In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution defined by a mixture of the multivariate normal distribution and the tempered stable subordinator. This distribution is able to capture two stylized facts: fat-tails and asymmetric tails , that have been empirically observed for asset return distributions. On the new market model,  a new portfolio optimization method, which is an extension of Markowitz's mean-variance optimization , is discussed . The new optimization method considers not only reward and dispersion but also asymmetry. The efficient frontier is also extended from the mean-variance curve to a surface on three dimensional space of reward, dispersion, and asymmetry. We also propose a new performance measure which is an extension of  Sharpe Ratio. Moreover, we derive closed-form solutions for two important measures used by portfolio managers in portfolio construction: the marginal Value-at-Risk (VaR) and the marginal Conditional VaR (CVaR). We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average. First  perform the new portfolio optimization and then demonstrating how the marginal VaR and marginal CVaR can be used for portfolio optimization using the model. Based on the empirical evidence presented in this paper, our framework offers realistic portfolio optimization and tractable methods for portfolio risk management.
In this paper, we propose a market model with returns assumed to follow a multivariate normal tempered stable distribution defined by a mixture of the multivariate normal distribution and the tempered stable subordinator. This distribution is able to capture two stylized facts: fat-tails and asymmetry , that have been empirically observed for asset return distributions. On the new market model, we discuss a new portfolio optimization method, which is an extension of Markowitz's mean-variance optimization . The new optimization method considers not only reward and dispersion but also asymmetry. The efficient frontier is also extended to a curved surface on three-dimensional space of reward, dispersion, and asymmetry. We also propose a new performance measure which is an extension of the Sharpe Ratio. Moreover, we derive closed-form solutions for two important measures used by portfolio managers in portfolio construction: the marginal Value-at-Risk (VaR) and the marginal Conditional VaR (CVaR). We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average. First , perform the new portfolio optimization and then demonstrating how the marginal VaR and marginal CVaR can be used for portfolio optimization under the model. Based on the empirical evidence presented in this paper, our framework offers realistic portfolio optimization and tractable methods for portfolio risk management.
D
multivariate
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asymmetric tails
asymmetry
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we discuss
D
, is discussed
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from the mean-variance curve to a surface on three dimensional
to a curved surface on three-dimensional
A
the
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,
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using
under