• The Pure and Applied Mathematics
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• v.19 no.4
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• pp.363-381
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• 2012

Shortfall risk is considered by taking some exposed risks because the superhedging price is too expensive to be used in practice. Minimizing shortfall risk can be reduced to the problem of finding a randomized test ${\psi}$ in the static problem. The optimization problem can be solved via the classical Neyman-Pearson theory, and can be also explained in terms of hypothesis testing. We introduce the classical Neyman-Pearson lemma expressed in terms of mathematics and see how it is applied to shortfall risk in finance.

• The Pure and Applied Mathematics
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• v.19 no.2
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• pp.179-192
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• 2012

We find the solution minimizing the shortfall risk by using the Lagrange-multiplier method. The conventional duality method in the expected utility maximization problem is used and we get the same results as in the paper [21].

• The Korean Journal of Applied Statistics
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• v.29 no.1
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• pp.171-180
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• 2016

Basel committee suggests using Value-at-Risk (VaR) and expected shortfall (ES) as a measurement for market risk. Various estimation methods of VaR and ES have been studied in the literature. This paper compares semi-parametric methods, such as conditional autoregressive value at risk (CAViaR) and conditional autoregressive expectile (CARE) methods, and a Gaussian quasi-maximum likelihood estimator (QMLE)-based method through back-testing methods. We use unconditional coverage (UC) and conditional coverage (CC) tests for VaR, and a bootstrap test for ES to check the adequacy. A real data analysis is conducted for S&P 500 index and Hyundai Motor Co. stock price index data sets.

• Journal of the Korean Data and Information Science Society
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• v.24 no.6
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• pp.1503-1520
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• 2013

For multi-line insurance companies, allocating the risk capital to each line is a widely-accepted risk management exercise. In this article we consider several applications of the Euler capital allocation. First, we propose visual tools to present the diversification and the line-wise performance for a given loss portfolio so that the risk managers can understand the interactions among the lines. Secondly, on theoretical side, we prove that the Euler allocation is the directional derivative of the marginal or incremental allocation method, an alternative capital allocation rule in the literature. Lastly, we establish the equivalence between the mean-shortfall optimization and the RORAC optimization when the risk adjusted capital is the expected shortfall, and show how to construct the optimal insurance business mix that maximizes the portfolio RORAC. An actual loss sample of an insurance portfolio is used for numerical illustrations.

• The Korean Journal of Applied Statistics
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• v.34 no.1
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• pp.39-56
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• 2021

Many researches have been done on portfolio optimization since Markowitz (1952) published a diversified investment model. Markowitz's mean-variance portfolio optimization problem is established under the assumption that the distribution of returns follows a normal distribution. However, in real life, the distribution of returns does not follow a normal distribution, and variance is not a robust statistic as it is heavily influenced by outliers. To overcome these potential issues, mean-shortfall portfolio model was proposed that utilized downside risk, shortfall, as a risk index. In this paper, we propose a perturbation method that uses the shortfall as a risk index of the portfolio. The proposed portfolio utilizes an adaptive Lasso to obtain a sparse and stable asset selection because it can reduce management and transaction costs. The proposed optimization is easily applicable as it can be computed using an efficient linear programming. In our real data analysis, we show the validity of the proposed perturbation method.

• Journal of the Korea Society for Simulation
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• v.23 no.4
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• pp.163-170
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• 2014

This study calculates the employee receives severance pay scale are paid from the company in the DC system. In addition, by utilizing the reserve growth model were studied in accordance with shortfall risk levels generated by stochastic asset allocation. For the analysis, from 2004 to 2013 using the KOSPI returns and total bond yields were simulated. Scenario 1 is when compared to the severance reserve is insufficient. Scenario 2 is the same as if toy reserve this severance pay. During one period, depending on the asset allocation of stocks and bonds was confirmed that the probability pension risk does not occur. And we suggest that members of DC pension risk endeavor with the government and companies to avoid.

• Journal of the Korean Data and Information Science Society
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• v.25 no.6
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• pp.1171-1180
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• 2014

We considered saddlepoint approximations to VaR (value at risk) and ES (expected shortfall) which frequently encountered in finance and insurance as the measures of risk management. In this paper we supposed univariate and multivariate skew-normal distributions, instead of traditional normal class distributions, as underlying distribution of linear portfolios. Simulation results are provided and showed the suggested saddlepoint approximations are very accurate than normal approximations.

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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• pp.413-428
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• 2012

Every contingent claim is unable to be replicated in the incomplete markets. Shortfall risk is considered with some risk exposure. We show how the dynamic optimization problem with the capital constraint can be reduced to the problem to find an optimal modified claim $\tilde{\psi}H$ where$\tilde{\psi}H$ is a randomized test in the static problem. Convex and coherent risk measures defined in the Orlicz hearts spaces, $M^{\Phi}$, are used as risk measure. It can be shown that we have the same results as in [21, 22] even though convex and coherent risk measures defined in the Orlicz hearts spaces, $M^{\Phi}$, are used. In this paper, we use Fenchel duality Theorem in the literature to deduce necessary and sufficient optimality conditions for the static optimization problem using convex duality methods.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.389-407
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• 2006

Extreme value theory has been used widely in many areas of science and engineering to deal with the assessment of extreme events which are rare but have catastrophic consequences. The potential of extreme value theory has only been recognized recently in finance area. In this paper, we provide an overview of extreme value theory for estimating and assessing value at risk and expected shortfall which are the methods for modelling and measuring the extreme financial risks. We illustrate that the approach based on extreme value theory is very useful for estimating tail related risk measures through backtesting of an empirical data.

• Journal of the Korean Data and Information Science Society
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• v.27 no.4
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• pp.959-967
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• 2016

Distributional assumptions on equity returns play a key role in valuation theories for derivative securities. Elberlein and Keller (1995) investigated the distributional form of compound returns and found that some of standard assumptions can not be justified. Instead, Generalized Hyperbolic (GH) distribution fit the empirical returns with high accuracy. Hu and Kercheval (2007) also show that the normal distribution leads to VaR (Value at Risk) estimate that significantly underestimate the realized empirical values, while the GH distributions do not. We consider saddlepoint approximations to estimate the VaR and the ES (Expected Shortfall) which frequently encountered in finance and insurance as measures of risk management. We supposed GH distributions instead of normal ones, as underlying distribution of linear portfolios. Simulation results show the saddlepoint approximations are very accurate than normal ones.