Risks, Volume 8, Issue 4 (December 2020) – 38 articles

We propose an Explainable AI model that can be employed in order to explain why a customer buys or abandons a non-life insurance coverage. The method consists in applying similarity clustering to the Shapley values that were obtained from a highly accurate XGBoost predictive classification algorithm. Our proposed method can be embedded into a technologically-based insurance service (Insurtech), allowing to understand, in real time, the factors that most contribute to customers’ decisions, thereby gaining proactive insights on their needs. We prove the validity of our model with an empirical analysis that was conducted on data regarding purchases of insurance micro-policies. Two aspects are investigated: the propensity to buy an insurance policy and the risk of churn of an existing customer. The results from the analysis reveal that customers can be effectively and quickly grouped according to a similar set of characteristics, which can predict their buying or churn behaviour well. Full article

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks. Full article

Risk taking is an inherent element of the banking business. Banks make conscious decisions regarding risk taking as they expect to make more return if they take more risk. The primary objective of this study is to empirically investigate the efficiency of Indian banks in generating return relative to the risk they take. If the efficiency measurement is not adjusted for different risk preferences, then a bank earning lower return at lower risk may be misclassified as less efficient compared to peers earning the same level of return, but operating at a higher level of risk. This paper uses measures of liquidity risk, credit risk, market risk, and insolvency risk to develop a risk-return stochastic frontier in order to examine the risk efficiency of banks, a novel attempt in the Indian context. The paper further analyzes the efficiency of banks with respect to bank specific characteristics and risk management regimes. The models are estimated using data from a sample of 47 major banks for the period 2009–2018. The study reveals that Indian banks, on average, exhibited lower efficiency in trading risk against return during the sample period. Full article

The reform of 2013 represented a qualitative leap in the reform of the Spanish pension system. Unlike its predecessors, it introduced two automatic resetting mechanisms similar to those of other European countries. The first is the sustainability factor, scheduled to come into effect in 2019 but delayed until 2023, and its ultimate reversal cannot be ruled out. The objective of this study was to quantify the savings, or the lowest expenditure, that can be achieved in the Spanish public contributory pension system by applying it. These savings are measured in terms of cash—of annual expenditure—and in terms of accrual by calculating its present actuarial value. Combining these two methods is one of the contributions of this work. This work was only intended to analyze the impact of the Sustainability Factor, therefore, it did not take into account the impact of the Pension Revaluation Index, which is the second mechanism introduced in the reform of the pension to 2013. An ad hoc projection method was used, combining microdata from the Continuous Sample of Working Lives (MCVL), aggregate data from the pension system, the financial-actuarial projection method, and actuarial techniques. The diversity of the data used is the second contribution of this work. The application of the sustainability factor would improve the viability of the system, since the savings that could be achieved, measured in terms of GDP for each year, would be 1.029% by 2050; 1.094% in 2057, the maximum; and 1.026% in the last year of projection. In terms of the present actuarial value and as a function of annual GDP, in 2050, the savings would be 1.27%, 1.40% in 2044, the maximum, and in 2067 it would decrease to 0.98%. Full article

We study price formation in intraday electricity markets in the presence of intermittent renewable generation. We consider the setting where a major producer may interact strategically with a large number of small producers. Using stochastic control theory, we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in a closed form in the asymptotic framework of mean field games with a major player. Full article

This paper models the gradual elements of catastrophic events on non-life insurance capital with a particular focus on the impact of pandemics, such as COVID-19. A combination of actuarial and epidemiological models are handled by the Markovian probabilistic approach, with Feynman’s path calculation and Dirac notations, in order to observe how a pandemic risk may affect an insurer via reduced business. We also examine how the effects of a pandemic can be taken into account both during and at the end of the process. Examples are also provided showing the potential effects of a pandemic on different types of insurance product. Full article

Specially in the case of scenarios under uncertainty, the efficient management of risk when matching assets and liabilities is a relevant issue for most insurance companies. This paper considers such a scenario, where different assets can be aggregated to better match a liability (or the other way around), and the goal is to find the asset-liability assignments that maximises the overall benefit over a time horizon. To solve this stochastic optimisation problem, a simulation-optimisation methodology is proposed. We use integer programming to generate efficient asset-to-liability assignments, and Monte-Carlo simulation is employed to estimate the risk of failing to pay due liabilities. The simulation results allow us to set a safety margin parameter for the integer program, which encourage the generation of solutions satisfying a minimum reliability threshold. A series of computational experiments contribute to illustrate the proposed methodology and its utility in practical risk management. Full article

According to a statement made in the BP Energy Outlook report in 2017, most of the world’s liquid fuel (petroleum) is being consumed by the transportation industry. The mechanisms used to stimulate changes in the energy markets are affected by government policies that act in more ambitious ways than purely market-driven forces; different governments have promoted incentives involving electric mobility, especially in urban areas. The substitution for crude oil by renewable energy inputs in the transport sector is a major concern for oil producers. Among the different types of clean energies, lithium (Li) is currently assuming an increasingly strategic role. The goals of this paper are two-fold: First, we study the dynamics of the lithium industry and then the beta risk behavior of the 10 largest oil companies in the world for the time period between 11 February 2008 and 10 January 2019. We use an approach based on the continuous wavelet transform (CWT) method. The results indicate that there is a period of dependence between late 2013 and 2016 that occurs in the long-run frequencies of between 32 and 198 days for all cases, except for in the case of PetroChina, thereby demonstrating that the beta term is time-varying. We also find evidence that the beta term reflects and advances oil companies’ responsiveness to movements in the lithium market. In the second part of the paper, we study the dynamics of the beta series by using long-run dependence approaches. The results indicate that the betas are highly persistent, with the order of integration found to be significantly above 1 in all cases. Full article

A major concern of the adoption and scalability of Blockchain technologies refers to their efficient use for payments. In this work, we analyze how Lightning Network (LN), which represents a relevant infrastructural novelty, is influenced by the market dynamics of its referring cryptocurrency, namely Bitcoin. In so doing, we focus on how the LN is efficient in performing transactions and we relate this feature to the market conditions of Bitcoin. By applying the Toda–Yamamoto variant of Granger-causality, we note that market conditions of Bitcoin do not significantly influence the topological configuration of the LN. Hence, although the LN represents a second layer on the Bitcoin blockchain, our findings suggest that its efficient functioning does not appear to be related to the simple market performance of its underlying cryptocurrency and, in particular, of its volatile market fluctuations. This result may therefore contribute to shed light on the practical usage of the LN as a blockchain technology to favor transactions. Full article

An array of developments impacting the financial services industry, such as increasing complexity, interconnectedness, third party dependencies and digitalization, means operational resilience will remain a significant area of concern for policy makers, investors and customers. The purpose of this study is to evaluate if banks are disclosing information on their operational resilience risk. The study initially reviews the regulatory landscape for operational resiliency. The recent annual reports of the GSIB banks are reviewed to identify if they have made references to operational resilience. Through text mining, a frequency analysis of terms related to operational resilience was done, followed by an evaluation to understand the existence of relationships between these terms. The study shows that the regulatory guidance for operational resilience is still evolving with much of the current impetus on cybersecurity. There is a notable gap between banks that have reported on operational resiliency and those that have not, with a few patterns visible. Research in the area of operational resilience is relatively new and limited, and this research for the first time analyses the disclosures related to operational resilience in annual reports. Further, for policymakers, it highlights the disparity in disclosures around this relatively new area of risk, thus calling for additional regulatory guidance. Full article

In this paper, we study the Parisian time of a reflected Brownian motion with drift on a finite collection of rays. We derive the Laplace transform of the Parisian time using a recursive method, and provide an exact simulation algorithm to sample from the distribution of the Parisian time. The paper was motivated by the settlement delay in the real-time gross settlement (RTGS) system. Both the central bank and the participating banks in the system are concerned about the liquidity risk, and are interested in the first time that the duration of settlement delay exceeds a predefined limit. We reduce this problem to the calculation of the Parisian time. The Parisian time is also crucial in the pricing of Parisian type options; to this end, we will compare our results to the existing literature. Full article

We aim to construct portfolios by employing different risk models and compare their performance in order to understand their appropriateness for effective portfolio management for investors. Mean variance (MV), semi variance (SV), mean absolute deviation (MaD) and conditional value at risk (CVaR) are considered as risk measures. The price data were extracted from the Pakistan stock exchange, Bombay stock exchange and Dhaka stock exchange under diverse economic conditions such as crisis, recovery and growth. We take the average of GDP of the selected period of each country as a cut-off point to make three economic scenarios. We use 40 stocks from the Pakistan stock exchange, 92 stocks from the Bombay stock exchange and 30 stocks from the Dhaka stock exchange. We compute optimal weights using global minimum variance portfolio (GMVP) for all stocks to construct optimal portfolios and analyze the data by using MV, SV, MaD and CVaR models for each subperiod. We find that CVaR (95%) gives better results in each scenario for all three countries and performance of portfolios is inconsistent in different scenarios. Full article

The goal of this paper was to exploit the Bayesian approach and MCMC procedures to structure an internal model to quantify the reserve risk of a non-life insurer under Solvency II regulation. To this aim, we provide an extension of the Correlated Chain Ladder (CCL) model to the one-year time horizon. In this way, we obtain the predictive distribution of the next year obligations and we are able to assess a capital requirement compliant with Solvency II framework. Numerical results compare the one-year CCL with other traditional approaches, such as Re-Reserving and the Merz and Wüthrich formula. One-year CCL proves to be a legitimate alternative, providing values comparable with the more traditional approaches and more robust and accurate risk estimations, that embed external knowledge not present in the data and allow for a more precise and tailored representation of the risk profile of the insurer. Full article

We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena such as volatility clustering or long memory. After recalling recent results on option pricing in subordinated market models, we establish several analytical formulas for market sensitivities and portfolio performance in this class of models, and discuss some useful approximations when options are not far from the money. We also provide some tools for volatility modelling and delta hedging, as well as comparisons with numerical Fourier techniques. Full article

This paper develops an approach based on Gram–Charlier-like expansions for modeling financial series to take in due account features such as leptokurtosis. A Gram–Charlier-like expansion adjusts the moments of interest of a given distribution via its own orthogonal polynomials. This approach, formerly adopted for univariate series, is here extended to a multivariate context by means of spherical densities. Previous works proposed the Gram–Charlier of the multivariate Gaussian, obtained by using Hermite polynomials. This work shows how polynomial expansions of an entire class of spherical laws can be worked out with the aim of obtaining a wide set of leptokurtic multivariate distributions. A Gram–Charlier-like expansion is a distribution characterized by an additional parameter with respect to the parent spherical law. This parameter, which measures the increase in kurtosis due to the polynomial expansion, can be estimated so as to make the resulting distribution capable of describing the empirical kurtosis found in the data. An application of the Gram–Charlier-like expansions to a set of financial assets proves their effectiveness in modeling multivariate financial series and assessing risk measures, such as the value at risk and the expected shortfall. Full article

The COVID-19 pandemic has caused a significant slowdown in the development of almost all economies in the world. In this context, the main goal of this research is to try to present changes in the value of fiscal, investment and export multipliers as a result of the COVID-19 pandemic. The research was conducted in selected European Union countries. They are France, Germany, Italy, Poland, Portugal and Spain. This research is based on the theory of effective demand. The values of feeds and leakages of total demand in the period from 2015 to 2020 were examined and calculated. On this basis, the individual multipliers of autonomous spending were analyzed and their changes in the first period of the COVID-19 pandemic were presented. The analyses led to a surprising conclusion: it found that the autonomous spending multipliers in some economies increased. This means that they have become ‘security buffers’ for the health of economies. This means that the increase in their value weakened the negative effects of changes in autonomous expenditure on gross domestic product. Full article

We propose a multi-cohort model that is able to capture the mortality correlation between different cohorts. The model is based on the Hull and White process to which we incorporate inter-generational risk factors, by modifying its stochastic part. We provide a pricing framework for a new survival forward contract under the Cost of Capital, risk-neutral and Sharpe approaches, allowing to cover the global multi-cohort longevity risk. We give numerical illustrations for Belgian cohorts, and we compute the price of the longevity derivative under the proposed methods, for different correlation levels. Full article

We propose a way to compute the hedging Delta using the Malliavin weight method. Our approach, which we name the $\lambda$-method, generally outperforms the standard Monte Carlo finite difference method, especially for discontinuous payoffs. Furthermore, our approach is nonparametric, as we only assume a general local volatility model and we substitute the volatility and the other processes involved in the Greek formula with quantities that can be nonparametrically estimated from a given time series of observed prices. Full article

With the emergence of global digital service providers, concerns about digital oligopolies have increased, with a wide range of potentially harmful effects being discussed. One of these relates to cyber security, where it has been argued that market concentration can increase cyber risk. Such a state of affairs could have dire consequences for insurers and reinsurers, who underwrite cyber risk and are already very concerned about accumulation risk. Against this background, the paper develops some theory about how convex cyber risk affects Cournot oligopoly markets of data storage. It is demonstrated that with constant or increasing marginal production cost, the addition of increasing marginal cyber risk cost decreases the differences between the optimal numbers of records stored by the oligopolists, in effect offsetting the advantage of lower marginal production cost. Furthermore, based on the empirical literature on data breach cost, two possibilities are found: (i) that such cyber risk exhibits decreasing marginal cost in the number of records stored and (ii) the opposite possibility that such cyber risk instead exhibits increasing marginal cost in the number of records stored. The article is concluded with a discussion of the findings and some directions for future research. Full article

This paper focuses on weather derivatives as efficient risk management instruments and proposes a more advanced approach for their pricing. An “hybrid” contract is introduced, combining insurance properties, specifically tailored for the region under study and introducing Value-at-Risk (VaR) and Expected Shortfall (ES) as appropriate measures for the strike price. The numerical results show that VaR and ES are both efficient ways for managing the so-called Tail Risk; further, being ES more conservative than VaR and due to its subadditivity property, it can be seen that seasonal contracts are generally better off than monthly contracts in reducing global risk. Full article

The lifestyles and backgrounds of individuals across the United States differ widely. Some of these differences are easily measurable (ethnicity, age, income, etc.) while others are not (stress levels, empathy, diet, exercise, etc.). Though every person is unique, individuals living closer together likely have more similar lifestyles than individuals living hundreds of miles apart. Because lifestyle and environmental factors contribute to mortality, spatial correlation may be an important feature in mortality modeling. However, many of the current mortality models fail to account for spatial relationships. This paper introduces spatio-temporal trends into traditional mortality modeling using Bayesian hierarchical models with conditional auto-regressive (CAR) priors. We show that these priors, commonly used for areal data, are appropriate for modeling county-level spatial trends in mortality data covering the contiguous United States. We find that mortality rates of neighboring counties are highly correlated. Additionally, we find that mortality improvement or deterioration trends between neighboring counties are also highly correlated. Full article

The least-squares Monte Carlo method has proved to be a suitable approximation technique for the calculation of a life insurer’s solvency capital requirements. We suggest to enhance it by the use of a neural network based approach to construct the proxy function that models the insurer’s loss with respect to the risk factors the insurance business is exposed to. After giving a mathematical introduction to feed forward neural networks and describing the involved hyperparameters, we apply this popular form of neural networks to a slightly disguised data set from a German life insurer. Thereby, we demonstrate all practical aspects, such as the hyperparameter choice, to obtain our candidate neural networks by bruteforce, the calibration (“training”) and validation (“testing”) of the neural networks and judging their approximation performance. Compared to adaptive OLS, GLM, GAM and FGLS regression approaches, an ensemble built of the 10 best derived neural networks shows an excellent performance. Through a comparison with the results obtained by every single neural network, we point out the significance of the ensemble-based approach. Lastly, we comment on the interpretability of neural networks compared to polynomials for sensitivity analyses. Full article

The crisis caused by the outbreak of COVID-19 revealed the global unpreparedness for handling the impact of a pandemic. In this paper, we present a first quarter chronicle of COVID-19 in Hubei China, Italy and Spain, particularly focusing on infection speed, death and fatality rates. By analysing the parameters of the best fitting distributions of the available data for the three rates in each of the three regions, we illustrate the pandemic’s evolution in relation to government measures. We compared the effectiveness of lockdown measures by observing the true situation in each dataset, without proposing a mathematical model. The feasibility of obtaining a firm conclusion in regard to the best solution for containing COVID-19 is limited, with a universal solution failing to exist due to globally varying culture, mentality and behaviours. Our method provides valid insights into the individual and national actions implemented and adhered to in order to slow the effect of the pandemic during the first-wave of COVID-19. Full article

In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing bounds obtained from the Value-at-Risk admit a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller’s bound for a call option is smaller than the buyer’s bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between good-deal bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of risk measures based on multiple eligible assets in our setting. Full article

In traditional Reinforcement Learning (RL), agents learn to optimize actions in a dynamic context based on recursive estimation of expected values. We show that this form of machine learning fails when rewards (returns) are affected by tail risk, i.e., leptokurtosis. Here, we adapt a recent extension of RL, called distributional RL (disRL), and introduce estimation efficiency, while properly adjusting for differential impact of outliers on the two terms of the RL prediction error in the updating equations. We show that the resulting “efficient distributional RL” (e-disRL) learns much faster, and is robust once it settles on a policy. Our paper also provides a brief, nontechnical overview of machine learning, focusing on RL. Full article

This paper follows the recent literature on real estate price prediction and proposes to take advantage of machine learning techniques to better explain which variables are more important in describing the real estate market evolution. We apply the random forest algorithm on London real estate data and analyze the local variables that influence the interaction between housing demand, supply and price. The variables choice is based on an urban point of view, where the main force driving the market is the interaction between local factors like population growth, net migration, new buildings and net supply. Full article

This article introduces a novel use of the vine copula which captures dependence among multi-line claim triangles, especially when an insurance portfolio consists of more than two lines of business. First, we suggest a way to choose an optimal joint loss development model for multiple lines of business that considers marginal distribution, vine copula structure, and choice of family for each pair of copulas. The performance of the model is also demonstrated with Bayesian model diagnostics and out-of-sample validation measures. Finally, we provide an implication of the dependence modeling, which allows a company to analyze and establish the risk capital for whole portfolio. Full article

This work considered the economic security of an enterprise with regard to the valuation of intangible assets according to the International Valuation Standards (IVS). This study is essential due to a growing number of companies with intangible assets (trademarks, patents, know-how, etc.) as their main value. This study included analysis of the impact created by the value of intangible assets and intellectual property on company capitalization and economic security plus a regression model. An algorithm was developed to determine the economic security of a business based on the valuation of intangible assets according to the IVS. The suggested algorithm can allow a company to manage its intangible assets effectively using the IVS, which, in turn, will provide the required level of economic security for further development and achievement of strategic goals by the business entity. Full article

The prediction of human longevity levels in the future by direct forecasting of life expectancy offers numerous advantages, compared to methods based on extrapolation of age-specific death rates. However, the reconstruction of accurate life tables starting from a given level of life expectancy at birth, or any other age, is not straightforward. Model life tables have been extensively used for estimating age patterns of mortality in poor-data countries. We propose a new model inspired by indirect estimation techniques applied in demography, which can be used to estimate full life tables at any point in time, based on a given value of life expectancy at birth. Our model relies on the existing high correlations between levels of life expectancy and death rates across ages. The methods presented in this paper are implemented in a publicly available R package. Full article

Motivated by new financial markets where there is no canonical choice of a risk-neutral measure, we compared two different methods for pricing options: calibration with an entropic penalty term and valuation by the Esscher measure. The main aim of this paper is to contrast the outcomes of those two methods with real-traded call option prices in a liquid market like NASDAQ stock exchange, using data referring to the period 2019–2020. Although the Esscher measure method slightly underperforms the calibration method in terms of absolute values of the percentage difference between real and model prices, it could be the only feasible choice if there are not many liquidly traded derivatives in the market. Full article