AI 中文总结
该研究将经典马克维茨投资组合优化的方差风险测度推广至严格凸二次风险测度，推导了多种投资组合的闭式公式，发现切线投资组合与最大夏普比率投资组合不重合的新几何现象并通过数值示例验证。
AI 中文摘要
我们证明，最初以方差作为风险测度构建的经典马克维茨投资组合选择理论的关键优化结果，在更广泛的严格凸二次风险测度类下仍以显式闭式形式存在。所提框架将协方差矩阵替换为任意对称正定矩阵，并允许添加额外的线性项和常数项，因此涵盖了交易成本优化、基准相对优化、协方差正则化及因子模型中出现的各类模型。我们为有效前沿、全局最小风险投资组合、最大夏普比率投资组合、资本市场曲线、切线投资组合及最大效用投资组合推导了闭式公式。与经典马克维茨模型不同，切线投资组合与最大夏普比率投资组合并不重合，揭示了一种新的几何现象。一个数值示例验证了所推导的公式。
英文摘要
We show that the key optimization results of the classical Markowitz portfolio selection theory, originally formulated for variance as the risk measure, remain available in explicit closed form under a broader class of strictly convex quadratic risk measures. The proposed framework replaces the covariance matrix with an arbitrary symmetric positive definite matrix and allows additional linear and constant terms, thereby containing various models arising in transaction cost optimization, benchmark relative optimization, covariance regularization, and factor models. Closed-form formulas are obtained for the efficient frontier, the global minimum risk portfolio, the maximum Sharpe ratio portfolio, the Capital Market Curve, the tangency portfolio, and the maximum utility portfolio. In contrast to the classical Markowitz model, the tangency portfolio does not coincide with the maximum Sharpe ratio portfolio, revealing a new geometric phenomenon. A numerical example confirms the derived formulas.
