摘要：
我們給一個新的效用函數的定義，叫分位數相依的效用函數，可以用來解釋一些在展望理論中的一些行為，傳統的效用函數並不適用在勞倫斯優佔，但分位數相依的效用函數卻可以。由於分位數相依效用函數的發明，我們將勞倫斯優佔推廣到一階勞倫斯優佔和幾乎式勞倫斯優佔，相較於幾乎式隨機優佔，幾乎式勞倫斯優佔有許多好處。一個長久的辯論有關於"長時間而言究竟股票是否會優於債卷"，我們得到了長期而言股票會以一階及二階幾乎勞倫斯優佔債卷的結論。

Abstract:
We give utility function a new definition, called quantile-dependent utility. It can be used to explain some behaviors in the prospect theory. Traditional utility is not applicable to Lorenz dominance but quantlie-dependent utility is applicable. Because of the invention of quantile-dependent utility, we extend Lorenz dominance from second-degree to first-degree, and also extent Lorenz dominance to almost Lorenz dominance. Comparing almost stochastic dominance, there are many advantages in almost Lorenz dominance. The debate of "Is stocks dominate bonds for the long run" cannot be solved by almost stochastic dominance but can be solved by almost Lorenz dominance. We conclude that stocks dominate bonds by first-degree and second degree almost Lorenz dominance for sufficient long run.

目錄
摘要............................................................................1

誌謝............................................................................2

第一章 introduction.............................................................4

第二章 Quantile-dependent utility function......................................6

第三章 Lorenz dominance and stochastic dominance................................9

第四章Almost Lorenz dominance..................................................12

第五章Is stocks better than bonds for the long run ?...........................16

第六章Empirical................................................................18

第七章Conclusion...............................................................22

Appendix........................................................................23

Reference.......................................................................29

Reference:
[1]. Bali, T. G., Demirtas, K. O., Levy, H., & Wolf, A. (2009). Bonds versus stocks: Investors’ age and risk taking. Journal of Monetary Economics, 56(6), 817-830.
[2]. Bernstein, P. L. (1976). The Boomerang. The journal of portfolio management, 2(3), 4.
[3]. Hadar, J., & Russell, W. R. (1969). Rules for ordering uncertain prospects. The American economic review, 59(1), 25-34.
[4]. Hanoch, G., & Levy, H. (1969). The efficiency analysis of choices involving risk. The Review of Economic Studies, 36(3), 335-346.
[5]. Huang, R. J., Tzeng, L. Y., Wang, J. Y., & Zhao, L. (2016). Asymptotic Stochastic Dominance.
[6] Kahneman, D., & Tversky, A. (2013). Prospect theory: An analysis of decision under risk. In HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING: Part I(pp. 99-127).
[7]. Leshno, M., & Levy, H. (2002). Preferred by “all” and preferred by “most” decision makers: Almost stochastic dominance. Management Science, 48(8), 1074-1085.
[8]. Levy, H. (2015). Aging population, retirement, and risk taking. Management Science, 62(5), 1415-1430.
[9]. Levy, M. (2009). Almost stochastic dominance and stocks for the long run. European Journal of Operational Research, 194(1), 250-257.
[10]. Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association, 9(70), 209-219.
[11]. Markowitz, H. M. (1976). Investment for the long run: New evidence for an old rule. The Journal of Finance, 31(5), 1273-1286.
[12].Markowitz, H., (2006). Samuelson and investment for the long run. In: Szenberg, M. (Ed.), Samuelsonian Economics and the 21st Century.Oxford University Press.
[13]. Price, K., Price, B., & Nantell, T. J. (1982). Variance and lower partial moment measures of systematic risk: some analytical and empirical results. The Journal of Finance, 37(3), 843-855.
[14].Rothschild, M., Stiglitz, J., (1970). Increasing risk: I. A Defintion. Journal of Economic Theory 2, 225–243.
[15].Samuelson, P.A., (1989a). The judgment of economic science on rational portfolio management. Journal of Portfolio Management 16 (1), 4–12.
[16]. Samuelson, P. A. (1989b). A case at last for age-phased reduction in equity. Proceedings of the national academy of sciences, 86(22), 9048-9051.
[17].Samuelson, P.A., (1994). The long-run case for equities and how it can be oversold. Journal of Portfolio Management 21 (1), 15–24.
[18].Shalit, H. (2014). Portfolio Risk Management Using the Lorenz Curve. The Journal of Portfolio Management, 40(3), 152-159.
[19]. Shorrocks, A. F. (1983). Ranking income distributions. Economica, 50(197), 3-17.
[20]. Tzeng, L. Y., Huang, R. J., & Shih, P. T. (2013). Revisiting almost second-degree stochastic dominance. management Science, 59(5), 1250-1254.
[21]. Von Neumann, J., & Morgenstern, O. (2007). Theory of games and economic behavior. Princeton university press.
[22]. Zheng, B. (2016). Almost Lorenz Dominance.