidence that the counterfactual explanation can capture the nature of the real statement features that changed between the current quarter and the following quarter when ratings improved. The empirical results show that the higher the rating of a company the greater the "effort" required to further improve credit rating.

We discuss a class of debt management problems in a stochastic environment model. We propose a model for the debt-to-GDP (Gross Domestic Product) ratio where the government interventions via fiscal policies affect the public debt and the GDP growth rate at the same time. We allow for stochastic interest rate and possible correlation with the GDP growth rate through the dependence of both the processes (interest rate and GDP growth rate) on a stochastic factor which may represent any relevant macroeconomic variable, such as the state of economy. We tackle the problem of a government whose goal is to determine the fiscal policy in order to minimize a general functional cost. We prove that the value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation and provide a Verification Theorem based on classical solutions. We investigate the form of the candidate optimal fiscal policy in many cases of interest, providing interesting policy insights. Finally, we discuss two applications to the debt reduction problem and debt smoothing, providing explicit expressions of the value function and the optimal policy in some special cases.

Cryptocurrencies return cross-predictability and technological similarity yield information on risk propagation and market segmentation. To investigate these effects, we build a time-varying network for cryptocurrencies, based on the evolution of return cross-predictability and technological similarities. We develop a dynamic covariate-assisted spectral clustering method to consistently estimate the latent community structure of cryptocurrencies network that accounts for both sets of information. We demonstrate that investors can achieve better risk diversification by investing in cryptocurrencies from different communities. A cross-sectional portfolio that implements an inter-crypto momentum trading strategy earns a 1.08% daily return. By dissecting the portfolio returns on behavioral factors, we confirm that our results are not driven by behavioral mechanisms.

Protection of creditors is a key objective of financial regulation. Where the protection needs are high, i.e., in banking and insurance, regulatory solvency requirements are an instrument to prevent that creditors incur losses on their claims. The current regulatory requirements based on Value at Risk and Average Value at Risk limit the probability of default of financial institutions, but they fail to control the size of recovery on creditors' claims in the case of default. We resolve this failure by developing a novel risk measure, Recovery Value at Risk. Our conceptual approach can flexibly be extended and allows the construction of general recovery risk measures for various risk management purposes. By design, these risk measures control recovery on creditors' claims and integrate the protection needs of creditors into the incentive structure of the management. We provide detailed case studies and applications: We analyze how recovery risk measures react to the joint distributions of assets and liabilities on firms' balance sheets and compare the corresponding capital requirements with the current regulatory benchmarks based on Value at Risk and Average Value at Risk. We discuss how to calibrate recovery risk measures to historic regulatory standards. Finally, we show that recovery risk measures can be applied to performance-based management of business divisions of firms and that they allow for a tractable characterization of optimal tradeoffs between risk and return in the context of investment management.

Valuation adjustments, collectively named XVA, play an important role in modern derivatives pricing. XVA are an exotic pricing component since they require the forward simulation of multiple risk factors in order to compute the portfolio exposure including collateral, leading to a significant model risk and computational effort, even in case of plain vanilla trades. This work analyses the most critical model risk factors, meant as those to which XVA are most sensitive, finding an acceptable compromise between accuracy and performance. This task has been conducted in a complete context including a market standard multi-curve G2++ model calibrated on real market data, both Variation Margin and ISDA-SIMM dynamic Initial Margin, different collateralization schemes, and the most common linear and non-linear interest rates derivatives. Moreover, we considered an alternative analytical approach for XVA in case of uncollateralized Swaps. We show that a crucial element is the construction of a parsimonious time grid capable of capturing all periodical spikes arising in collateralized exposure during the Margin Period of Risk. To this end, we propose a workaround to efficiently capture all spikes. Moreover, we show that there exists a parameterization which allows to obtain accurate results in a reasonable time, which is a very important feature for practical applications. In order to address the valuation uncertainty linked to the existence of a range of different parameterizations, we calculate the Model Risk AVA (Additional Valuation Adjustment) for XVA according to the provisions of the EU Prudent Valuation regulation. Finally, this work can serve as an handbook containing step-by-step instructions for the implementation of a complete, realistic and robust modelling framework of collateralized exposure and XVA.

Background: During the last years, there has been a lot of discussion and estimations on the energy consumption of Bitcoin miners. However, most of the studies are focused on estimating energy consumption, not in exploring the factors that determine it.

Goal: To explore the factors that determine maximum energy consumption of Bitcoin miners. In particular, analyze the limits of energy consumption, and to which extent variations of the factors could produce its reduction.

Method: Estimate the overall profit of all Bitcoin miners during a certain period of time, and the costs (including energy) that they face during that time, because of the mining activity. The underlying assumptions is that miners will only consume energy to mine Bitcoin if they have the expectation of profit, and at the same time they are competitive with respect of each other. Therefore, they will operate as a group in the point where profits balance expenditures.

Results: We show a basic equation that determines energy consumption based on some specific factors: minting, transaction fees, exchange rate, energy price, and amortization cost. We also define the Amortization Factor, which can be computed for mining devices based on their cost and energy consumption, helps to understand how the cost of equipment influences total energy consumption.

Conclusions: The factors driving energy consumption are identified, and from them, some ways in which Bitcoin energy consumption could be reduced are discussed. Some of these ways do not reduce the most important properties of Bitcoin, such as the chances of control of the aggregated hashpower, or the fundamentals of the proof of work mechanism. In general, the methods presented can help to predict energy consumption in different scenarios, based on factors that can be calculated from available data, or assumed in scenarios.

We study financial systems from a game-theoretic standpoint. A financial system is represented by a network, where nodes correspond to firms, and directed labeled edges correspond to debt contracts between them. The existence of cycles in the network indicates that a payment of a firm to one of its lenders might result to some incoming payment. So, if a firm cannot fully repay its debt, then the exact (partial) payments it makes to each of its creditors can affect the cash inflow back to itself. We naturally assume that the firms are interested in their financial well-being (utility) which is aligned with the amount of incoming payments they receive from the network. This defines a game among the firms, that can be seen as utility-maximizing agents who can strategize over their payments.

We are the first to study financial network games that arise under a natural set of payment strategies called priority-proportional payments. We investigate the existence and (in)efficiency of equilibrium strategies, under different assumptions on how the firms' utility is defined, on the types of debt contracts allowed between the firms, and on the presence of other financial features that commonly arise in practice. Surprisingly, even if all firms' strategies are fixed, the existence of a unique payment profile is not guaranteed. So, we also investigate the existence and computation of valid payment profiles for fixed payment strategies.

We define cooperative games on general graphs and generalize Lloyd S. Shapley's celebrated allocation formula for those games in terms of stochastic path integral driven by the associated Markov chain on each graph. We then show that the value allocation operator, one for each player defined by the stochastic path integral, coincides with the player's component game which is the solution to the least squares (or Poisson's) equation, in light of the combinatorial Hodge decomposition on general weighted graphs. Several motivational examples and applications are also presented.

Can online education enable all students to participate in and benefit from it equally? Massive online education without addressing the huge access gap and disparities in digital infrastructure would not only exclude a vast majority of students from learning opportunities but also exacerbate the existing socio-economic disparities in educational opportunities.

Portfolio managers often evaluate performance relative to benchmark, usually taken to be the Standard & Poor 500 stock index fund. This relative portfolio wealth is defined as the absolute portfolio wealth divided by wealth from investing in the benchmark (including reinvested dividends). The classic Merton problem for portfolio optimization considers absolute portfolio wealth. We combine absolute and relative wealth in our new utility function. We also consider the case of multiple benchmarks. To both absolute and relative wealth, we apply power utility functions, possibly with different exponents. We obtain an explicit solution and compare it to the classic Merton solution. We apply our results to the Capital Asset Pricing Model setting.

Motivated reasoning posits that people distort how they process new information in the direction of beliefs they find more attractive. This paper introduces a novel experimental paradigm that is able to portably identify motivated reasoning from Bayesian updating across a variety of factual questions; the paradigm analyzes how subjects assess the veracity of information sources that tell them the median of their belief distribution is too high or too low. A Bayesian would infer nothing about the source veracity from this message, but motivated reasoners would infer that the source were more truthful if it reported the direction that they find more attractive. I find novel evidence for politically-motivated reasoning about immigration, income mobility, crime, racial discrimination, gender, climate change, gun laws, and the performance of other subjects. Motivated reasoning from messages on these topics leads people's beliefs to become more polarized, even though the messages are uninformative.

This study examines how housing sector volatilities affect real estate investment trust (REIT) equity return in the United States. I argue that unexpected changes in housing variables can be a source of aggregate housing risk, and the first principal component extracted from the volatilities of U.S. housing variables can predict the expected REIT equity returns. I propose and construct a factor-based housing risk index as an additional factor in asset price models that uses the time-varying conditional volatility of housing variables within the U.S. housing sector. The findings show that the proposed housing risk index is economically and theoretically consistent with the risk-return relationship of the conditional Intertemporal Capital Asset Pricing Model (ICAPM) of Merton (1973), which predicts an average maximum of 5.6 percent of risk premium in REIT equity return. In subsample analyses, the positive relationship is not affected by sample periods' choice but shows higher housing risk beta values for the 2009-18 sample period. The relationship remains significant after controlling for VIX, Fama-French three factors, and a broad set of macroeconomic and financial variables. Moreover, the proposed housing beta also accurately forecasts U.S. macroeconomic and financial conditions.

We propose a methodology to approximate conditional distributions in the elliptope of correlation matrices based on conditional generative adversarial networks. We illustrate the methodology with an application from quantitative finance: Monte Carlo simulations of correlated returns to compare risk-based portfolio construction methods. Finally, we discuss about current limitations and advocate for further exploration of the elliptope geometry to improve results.