Risks
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Latest open access articles published in Risks at http://www.mdpi.com/journal/risks<![CDATA[Risks, Vol. 5, Pages 4: Acknowledgement to Reviewers of Risks in 2016]]>
http://www.mdpi.com/2227-9091/5/1/4
The editors of Risks would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016. [...]Risks2017-01-1251Editorial10.3390/risks501000442227-90912017-01-12doi: 10.3390/risks5010004 Risks Editorial Office<![CDATA[Risks, Vol. 5, Pages 3: The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation]]>
http://www.mdpi.com/2227-9091/5/1/3
We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this.Risks2017-01-0651Article10.3390/risks501000332227-90912017-01-06doi: 10.3390/risks5010003Yuguang FanPhilip GriffinRoss MallerAlexander SzimayerTiandong Wang<![CDATA[Risks, Vol. 5, Pages 2: On Comparison of Stochastic Reserving Methods with Bootstrapping]]>
http://www.mdpi.com/2227-9091/5/1/2
We consider the well-known stochastic reserve estimation methods on the basis of generalized linear models, such as the (over-dispersed) Poisson model, the gamma model and the log-normal model. For the likely variability of the claims reserve, bootstrap method is considered. In the bootstrapping framework, we discuss the choice of residuals, namely the Pearson residuals, the deviance residuals and the Anscombe residuals. In addition, several possible residual adjustments are discussed and compared in a case study. We carry out a practical implementation and comparison of methods using real-life insurance data to estimate reserves and their prediction errors. We propose to consider proper scoring rules for model validation, and the assessments will be drawn from an extensive case study.Risks2017-01-0451Article10.3390/risks501000222227-90912017-01-04doi: 10.3390/risks5010002Liivika TeeMeelis KäärikRauno Viin<![CDATA[Risks, Vol. 5, Pages 1: Optimal Retention Level for Infinite Time Horizons under MADM]]>
http://www.mdpi.com/2227-9091/5/1/1
In this paper, we approximate the aggregate claims process by using the translated gamma process under the classical risk model assumptions, and we investigate the ultimate ruin probability. We consider optimal reinsurance under the minimum ultimate ruin probability, as well as the maximum benefit criteria: released capital, expected profit and exponential-fractional-logarithmic utility from the insurer’s point of view. Numerical examples are presented to explain how the optimal initial surplus and retention level are changed according to the individual claim amounts, loading factors and weights of the criteria. In the decision making process, we use The Analytical Hierarchy Process (AHP) and The Technique for Order of Preference by Similarity to ideal Solution (TOPSIS) methods as the Multi-Attribute Decision Making methods (MADM) and compare our results considering different combinations of loading factors for both exponential and Pareto individual claims.Risks2016-12-2751Article10.3390/risks501000112227-90912016-12-27doi: 10.3390/risks5010001Başak Bulut KarageyikŞule Şahin<![CDATA[Risks, Vol. 4, Pages 48: How Does Reinsurance Create Value to an Insurer? A Cost-Benefit Analysis Incorporating Default Risk]]>
http://www.mdpi.com/2227-9091/4/4/48
Reinsurance is often empirically hailed as a value-adding risk management strategy which an insurer can utilize to achieve various business objectives. In the context of a distortion-risk-measure-based three-party model incorporating a policyholder, insurer and reinsurer, this article formulates explicitly the optimal insurance–reinsurance strategies from the perspective of the insurer. Our analytic solutions are complemented by intuitive but scientifically rigorous explanations on the marginal cost and benefit considerations underlying the optimal insurance–reinsurance decisions. These cost-benefit discussions not only cast light on the economic motivations for an insurer to engage in insurance with the policyholder and in reinsurance with the reinsurer, but also mathematically formalize the value created by reinsurance with respect to stabilizing the loss portfolio and enlarging the underwriting capacity of an insurer. Our model also allows for the reinsurer’s failure to deliver on its promised indemnity when the regulatory capital of the reinsurer is depleted by the reinsured loss. The reduction in the benefits of reinsurance to the insurer as a result of the reinsurer’s default is quantified, and its influence on the optimal insurance–reinsurance policies analyzed.Risks2016-12-1644Article10.3390/risks4040048482227-90912016-12-16doi: 10.3390/risks4040048Ambrose Lo<![CDATA[Risks, Vol. 4, Pages 50: Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle]]>
http://www.mdpi.com/2227-9091/4/4/50
In this paper, we study the optimal reinsurance problem where risks of the insurer are measured by general law-invariant risk measures and premiums are calculated under the TVaR premium principle, which extends the work of the expected premium principle. Our objective is to characterize the optimal reinsurance strategy which minimizes the insurer’s risk measure of its total loss. Our calculations show that the optimal reinsurance strategy is of the multi-layer form, i.e., f * ( x ) = x ∧ c * + ( x - d * ) + with c * and d * being constants such that 0 ≤ c * ≤ d * .Risks2016-12-1644Article10.3390/risks4040050502227-90912016-12-16doi: 10.3390/risks4040050Mi ChenWenyuan WangRuixing Ming<![CDATA[Risks, Vol. 4, Pages 51: Bayesian Option Pricing Framework with Stochastic Volatility for FX Data]]>
http://www.mdpi.com/2227-9091/4/4/51
The application of stochastic volatility (SV) models in the option pricing literature usually assumes that the market has sufficient option data to calibrate the model’s risk-neutral parameters. When option data are insufficient or unavailable, market practitioners must estimate the model from the historical returns of the underlying asset and then transform the resulting model into its risk-neutral equivalent. However, the likelihood function of an SV model can only be expressed in a high-dimensional integration, which makes the estimation a highly challenging task. The Bayesian approach has been the classical way to estimate SV models under the data-generating (physical) probability measure, but the transformation from the estimated physical dynamic into its risk-neutral counterpart has not been addressed. Inspired by the generalized autoregressive conditional heteroskedasticity (GARCH) option pricing approach by Duan in 1995, we propose an SV model that enables us to simultaneously and conveniently perform Bayesian inference and transformation into risk-neutral dynamics. Our model relaxes the normality assumption on innovations of both return and volatility processes, and our empirical study shows that the estimated option prices generate realistic implied volatility smile shapes. In addition, the volatility premium is almost flat across strike prices, so adding a few option data to the historical time series of the underlying asset can greatly improve the estimation of option prices.Risks2016-12-1644Article10.3390/risks4040051512227-90912016-12-16doi: 10.3390/risks4040051Ying WangSai ChoyHoi Wong<![CDATA[Risks, Vol. 4, Pages 49: Compositions of Conditional Risk Measures and Solvency Capital]]>
http://www.mdpi.com/2227-9091/4/4/49
In this paper, we consider compositions of conditional risk measures in order to obtain time-consistent dynamic risk measures and determine the solvency capital of a life insurer selling pension liabilities or a pension fund with a single cash-flow at maturity. We first recall the notion of conditional, dynamic and time-consistent risk measures. We link the latter with its iterated property, which gives us a way to construct time-consistent dynamic risk measures from a backward iteration scheme with the composition of conditional risk measures. We then consider particular cases with the conditional version of the value at risk, tail value at risk and conditional expectation measures. We finally give an application of these measures with the determination of the solvency capital of a pension liability, which offers a fixed guaranteed rate without any intermediate cash-flow. We assume that the company is fully hedged against the mortality and underwriting risks.Risks2016-12-1644Article10.3390/risks4040049492227-90912016-12-16doi: 10.3390/risks4040049Pierre DevolderAdrien Lebègue<![CDATA[Risks, Vol. 4, Pages 47: Macroprudential Insurance Regulation: A Swiss Case Study]]>
http://www.mdpi.com/2227-9091/4/4/47
This article provides a case study that analyzes national macroprudential insurance regulation in Switzerland. We consider an insurance market that is based on data from the Swiss private insurance industry. We stress this market with several scenarios related to financial and insurance risks, and we analyze the resulting risk capitals of the insurance companies. This stress-test analysis provides insights into the vulnerability of the Swiss private insurance sector to different risks and shocks.Risks2016-12-1544Article10.3390/risks4040047472227-90912016-12-15doi: 10.3390/risks4040047Philippe DeprezMario Wüthrich<![CDATA[Risks, Vol. 4, Pages 46: Deflation Risk and Implications for Life Insurers]]>
http://www.mdpi.com/2227-9091/4/4/46
Life insurers are exposed to deflation risk: falling prices could lead to insufficient investment returns, and inflation-indexed protections could make insurers vulnerable to deflation. In this spirit, this paper proposes a market-based methodology for measuring deflation risk based on a discrete framework: the latter accounts for the real interest rate, the inflation index level, its conditional variance, and the expected inflation rate. US inflation data are then used to estimate the model and show the importance of deflation risk. Specifically, the distribution of a fictitious life insurer’s future payments is investigated. We find that the proposed inflation model yields higher risk measures than the ones obtained using competing models, stressing the need for dynamic and market-consistent inflation modelling in the life insurance industry.Risks2016-12-0344Article10.3390/risks4040046462227-90912016-12-03doi: 10.3390/risks4040046Jean-François Bégin<![CDATA[Risks, Vol. 4, Pages 45: Predicting Human Mortality: Quantitative Evaluation of Four Stochastic Models]]>
http://www.mdpi.com/2227-9091/4/4/45
In this paper, we quantitatively compare the forecasts from four different mortality models. We consider one discrete-time model proposed by Lee and Carter (1992) and three continuous-time models: the Wills and Sherris (2011) model, the Feller process and the Ornstein-Uhlenbeck (OU) process. The first two models estimate the whole surface of mortality simultaneously, while in the latter two, each generation is modelled and calibrated separately. We calibrate the models to UK and Australian population data. We find that all the models show relatively similar absolute total error for a given dataset, except the Lee-Carter model, whose performance differs significantly. To evaluate the forecasting performance we therefore look at two alternative measures: the relative error between the forecasted and the actual mortality rates and the percentage of actual mortality rates which fall within a prediction interval. In terms of the prediction intervals, the results are more divergent since each model implies a different structure for the variance of mortality rates. According to our experiments, the Wills and Sherris model produces superior results in terms of the prediction intervals. However, in terms of the mean absolute error, the OU and the Feller processes perform better. The forecasting performance of the Lee Carter model is mostly dependent on the choice of the dataset.Risks2016-12-0244Article10.3390/risks4040045452227-90912016-12-02doi: 10.3390/risks4040045Anastasia Novokreshchenova<![CDATA[Risks, Vol. 4, Pages 44: Estimation of Star-Shaped Distributions]]>
http://www.mdpi.com/2227-9091/4/4/44
Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the ﬂexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and ﬁnancial data sets.Risks2016-11-3044Article10.3390/risks4040044442227-90912016-11-30doi: 10.3390/risks4040044Eckhard LiebscherWolf-Dieter Richter<![CDATA[Risks, Vol. 4, Pages 43: Parameter Estimation in Stable Law]]>
http://www.mdpi.com/2227-9091/4/4/43
For general stable distribution, cumulant function based parameter estimators are proposed. Extensive simulation experiments are carried out to validate the effectiveness of the estimates over the entire parameter space. An application to non-life insurance losses distribution is made.Risks2016-11-2544Article10.3390/risks4040043432227-90912016-11-25doi: 10.3390/risks4040043Annika Krutto<![CDATA[Risks, Vol. 4, Pages 42: Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics]]>
http://www.mdpi.com/2227-9091/4/4/42
An insurance company offers an insurance contract ( p , K ) , consisting of a premium p and a deductible K. In this paper, we consider the problem of choosing the premium optimally as a function of the deductible. The insurance company is facing a market of N customers, each characterized by their personal claim frequency, α, and risk aversion, β. When a customer is offered an insurance contract, she/he will, based on these characteristics, choose whether or not to insure. The decision process of the customer is analyzed in detail. Since the customer characteristics are unknown to the company, it models them as i.i.d. random variables; A 1 , … , A N for the claim frequencies and B 1 , … , B N for the risk aversions. Depending on the distributions of A i and B i , expressions for the portfolio size n ( p ; K ) ∈ [ 0 , N ] and average claim frequency α ( p ; K ) in the portfolio are obtained. Knowing these, the company can choose the premium optimally, mainly by minimizing the ruin probability.Risks2016-11-0944Article10.3390/risks4040042422227-90912016-11-09doi: 10.3390/risks4040042Julie Thøgersen<![CDATA[Risks, Vol. 4, Pages 41: Incorporation of Stochastic Policyholder Behavior in Analytical Pricing of GMABs and GMDBs]]>
http://www.mdpi.com/2227-9091/4/4/41
Variable annuities represent certain unit-linked life insurance products offering different types of protection commonly referred to as guaranteed minimum benefits (GMXBs). They are designed for the increasing demand of the customers for private pension provision. In this paper we analytically price variable annuities with guaranteed minimum repayments at maturity and in case of the insured’s death. If the contract is prematurely surrendered, the policyholder is entitled to the current value of the fund account reduced by the prevailing surrender fee. The financial market and the mortality model are affine linear. For the surrender model, a Cox process is deployed whose intensity is given by a deterministic function (s-curve) with stochastic inputs from the financial market. So, the policyholders’ surrender behavior depends on the performance of the financial market and is stochastic. The presented pricing scheme incorporates the stochastic surrender behavior of the policyholders and is only based on suitable closed-form approximations.Risks2016-11-0844Article10.3390/risks4040041412227-90912016-11-08doi: 10.3390/risks4040041Marcos EscobarMikhail KrayzlerFranz RamsauerDavid SaundersRudi Zagst<![CDATA[Risks, Vol. 4, Pages 40: A Note on Upper Tail Behavior of Liouville Copulas]]>
http://www.mdpi.com/2227-9091/4/4/40
The family of Liouville copulas is defined as the survival copulas of multivariate Liouville distributions, and it covers the Archimedean copulas constructed by Williamson’s d-transform. Liouville copulas provide a very wide range of dependence ranging from positive to negative dependence in the upper tails, and they can be useful in modeling tail risks. In this article, we study the upper tail behavior of Liouville copulas through their upper tail orders. Tail orders of a more general scale mixture model that covers Liouville distributions is first derived, and then tail order functions and tail order density functions of Liouville copulas are derived. Concrete examples are given after the main results.Risks2016-11-0844Article10.3390/risks4040040402227-90912016-11-08doi: 10.3390/risks4040040Lei Hua<![CDATA[Risks, Vol. 4, Pages 39: Frailty and Risk Classification for Life Annuity Portfolios]]>
http://www.mdpi.com/2227-9091/4/4/39
Life annuities are attractive mainly for healthy people. In order to expand their business, in recent years, some insurers have started offering higher annuity rates to those whose health conditions are critical. Life annuity portfolios are then supposed to become larger and more heterogeneous. With respect to the insurer’s risk profile, there is a trade-off between portfolio size and heterogeneity that we intend to investigate. In performing this, there is a second and possibly more important issue that we address. In actuarial practice, the different mortality levels of the several risk classes are obtained by applying adjustment coefficients to population mortality rates. Such a choice is not supported by a rigorous model. On the other hand, the heterogeneity of a population with respect to mortality can formally be described with a frailty model. We suggest adopting a frailty model for risk classification. We identify risk groups (or classes) within the population by assigning specific ranges of values to the frailty within each group. The different levels of mortality of the various groups are based on the conditional probability distributions of the frailty. Annuity rates for each class then can be easily justified, and a comprehensive investigation of insurer’s liabilities can be performed.Risks2016-10-2644Article10.3390/risks4040039392227-90912016-10-26doi: 10.3390/risks4040039Annamaria OlivieriErmanno Pitacco<![CDATA[Risks, Vol. 4, Pages 38: A Note on Health Insurance under Ex Post Moral Hazard]]>
http://www.mdpi.com/2227-9091/4/4/38
In the linear coinsurance problem, examined first by Mossin (1968), a higher absolute risk aversion with respect to wealth in the sense of Arrow–Pratt implies a higher optimal coinsurance rate. We show that this property does not hold for health insurance under ex post moral hazard; i.e., when illness severity cannot be observed by insurers, and policyholders decide on their health expenditures. The optimal coinsurance rate trades off a risk-sharing effect and an incentive effect, both related to risk aversion.Risks2016-10-2544Article10.3390/risks4040038382227-90912016-10-25doi: 10.3390/risks4040038Pierre Picard<![CDATA[Risks, Vol. 4, Pages 37: A Note on Realistic Dividends in Actuarial Surplus Models]]>
http://www.mdpi.com/2227-9091/4/4/37
Because of the profitable nature of risk businesses in the long term, de Finetti suggested that surplus models should allow for cash leakages, as otherwise the surplus would unrealistically grow (on average) to infinity. These leakages were interpreted as ‘dividends’. Subsequent literature on actuarial surplus models with dividend distribution has mainly focussed on dividend strategies that either maximise the expected present value of dividends until ruin or lead to a probability of ruin that is less than one (see Albrecher and Thonhauser, Avanzi for reviews). An increasing number of papers are directly interested in modelling dividend policies that are consistent with actual practice in financial markets. In this short note, we review the corporate finance literature with the specific aim of fleshing out properties that dividend strategies should ideally satisfy, if one wants to model behaviour that is consistent with practice.Risks2016-10-2044Article10.3390/risks4040037372227-90912016-10-20doi: 10.3390/risks4040037Benjamin AvanziVincent TuBernard Wong<![CDATA[Risks, Vol. 4, Pages 36: Nested MC-Based Risk Measurement of Complex Portfolios: Acceleration and Energy Efficiency]]>
http://www.mdpi.com/2227-9091/4/4/36
Risk analysis and management currently have a strong presence in financial institutions, where high performance and energy efficiency are key requirements for acceleration systems, especially when it comes to intraday analysis. In this regard, we approach the estimation of the widely-employed portfolio risk metrics value-at-risk (VaR) and conditional value-at-risk (cVaR) by means of nested Monte Carlo (MC) simulations. We do so by combining theory and software/hardware implementation. This allows us for the first time to investigate their performance on heterogeneous compute systems and across different compute platforms, namely central processing unit (CPU), many integrated core (MIC) architecture XeonPhi, graphics processing unit (GPU), and field-programmable gate array (FPGA). To this end, the OpenCL framework is employed to generate portable code, and the size of the simulations is scaled in order to evaluate variations in performance. Furthermore, we assess different parallelization schemes, and the targeted platforms are evaluated and compared in terms of runtime and energy efficiency. Our implementation also allowed us to derive a new algorithmic optimization regarding the generation of the required random number sequences. Moreover, we provide specific guidelines on how to properly handle these sequences in portable code, and on how to efficiently implement nested MC-based VaR and cVaR simulations on heterogeneous compute systems.Risks2016-10-1844Article10.3390/risks4040036362227-90912016-10-18doi: 10.3390/risks4040036Sascha DesmettreRalf KornJavier VarelaNorbert Wehn<![CDATA[Risks, Vol. 4, Pages 35: A Note on the Impact of Parameter Uncertainty on Barrier Derivatives]]>
http://www.mdpi.com/2227-9091/4/4/35
This paper presents a comprehensive extension of pricing two-dimensional derivatives depending on two barrier constraints. We assume randomness on the covariance matrix as a way of generalizing. We analyse common barrier derivatives, enabling us to study parameter uncertainty and the risk related to the estimation procedure (estimation risk). In particular, we use the distribution of empirical parameters from IBM and EURO STOXX50. The evidence suggests that estimation risk should not be neglected in the context of multidimensional barrier derivatives, as it could cause price differences of up to 70%.Risks2016-09-2944Article10.3390/risks4040035352227-90912016-09-29doi: 10.3390/risks4040035Marcos EscobarSven Panz<![CDATA[Risks, Vol. 4, Pages 34: Sharp Convex Bounds on the Aggregate Sums–An Alternative Proof]]>
http://www.mdpi.com/2227-9091/4/4/34
It is well known that a random vector with given marginals is comonotonic if and only if it has the largest convex sum, and that a random vector with given marginals (under an additional condition) is mutually exclusive if and only if it has the minimal convex sum. This paper provides an alternative proof of these two results using the theories of distortion risk measure and expected utility.Risks2016-09-2944Article10.3390/risks4040034342227-90912016-09-29doi: 10.3390/risks4040034Chuancun YinDan Zhu<![CDATA[Risks, Vol. 4, Pages 33: Multivariate TVaR-Based Risk Decomposition for Vector-Valued Portfolios]]>
http://www.mdpi.com/2227-9091/4/4/33
In order to protect stakeholders of insurance companies and financial institutions against adverse outcomes of risky businesses, regulators and senior management use capital allocation techniques. For enterprise-wide risk management, it has become important to calculate the contribution of each risk within a portfolio. For that purpose, bivariate lower and upper orthant tail value-at-risk can be used for capital allocation. In this paper, we present multivariate value-at-risk and tail-value-at-risk for d ≥ 2 , and we focus on three different methods to calculate optimal values for the contribution of each risk within the sums of random vectors to the overall portfolio, which could particularly apply to insurance and financial portfolios.Risks2016-09-2344Article10.3390/risks4040033332227-90912016-09-23doi: 10.3390/risks4040033Mélina MailhotMhamed Mesfioui<![CDATA[Risks, Vol. 4, Pages 32: The Wasserstein Metric and Robustness in Risk Management]]>
http://www.mdpi.com/2227-9091/4/3/32
In the aftermath of the financial crisis, it was realized that the mathematical models used for the valuation of financial instruments and the quantification of risk inherent in portfolios consisting of these financial instruments exhibit a substantial model risk. Consequently, regulators and other stakeholders have started to require that the internal models used by financial institutions are robust. We present an approach to consistently incorporate the robustness requirements into the quantitative risk management process of a financial institution, with a special focus on insurance. We advocate the Wasserstein metric as the canonical metric for approximations in robust risk management and present supporting arguments. Representing risk measures as statistical functionals, we relate risk measures with the concept of robustness and hence continuity with respect to the Wasserstein metric. This allows us to use results from robust statistics concerning continuity and differentiability of functionals. Finally, we illustrate our approach via practical applications.Risks2016-08-3143Article10.3390/risks4030032322227-90912016-08-31doi: 10.3390/risks4030032Rüdiger KieselRobin RühlickeGerhard StahlJinsong Zheng<![CDATA[Risks, Vol. 4, Pages 31: Choosing Markovian Credit Migration Matrices by Nonlinear Optimization]]>
http://www.mdpi.com/2227-9091/4/3/31
Transition matrices, containing credit risk information in the form of ratings based on discrete observations, are published annually by rating agencies. A substantial issue arises, as for higher rating classes practically no defaults are observed yielding default probabilities of zero. This does not always reflect reality. To circumvent this shortcoming, estimation techniques in continuous-time can be applied. However, raw default data may not be available at all or not in the desired granularity, leaving the practitioner to rely on given one-year transition matrices. Then, it becomes necessary to transform the one-year transition matrix to a generator matrix. This is known as the embedding problem and can be formulated as a nonlinear optimization problem, minimizing the distance between the exponential of a potential generator matrix and the annual transition matrix. So far, in credit risk-related literature, solving this problem directly has been avoided, but approximations have been preferred instead. In this paper, we show that this problem can be solved numerically with sufficient accuracy, thus rendering approximations unnecessary. Our direct approach via nonlinear optimization allows one to consider further credit risk-relevant constraints. We demonstrate that it is thus possible to choose a proper generator matrix with additional structural properties.Risks2016-08-3043Article10.3390/risks4030031312227-90912016-08-30doi: 10.3390/risks4030031Maximilian HughesRalf Werner<![CDATA[Risks, Vol. 4, Pages 30: On the Capital Allocation Problem for a New Coherent Risk Measure in Collective Risk Theory]]>
http://www.mdpi.com/2227-9091/4/3/30
In this paper we introduce a new coherent cumulative risk measure on a subclass in the space of càdlàg processes. This new coherent risk measure turns out to be tractable enough within a class of models where the aggregate claims is driven by a spectrally positive Lévy process. We focus our motivation and discussion on the problem of capital allocation. Indeed, this risk measure is well-suited to address the problem of capital allocation in an insurance context. We show that the capital allocation problem for this risk measure has a unique solution determined by the Euler allocation method. Some examples and connections with existing results as well as practical implications are also discussed.Risks2016-08-1643Article10.3390/risks4030030302227-90912016-08-16doi: 10.3390/risks4030030Hirbod AssaManuel MoralesHassan Omidi Firouzi<![CDATA[Risks, Vol. 4, Pages 29: Optimal Insurance with Heterogeneous Beliefs and Disagreement about Zero-Probability Events]]>
http://www.mdpi.com/2227-9091/4/3/29
In problems of optimal insurance design, Arrow’s classical result on the optimality of the deductible indemnity schedule holds in a situation where the insurer is a risk-neutral Expected-Utility (EU) maximizer, the insured is a risk-averse EU-maximizer, and the two parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. Recently, Ghossoub re-examined Arrow’s problem in a setting where the two parties have different subjective beliefs about the realizations of the insurable random loss, and he showed that if these beliefs satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub only gave a characterization of these optimal indemnity schedules in the special case of an MLR. In this paper, we consider the general case, allowing for disagreement about zero-probability events. We fully characterize the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the insured’s probability measure, and we obtain Arrow’s classical result, as well as one of the results of Ghossoub as corollaries. Finally, we formalize Marshall’s argument that, in a setting of belief heterogeneity, an optimal indemnity schedule may take “any”shape.Risks2016-08-0543Article10.3390/risks4030029292227-90912016-08-05doi: 10.3390/risks4030029Mario Ghossoub<![CDATA[Risks, Vol. 4, Pages 28: Using Climate and Weather Data to Support Regional Vulnerability Screening Assessments of Transportation Infrastructure]]>
http://www.mdpi.com/2227-9091/4/3/28
Extreme weather and climate change can have a significant impact on all types of infrastructure and assets, regardless of location, with the potential for human casualties, physical damage to assets, disruption of operations, economic and community distress, and environmental degradation. This paper describes a methodology for using extreme weather and climate data to identify climate-related risks and to quantify the potential impact of extreme weather events on certain types of transportation infrastructure as part of a vulnerability screening assessment. This screening assessment can be especially useful when a large number of assets or large geographical areas are being studied, with the results enabling planners and asset managers to undertake a more detailed assessment of vulnerability on a more targeted number of assets or locations. The methodology combines climate, weather, and impact data to identify vulnerabilities to a range of weather and climate related risks over a multi-decadal planning period. The paper applies the methodology to perform an extreme weather and climate change vulnerability screening assessment on transportation infrastructure assets for the State of Tennessee. This paper represents the results of one of the first efforts at spatial vulnerability assessments of transportation infrastructure and provides important insights for any organization considering the impact of climate and weather events on transportation or other critical infrastructure systems.Risks2016-08-0343Article10.3390/risks4030028282227-90912016-08-03doi: 10.3390/risks4030028Leah DundonKatherine NelsonJaney CampMark AbkowitzAlan Jones<![CDATA[Risks, Vol. 4, Pages 25: Understanding Reporting Delay in General Insurance]]>
http://www.mdpi.com/2227-9091/4/3/25
The aim of this paper is to understand and to model claims arrival and reporting delay in general insurance. We calibrate two real individual claims data sets to the statistical model of Jewell and Norberg. One data set considers property insurance and the other one casualty insurance. For our analysis we slightly relax the model assumptions of Jewell allowing for non-stationarity so that the model is able to cope with trends and with seasonal patterns. The performance of our individual claims data prediction is compared to the prediction based on aggregate data using the Poisson chain-ladder method.Risks2016-07-0843Article10.3390/risks4030025252227-90912016-07-08doi: 10.3390/risks4030025Richard VerrallMario Wüthrich<![CDATA[Risks, Vol. 4, Pages 27: Lead–Lag Relationship Using a Stop-and-Reverse-MinMax Process]]>
http://www.mdpi.com/2227-9091/4/3/27
The intermarket analysis, in particular the lead–lag relationship, plays an important role within financial markets. Therefore, a mathematical approach to be able to find interrelations between the price development of two different financial instruments is developed in this paper. Computing the differences of the relative positions of relevant local extrema of two charts, i.e., the local phase shifts of these price developments, gives us an empirical distribution on the unit circle. With the aid of directional statistics, such angular distributions are studied for many pairs of markets. It is shown that there are several very strongly correlated financial instruments in the field of foreign exchange, commodities and indexes. In some cases, one of the two markets is significantly ahead with respect to the relevant local extrema, i.e., there is a phase shift unequal to zero between them.Risks2016-07-0743Article10.3390/risks4030027272227-90912016-07-07doi: 10.3390/risks4030027Stanislaus Maier-PaapeAndreas Platen<![CDATA[Risks, Vol. 4, Pages 26: Optimal Reinsurance with Heterogeneous Reference Probabilities]]>
http://www.mdpi.com/2227-9091/4/3/26
This paper studies the problem of optimal reinsurance contract design. We let the insurer use dual utility, and the premium is an extended Wang’s premium principle. The novel contribution is that we allow for heterogeneity in the beliefs regarding the underlying probability distribution. We characterize layer-reinsurance as an optimal reinsurance contract. Moreover, we characterize layer-reinsurance as optimal contracts when the insurer faces costs of holding regulatory capital. We illustrate this in cases where both firms use the Value-at-Risk or the conditional Value-at-Risk.Risks2016-07-0743Article10.3390/risks4030026262227-90912016-07-07doi: 10.3390/risks4030026Tim Boonen<![CDATA[Risks, Vol. 4, Pages 23: Risk Minimization for Insurance Products via F-Doubly Stochastic Markov Chains]]>
http://www.mdpi.com/2227-9091/4/3/23
We study risk-minimization for a large class of insurance contracts. Given that the individual progress in time of visiting an insurance policy’s states follows an F -doubly stochastic Markov chain, we describe different state-dependent types of insurance benefits. These cover single payments at maturity, annuity-type payments and payments at the time of a transition. Based on the intensity of the F -doubly stochastic Markov chain, we provide the Galtchouk-Kunita-Watanabe decomposition for a general insurance contract and specify risk-minimizing strategies in a Brownian financial market setting. The results are further illustrated explicitly within an affine structure for the intensity.Risks2016-07-0743Article10.3390/risks4030023232227-90912016-07-07doi: 10.3390/risks4030023Francesca BiaginiAndreas GrollJan Widenmann<![CDATA[Risks, Vol. 4, Pages 24: Superforecasting: The Art and Science of Prediction. By Philip Tetlock and Dan Gardner]]>
http://www.mdpi.com/2227-9091/4/3/24
Let me say from the outset that this is an excellent book to read. It is not only informative, as it should be for a book on forecasting, but it is highly entertaining.[...]Risks2016-07-0543Book Review10.3390/risks4030024242227-90912016-07-05doi: 10.3390/risks4030024Daniel Buncic<![CDATA[Risks, Vol. 4, Pages 22: A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework]]>
http://www.mdpi.com/2227-9091/4/3/22
In this paper, we review pricing of the variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework and review the existing numerical methods. We also discuss pricing under the complete/incomplete financial market models, stochastic mortality and optimal/sub-optimal policyholder behavior, and in the presence of taxes. For numerical valuation of these contracts in the case of simple risky asset process, we develop a direct integration method based on the Gauss-Hermite quadratures with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is easier to implement and faster when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a numerical benchmark for practitioners and researchers developing pricing of variable annuity guarantees to assess the accuracy of their numerical implementation.Risks2016-07-0543Article10.3390/risks4030022222227-90912016-07-05doi: 10.3390/risks4030022Pavel ShevchenkoXiaolin Luo<![CDATA[Risks, Vol. 4, Pages 21: The Myth of Methuselah and the Uncertainty of Death: The Mortality Fan Charts]]>
http://www.mdpi.com/2227-9091/4/3/21
This paper uses mortality fan charts to illustrate prospective future male mortality. These fan charts show both the most likely path of male mortality and the bands of uncertainty surrounding that path. The fan charts are based on a model of male mortality that is known to provide a good fit to UK mortality data. The fan charts suggest that there are clear limits to longevity—that future mortality rates are very uncertain and tend to become more uncertain the further ahead the forecast—and that forecasts of future mortality uncertainty must also take account of uncertainty in the parameters of the underlying mortality model.Risks2016-07-0443Article10.3390/risks4030021212227-90912016-07-04doi: 10.3390/risks4030021Kevin DowdDavid BlakeAndrew Cairns<![CDATA[Risks, Vol. 4, Pages 20: Survey on Log-Normally Distributed Market-Technical Trend Data]]>
http://www.mdpi.com/2227-9091/4/3/20
In this survey, a short introduction of the recent discovery of log-normally-distributed market-technical trend data will be given. The results of the statistical evaluation of typical market-technical trend variables will be presented. It will be shown that the log-normal assumption fits better to empirical trend data than to daily returns of stock prices. This enables one to mathematically evaluate trading systems depending on such variables. In this manner, a basic approach to an anti-cyclic trading system will be given as an example.Risks2016-07-0443Article10.3390/risks4030020202227-90912016-07-04doi: 10.3390/risks4030020René BrennerStanislaus Maier-Paape<![CDATA[Risks, Vol. 4, Pages 19: An Optimal Turkish Private Pension Plan with a Guarantee Feature]]>
http://www.mdpi.com/2227-9091/4/3/19
The Turkish Private Pension System is an investment system which aims to generate income for future consumption. This is a volunteer system, and the contributions are held in individual portfolios. Therefore, management of the funds is an important issue for both the participants and the insurance company. In this study, we propose an optimal private pension plan with a guarantee feature that is based on Constant Proportion Portfolio Insurance (CPPI). We derive a closed form formula for the optimal strategy with the help of dynamic programming. Moreover, our model is evaluated with numerical examples, and we compare its performance by implementing a sensitivity analysis.Risks2016-06-2743Article10.3390/risks4030019192227-90912016-06-27doi: 10.3390/risks4030019Ayşegül İşcanog̃lu-Çekiç<![CDATA[Risks, Vol. 4, Pages 18: Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model]]>
http://www.mdpi.com/2227-9091/4/3/18
The discrete-time multifactor Vasiček model is a tractable Gaussian spot rate model. Typically, two- or three-factor versions allow one to capture the dependence structure between yields with different times to maturity in an appropriate way. In practice, re-calibration of the model to the prevailing market conditions leads to model parameters that change over time. Therefore, the model parameters should be understood as being time-dependent or even stochastic. Following the consistent re-calibration (CRC) approach, we construct models as concatenations of yield curve increments of Hull–White extended multifactor Vasiček models with different parameters. The CRC approach provides attractive tractable models that preserve the no-arbitrage premise. As a numerical example, we fit Swiss interest rates using CRC multifactor Vasiček models.Risks2016-06-2343Article10.3390/risks4030018182227-90912016-06-23doi: 10.3390/risks4030018Philipp HarmsDavid StefanovitsJosef TeichmannMario Wüthrich<![CDATA[Risks, Vol. 4, Pages 17: Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window]]>
http://www.mdpi.com/2227-9091/4/2/17
We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival times depending on the claims that arrive within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the bonus-malus (BM) feature of pricing car insurance. We discuss first the asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process, and via an appropriate change of measure, ruin probabilities could be computed to a closed-form formulae. Additionally, we employ the importance sampling simulations to derive ruin probabilities, which further permit an in-depth analysis of a few concrete cases.Risks2016-06-1542Article10.3390/risks4020017172227-90912016-06-15doi: 10.3390/risks4020017Corina ConstantinescuSuhang DaiWeihong NiZbigniew Palmowski<![CDATA[Risks, Vol. 4, Pages 16: Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities]]>
http://www.mdpi.com/2227-9091/4/2/16
This paper studies the dependence between coupled lives, i.e., the spouses’ dependence, across different generations, and its effects on prices of reversionary annuities in the presence of longevity risk. Longevity risk is represented via a stochastic mortality intensity. We find that a generation-based model is important, since spouses’ dependence decreases when passing from older generations to younger generations. The independence assumption produces quantifiable mispricing of reversionary annuities, with different effects on different generations. The research is conducted using a well-known dataset of double life contracts.Risks2016-05-2542Article10.3390/risks4020016162227-90912016-05-25doi: 10.3390/risks4020016Elisa LucianoJaap SpreeuwElena Vigna<![CDATA[Risks, Vol. 4, Pages 15: Improving Convergence of Binomial Schemes and the Edgeworth Expansion]]>
http://www.mdpi.com/2227-9091/4/2/15
Binomial trees are very popular in both theory and applications of option pricing. As they often suffer from an irregular convergence behavior, improving this is an important task. We build upon a new version of the Edgeworth expansion for lattice models to construct new and quickly converging binomial schemes with a particular application to barrier options.Risks2016-05-2342Article10.3390/risks4020015152227-90912016-05-23doi: 10.3390/risks4020015Alona BockRalf Korn<![CDATA[Risks, Vol. 4, Pages 14: Estimating Quantile Families of Loss Distributions for Non-Life Insurance Modelling via L-Moments]]>
http://www.mdpi.com/2227-9091/4/2/14
This paper discusses different classes of loss models in non-life insurance settings. It then overviews the class of Tukey transform loss models that have not yet been widely considered in non-life insurance modelling, but offer opportunities to produce flexible skewness and kurtosis features often required in loss modelling. In addition, these loss models admit explicit quantile specifications which make them directly relevant for quantile based risk measure calculations. We detail various parameterisations and sub-families of the Tukey transform based models, such as the g-and-h, g-and-k and g-and-j models, including their properties of relevance to loss modelling. One of the challenges that are amenable to practitioners when fitting such models is to perform robust estimation of the model parameters. In this paper we develop a novel, efficient, and robust procedure for estimating the parameters of this family of Tukey transform models, based on L-moments. It is shown to be more efficient than the current state of the art estimation methods for such families of loss models while being simple to implement for practical purposes.Risks2016-05-2042Article10.3390/risks4020014142227-90912016-05-20doi: 10.3390/risks4020014Gareth PetersWilson ChenRichard Gerlach<![CDATA[Risks, Vol. 4, Pages 12: Macro vs. Micro Methods in Non-Life Claims Reserving (an Econometric Perspective)]]>
http://www.mdpi.com/2227-9091/4/2/12
Traditionally, actuaries have used run-off triangles to estimate reserve (“macro” models, on aggregated data). However, it is possible to model payments related to individual claims. If those models provide similar estimations, we investigate uncertainty related to reserves with “macro” and “micro” models. We study theoretical properties of econometric models (Gaussian, Poisson and quasi-Poisson) on individual data, and clustered data. Finally, applications in claims reserving are considered.Risks2016-05-1442Article10.3390/risks4020012122227-90912016-05-14doi: 10.3390/risks4020012Arthur CharpentierMathieu Pigeon<![CDATA[Risks, Vol. 4, Pages 13: Community Analysis of Global Financial Markets]]>
http://www.mdpi.com/2227-9091/4/2/13
We analyze the daily returns of stock market indices and currencies of 56 countries over the period of 2002–2012. We build a network model consisting of two layers, one being the stock market indices and the other the foreign exchange markets. Synchronous and lagged correlations are used as measures of connectivity and causality among different parts of the global economic system for two different time intervals: non-crisis (2002–2006) and crisis (2007–2012) periods. We study community formations within the network to understand the influences and vulnerabilities of specific countries or groups of countries. We observe different behavior of the cross correlations and communities for crisis vs. non-crisis periods. For example, the overall correlation of stock markets increases during crisis while the overall correlation in the foreign exchange market and the correlation between stock and foreign exchange markets decrease, which leads to different community structures. We observe that the euro, while being central during the relatively calm period, loses its dominant role during crisis. Furthermore we discover that the troubled Eurozone countries, Portugal, Italy, Greece and Spain, form their own cluster during the crisis period.Risks2016-05-1342Article10.3390/risks4020013132227-90912016-05-13doi: 10.3390/risks4020013Irena VodenskaAlexander BeckerDi ZhouDror KenettH. StanleyShlomo Havlin<![CDATA[Risks, Vol. 4, Pages 11: Participating Life Insurance Products with Alternative Guarantees: Reconciling Policyholders’ and Insurers’ Interests]]>
http://www.mdpi.com/2227-9091/4/2/11
Traditional participating life insurance contracts with year-to-year (cliquet-style) guarantees have come under pressure in the current situation of low interest rates and volatile capital markets, in particular when priced in a market-consistent valuation framework. In addition, such guarantees lead to rather high capital requirements under risk-based solvency frameworks such as Solvency II or the Swiss Solvency Test (SST). Therefore, insurers in several countries have developed new forms of participating products with alternative (typically weaker and/or lower) guarantees that are less risky for the insurer. In a previous paper, it has been shown that such alternative product designs can lead to higher capital efficiency, i.e., higher and more stable profits and reduced capital requirements. As a result, the financial risk for the insurer is significantly reduced while the main guarantee features perceived and requested by the policyholder are preserved. Based on these findings, this paper now combines the insurer’s and the policyholder’s perspective by analyzing product versions that compensate policyholders for the less valuable guarantees. We particularly identify combinations of asset allocation and profit participation rate for the different product designs that lead to an identical expected profit for the insurer (and identical risk-neutral value for the policyholder), but differ with respect to the insurer’s risk and solvency capital requirements as well as with respect to the real-world return distribution for the policyholder. We show that alternative products can be designed in a way that the insurer’s expected profitability remains unchanged, the insurer’s risk and hence capital requirement is substantially reduced and the policyholder’s expected return is increased. This illustrates that such products might be able to reconcile insurers’ and policyholders’ interests and serve as an alternative to the rather risky cliquet-style products.Risks2016-05-0542Article10.3390/risks4020011112227-90912016-05-05doi: 10.3390/risks4020011Andreas ReußJochen RußJochen Wieland<![CDATA[Risks, Vol. 4, Pages 10: Telematics and Gender Discrimination: Some Usage-Based Evidence on Whether Men’s Risk of Accidents Differs from Women’s]]>
http://www.mdpi.com/2227-9091/4/2/10
Pay-as-you-drive (PAYD), or usage-based automobile insurance (UBI), is a policy agreement tied to vehicle usage. In this paper we analyze the effect of the distance traveled on the risk of accidents among young drivers with a PAYD policy. We use regression models for survival data to estimate how long it takes them to have their first accident at fault during the coverage period. Our empirical application with real data is presented and shows that gender differences are mainly attributable to the intensity of use. Indeed, although gender has a significant effect in explaining the time to the first crash, this effect is no longer significant when the average distance traveled per day is introduced in the model. This suggests that gender differences in the risk of accidents are, to a large extent, attributable to the fact that men drive more often than women. Estimates of the time to the first accident for different driver risk types are presented. We conclude that no gender discrimination is necessary if telematics provides enough information on driving habits.Risks2016-04-0842Article10.3390/risks4020010102227-90912016-04-08doi: 10.3390/risks4020010Mercedes AyusoMontserrat GuillenAna Pérez-Marín<![CDATA[Risks, Vol. 4, Pages 9: Inflation Protected Investment Strategies]]>
http://www.mdpi.com/2227-9091/4/2/9
In this paper, a dynamic inflation-protected investment strategy is presented, which is based on traditional asset classes and Markov-switching models. Different stock market, as well as inflation regimes are identified, and within those regimes, the inflation hedging potential of stocks, bonds, real estate, commodities and gold are investigated. Within each regime, we determine optimal investment portfolios driven by the investment idea of protection from losses due to changing inflation if inflation is rising or high, but decoupling the performance from inflation if inflation is low. The results clearly indicate that these asset classes behave differently in different stock market and inflation regimes. Whereas in the long-run, we agree with the general opinion in the literature that stocks and bonds are a suitable hedge against inflation, we observe for short time horizons that the hedging potential of each asset class, especially of real estate and commodities, depend strongly on the state of the current market environment. Thus, our approach provides a possible explanation for different statements in the literature regarding the inflation hedging properties of these asset classes. A dynamic inflation-protected investment strategy is developed, which combines inflation protection and upside potential. This strategy outperforms standard buy-and-hold strategies, as well as the well-known 1 N -portfolio.Risks2016-03-2842Article10.3390/risks402000992227-90912016-03-28doi: 10.3390/risks4020009Mirco MahlstedtRudi Zagst<![CDATA[Risks, Vol. 4, Pages 8: Optimal Insurance for a Minimal Expected Retention: The Case of an Ambiguity-Seeking Insurer]]>
http://www.mdpi.com/2227-9091/4/1/8
In the classical expected utility framework, a problem of optimal insurance design with a premium constraint is equivalent to a problem of optimal insurance design with a minimum expected retention constraint. When the insurer has ambiguous beliefs represented by a non-additive probability measure, as in Schmeidler, this equivalence no longer holds. Recently, Amarante, Ghossoub and Phelps examined the problem of optimal insurance design with a premium constraint when the insurer has ambiguous beliefs. In particular, they showed that when the insurer is ambiguity-seeking, with a concave distortion of the insured’s probability measure, then the optimal indemnity schedule is a state-contingent deductible schedule, in which the deductible depends on the state of the world only through the insurer’s distortion function. In this paper, we examine the problem of optimal insurance design with a minimum expected retention constraint, in the case where the insurer is ambiguity-seeking. We obtain the aforementioned result of Amarante, Ghossoub and Phelps and the classical result of Arrow as special cases.Risks2016-03-2141Article10.3390/risks401000882227-90912016-03-21doi: 10.3390/risks4010008Massimiliano AmaranteMario Ghossoub<![CDATA[Risks, Vol. 4, Pages 7: Nonlinear Time Series and Neural-Network Models of Exchange Rates between the US Dollar and Major Currencies]]>
http://www.mdpi.com/2227-9091/4/1/7
This paper features an analysis of major currency exchange rate movements in relation to the US dollar, as constituted in US dollar terms. Euro, British pound, Chinese yuan, and Japanese yen are modelled using a variety of non-linear models, including smooth transition regression models, logistic smooth transition regressions models, threshold autoregressive models, nonlinear autoregressive models, and additive nonlinear autoregressive models, plus Neural Network models. The models are evaluated on the basis of error metrics for twenty day out-of-sample forecasts using the mean average percentage errors (MAPE). The results suggest that there is no dominating class of time series models, and the different currency pairs relationships with the US dollar are captured best by neural net regression models, over the ten year sample of daily exchange rate returns data, from August 2005 to August 2015.Risks2016-03-1641Article10.3390/risks401000772227-90912016-03-16doi: 10.3390/risks4010007David AllenMichael McAleerShelton PeirisAbhay Singh<![CDATA[Risks, Vol. 4, Pages 6: Analysis of Insurance Claim Settlement Process with Markovian Arrival Processes]]>
http://www.mdpi.com/2227-9091/4/1/6
This paper proposes a model for the claim occurrence, reporting, and handling process of insurance companies. It is assumed that insurance claims occur according to a Markovian arrival process. An incurred claim goes through some stages of a claim reporting and handling process, such as Incurred But Not Reported (IBNR), Reported But Not Settled (RBNS) and Settled (S). We derive formulas for the joint distribution and the joint moments for the amount of INBR, RBNS and Settled claims. This model generalizes previous ones in the literature, which generally assume Poisson claim arrivals. Due to the flexibility of the Markovian arrival process, the model can be used to evaluate how the claim occurring, reporting, and handling mechanisms may affect the volatilities of the amount of IBNR, RBNS and Settled claims, and the interdependencies among them.Risks2016-03-1141Article10.3390/risks401000662227-90912016-03-11doi: 10.3390/risks4010006Jiandong Ren<![CDATA[Risks, Vol. 4, Pages 5: High-Frequency Financial Econometrics]]>
http://www.mdpi.com/2227-9091/4/1/5
This book is fundamentally about the estimation of risk.[...]Risks2016-02-2641Book Review10.3390/risks401000552227-90912016-02-26doi: 10.3390/risks4010005Harley Thompson<![CDATA[Risks, Vol. 4, Pages 4: Multivariate Frequency-Severity Regression Models in Insurance]]>
http://www.mdpi.com/2227-9091/4/1/4
In insurance and related industries including healthcare, it is common to have several outcome measures that the analyst wishes to understand using explanatory variables. For example, in automobile insurance, an accident may result in payments for damage to one’s own vehicle, damage to another party’s vehicle, or personal injury. It is also common to be interested in the frequency of accidents in addition to the severity of the claim amounts. This paper synthesizes and extends the literature on multivariate frequency-severity regression modeling with a focus on insurance industry applications. Regression models for understanding the distribution of each outcome continue to be developed yet there now exists a solid body of literature for the marginal outcomes. This paper contributes to this body of literature by focusing on the use of a copula for modeling the dependence among these outcomes; a major advantage of this tool is that it preserves the body of work established for marginal models. We illustrate this approach using data from the Wisconsin Local Government Property Insurance Fund. This fund offers insurance protection for (i) property; (ii) motor vehicle; and (iii) contractors’ equipment claims. In addition to several claim types and frequency-severity components, outcomes can be further categorized by time and space, requiring complex dependency modeling. We find significant dependencies for these data; specifically, we find that dependencies among lines are stronger than the dependencies between the frequency and average severity within each line.Risks2016-02-2541Article10.3390/risks401000442227-90912016-02-25doi: 10.3390/risks4010004Edward FreesGee LeeLu Yang<![CDATA[Risks, Vol. 4, Pages 3: Premiums for Long-Term Care Insurance Packages: Sensitivity with Respect to Biometric Assumptions]]>
http://www.mdpi.com/2227-9091/4/1/3
Long-term care insurance (LTCI) covers are rather recent products, in the framework of health insurance. It follows that specific biometric data are scanty; pricing and reserving problems then arise because of difficulties in the choice of appropriate technical bases. Different benefit structures imply different sensitivity degrees with respect to changes in biometric assumptions. Hence, an accurate sensitivity analysis can help in designing LTCI products and, in particular, in comparing stand-alone products to combined products, i.e., packages including LTCI benefits and other lifetime-related benefits. Numerical examples show, in particular, that the stand-alone cover is much riskier than all of the LTCI combined products that we have considered. As a consequence, the LTCI stand-alone cover is a highly “absorbing” product as regards capital requirements for solvency purposes.Risks2016-02-2241Article10.3390/risks401000332227-90912016-02-22doi: 10.3390/risks4010003Ermanno Pitacco<![CDATA[Risks, Vol. 4, Pages 2: Ruin Analysis of a Discrete-Time Dependent Sparre Andersen Model with External Financial Activities and Randomized Dividends]]>
http://www.mdpi.com/2227-9091/4/1/2
We consider a discrete-time dependent Sparre Andersen risk model which incorporates multiple threshold levels characterizing an insurer’s minimal capital requirement, dividend paying situations, and external financial activities. We focus on the development of a recursive computational procedure to calculate the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin associated with this model. We investigate several numerical examples and make some observations concerning the impact our threshold levels have on the finite-time ruin probabilities and expected total discounted dividends paid prior to ruin.Risks2016-02-0341Article10.3390/risks401000222227-90912016-02-03doi: 10.3390/risks4010002Sung KimSteve Drekic<![CDATA[Risks, Vol. 4, Pages 1: Acknowledgement to Reviewers of Risks in 2015]]>
http://www.mdpi.com/2227-9091/4/1/1
The editors of Risks would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2015. [...]Risks2016-01-2141Editorial10.3390/risks401000112227-90912016-01-21doi: 10.3390/risks4010001 Risks Editorial Office<![CDATA[Risks, Vol. 3, Pages 624-646: Modified Munich Chain-Ladder Method]]>
http://www.mdpi.com/2227-9091/3/4/624
The Munich chain-ladder method for claims reserving was introduced by Quarg and Mack on an axiomatic basis. We analyze these axioms, and we define a modified Munich chain-ladder method which is based on an explicit stochastic model. This stochastic model then allows us to consider claims prediction and prediction uncertainty for the Munich chain-ladder method in a consistent way.Risks2015-12-2134Article10.3390/risks30406246246462227-90912015-12-21doi: 10.3390/risks3040624Michael MerzMario Wüthrich<![CDATA[Risks, Vol. 3, Pages 599-623: Dependence Uncertainty Bounds for the Expectile of a Portfolio]]>
http://www.mdpi.com/2227-9091/3/4/599
We study upper and lower bounds on the expectile risk measure of risky portfolios when the joint distribution of the risky components is not fully specified. First, we summarize methods for obtaining bounds when only the marginal distributions of the components are known, but not their interdependence (unconstrained bounds). In particular, we provide the best-possible upper bound and the best-possible lower bound (under some conditions), as well as numerical procedures to compute them. We also derive simple analytic bounds that appear adequate in various situations of interest. Second, we study bounds when some information on interdependence is available (constrained bounds). When the variance of the portfolio is known, a simple-to-compute upper bound is provided, and we illustrate that it may significantly improve the unconstrained upper bound. We also show that the unconstrained lower bound cannot be readily improved using variance information. Next, we derive improved bounds when the bivariate distributions of each of the risky components and a risk factor are known. When the factor induces a positive dependence among the components, it is typically possible to improve the unconstrained lower bound. Finally, the unconstrained dependence uncertainty spreads of expected shortfall, value-at-risk and the expectile are compared.Risks2015-12-1034Article10.3390/risks30405995996232227-90912015-12-10doi: 10.3390/risks3040599Edgars JakobsonsSteven Vanduffel<![CDATA[Risks, Vol. 3, Pages 573-598: Information-Based Trade in German Real Estate and Equity Markets]]>
http://www.mdpi.com/2227-9091/3/4/573
This paper employs four established market microstructure measures on information-based trade in financial markets. A set of German mid and small caps is used to analyze potential differential information content in real estate stocks compared to other asset classes. After linking substantially lower amounts of information-based trade in real estate stocks to higher liquidity premia, it is found that the evolution of the information content in real estate and other assets follows similar trends. Consequently, interdependence is tested for rolling time windows, revealing strong informational links between real estate and other assets. Particularly, small caps, financials, as well as companies offering consumer goods and services show a close relationship to real estate. Depending on the choice of the measure of information-based trade, up to 75% of the variation in the information content in real estate shares is related to other asset classes, pointing to the notion of high dependence.Risks2015-12-0734Article10.3390/risks30405735735982227-90912015-12-07doi: 10.3390/risks3040573Marco Wölfle<![CDATA[Risks, Vol. 3, Pages 553-572: Stochastic Optimal Control for Online Seller under Reputational Mechanisms]]>
http://www.mdpi.com/2227-9091/3/4/553
In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller’s reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from [1] and used effectively in the Stochastic Optimal Control of Advertising [2]. This model of reputation is combined with the effect of online reputation on sales price empirically verified in [3]. We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller’s reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct.Risks2015-12-0434Article10.3390/risks30405535535722227-90912015-12-04doi: 10.3390/risks3040553Milan BradonjićMatthew CausleyAlbert Cohen<![CDATA[Risks, Vol. 3, Pages 543-552: Production Flexibility and Hedging]]>
http://www.mdpi.com/2227-9091/3/4/543
We extend the analysis on hedging with price and output uncertainty by endogenizing the output decision. Specifically, we consider the joint determination of output and hedging in the case of flexibility in production. We show that the risk-averse firm always maintains a short position in the futures market when the futures price is actuarially fair. Moreover, in the context of an example, we show that the presence of production flexibility reduces the incentive to hedge for all risk averse agents.Risks2015-12-0434Article10.3390/risks30405435435522227-90912015-12-04doi: 10.3390/risks3040543Georges DionneMarc Santugini<![CDATA[Risks, Vol. 3, Pages 515-542: The Impact of Guarantees on the Performance of Pension Saving Schemes: Insights from the Literature]]>
http://www.mdpi.com/2227-9091/3/4/515
Guarantees are often seen as the key characteristics of pension saving products, but securing them can become costly and is of central relevance especially in the course of the current low interest rate environment. In this article, we deal with the question of how costly the typical types of guarantees are, in the sense that they reduce a pension saving scheme’s financial performance over time. In this context, we aim to provide a presentation of insights from selected literature studying the impact of point-to-point guarantees and cliquet-style interest rate guarantees on the performance of pension contracts. The comparative analysis emphasizes that, in most cases, guarantee costs are not negligible with regard to a contract’s financial performance, especially compared to benchmarks, and that customers knowingly opt for such guarantees (or not) is, thus, indispensable. To further investigate the willingness-to-pay for guarantees in life insurance is an area for future research, in particular for innovative contract design.Risks2015-11-2034Article10.3390/risks30405155155422227-90912015-11-20doi: 10.3390/risks3040515Alexander Bohnert<![CDATA[Risks, Vol. 3, Pages 491-514: On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy]]>
http://www.mdpi.com/2227-9091/3/4/491
In the compound Poisson insurance risk model under a dividend barrier strategy, this paper aims to analyze jointly the aggregate discounted claim amounts until ruin and the total discounted dividends until ruin, which represent the insurer’s payments to its policyholders and shareholders, respectively. To this end, we introduce a Gerber–Shiu-type function, which further incorporates the higher moments of these two quantities. This not only unifies the individual study of various ruin-related quantities, but also allows for new measures concerning covariances to be calculated. The integro-differential equation satisfied by the generalized Gerber–Shiu function and the boundary condition are derived. In particular, when the claim severity is distributed as a combination of exponentials, explicit expressions for this Gerber–Shiu function in some special cases are given. Numerical examples involving the covariances between any two of (i) the aggregate discounted claims until ruin, (ii) the discounted dividend payments until ruin and (iii) the time of ruin are presented along with some interpretations.Risks2015-11-1034Article10.3390/risks30404914915142227-90912015-11-10doi: 10.3390/risks3040491Eric CheungHaibo LiuJae-Kyung Woo<![CDATA[Risks, Vol. 3, Pages 474-490: Combining Alphas via Bounded Regression]]>
http://www.mdpi.com/2227-9091/3/4/474
We give an explicit algorithm and source code for combining alpha streams via bounded regression. In practical applications, typically, there is insufficient history to compute a sample covariance matrix (SCM) for a large number of alphas. To compute alpha allocation weights, one then resorts to (weighted) regression over SCM principal components. Regression often produces alpha weights with insufficient diversification and/or skewed distribution against, e.g., turnover. This can be rectified by imposing bounds on alpha weights within the regression procedure. Bounded regression can also be applied to stock and other asset portfolio construction. We discuss illustrative examples.Risks2015-11-0434Article10.3390/risks30404744744902227-90912015-11-04doi: 10.3390/risks3040474Zura Kakushadze<![CDATA[Risks, Vol. 3, Pages 455-473: Hidden Markov Model for Stock Selection]]>
http://www.mdpi.com/2227-9091/3/4/455
The hidden Markov model (HMM) is typically used to predict the hidden regimes of observation data. Therefore, this model finds applications in many different areas, such as speech recognition systems, computational molecular biology and financial market predictions. In this paper, we use HMM for stock selection. We first use HMM to make monthly regime predictions for the four macroeconomic variables: inflation (consumer price index (CPI)), industrial production index (INDPRO), stock market index (S&amp;P 500) and market volatility (VIX). At the end of each month, we calibrate HMM’s parameters for each of these economic variables and predict its regimes for the next month. We then look back into historical data to find the time periods for which the four variables had similar regimes with the forecasted regimes. Within those similar periods, we analyze all of the S&amp;P 500 stocks to identify which stock characteristics have been well rewarded during the time periods and assign scores and corresponding weights for each of the stock characteristics. A composite score of each stock is calculated based on the scores and weights of its features. Based on this algorithm, we choose the 50 top ranking stocks to buy. We compare the performances of the portfolio with the benchmark index, S&amp;P 500. With an initial investment of $100 in December 1999, over 15 years, in December 2014, our portfolio had an average gain per annum of 14.9% versus 2.3% for the S&amp;P 500.Risks2015-10-2934Article10.3390/risks30404554554732227-90912015-10-29doi: 10.3390/risks3040455Nguyet NguyenDung Nguyen<![CDATA[Risks, Vol. 3, Pages 445-454: Risk Classification Efficiency and the Insurance Market Regulation]]>
http://www.mdpi.com/2227-9091/3/4/445
Given that the insurance market is characterized by asymmetric information, its efficiency has traditionally been based to a large extent on risk classification. In certain regulations, however, we can find restrictions on these differentiations, primarily the ban on those considered to be “discriminatory”. In 2011, following the European Union Directive 2004/113/EC, the European Court of Justice concluded that any gender-based discrimination was prohibited, meaning that gender equality in the European Union had to be ensured from 21 December 2012. Another restriction was imposed by EU and national competition regulation on the exchange of information considered as anti-competitive behavior. This paper aims to contribute to the recent policy debate in the EU, evaluating the negative economic consequences of these regulatory restrictions in terms of market efficiency.Risks2015-09-2534Article10.3390/risks30404454454542227-90912015-09-25doi: 10.3390/risks3040445Donatella Porrini<![CDATA[Risks, Vol. 3, Pages 420-444: The Financial Stress Index: Identification of Systemic Risk Conditions]]>
http://www.mdpi.com/2227-9091/3/3/420
This paper develops a financial stress measure for the United States, the Cleveland Financial Stress Index (CFSI). The index is based on publicly available data describing a six-market partition of the financial system comprising credit, funding, real estate, securitization, foreign exchange, and equity markets. This paper improves upon existing stress measures by objectively selecting between several index weighting methodologies across a variety of monitoring frequencies through comparison against a volatility-based benchmark series. The resulting measure facilitates the decomposition of stress to identify disruptions in specific markets and provides insight into historical stress regimes.Risks2015-09-1633Article10.3390/risks30304204204442227-90912015-09-16doi: 10.3390/risks3030420Mikhail OetJohn DooleyStephen Ong<![CDATA[Risks, Vol. 3, Pages 390-419: Multi-Objective Stochastic Optimization Programs for a Non-Life Insurance Company under Solvency Constraints]]>
http://www.mdpi.com/2227-9091/3/3/390
In the paper, we introduce a multi-objective scenario-based optimization approach for chance-constrained portfolio selection problems. More specifically, a modified version of the normal constraint method is implemented with a global solver in order to generate a dotted approximation of the Pareto frontier for bi- and tri-objective programming problems. Numerical experiments are carried out on a set of portfolios to be optimized for an EU-based non-life insurance company. Both performance indicators and risk measures are managed as objectives. Results show that this procedure is effective and readily applicable to achieve suitable risk-reward tradeoff analysis.Risks2015-09-1533Article10.3390/risks30303903904192227-90912015-09-15doi: 10.3390/risks3030390Massimiliano KaucicRoberto Daris<![CDATA[Risks, Vol. 3, Pages 365-389: Supervising System Stress in Multiple Markets]]>
http://www.mdpi.com/2227-9091/3/3/365
This paper develops an extended financial stress measure that considers the supervisory objective of identifying risks to the stability of the financial system. The measure provides a continuous and bounded signal of financial stress using daily public market data. Broad coverage of material financial system markets over time is achieved by leveraging dynamic credit weights. We consider how this measure can be used to monitor, analyze, and alert financial system stress.Risks2015-09-1433Article10.3390/risks30303653653892227-90912015-09-14doi: 10.3390/risks3030365Mikhail OetJohn DooleyAmanda JanoskoDieter GramlichStephen Ong<![CDATA[Risks, Vol. 3, Pages 338-364: Valuation of Index-Linked Cash Flows in a Heath–Jarrow–Morton Framework]]>
http://www.mdpi.com/2227-9091/3/3/338
In this paper, we study the valuation of stochastic cash flows that exhibit dependence on interest rates. We focus on insurance liability cash flows linked to an index, such as a consumer price index or wage index, where changes in the index value can be partially understood in terms of changes in the term structure of interest rates. Insurance liability cash flows that are not explicitly linked to an index may still be valued in our framework by interpreting index returns as so-called claims inflation, i.e., an increase in claims cost per sold insurance contract. We focus primarily on the case when a deep and liquid market for index-linked contracts is absent or when the market price data are unreliable. Firstly, we present an approach for assigning a monetary value to a stochastic cash flow that does not require full knowledge of the joint dynamics of the cash flow and the term structure of interest rates. Secondly, we investigate in detail model selection, estimation and validation in a Heath–Jarrow–Morton framework. Finally, we analyze the effects of model uncertainty on the valuation of the cash flows and how forecasts of cash flows and interest rates translate into model parameters and affect the valuation.Risks2015-09-1033Article10.3390/risks30303383383642227-90912015-09-10doi: 10.3390/risks3030338Jonas AlmFilip Lindskog<![CDATA[Risks, Vol. 3, Pages 318-337: Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs]]>
http://www.mdpi.com/2227-9091/3/3/318
For pension-savers, a low payoff is a financial disaster. Such investors will most likely prefer left-skewed payoff distributions over right-skewed payoff distributions. We explore how such distributions can be delivered. Cautious-relaxed utility measures are cautious in ensuring that payoffs don’t fall much below a reference value, but relaxed about exceeding it. We find that the payoff distribution delivered by a cautious-relaxed utility measure has appealing features which payoff distributions delivered by traditional utility functions don’t. In particular, cautious-relaxed distributions can have the mass concentrated on the left, hence be left-skewed. However, cautious-relaxed strategies prescribe frequent portfolio adjustments which may be expensive if transaction costs are charged. In contrast, more traditional strategies can be time-invariant. Thus we investigate the impact of transaction costs on the appeal of cautious-relaxed strategies. We find that relatively high transaction fees are required for the cautious-relaxed strategy to lose its appeal. This paper contributes to the literature which compares utility measures by the payoff distributions they produce and finds that a cautious-relaxed utility measure will deliver payoffs that many investors will prefer.Risks2015-08-2133Article10.3390/risks30303183183372227-90912015-08-21doi: 10.3390/risks3030318Jacek Krawczyk<![CDATA[Risks, Vol. 3, Pages 290-317: Life Insurance Cash Flows with Policyholder Behavior]]>
http://www.mdpi.com/2227-9091/3/3/290
The problem of the valuation of life insurance payments with policyholder behavior is studied. First, a simple survival model is considered, and it is shown how cash flows without policyholder behavior can be modified to include surrender and free policy behavior by calculation of simple integrals. In the second part, a more general disability model with recovery is studied. Here, cash flows are determined by solving a modified Kolmogorov forward differential equation. We conclude the paper with numerical examples illustrating the methods proposed and the impact of policyholder behavior.Risks2015-07-2433Article10.3390/risks30302902903172227-90912015-07-24doi: 10.3390/risks3030290Kristian BuchardtThomas Møller<![CDATA[Risks, Vol. 3, Pages 277-289: Monopolistic Insurance and the Value of Information]]>
http://www.mdpi.com/2227-9091/3/3/277
The value of information regarding risk class for a monopoly insurer and its customers is examined in both symmetric and asymmetric information environments. A monopolist always prefers contracting with uninformed customers as this maximizes the rent extracted under symmetric information while also avoiding the cost of adverse selection when information is held asymmetrically. Although customers are indifferent to symmetric information when they are initially uninformed, they prefer contracting with hidden knowledge rather than symmetric information since the monopoly responds to adverse selection by sharing gains from trade with high-risk customers when low risks are predominant in the insurance pool. However, utilitarian social welfare is highest when customers are uninformed, and is higher when information is symmetric rather than asymmetric.Risks2015-07-2433Article10.3390/risks30302772772892227-90912015-07-24doi: 10.3390/risks3030277Arthur Snow<![CDATA[Risks, Vol. 3, Pages 250-276: Best-Estimates in Bond Markets with Reinvestment Risk]]>
http://www.mdpi.com/2227-9091/3/3/250
The concept of best-estimate, prescribed by regulators to value insurance liabilities for accounting and solvency purposes, has recently been discussed extensively in the industry and related academic literature. To differentiate hedgeable and non-hedgeable risks in a general case, recent literature defines best-estimates using orthogonal projections of a claim on the space of replicable payoffs. In this paper, we apply this concept of best-estimate to long-maturity claims in a market with reinvestment risk, since in this case the total liability cannot easily be separated into hedgeable and non-hedgeable parts. We assume that a limited number of short-maturity bonds are traded, and derive the best-estimate price of bonds with longer maturities, thus obtaining a best-estimate yield curve. We therefore use the multifactor Vasiˇcek model and derive within this framework closed-form expressions for the best-estimate prices of long-term bonds.Risks2015-07-1633Article10.3390/risks30302502502762227-90912015-07-16doi: 10.3390/risks3030250Anne MacKayMario Wüthrich<![CDATA[Risks, Vol. 3, Pages 234-249: Options with Extreme Strikes]]>
http://www.mdpi.com/2227-9091/3/3/234
In this short paper, we study the asymptotics for the price of call options for very large strikes and put options for very small strikes. The stock price is assumed to follow the Black–Scholes models. We analyze European, Asian, American, Parisian and perpetual options and conclude that the tail asymptotics for these option types fall into four scenarios.Risks2015-07-0833Article10.3390/risks30302342342492227-90912015-07-08doi: 10.3390/risks3030234Lingjiong Zhu<![CDATA[Risks, Vol. 3, Pages 219-233: Multiscale Analysis of the Predictability of Stock Returns]]>
http://www.mdpi.com/2227-9091/3/2/219
Due to the strong complexity of financial markets, economics does not have a unified theory of price formation in financial markets. The most common assumption is the Efficient-Market Hypothesis, which has been attacked by a number of researchers, using different tools. There were varying degrees to which these tools complied with the formal definitions of efficiency and predictability. In our earlier work, we analysed the predictability of stock returns at two time scales using the entropy rate, which can be directly linked to the mathematical definition of predictability. Nonetheless, none of the above-mentioned studies allow any general understanding of how the financial markets work, beyond disproving the Efficient-Market Hypothesis. In our previous study, we proposed the Maximum Entropy Production Principle, which uses the entropy rate to create a general principle underlying the price formation processes. Both of these studies show that the predictability of price changes is higher at the transaction level intraday scale than the scale of daily returns, but ignore all scales in between. In this study we extend these ideas using the multiscale entropy analysis framework to enhance our understanding of the predictability of price formation processes at various time scales.Risks2015-06-0832Article10.3390/risks30202192192332227-90912015-06-08doi: 10.3390/risks3020219Paweł Fiedor<![CDATA[Risks, Vol. 3, Pages 183-218: A Two-Account Life Insurance Model for Scenario-Based Valuation Including Event Risk]]>
http://www.mdpi.com/2227-9091/3/2/183
Using a two-account model with event risk, we model life insurance contracts taking into account both guaranteed and non-guaranteed payments in participating life insurance as well as in unit-linked insurance. Here, event risk is used as a generic term for life insurance events, such as death, disability, etc. In our treatment of participating life insurance, we have special focus on the bonus schemes “consolidation” and “additional benefits”, and one goal is to formalize how these work and interact. Another goal is to describe similarities and differences between participating life insurance and unit-linked insurance. By use of a two-account model, we are able to illustrate general concepts without making the model too abstract. To allow for complicated financial markets without dramatically increasing the mathematical complexity, we focus on economic scenarios. We illustrate the use of our model by conducting scenario analysis based on Monte Carlo simulation, but the model applies to scenarios in general and to worst-case and best-estimate scenarios in particular. In addition to easy computations, our model offers a common framework for the valuation of life insurance payments across product types. This enables comparison of participating life insurance products and unit-linked insurance products, thus building a bridge between the two different ways of formalizing life insurance products. Finally, our model distinguishes itself from the existing literature by taking into account the Markov model for the state of the policyholder and, hereby, facilitating event risk.Risks2015-06-0432Article10.3390/risks30201831832182227-90912015-06-04doi: 10.3390/risks3020183Ninna JensenKristian Schomacker<![CDATA[Risks, Vol. 3, Pages 164-182: The Impact of Reinsurance Strategies on Capital Requirements for Premium Risk in Insurance]]>
http://www.mdpi.com/2227-9091/3/2/164
New risk-based solvency requirements for insurance companies across European markets have been introduced by Solvency II and will come in force from 1 January 2016. These requirements, derived by a Standard Formula or an Internal Model, will be by far more risk-sensitive than the required solvency margin provided by the current legislation. In this regard, a Partial Internal Model for Premium Risk is developed here for a multi-line Non-Life insurer. We follow a classical approach based on a Collective Risk Model properly extended in order to consider not only the volatility of aggregate claim amounts but also expense volatility. To measure the effect of risk mitigation, suitable reinsurance strategies are pursued. We analyze how naïve coverage as conventional Quota Share and Excess of Loss reinsurance may modify the exact moments of the distribution of technical results. Furthermore, we investigate how alternative choices of commission rates in proportional treaties may affect the variability of distribution. Numerical results are also figured out in the last part of the paper with evidence of different effects for small and large companies. The main reasons for these differences are pointed out.Risks2015-06-0332Article10.3390/risks30201641641822227-90912015-06-03doi: 10.3390/risks3020164Gian ClementeNino SavelliDiego Zappa<![CDATA[Risks, Vol. 3, Pages 139-163: Interconnectedness of Financial Conglomerates]]>
http://www.mdpi.com/2227-9091/3/2/139
Being active in both the insurance sector and the banking sector, financial conglomerates intrinsically increase the interconnections between the banking sector and the insurance sector. We address two main concerns about financial conglomerates using a unique database on bilateral exposures between 21 French financial institutions. First, we investigate to what extent to which the insurers that are part of financial conglomerates differ from pure insurers. Second, we show that in the presence of sovereign risk, the components of a financial conglomerate are better off than if they were distinct entities. Our empirical findings bring a new perspective to the previous results of the literature based on using different types of data.Risks2015-05-2132Article10.3390/risks30201391391632227-90912015-05-21doi: 10.3390/risks3020139Gaël HautonJean-Cyprien Héam<![CDATA[Risks, Vol. 3, Pages 112-138: Custom v. Standardized Risk Models]]>
http://www.mdpi.com/2227-9091/3/2/112
We discuss when and why custom multi-factor risk models are warranted and give source code for computing some risk factors. Pension/mutual funds do not require customization but standardization. However, using standardized risk models in quant trading with much shorter holding horizons is suboptimal: (1) longer horizon risk factors (value, growth, etc.) increase noise trades and trading costs; (2) arbitrary risk factors can neutralize alpha; (3) “standardized” industries are artificial and insufficiently granular; (4) normalization of style risk factors is lost for the trading universe; (5) diversifying risk models lowers P&amp;L correlations, reduces turnover and market impact, and increases capacity. We discuss various aspects of custom risk model building.Risks2015-05-2032Article10.3390/risks30201121121382227-90912015-05-20doi: 10.3390/risks3020112Zura KakushadzeJim Liew<![CDATA[Risks, Vol. 3, Pages 103-111: Rationality Parameter for Exercising American Put]]>
http://www.mdpi.com/2227-9091/3/2/103
In this paper, irrational exercise behavior of the buyer of an American put is characterized by a single parameter. We model irrational exercise rules as the first jump time of a point processes with stochastic intensity. By the rationality parameter, we parameterize a family of stochastic intensities that depends on the value of the put itself. We present a probabilistic proof that the value of the American put using the irrational exercise rule converges to the arbitrage-free price as the rationality parameter converges to infinity. Another application of this result is the penalty method for approximating the price of an American put.Risks2015-05-2032Article10.3390/risks30201031031112227-90912015-05-20doi: 10.3390/risks3020103Kamille GadJesper Pedersen<![CDATA[Risks, Vol. 3, Pages 77-102: Portability, Salary and Asset Price Risk: A Continuous-Time Expected Utility Comparison of DB and DC Pension Plans]]>
http://www.mdpi.com/2227-9091/3/1/77
This paper compares two different types of private retirement plans from the perspective of a representative beneficiary: a defined benefit (DB) and a defined contribution (DC) plan. While salary risk is the main common risk factor in DB and DC pension plans, one of the key differences is that DB plans carry portability risks, whereas DC plans bear asset price risk. We model these tradeoffs explicitly in this paper and compare these two plans in a utility-based framework. Our numerical analysis focuses on answering the question of when the beneficiary is indifferent between the DB and DC plan. Most of our results confirm the findings in the existing literature, among which, e.g., portability losses considerably reduce the relative attractiveness of the DB plan. However, we also find that the attractiveness of the DB plan can decrease in the level of risk aversion, which is inconsistent with the existing literature.Risks2015-03-1331Article10.3390/risks3010077771022227-90912015-03-13doi: 10.3390/risks3010077An ChenFilip Uzelac<![CDATA[Risks, Vol. 3, Pages 61-76: Double Crowding-Out Effects of Means-Tested Public Provision for Long-Term Care]]>
http://www.mdpi.com/2227-9091/3/1/61
Publicly provided long-term care (LTC) insurance with means-tested benefits is suspected to crowd out either private saving or informal care. This contribution predicts crowding-out effects for both private saving and informal care for policy measures designed to relieve the public purse from LTC expenditure such as more stringent means testing and increased taxation of inheritance. These effects result from the interaction of a parent who decides on the amount of saving in retirement and a caregiver who decides on the effort devoted to informal care which lowers the probability of admission to a nursing home. Double crowding-out effects are also found to be the consequence of exogenous influences, notably a higher opportunity cost of caregiving.Risks2015-02-2531Article10.3390/risks301006161762227-90912015-02-25doi: 10.3390/risks3010061Christophe CourbagePeter Zweifel<![CDATA[Risks, Vol. 3, Pages 35-60: Safety Margins for Systematic Biometric and Financial Risk in a Semi-Markov Life Insurance Framework]]>
http://www.mdpi.com/2227-9091/3/1/35
Insurance companies use conservative first order valuation bases to calculate insurance premiums and reserves. These valuation bases have a significant impact on the insurer’s solvency and on the premiums of the insurance products. Safety margins for systematic biometric and financial risk are in practice typically chosen as time-constant percentages on top of the best estimate transition intensities. We develop a risk-oriented method for the allocation of a total safety margin to the single safety margins at each point in time and each state. In a case study, we demonstrate the suitability of the proposed method in different frameworks. The results show that the traditional method yields an unwanted variability of the safety level with respect to time, whereas the variability can be significantly reduced by the new method. Furthermore, the case study supports the German 60 percent rule for the technical interest rate.Risks2015-01-1931Article10.3390/risks301003535602227-90912015-01-19doi: 10.3390/risks3010035Andreas Niemeyer<![CDATA[Risks, Vol. 3, Pages 26-34: Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems]]>
http://www.mdpi.com/2227-9091/3/1/26
We identify restrictions on a decision maker’s utility function that are both necessary and sufficient to preserve dominance reasoning in each of two versions of the Two-Envelope Paradox (TEP). For the classical TEP, the utility function must satisfy a certain recurrence inequality. For the St. Petersburg TEP, the utility function must be bounded above asymptotically by a power function, which can be tightened to a constant. By determining the weakest conditions for dominance reasoning to hold, the article settles an open question in the research literature. Remarkably, neither constant-bounded utility nor finite expected utility is necessary for resolving the classical TEP; instead, finite expected utility is both necessary and sufficient for resolving the St. Petersburg TEP.Risks2015-01-1931Article10.3390/risks301002626342227-90912015-01-19doi: 10.3390/risks3010026Michael Powers<![CDATA[Risks, Vol. 3, Pages 24-25: Acknowledgement to Reviewers of Risks in 2014]]>
http://www.mdpi.com/2227-9091/3/1/24
The editors of Risks would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...]Risks2015-01-0831Editorial10.3390/risks301002424252227-90912015-01-08doi: 10.3390/risks3010024 Risks Editorial Office<![CDATA[Risks, Vol. 3, Pages 1-23: Inhomogeneous Long-Range Percolation for Real-Life Network Modeling]]>
http://www.mdpi.com/2227-9091/3/1/1
The study of random graphs has become very popular for real-life network modeling, such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice Zd, d ≥ 1, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model, various geometric properties, such as the percolation behavior, the degree distribution and graph distances, have been analyzed. In the present paper, we complement the picture of graph distances and we prove continuity of the percolation probability in the phase transition point. We also provide an illustration of the model connected to financial networks.Risks2015-01-0631Article10.3390/risks30100011232227-90912015-01-06doi: 10.3390/risks3010001Philippe DeprezRajat HazraMario Wüthrich<![CDATA[Risks, Vol. 2, Pages 469-488: Worst-Case Portfolio Optimization under Stochastic Interest Rate Risk]]>
http://www.mdpi.com/2227-9091/2/4/469
We investigate a portfolio optimization problem under the threat of a market crash, where the interest rate of the bond is modeled as a Vasicek process, which is correlated with the stock price process. We adopt a non-probabilistic worst-case approach for the height and time of the market crash. On a given time horizon [0; T], we then maximize the investor’s expected utility of terminal wealth in the worst-case crash scenario. Our main result is an explicit characterization of the worst-case optimal portfolio strategy for the class of HARA (hyperbolic absolute risk aversion) utility functions.Risks2014-12-0124Article10.3390/risks20404694694882227-90912014-12-01doi: 10.3390/risks2040469Tina EnglerRalf Korn<![CDATA[Risks, Vol. 2, Pages 467-468: Special Issue on Risk Management Techniques for Catastrophic and Heavy-Tailed Risks]]>
http://www.mdpi.com/2227-9091/2/4/467
The publication of several special issues was part of the initiatives taken in 2013 to launch Risks as a new online journal. It seemed natural to devote one to this important, concrete and complex problem of managing catastrophic and heavy tailed risks. We received an enthusiastic response last spring to the call for invited and contributed research papers and are proud of the special issue now being published. The emphasis was put on quality rather than quantity; this special issue contains three invited and two contributed research papers.Risks2014-11-1424Editorial10.3390/risks20404674674682227-90912014-11-14doi: 10.3390/risks2040467Alejandro BalbásJosé Garrido<![CDATA[Risks, Vol. 2, Pages 456-466: A Duality Result for the Generalized Erlang Risk Model]]>
http://www.mdpi.com/2227-9091/2/4/456
In this article, we consider the generalized Erlang risk model and its dual model. By using a conditional measure-preserving correspondence between the two models, we derive an identity for two interesting conditional probabilities. Applications to the discounted joint density of the surplus prior to ruin and the deficit at ruin are also discussed.Risks2014-11-0624Article10.3390/risks20404564564662227-90912014-11-06doi: 10.3390/risks2040456Lanpeng JiChunsheng Zhang<![CDATA[Risks, Vol. 2, Pages 434-455: A Markov Chain Model for Contagion]]>
http://www.mdpi.com/2227-9091/2/4/434
We introduce a bivariate Markov chain counting process with contagion for modelling the clustering arrival of loss claims with delayed settlement for an insurance company. It is a general continuous-time model framework that also has the potential to be applicable to modelling the clustering arrival of events, such as jumps, bankruptcies, crises and catastrophes in finance, insurance and economics with both internal contagion risk and external common risk. Key distributional properties, such as the moments and probability generating functions, for this process are derived. Some special cases with explicit results and numerical examples and the motivation for further actuarial applications are also discussed. The model can be considered a generalisation of the dynamic contagion process introduced by Dassios and Zhao (2011).Risks2014-11-0524Article10.3390/risks20404344344552227-90912014-11-05doi: 10.3390/risks2040434Angelos DassiosHongbiao Zhao<![CDATA[Risks, Vol. 2, Pages 425-433: A Note on the Fundamental Theorem of Asset Pricing under Model Uncertainty]]>
http://www.mdpi.com/2227-9091/2/4/425
We show that the recent results on the Fundamental Theorem of Asset Pricing and the super-hedging theorem in the context of model uncertainty can be extended to the case in which the options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.Risks2014-10-1024Article10.3390/risks20404254254332227-90912014-10-10doi: 10.3390/risks2040425Erhan BayraktarYuchong ZhangZhou Zhou<![CDATA[Risks, Vol. 2, Pages 411-424: Measuring Risk When Expected Losses Are Unbounded]]>
http://www.mdpi.com/2227-9091/2/4/411
This paper proposes a new method to introduce coherent risk measures for risks with infinite expectation, such as those characterized by some Pareto distributions. Extensions of the conditional value at risk, the weighted conditional value at risk and other examples are given. Actuarial applications are analyzed, such as extensions of the expected value premium principle when expected losses are unbounded.Risks2014-09-3024Article10.3390/risks20404114114242227-90912014-09-30doi: 10.3390/risks2040411Alejandro BalbásIván BlancoJosé Garrido<![CDATA[Risks, Vol. 2, Pages 393-410: Tail Risk in Commercial Property Insurance]]>
http://www.mdpi.com/2227-9091/2/4/393
We present some new evidence on the tail distribution of commercial property losses based on a recently constructed dataset on large commercial risks. The dataset is based on contributions from Lloyd’s of London syndicates, and provides information on over three thousand claims occurred during the period 2000–2012, including detailed information on exposures. We use occupancy characteristics to compare the tail risk profiles of different commercial property exposures, and find evidence of substantial heterogeneity in tail behavior. The results demonstrate the benefits of aggregating granular information on both claims and exposures from different data sources, and provide warning against the use of reserving and capital modeling approaches that are not robust to heavy tails.Risks2014-09-2924Article10.3390/risks20403933934102227-90912014-09-29doi: 10.3390/risks2040393Enrico BiffisErik Chavez<![CDATA[Risks, Vol. 2, Pages 349-392: An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality]]>
http://www.mdpi.com/2227-9091/2/3/349
A spectrum of upper bounds (Qα(X ; p) αε[0,∞] on the (largest) (1-p)-quantile Q(X;p) of an arbitrary random variable X is introduced and shown to be stable and monotonic in α, p, and X , with Q0(X ;p) = Q(X;p). If p is small enough and the distribution of X is regular enough, then Qα(X ; p) is rather close to Q(X ; p). Moreover, these quantile bounds are coherent measures of risk. Furthermore, Qα(X ; p) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, Q0(X;p) and Q1(X ; p) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds Qα(X ; p) can also be used as measures of economic inequality. The spectrum parameter α plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.Risks2014-09-2323Article10.3390/risks20303493493922227-90912014-09-23doi: 10.3390/risks2030349Iosif Pinelis<![CDATA[Risks, Vol. 2, Pages 315-348: Model Risk in Portfolio Optimization]]>
http://www.mdpi.com/2227-9091/2/3/315
We consider a one-period portfolio optimization problem under model uncertainty. For this purpose, we introduce a measure of model risk. We derive analytical results for this measure of model risk in the mean-variance problem assuming we have observations drawn from a normal variance mixture model. This model allows for heavy tails, tail dependence and leptokurtosis of marginals. The results show that mean-variance optimization is seriously compromised by model uncertainty, in particular, for non-Gaussian data and small sample sizes. To mitigate these shortcomings, we propose a method to adjust the sample covariance matrix in order to reduce model risk.Risks2014-08-0623Article10.3390/risks20303153153482227-90912014-08-06doi: 10.3390/risks2030315David StefanovitsUrs SchubigerMario Wüthrich<![CDATA[Risks, Vol. 2, Pages 289-314: Joint Asymptotic Distributions of Smallest and Largest Insurance Claims]]>
http://www.mdpi.com/2227-9091/2/3/289
Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalised sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.Risks2014-07-3123Article10.3390/risks20302892893142227-90912014-07-31doi: 10.3390/risks2030289Hansjörg AlbrecherChristian RobertJef Teugels<![CDATA[Risks, Vol. 2, Pages 277-288: Random Shifting and Scaling of Insurance Risks]]>
http://www.mdpi.com/2227-9091/2/3/277
Random shifting typically appears in credibility models whereas random scaling is often encountered in stochastic models for claim sizes reflecting the time-value property of money. In this article we discuss some aspects of random shifting and random scaling of insurance risks focusing in particular on credibility models, dependence structure of claim sizes in collective risk models, and extreme value models for the joint dependence of large losses. We show that specifying certain actuarial models using random shifting or scaling has some advantages for both theoretical treatments and practical applications.Risks2014-07-2223Article10.3390/risks20302772772882227-90912014-07-22doi: 10.3390/risks2030277Enkelejd HashorvaLanpeng Ji<![CDATA[Risks, Vol. 2, Pages 260-276: The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio]]>
http://www.mdpi.com/2227-9091/2/3/260
Risk diversification is the basis of insurance and investment. It is thus crucial to study the effects that could limit it. One of them is the existence of systemic risk that affects all of the policies at the same time. We introduce here a probabilistic approach to examine the consequences of its presence on the risk loading of the premium of a portfolio of insurance policies. This approach could be easily generalized for investment risk. We see that, even with a small probability of occurrence, systemic risk can reduce dramatically the diversification benefits. It is clearly revealed via a non-diversifiable term that appears in the analytical expression of the variance of our models. We propose two ways of introducing it and discuss their advantages and limitations. By using both VaR and TVaR to compute the loading, we see that only the latter captures the full effect of systemic risk when its probability to occur is low.Risks2014-07-0923Article10.3390/risks20302602602762227-90912014-07-09doi: 10.3390/risks2030260Marc BusseMichel DacorognaMarie Kratz<![CDATA[Risks, Vol. 2, Pages 249-259: Elementary Bounds on the Ruin Capital in a Diffusion Model of Risk]]>
http://www.mdpi.com/2227-9091/2/3/249
In a diffusion model of risk, we focus on the initial capital needed to make the probability of ruin within finite time equal to a prescribed value. It is defined as a solution of a nonlinear equation. The endeavor to write down and to investigate analytically this solution as a function of the premium rate seems not technically feasible. Instead, we obtain informative bounds for this capital in terms of elementary functions.Risks2014-07-0823Article10.3390/risks20302492492592227-90912014-07-08doi: 10.3390/risks2030249Vsevolod Malinovskii<![CDATA[Risks, Vol. 2, Pages 226-248: Demand of Insurance under the Cost-of-Capital Premium Calculation Principle]]>
http://www.mdpi.com/2227-9091/2/2/226
We study the optimal insurance design problem. This is a risk sharing problem between an insured and an insurer. The main novelty in this paper is that we study this optimization problem under a risk-adjusted premium calculation principle for the insurance cover. This risk-adjusted premium calculation principle uses the cost-of-capital approach as it is suggested (and used) by the regulator and the insurance industry.Risks2014-06-1722Article10.3390/risks20202262262482227-90912014-06-17doi: 10.3390/risks2020226Michael MerzMario Wüthrich