在報酬呈不對稱分布時,由於變異數是以計算報酬波動變異的方式來衡量投資風險,相較於低偏動差 (lower partial moment) 僅衡量投資人的下方風險 (downside risk),變異數明顯無法較低偏動差準確的衡量投資人面臨的投資風險。為了避免投資人在不對稱的報酬分配下,以變異數衡量投資風險造成錯誤的投資決策,本研究提出「對稱變異統計量」 (symmetric variation statistic) 進行「對稱變異檢定」 (symmetric variation test),利用蒙地卡羅模擬法 (Monte Carlo simulation) 產生對稱變異檢定的臨界值表,協助投資人在進行投資決策之前,檢定變異數與低偏動差對於投資風險的衡量是否達到顯著的差異,並藉此判斷變異數高估或低估投資風險,進而決定是否該調整投資決策,或以低偏動差代替變異數重新衡量投資風險,擬定新的投資策略。另外,本研究以模擬的方式檢測對稱變異檢定的型一誤差與檢定力後發現,參數的變化並不會影響對稱變異檢定的型一誤差與檢定力,且隨著樣本數的增加,對稱變異檢定的型一誤差扭曲 (size distortion) 逐漸趨緩,但是檢定力卻隨之減小。
Variance measures investment risk as calculating the volatile variation of return, and lower partial moment (LPM) only measures investment risk as downside risk. When the distribution of return is asymmetric, LPM is more suitable than variance to measure investment risk. To avoid the wrong investment decision by variance estimation under the asymmetric distribution of return, we propose symmetric variation statistic and use Monte Carlo to simulate the critical value table of symmetric variation test. This can examine if the variance and LPM exist the significant difference of measuring investment risk. If the difference is significant, the portfolio weight or performance evaluation should be adjusted. Indeed, we can use LPM to re-estimate the investment risk to formulate a new investment strategy. We found that the variation of parameters doesn’t influence the size and power of symmetric variation test. And as the sample increase, the size distortion of symmetric variation test will reduce, but the power of symmetric variation test will also lower.