Risks
http://www.mdpi.com/journal/risks
Latest open access articles published in Risks at http://www.mdpi.com/journal/risks<![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 <![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<![CDATA[Risks, Vol. 2, Pages 211-225: When the U.S. Stock Market Becomes Extreme?]]>
http://www.mdpi.com/2227-9091/2/2/211
Over the last three decades, the world economy has been facing stock market crashes, currency crisis, the dot-com and real estate bubble burst, credit crunch and banking panics. As a response, extreme value theory (EVT) provides a set of ready-made approaches to risk management analysis. However, EVT is usually applied to standardized returns to offer more reliable results, but remains difficult to interpret in the real world. This paper proposes a quantile regression to transform standardized returns into theoretical raw returns making them economically interpretable. An empirical test is carried out on the S&amp;P500 stock index from 1950 to 2013. The main results indicate that the U.S stock market becomes extreme from a price variation of ±1.5% and the largest one-day decline of the 2007–2008 period is likely, on average, to be exceeded one every 27 years.Risks2014-05-2822Article10.3390/risks20202112112252227-90912014-05-28doi: 10.3390/risks2020211Sofiane Aboura<![CDATA[Risks, Vol. 2, Pages 195-210: Neumann Series on the Recursive Moments of Copula-Dependent Aggregate Discounted Claims]]>
http://www.mdpi.com/2227-9091/2/2/195
We study the recursive moments of aggregate discounted claims, where the dependence between the inter-claim time and the subsequent claim size is considered. Using the general expression for the m-th order moment proposed by Léveillé and Garrido (Scand. Actuar. J. 2001, 2, 98–110), which takes the form of the Volterra integral equation (VIE), we used the method of successive approximation to derive the Neumann series of the recursive moments. We then compute the first two moments of aggregate discounted claims, i.e., its mean and variance, based on the Neumann series expression, where the dependence structure is captured by a Farlie–Gumbel–Morgenstern (FGM) copula, a Gaussian copula and a Gumbel copula with exponential marginal distributions. Insurance premium calculations with their figures are also illustrated.Risks2014-05-2722Article10.3390/risks20201951952102227-90912014-05-27doi: 10.3390/risks2020195Siti Mohd RamliJiwook Jang<![CDATA[Risks, Vol. 2, Pages 171-194: Optimal Consumption and Investment with Labor Income and European/American Capital Guarantee]]>
http://www.mdpi.com/2227-9091/2/2/171
We present the optimal consumption and investment strategy for an investor, endowed with labor income, searching to maximize utility from consumption and terminal wealth when facing a binding capital constraint of a European (constraint on terminal wealth) or an American (constraint on the wealth process) type. In both cases, the optimal strategy is proven to be of the option-based portfolio insurance type. The optimal strategy combines a long position in the optimal unrestricted allocation with a put option. In the American case, where the investor is restricted to fulfill a capital guarantee at every intermediate time point over the interval of optimization, we prove that the investor optimally changes his budget constraint for the unrestricted allocation whenever the constraint is active. The strategy is explained in a step-by-step manner, and numerical illustrations are presented in order to support intuition and to compare the restricted optimal strategy with the unrestricted optimal counterpart.Risks2014-05-1622Article10.3390/risks20201711711942227-90912014-05-16doi: 10.3390/risks2020171Morten Kronborg<![CDATA[Risks, Vol. 2, Pages 146-170: Attracting Health Insurance Buyers through Selective Contracting: Results of a Discrete-Choice Experiment among Users of Hospital Services in the Netherlands]]>
http://www.mdpi.com/2227-9091/2/2/146
In 2006, the Netherlands commenced market based reforms in its health care system. The reforms included selective contracting of health care providers by health insurers. This paper focuses on how health insurers may increase their market share on the health insurance market through selective contracting of health care providers. Selective contracting is studied by eliciting the preferences of health care consumers for attributes of health care services that an insurer could negotiate on behalf of its clients with health care providers. Selective contracting may provide incentives for health care providers to deliver the quality that consumers need and demand. Selective contracting also enables health insurers to steer individual patients towards selected health care providers. We used a stated preference technique known as a discrete choice experiment to collect and analyze the data. Results indicate that consumers care about both costs and quality of care, with healthy consumers placing greater emphasis on costs and consumers with poorer health placing greater emphasis on quality of care. It is possible for an insurer to satisfy both of these criteria by selective contracting health care providers who consequently purchase health care that is both efficient and of good quality.Risks2014-04-1522Article10.3390/risks20201461461702227-90912014-04-15doi: 10.3390/risks2020146Evelien BergrathMilena PavlovaWim Groot<![CDATA[Risks, Vol. 2, Pages 132-145: Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms]]>
http://www.mdpi.com/2227-9091/2/2/132
This paper is focused on solving different hard optimization problems that arise in the field of insurance and, more specifically, in reinsurance problems. In this area, the complexity of the models and assumptions considered in the definition of the reinsurance rules and conditions produces hard black-box optimization problems (problems in which the objective function does not have an algebraic expression, but it is the output of a system (usually a computer program)), which must be solved in order to obtain the optimal output of the reinsurance. The application of traditional optimization approaches is not possible in this kind of mathematical problem, so new computational paradigms must be applied to solve these problems. In this paper, we show the performance of two evolutionary and swarm intelligence techniques (evolutionary programming and particle swarm optimization). We provide an analysis in three black-box optimization problems in reinsurance, where the proposed approaches exhibit an excellent behavior, finding the optimal solution within a fraction of the computational cost used by inspection or enumeration methods.Risks2014-04-0122Article10.3390/risks20201321321452227-90912014-04-01doi: 10.3390/risks2020132Sancho Salcedo-SanzLeo Carro-CalvoMercè ClaramuntAna CastañerMaite Mármol<![CDATA[Risks, Vol. 2, Pages 103-131: 1980–2008: The Illusion of the Perpetual Money Machine and What It Bodes for the Future]]>
http://www.mdpi.com/2227-9091/2/2/103
We argue that the present crisis and stalling economy that have been ongoing since 2007 are rooted in the delusionary belief in policies based on a “perpetual money machine” type of thinking. We document strong evidence that, since the early 1980s, consumption has been increasingly funded by smaller savings, booming financial profits, wealth extracted from house price appreciation and explosive debt. This is in stark contrast with the productivity-fueled growth that was seen in the 1950s and 1960s. We describe the transition, in gestation in the 1970s, towards the regime of the “illusion of the perpetual money machine”, which started at full speed in the early 1980s and developed until 2008. This regime was further supported by a climate of deregulation and a massive growth in financial derivatives designed to spread and diversify the risks globally. The result has been a succession of bubbles and crashes, including the worldwide stock market bubble and great crash of October 1987, the savings and loans crisis of the 1980s, the burst in 1991 of the enormous Japanese real estate and stock market bubbles, the emerging markets bubbles and crashes in 1994 and 1997, the Long-Term Capital Management (LTCM) crisis of 1998, the dotcom bubble bursting in 2000, the recent house price bubbles, the financialization bubble via special investment vehicles, the stock market bubble, the commodity and oil bubbles and the current debt bubble, all developing jointly and feeding on each other until 2008. This situation may be further aggravated in the next decade by an increase in financialization, through exchange-traded-funds (ETFs), speed and automation, through algorithmic trading and public debt, and through growing unfunded liabilities. We conclude that, to get out of this catch 22 situation, we should better manage and understand the incentive structures in our society, we need to focus our efforts on our real economy and we have to respect and master the art of planning and prediction. Only gradual change, with a clear long term planning, can steer our financial and economic system from the turbulence associated with the perpetual money machine to calmer and more sustainable waters. Risks2014-04-0122Article10.3390/risks20201031031312227-90912014-04-01doi: 10.3390/risks2020103Didier SornettePeter Cauwels<![CDATA[Risks, Vol. 2, Pages 89-102: Initial Investigations of Intra-Day News Flow of S&P500 Constituents]]>
http://www.mdpi.com/2227-9091/2/2/89
In this work, we examine Thomas Reuters News Analytics (TRNA) data. We found several fascinating discoveries. First, we document the phenomenon that we label “Jam-the-Close”: The last half hour of trading (15:30 to 16:00 EST) contains a substantial and statistically significant amount of news sentiment releases. This finding is robust across years and months of the year. Next, upon further investigations we found that the “novelty” score is on average 0.67 in this period vs. 2.09 prior to midday. This indicates that “new” news is flowing at a rapid pace prior to the close. Finally, we discuss the implication of such phenomena in the context of existing financial literature.Risks2014-04-0122Article10.3390/risks2020089891022227-90912014-04-01doi: 10.3390/risks2020089Jim LiewZhechao Zhou<![CDATA[Risks, Vol. 2, Pages 74-88: Modeling Cycle Dependence in Credit Insurance]]>
http://www.mdpi.com/2227-9091/2/1/74
Business and credit cycles have an impact on credit insurance, as they do on other businesses. Nevertheless, in credit insurance, the impact of the systemic risk is even more important and can lead to major losses during a crisis. Because of this, the insurer surveils and manages policies almost continuously. The management actions it takes limit the consequences of a downturning cycle. However, the traditional modeling of economic capital does not take into account this important feature of credit insurance. This paper proposes a model aiming to estimate future losses of a credit insurance portfolio, while taking into account the insurer’s management actions. The model considers the capacity of the credit insurer to take on less risk in the case of a cycle downturn, but also the inverse, in the case of a cycle upturn; so, losses are predicted with a more dynamic perspective. According to our results, the economic capital is over-estimated when not considering the management actions of the insurer.Risks2014-03-1421Article10.3390/risks201007474882227-90912014-03-14doi: 10.3390/risks2010074Anisa CajaFrédéric Planchet<![CDATA[Risks, Vol. 2, Pages 49-73: Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction]]>
http://www.mdpi.com/2227-9091/2/1/49
In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most of the literature measures performance of the system in terms of the stationary characteristics of this Markov chain. However, the rate of convergence to stationarity may be slow in comparison to the typical sojourn time of a customer in the portfolio. We suggest an age-correction to the stationary distribution and present an extensive numerical study of its effects. An important feature of the modeling is a Bayesian view, where the Poisson rate according to which claims are generated for a customer is the outcome of a random variable specific to the customer.Risks2014-03-1121Article10.3390/risks201004949732227-90912014-03-11doi: 10.3390/risks2010049Søren Asmussen<![CDATA[Risks, Vol. 2, Pages 25-48: An Academic Response to Basel 3.5]]>
http://www.mdpi.com/2227-9091/2/1/25
Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we highlight some of the underlying issues, both methodologically, as well as through examples. In particular, we frame this discussion in the context of two recent regulatory documents we refer to as Basel 3.5.Risks2014-02-2721Article10.3390/risks201002525482227-90912014-02-27doi: 10.3390/risks2010025Paul EmbrechtsGiovanni PuccettiLudger RüschendorfRuodu WangAntonela Beleraj<![CDATA[Risks, Vol. 2, Pages 3-24: Catastrophe Insurance Modeled by Shot-Noise Processes]]>
http://www.mdpi.com/2227-9091/2/1/3
Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed.Risks2014-02-2121Article10.3390/risks20100033242227-90912014-02-21doi: 10.3390/risks2010003Thorsten Schmidt<![CDATA[Risks, Vol. 2, Pages 1-2: Publishing Risks]]>
http://www.mdpi.com/2227-9091/2/1/1
“What is complicated is not necessarily insightful and what is insightful is not necessarily complicated: Risks welcomes simple manuscripts that contribute with insight, outlook, understanding and overview”—a quote from the first editorial of this journal [1]. Good articles are not characterized by their level of complication but by their level of imagination, innovation, and power of penetration. Creativity sessions and innovative tasks are most elegant and powerful when they are delicately simple. This is why the articles you most remember are not the complicated ones that you struggled to digest, but the simpler ones you enjoyed swallowing. [...]Risks2014-02-2121Editorial10.3390/risks2010001122227-90912014-02-21doi: 10.3390/risks2010001Mogens Steffensen<![CDATA[Risks, Vol. 1, Pages 192-212: Ruin Time and Severity for a Lévy Subordinator Claim Process: A Simple Approach]]>
http://www.mdpi.com/2227-9091/1/3/192
This paper is concerned with an insurance risk model whose claim process is described by a Lévy subordinator process. Lévy-type risk models have been the object of much research in recent years. Our purpose is to present, in the case of a subordinator, a simple and direct method for determining the finite time (and ultimate) ruin probabilities, the distribution of the ruin severity, the reserves prior to ruin, and the Laplace transform of the ruin time. Interestingly, the usual net profit condition will be essentially relaxed. Most results generalize those known for the compound Poisson claim process.Risks2013-12-1313Article10.3390/risks10301921922122227-90912013-12-13doi: 10.3390/risks1030192Claude LefèvrePhilippe Picard<![CDATA[Risks, Vol. 1, Pages 176-191: Impact of Climate Change on Heat Wave Risk]]>
http://www.mdpi.com/2227-9091/1/3/176
We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring the potential impact of climate change on heat wave risk. Numerical illustrations are given.Risks2013-12-1213Article10.3390/risks10301761761912227-90912013-12-12doi: 10.3390/risks1030176Romain BiardChristophette Blanchet-ScallietAnne Eyraud-LoiselStéphane Loisel<![CDATA[Risks, Vol. 1, Pages 162-175: U.S. Equity Mean-Reversion Examined]]>
http://www.mdpi.com/2227-9091/1/3/162
In this paper we introduce an intra-sector dynamic trading strategy that captures mean-reversion opportunities across liquid U.S. stocks. Our strategy combines the Avellaneda and Lee methodology (AL; Quant. Financ. 2010, 10, 761–782) within the Black and Litterman framework (BL; J. Fixed Income, 1991, 1, 7–18; Financ. Anal. J. 1992, 48, 28–43). In particular, we incorporate the s-scores and the conditional mean returns from the Orstein and Ulhembeck (Phys. Rev. 1930, 36, 823–841) process into BL. We find that our combined strategy ALBL has generated a 45% increase in Sharpe Ratio when compared to the uncombined AL strategy over the period from January 2, 2001 to May 27, 2010. These new indices, built to capture dynamic trading strategies, will definitely be an interesting addition to the growing hedge fund index offerings. This paper introduces our first “focused-core” strategy, namely, U.S. Equity Mean-Reversion.Risks2013-12-0413Article10.3390/risks10301621621752227-90912013-12-04doi: 10.3390/risks1030162Jim LiewRyan Roberts<![CDATA[Risks, Vol. 1, Pages 148-161: A Risk Model with an Observer in a Markov Environment]]>
http://www.mdpi.com/2227-9091/1/3/148
We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration.Risks2013-11-1113Article10.3390/risks10301481481612227-90912013-11-11doi: 10.3390/risks1030148Hansjörg AlbrecherJevgenijs Ivanovs<![CDATA[Risks, Vol. 1, Pages 119-147: Optimal Dynamic Portfolio with Mean-CVaR Criterion]]>
http://www.mdpi.com/2227-9091/1/3/119
Value-at-risk (VaR) and conditional value-at-risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of a Neyman–Pearson type binary solution. We add a constraint on expected return to investigate the mean-CVaR portfolio selection problem in a dynamic setting: the investor is faced with a Markowitz type of risk reward problem at the final horizon, where variance as a measure of risk is replaced by CVaR. Based on the complete market assumption, we give an analytical solution in general. The novelty of our solution is that it is no longer the Neyman–Pearson type, in which the final optimal portfolio takes only two values. Instead, in the case in which the portfolio value is required to be bounded from above, the optimal solution takes three values; while in the case in which there is no upper bound, the optimal investment portfolio does not exist, though a three-level portfolio still provides a sub-optimal solution.Risks2013-11-1113Article10.3390/risks10301191191472227-90912013-11-11doi: 10.3390/risks1030119Jing LiMingxin Xu<![CDATA[Risks, Vol. 1, Pages 101-118: Optimal Deterministic Investment Strategies for Insurers]]>
http://www.mdpi.com/2227-9091/1/3/101
We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility and the probability of ruin. Following recent research, we assume that investment strategies have to be deterministic. This leads to deterministic control problems, which are quite easy to solve. Moreover, it turns out that there are some interesting links between the optimal investment strategies of these problems. Finally, we also show that this approach works in the Lévy process framework.Risks2013-11-0713Article10.3390/risks10301011011182227-90912013-11-07doi: 10.3390/risks1030101Nicole BäuerleUlrich Rieder<![CDATA[Risks, Vol. 1, Pages 81-100: Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates]]>
http://www.mdpi.com/2227-9091/1/3/81
In the actuarial literature, it has become common practice to model future capital returns and mortality rates stochastically in order to capture market risk and forecasting risk. Although interest rates often should and mortality rates always have to be non-negative, many authors use stochastic diffusion models with an affine drift term and additive noise. As a result, the diffusion process is Gaussian and, thus, analytically tractable, but negative values occur with positive probability. The argument is that the class of Gaussian diffusions would be a good approximation of the real future development. We challenge that reasoning and study the asymptotics of diffusion processes with affine drift and a general noise term with corresponding diffusion processes with an affine drift term and an affine noise term or additive noise. Our study helps to quantify the error that is made by approximating diffusive interest and mortality rate models with Gaussian diffusions and affine diffusions. In particular, we discuss forward interest and forward mortality rates and the error that approximations cause on the valuation of life insurance claims.Risks2013-10-2513Article10.3390/risks1030081811002227-90912013-10-25doi: 10.3390/risks1030081Marcus Christiansen<![CDATA[Risks, Vol. 1, Pages 57-80: A Welfare Analysis of Capital Insurance]]>
http://www.mdpi.com/2227-9091/1/2/57
This paper presents a welfare analysis of several capital insurance programs in a rational expectation equilibrium setting. We first explicitly characterize the equilibrium of each capital insurance program. Then, we demonstrate that a capital insurance program based on aggregate loss is better than classical insurance, when big financial institutions have similar expected loss exposures. By contrast, classical insurance is more desirable when the bank’s individual risk is consistent with the expected loss in a precise way. Our analysis shows that a capital insurance program is a useful tool to hedge systemic risk from the regulatory perspective.Risks2013-09-1712Article10.3390/risks102005757802227-90912013-09-17doi: 10.3390/risks1020057Ekaterina PanttserWeidong Tian<![CDATA[Risks, Vol. 1, Pages 45-56: Optimal Reinsurance: A Risk Sharing Approach]]>
http://www.mdpi.com/2227-9091/1/2/45
This paper proposes risk sharing strategies, which allow insurers to cooperate and diversify non-systemic risk. We deal with both deviation measures and coherent risk measures and provide general mathematical methods applying to optimize them all. Numerical examples are given in order to illustrate how efficiently the non-systemic risk can be diversified and how effective the presented mathematical tools may be. It is also illustrated how the existence of huge disasters may lead to wrong solutions of our optimal risk sharing problem, in the sense that the involved risk measure could ignore the existence of a non-null probability of "global ruin" after the design of the optimal risk sharing strategy. To overcome this caveat, one can use more conservative risk measures. The stability in the large of the optimal sharing plan guarantees that "the global ruin caveat" may be also addressed and solved with the presented methods.Risks2013-08-0512Article10.3390/risks102004545562227-90912013-08-05doi: 10.3390/risks1020045Alejandro BalbasBeatriz BalbasRaquel Balbas<![CDATA[Risks, Vol. 1, Pages 43-44: Surrounding Risks]]>
http://www.mdpi.com/2227-9091/1/1/43
Research in insurance and finance was always intersecting although they were originally and generally viewed as separate disciplines. Insurance is about transferring risks between parties such that the burdens of risks are borne by those who can. This makes insurance transactions a beneficial activity for the society. It calls on detection, modelling, valuation, and controlling of risks. One of the main sources of control is diversification of risks and in that respect it becomes an issue in itself to clarify diversifiability of risks. However, many diversifiable risks are not, by nature or by contract design, separable from non-diversifiable risks that are, on the other hand, sometimes traded in financial markets and sometimes not. A key observation is that the economic risk came before the insurance contract: Mother earth destroys and kills incidentally and mercilessly, but the uncertainty of economic consequences can be more or less cleverly distributed by the introduction of an insurance market. [...]Risks2013-05-3011Editorial10.3390/risks101004343442227-90912013-05-30doi: 10.3390/risks1010043Mogens Steffensen<![CDATA[Risks, Vol. 1, Pages 34-42: Understanding the “Black Box” of Employer Decisions about Health Insurance Benefits: The Case of Depression Products]]>
http://www.mdpi.com/2227-9091/1/1/34
In a randomized trial of two interventions on employer health benefit decision-making, 156 employers in the evidence-based (EB) condition attended a two hour presentation reviewing scientific evidence demonstrating depression products that increase high quality treatment of depression in the workforce provide the employer a return on investment. One-hundred sixty-nine employers participating in the usual care (UC) condition attended a similar length presentation reviewing scientific evidence supporting healthcare effectiveness data and information set (HEDIS) monitoring. This study described the decision-making process in 264 (81.2%) employers completing 12 month follow-up. The EB intervention did not increase the proportion of employers who discussed depression products with others in the company (29.2% versus 32.1%, p &gt; 0.10), but it did significantly influence the content of the discussions that occurred. Discussion in EB companies promoted the capacity of a depression product to realize a return on investment (18.4% versus 4.7%, p = 0.05) and to improve productivity (47.4% versus 25.6%, p = 0.06) more often than discussions in UC companies. Almost half of EB and UC employers reported that return on investment has a large impact on health benefit decision-making. These results demonstrate the difficulty of influencing employer decisions about health benefits using group presentations.Risks2013-05-2911Article10.3390/risks101003434422227-90912013-05-29doi: 10.3390/risks1010034Kathryn RostAiria PapadopoulosSu WangDonna Marshall<![CDATA[Risks, Vol. 1, Pages 14-33: Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-II Model]]>
http://www.mdpi.com/2227-9091/1/1/14
Evaluating risk measures, premiums, and capital allocation based on dependent multi-losses is a notoriously difficult task. In this paper, we demonstrate how this can be successfully accomplished when losses follow the multivariate Pareto distribution of the second kind, which is an attractive model for multi-losses whose dependence and tail heaviness are influenced by a heavy-tailed background risk. A particular attention is given to the distortion and weighted risk measures and allocations, as well as their special cases such as the conditional layer expectation, tail value at risk, and the truncated tail value at risk. We derive formulas that are either of closed form or follow well-defined recursive procedures. In either case, their computational use is straightforward.Risks2013-03-0511Article10.3390/risks101001414332227-90912013-03-05doi: 10.3390/risks1010014Alexandru AsimitRaluca VernicRiċardas Zitikis<![CDATA[Risks, Vol. 1, Pages 1-13: Early Warning to Insolvency in the Pension Fund: The French Case]]>
http://www.mdpi.com/2227-9091/1/1/1
The financial equilibrium of pension funds relies on the appropriate computation of retirement benefits, taking account of future payments and discount rates. Short-term errors in the commitment for retirement benefits, ill-suited investment in the stock market, or improper mixture with pay-as-you-go payments have long-term consequences and may lead the pension fund to insolvency. The differential equation governing the current assets shows the respective weights associated with the error on the interest rate, the error on the extra bonus, and the error made in forecasting mortality. These weights are estimated through simulations. A short follow-up is sufficient to estimate the three errors. A threshold for the extra interest rate to be earned on the financial market is given to counter-balance the extra bonus when mortality is forecast correctly.Risks2013-01-1811Article10.3390/risks10100011132227-90912013-01-18doi: 10.3390/risks1010001Noël Bonneuil