Series expansions for convolutions of Pareto distributions

2015 ◽  
Vol 32 (1) ◽  
Author(s):  
Quang Huy Nguyen ◽  
Christian Y. Robert
Keyword(s):  
Infinite Series ◽  
Asymptotic Expansions ◽  
Value At Risk ◽  
Risk Measures ◽  
Random Variables ◽  
Series Expansions ◽  
Sums Of Random Variables ◽  
Scale Parameters ◽  
Pareto Distributions ◽  
Higher Order Terms

AbstractAsymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.

On risk measuring in the variance-gamma model

2018 ◽  
Vol 35 (1-2) ◽  
pp. 23-33 ◽  
Author(s):  
Roman V. Ivanov
Keyword(s):  
Value At Risk ◽  
Hypergeometric Functions ◽  
Risk Measures ◽  
Random Variables ◽  
Gamma Model ◽  
Variance Gamma ◽  
Stochastic Volatilities ◽  
Investment Returns ◽  
Variance Gamma Model ◽  
Analytical Expressions

AbstractIn this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.

scholarly journals Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

2007 ◽  
Vol 44 (3) ◽  
pp. 670-684 ◽  
Author(s):  
Ph. Barbe ◽  
W. P. McCormick ◽  
C. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Weibull Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Subexponential Distribution ◽  
Independent Random Variables ◽  
Sums Of Random Variables

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

scholarly journals Robustness regions for measures of risk aggregation

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Silvana M. Pesenti ◽  
Pietro Millossovich ◽  
Andreas Tsanakas
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Random Variables ◽  
Distribution Functions ◽  
Aggregation Function ◽  
Convex Risk Measures ◽  
Linear Growth Condition ◽  
Risk Aggregation ◽  
Definition Of

AbstractOne of risk measures’ key purposes is to consistently rank and distinguish between different risk profiles. From a practical perspective, a risk measure should also be robust, that is, insensitive to small perturbations in input assumptions. It is known in the literature [14, 39], that strong assumptions on the risk measure’s ability to distinguish between risks may lead to a lack of robustness. We address the trade-off between robustness and consistent risk ranking by specifying the regions in the space of distribution functions, where law-invariant convex risk measures are indeed robust. Examples include the set of random variables with bounded second moment and those that are less volatile (in convex order) than random variables in a given uniformly integrable set. Typically, a risk measure is evaluated on the output of an aggregation function defined on a set of random input vectors. Extending the definition of robustness to this setting, we find that law-invariant convex risk measures are robust for any aggregation function that satisfies a linear growth condition in the tail, provided that the set of possible marginals is uniformly integrable. Thus, we obtain that all law-invariant convex risk measures possess the aggregation-robustness property introduced by [26] and further studied by [40]. This is in contrast to the widely-used, non-convex, risk measure Value-at-Risk, whose robustness in a risk aggregation context requires restricting the possible dependence structures of the input vectors.

scholarly journals Distortion risk measures for sums of random variables

2004 ◽  
Vol 26 (4) ◽  
pp. 631-641
Author(s):  
Grzegorz Darkiewicz ◽  
Jan Dhaene ◽  
Marc Goovaerts
Keyword(s):  
Risk Measures ◽  
Random Variables ◽  
Sums Of Random Variables ◽  
Distortion Risk Measures

The sum of two independent polynomially-modified hyperbolic secant random variables with application in computational finance

2021 ◽  
pp. 2150030
Author(s):  
A. A. L. Zadeh ◽  
Hojatollah Zakerzadeh ◽  
Hamzeh Torabi
Keyword(s):  
At Risk ◽  
Value At Risk ◽  
Hermite Polynomial ◽  
Risk Measures ◽  
Random Variables ◽  
Expected Shortfall ◽  
Computational Finance ◽  
Sample Period ◽  
Proposed Model ◽  
Multiple Assets

In this paper, by reshaping the hyperbolic secant distribution using Hermite polynomial, we devise a polynomially-modified hyperbolic secant distribution which is more flexible than secant distribution to capture the skewness, heavy-tailedness and kurtosis of data. As a portfolio possibly consists of multiple assets, the distribution of the sum of independent polynomially-modified hyperbolic secant random variables is derived. In exceptional cases, we evaluate risk measures such as value at risk and expected shortfall (ES) for the sum of two independent polynomially-modified hyperbolic secant random variables. Finally, using real datasets from four international computers stocks, such as Adobe Systems, Microsoft, Nvidia and Symantec Corporations, the effectiveness of the proposed model is shown by the goodness of Gram–Charlier-like expansion of hyperbolic secant law, for performance of value at risk and ES estimation, both in and out of the sample period.

scholarly journals Asymptotic expansions for the probabilities of large deviations for sums of random variables related to a Markov chain

1970 ◽  
Vol 10 (2) ◽  
pp. 359-366
Author(s):  
L. Saulis ◽  
V. Statulevičius
Keyword(s):  
Markov Chain ◽  
Large Deviations ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Sums Of Random Variables ◽  
Probabilities Of Large Deviations

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Л. И. Саулис, В. А. Статулявичус. Асимптотическое разложение для вероятностей больших уклонений сумм случайных величин, связанных в цепь Маркова L. Saulis, V. Statulevičius. Atsitiktinių dydžių, surištų į Markovo grandinę, didžiųjų nukrypimų asimptotinis dėstinys

scholarly journals Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

2007 ◽  
Vol 44 (03) ◽  
pp. 670-684 ◽  
Author(s):  
Ph. Barbe ◽  
W. P. McCormick ◽  
C. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Weibull Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Subexponential Distribution ◽  
Independent Random Variables ◽  
Sums Of Random Variables

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

scholarly journals Early warning of systemic risk in global banking: eigen-pair R number for financial contagion and market price-based methods

2021 ◽  
Author(s):  
Sheri Markose ◽  
Simone Giansante ◽  
Nicolas A. Eterovic ◽  
Mateusz Gatkowski
Keyword(s):  
Early Warning ◽  
Systemic Risk ◽  
Value At Risk ◽  
Risk Measures ◽  
Market Price ◽  
System Stability ◽  
Conditional Value At Risk ◽  
Centrality Measures ◽  
Global Banking ◽  
Pair Method

AbstractWe analyse systemic risk in the core global banking system using a new network-based spectral eigen-pair method, which treats network failure as a dynamical system stability problem. This is compared with market price-based Systemic Risk Indexes, viz. Marginal Expected Shortfall, Delta Conditional Value-at-Risk, and Conditional Capital Shortfall Measure of Systemic Risk in a cross-border setting. Unlike paradoxical market price based risk measures, which underestimate risk during periods of asset price booms, the eigen-pair method based on bilateral balance sheet data gives early-warning of instability in terms of the tipping point that is analogous to the R number in epidemic models. For this regulatory capital thresholds are used. Furthermore, network centrality measures identify systemically important and vulnerable banking systems. Market price-based SRIs are contemporaneous with the crisis and they are found to covary with risk measures like VaR and betas.

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