Safety-first (SF) rules have been increasingly useful in particular for construction of optimal portfolios related to pension and other social insurance funds. How's the performance of the optimal portfolios constructed by different SF rules is an interesting practical question but yet less investigated theoretically. In this paper, we therefore analytically investigate the properties of the risky portfolios constructed by the three popular SF rules, denoted by the RSF, TSF and KSF, which are suggested and developed by A. D. Roy, L. G. Telser and S. Kataoka, respectively. Using Sharpe ratio as a measure of portfolio performance, we theoretically derive that the performance of an optimal portfolio constructed by the KSF approach depends on an acceptable level of extreme risk tolerance. The unique solution where the performance of the KSF portfolio is the same as that of the other two SF portfolios is found. By this we interestingly find that except this special case, under the finite optimal portfolios existent, the KSF portfolio always dominates the TSF portfolio in terms of the Sharpe ratio. In addition, in some market scenarios, even when the RSF and TSF portfolios do not exist in finite forms, the KSF rule can still apply to get a finite optimal portfolio. Moreover, in comparison with the RSF rule, a series of finite KSF portfolios can be interestingly constructed with their Sharpe ratios approaching to the maximum Sharpe ratio, which however cannot be reached by any corresponding finite RSF portfolio. Numerical comparisons of these rules by using a set of real data are further empirically demonstrated.

中文翻译：

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通过安全第一规则优化的投资组合的绩效如何：具有经验比较的理论

安全第一（SF）规则越来越有用，特别是在构建与养老金和其他社会保险基金有关的最佳投资组合时。由不同的SF规则构造的最优投资组合的绩效如何是一个有趣的实际问题，但在理论上却很少研究。因此，在本文中，我们分析性地研究了由三种流行的SF规则（分别由AD Roy，LG Telser和S. Kataoka提出和建议）以RSF，TSF和KSF表示的风险投资组合的性质。使用夏普比率作为投资组合绩效的一种度量，我们从理论上得出，通过KSF方法构建的最优投资组合的绩效取决于可接受的极端风险承受水平。找到了独特的解决方案，其中KSF产品组合的性能与其他两个SF产品组合的性能相同。由此我们有趣地发现，除了这种特殊情况外，在存在有限的最优投资组合的情况下，就夏普比率而言，KSF投资组合始终在TSF投资组合中占主导地位。此外，在某些市场情况下，即使RSF和TSF投资组合不存在有限形式，KSF规则仍然可以适用于获得有限的最优投资组合。此外，与RSF规则相比，有趣的是，可以构建一系列有限的KSF投资组合，使其Sharpe比率接近最大Sharpe比率，但是任何相应的有限RSF投资组合都无法达到。通过使用一组实际数据对这些规则进行数值比较，进一步通过经验证明。