<scp>Bounds for Right Tails of Deterministic and Stochastic Sums of Random Variables</scp>

Darkiewicz, Grzegorz; Deelstra, Griselda; Dhaene, Jan; Hoedemakers, Tom; Vanmaele, Michèle

doi:10.1111/j.1539-6975.2009.01322.x

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…For applications of this idea in various contexts, such as the modeling of annuities, option pricing, and portfolio selection problems, see Curran , Darkiewicz et al. , Vanduffel et al. , and Dhaene et al.…”

Section: Introductionmentioning

confidence: 99%

Value‐at‐Risk Bounds With Variance Constraints

Bernard¹,

Rüschendorf²,

Vanduffel³

2015

J of Risk & Insurance

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We study bounds on the Value-at-Risk (VaR) of a portfolio when besides the marginal distributions of the components its variance is also known, a situation that is of considerable interest in risk management. We discuss when the bounds are sharp (attainable) and also point out a new connection between the study of VaR bounds and the convex ordering of aggregate risk. This connection leads to the construction of an algorithm, called Extended Rearrangement Algorithm (ERA), that makes it possible to approximate sharp VaR bounds. We test the stability and the quality of the algorithm in several numerical examples. We apply the results to the case of credit risk portfolio models and verify that adding the variance constraint gives rise to significantly tighter bounds in all situations of interest. IntroductionIn this article, we study bounds on the Value-at-Risk (VaR) of sums of risks with known marginal distributions (describing the stand-alone risks) under the additional constraint that a bound on the variance of the sum is known. The variance of the sum partially describes the dependence among the risks, as it involves their correlations. This setting is of significant interest as in many practical situations it corresponds closely to the maximum information at hand when assessing the VaR of a portfolio. For example, in the context of credit risk portfolio models, one typically has knowledge regarding the marginal risks (through the so-called PD, EAD, and LGD models), and the variance of the aggregate risk (sum of the individual losses) is also often available, as obtained from default correlation models or through statistical analysis of observed

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Section: Introductionmentioning

confidence: 99%

Value‐at‐Risk Bounds With Variance Constraints

Bernard¹,

Rüschendorf²,

Vanduffel³

2015

J of Risk & Insurance

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“…Analytical results for comonotonic bounds of the present value function of a sum of discounted deterministic cash-flows are derived. Darkiewicz et al (2009) first investigate lower and upper bounds for right tails (stop-loss premiums) of deterministic and stochastic sums of non-independent random variables, using the concepts of Sections 1 and 2. Then, the performance of the presented approximations is investigated numerically for individual life annuity contracts as well as for life annuity portfolios.…”

Section: Life Insurance and Pensionsmentioning

confidence: 99%

An Overview of Comonotonicity and Its Applications in Finance and Insurance

Deelstra

Dhaene

Vanmaele

2011

Advanced Mathematical Methods for Finance

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Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this paper, we give an extensive bibliographic overview -without claiming to be complete -of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements over the last five years. These applications range from pricing and hedging of derivatives over risk management to life insurance.

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“…When the risks are comonotone, the risk‐minimizing problem can easily be dealt with. When the risks are not comonotone, we propose an approximation of the problem by replacing the noncomonotonic sum by comonotonic sums that are smaller or larger in convex order; see Dhaene et al (2002a) for the theory and, for example, Dhaene et al (2002b) and Darkiewicz et al (2009) for possible applications. The lower comonotonic sum is based on a conditioning random variable that has to be chosen in an appropriate way.…”

Section: Introductionmentioning

confidence: 99%

Minimizing the Risk of a Financial Product Using a Put Option

Deelstra

Vanmaele

Vyncke

2010

Journal of Risk and Insurance

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In this paper, we elaborate a method for determining the optimal strike price for a put option, used to hedge a position in a financial product such as a basket of shares and a bond. This strike price is optimal in the sense that it minimizes, for a given budget, a class of risk measures satisfying certain properties. Formulas are derived for one single underlying as well as for a weighted sum of underlyings. For the latter we will consider two cases depending on the dependence structure of the components in this weighted sum. Applications and numerical results are presented.

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Bounds for Right Tails of Deterministic and Stochastic Sums of Random Variables

Cited by 6 publications

References 27 publications

Value‐at‐Risk Bounds With Variance Constraints

Value‐at‐Risk Bounds With Variance Constraints

An Overview of Comonotonicity and Its Applications in Finance and Insurance

Minimizing the Risk of a Financial Product Using a Put Option

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