考慮極值分佈下控制尾端風險 之最佳化投資組合

Safety-First Portfolio Optimization under Approximated Extreme Tail Distribution

周玉婷(YU-TING CHOU)

指導教授 ： 張國華

中原大學/工學院/工業工程研究所/碩士(2008年)

https://doi.org/10.6840/cycu200900642

摘要

在Mean-Variance model 裡，資產及投資組合的績效指標是根據其所對應的 報酬率之期望值與變異數來衡量，再者其可透過非常簡單且有效率的數學模式來 求得最佳投資組合。然而這樣的資訊已無法滿足投資者，投資者希望透過更多方 面的資訊來評估投資組合，特別是報酬率分佈的下端風險。在此研究中，我們考 慮在控制報酬分佈的下端風險之下並以Safety-First模式為基礎，建構出能求得最 佳投資組合之模式。另一方面，為了能夠精確地估計報酬分佈的下端風險，我們 通常使用極值理論來估計，但是使用極值理論來求得最佳的投資組合是非常沒有 效率的。 在本研究中，我們使用極值copula 找出資產報酬分佈之間的相關性及報酬 率的極值分佈，根據這兩點我們模擬資產未來報酬分佈的情形，並篩選出其所有 資產報酬率皆為負值的狀況來當作我們模式裡所需的近似尾端分佈。另外，我們 將傳統的Safety-First 模式並結合線性規劃發展成符合我們目的的模式。最後實 證結果顯示，我們研究的結果是優於市場。

關鍵字

極值理論 ； 下端風險 ； 尾端分佈 ； Safety-First模式 ； 投資組合

並列摘要

In mean-variance model, the performances of assets or portfolios are evaluated
based on the mean and the variance of the corresponding return rates and the portfolio
selection problem can be formulated into a simple mathematical program which can
be solved very efficiently. However, the performances evaluated by investors are in
more various aspects. Investors would better look for more information, especially
the downside risk (tail-probability from the worse case), of the returns. In this study,
we model a safety-first portfolio selection problem considering the downside risks.
For better estimations of the downside risks, we usually try to use the extreme value
theory to estimate them, however, it is not efficient to obtain an optimal portfolio if
we adopt extreme value procedure.
In this study, we use extreme copula to estimate the dependency on the distributions
of the returns and estimate the extreme value distribution of the return
rates, based on which we simulate the returns and sample the joint worse cases (rare
tail events) to obtain the approximated conditional tail distributions used for our
model. By further presenting the corresponding evaluations in the fashion of linear
equations, the corresponding safety-first portfolio optimization problem can be formulated
as linear programs, which can be solved by simplex method. Test results of
ours performances as well as the market’s are compared.

並列關鍵字

Portfolio Selection Problem ； Safety-First ； Tail Distribution ； Downside Risk ； Extreme Value Theory.

參考文獻

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