Portfolio optimization and marginal contribution to risk on multivariate normal tempered stable model

This paper proposes 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 can capture two stylized facts: fat-tails and asymmetry, that have been empirically observed for asset return distributions. We discuss a new portfolio optimization method on the new market model, which is an extension of Markowitz’s mean-variance optimization. The new optimization method considers not only reward and dispersion but also asymmetry in tails. The efficient frontier is extended to a curved surface on three-dimensional space of reward, dispersion, and asymmetry in tails. We also propose a new performance measure, which is an extension of the Sharpe ratio. Moreover, we derive closed-form solutions for portfolio managers’ two important measures 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 this paper’s empirical evidence, our framework offers realistic portfolio optimization and tractable methods for portfolio risk management.