傳統投資組合理論係以變異數來衡量投資風險，不過以此來估計風險時，無論價格上漲或下跌皆視為風險，但其實價格下跌才是投資人真正想規避的風險。基於上述爭議， Bawa 和 Lindenberg ( 1977）以左偏動差(Lower Partial Moment)為損失風險的觀念發展出平均數-左偏動差模型，Rockafella和Uryasev（2000）以條件風險值(Conditional Value-at-Risk )為損失風險的觀念發展出平均數-條件風險值模型。近年來，Estrada ( 2008 )則是以損失風險的觀念發展出平均數-半變異數模型。再者，平均數-變異數（MV）模型在求解最適投資組合時，會因為共變異數矩陣複雜度的提升，不但提高了計算上的困難度，也增加了計算時間，而 Konno 和 Yamazaki (1991）所提出的平均數-平均絕對離差模型，由於不需要利用到共變異數矩陣，所以可避免上述的問越 • 還可保留MV氏模型的優點。
因此，本研究以左偏動差、半變異數、平均絕對離差、條件風險值來衡量投資組合的風險，與利用變異數來衡量風險作比較，分析其所求解出的最適投資組合之差異，以協助投資人找出適合自己的投資組合模型。研究中發現若投資者為風險喜好者 ， 無論所收集到的報酬資料是否服從多元常態分配，並且投資者傾向於關切損失風險時，本研究建議該投資者可參考 MCVaR 模型所提供的最適投資決策，但若投資者不僅關切損失風險，也在乎上方潛在的獲利風險時，則建議該投資者可參考 MMAD 模型所提供的最適投資決策;相對地，若投資者為風險趨避者，並且同時在乎上下方風險時，再加上所收集到的報酬資料不服從多元常態分配時，本研究建議該投資者可參考MV模型所提供的最適投資決策，但若報酬資料服從多元常態分配並且只關切損失風險時，則建議該投資者可參考MSV 模型所提供的最適投資決策。此外，本文也證實當投資者在面對選擇性較多的投資標的時，不論是何種投資組合模型所決策出來的最適投資組合，其投資績效都會隨之提高。

The investment risk is measured by the variance in traditional mean-variance (MV) portfolio theory. Both overperformance and underperformance are considered as risk under the variance. However most investors worry about underperformance
rather than overperformance. Hence, Bawa and Lindenberg (1977) employed the lower partial moment (LPM) to evaluate the downside risk. Rockafellar and Uryasev (2000) used the conditional value-at-risk (CVaR) to find efficient portfolios. Recently. Estrada (2008) introduced a heuristic approach for the mean semivariance (MSV) optimization. Furthermore, the complexity of covariance matrix results in the difficulty of implementation of MV model. Konno and Yamazaki (1991) proposed the mean-mean absolute deviation (MMAD) model and claimed that MMAD model retains all the advantages of the MV model. It results form that the covariance matrix is not used in their model. This research intends to compare efficient portfolios generated from different risk measures, such as variance, LPM, SV , CVaR. and MAD.
The result shows that whether the dataset is multinormally distributed or not, if the investor is risk lover and tend to concern about underperformance, this study recommends that the investor may refer to MCVaR model provided the optimal decision-making. And if they are both concern about underperformance and overperformance, this study recommends that the investor may refer to MMAD model provided the optimal decision-making. Relatively, if the investor is risk averser and both concern about underperformance and overperformance. When the dataset is not multinormally distributed, this study recommends that the investor may refer to MV model provided the optimal decision-making. When the dataset is multinormally distributed and the investors tend to concern about underperformance, this study recommends that the investor may refer to MSV model provided the optimal decision-making. In addition, our study demonstrates in any kind of portfolio model. the more investment instruments that investors have, the higher portfolio performance could be produce, no matter what kind of portfolio model.