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Risks, Volume 2, Issue 3 (September 2014), Pages 249-392

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Research

Open AccessArticle Elementary Bounds on the Ruin Capital in a Diffusion Model of Risk
Risks 2014, 2(3), 249-259; doi:10.3390/risks2030249
Received: 29 March 2014 / Revised: 19 June 2014 / Accepted: 30 June 2014 / Published: 8 July 2014
Cited by 1 | PDF Full-text (313 KB) | HTML Full-text | XML Full-text
Abstract
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
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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. Full article
Open AccessArticle The Impact of Systemic Risk on the Diversification Benefits of a Risk Portfolio
Risks 2014, 2(3), 260-276; doi:10.3390/risks2030260
Received: 2 December 2013 / Revised: 15 May 2014 / Accepted: 17 June 2014 / Published: 9 July 2014
Cited by 2 | PDF Full-text (296 KB) | HTML Full-text | XML Full-text
Abstract
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
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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. Full article
Open AccessArticle Random Shifting and Scaling of Insurance Risks
Risks 2014, 2(3), 277-288; doi:10.3390/risks2030277
Received: 15 April 2014 / Revised: 26 June 2014 / Accepted: 15 July 2014 / Published: 22 July 2014
Cited by 1 | PDF Full-text (241 KB) | HTML Full-text | XML Full-text
Abstract
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
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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. Full article
Open AccessArticle Joint Asymptotic Distributions of Smallest and Largest Insurance Claims
Risks 2014, 2(3), 289-314; doi:10.3390/risks2030289
Received: 25 February 2014 / Revised: 15 June 2014 / Accepted: 15 July 2014 / Published: 31 July 2014
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Abstract
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
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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. Full article
(This article belongs to the Special Issue Risk Management Techniques for Catastrophic and Heavy-Tailed Risks)
Open AccessArticle Model Risk in Portfolio Optimization
Risks 2014, 2(3), 315-348; doi:10.3390/risks2030315
Received: 19 February 2014 / Revised: 17 June 2014 / Accepted: 30 July 2014 / Published: 6 August 2014
Cited by 1 | PDF Full-text (391 KB) | HTML Full-text | XML Full-text
Abstract
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
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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. Full article
Open AccessArticle An Optimal Three-Way Stable and Monotonic Spectrum of Bounds on Quantiles: A Spectrum of Coherent Measures of Financial Risk and Economic Inequality
Risks 2014, 2(3), 349-392; doi:10.3390/risks2030349
Received: 15 June 2014 / Revised: 21 August 2014 / Accepted: 10 September 2014 / Published: 23 September 2014
Cited by 2 | PDF Full-text (651 KB) | HTML Full-text | XML Full-text
Abstract
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
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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. Full article
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