Leshno and Levy (2002) 將隨機優勢法 (Stochastic Dominance, SD) 延伸，對於理性的投資決策者提出了幾乎隨機優勢法 (Almost Stochastic Dominance, ASD) 用以比較投資組合優劣。Bali et al. (2009) 以幾乎隨機優勢法實證得出：對理性 (不包含病態效用函數) 投資人來說，長期而言股票優於債券；然而，Levy (2009) 對Bali等人的結論有所質疑，他指出在幾乎隨機優勢法的框架下，報酬率較高之投資標的累積機率分配，相較於報酬率較低之投資標的累積機率分配，會隨著投資期間拉長而右移，導致股票看似在長期優於債券的結果。
因此，為解決文獻看法的歧異，我們將Shorrocks (1983) 及Shalit (2014)建立之Lorenz 曲線應用於投資組合分析。除此之外，我們也建立了新的比較準則—幾乎Lorenz優勢 (Almost Lorenz Dominance)，用以找出隨著投資期間增長，股票占投資組合中的最佳比例。

Leshno and Levy (2002) proposed almost stochastic dominance (ASD) approach for decision makers with economically relevant preferences. Bali et al. (2009) applied ASD to their analysis and claimed that in the long run stocks will dominate bonds. However, Levy (2009) pointed out the drawbacks of ASD rules and argued that stocks should not be preferred over bonds over time.
As a result, we follow Shorrocks (1983) and Shalit (2014) and employ Lorenz curve in the portfolio risk analysis. Furthermore, we establish a new framework called almost Lorenz dominance (ALD) and manage to find out the optimal weight of stocks in a portfolio with longer investment horizons

Table of contents
1. Introduction 1
2. Stochastic Dominance and Almost Stochastic Dominance 3
2.1. First-Degree and Second-Degree SD 3
2.2. Almost First-Degree and Almost Second-Degree SD 3
3. Lorenz Curve and Almost Lorenz Dominance 5
3.1. Lorenz Curve 5
3.2. The advantages of using Lorenz Curve 7
3.3. Almost Lorenz Dominance 8
4. Data and Empirical Results 10
5. Conclusion 15
Appendix 16
References 21

Table of tables
Table 1. Descriptive statistics for stock and bond returns. 10
Table 2. The violation area for stock portfolio versus bond portfolio. 12
Table 3. The ALD of diversified stock/bond portfolio. 13
Table 4. The ALD of diversified stock/bond portfolio (other investment horizons). 19

Table of figures
Fig. 1. The Lorenz Dominance. 6
Fig. 2. The Lorenz curves cross with each other. 9
Fig. 3. Lorenz curves for stock portfolio versus bond portfolio. 11
Fig. 4. The violation area of diversified stock/bond portfolio 14
Fig. 5. Lorenz for 40% stock, 60% bond portfolio versus 100% bond portfolio 16
Fig. 6. Lorenz for 60% stock, 40% bond portfolio versus 100% bond portfolio 17
Fig. 7. Lorenz for 80% stock, 20% bond portfolio versus 100% bond portfolio 18

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