在傳統投資組合理論中，最著名的爲Markowitz(1952)所提出的均值－變異數模式，此模式係以變異數來衡量風險，不過以此來估計風險時，無論價格上漲或下跌皆視爲相同風險，但以此來測量風險，無法適當反應低機率事件的風險。基於上述觀點，Markowitz(1959)針對此現象作了修正，提出了半變異數(Semivariance)的觀念，而Estrada(2008)進而以此半變異數爲損失風險的觀念發展出一種較簡易的平均數－半變異數模型，其次，Bawa and Lindenberg(1977)以左偏動差(Lowe Partial Moment)做爲損失風險的觀念而發展出平均數－左偏動差模型。再者，Konno and Yamazaki(1991)另外提出了平均數－平均絕對離差模型，此模型不但節省計算時間，並且在求解最適投資組合時，也不需要共變異數矩陣，所以降低了計算上的困難度，最後，Rockafellar and Uryasev(2000)則以條件風險值(Conditional Value-at-Risk)爲損失風險的觀念發展出平均數－條件風險值模型。綜觀上述不同風險測量之投資組合模型，本研究以半變異數、左偏動差、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，分別在常態與非常態分配下，分析其所求解出的最適投資組合之關係與差異，並進行相似度分析。

Markowitz (1952) proposed the famous mean-variance (MV) model for portfolio selection. In the MV model, the risk of investment is measured by variance. However, from the view of measuring risk, the variance is not a satisfactory measure of risk since it penalizes gains and losses in the same way, and the variance is inappropriate to reflect the risk of low probability events. Due to above reason, many researchers had proposed different points of view to measure risks. For example, Markowitz (1959) proposed another risk measurement, semivariance (SV), to avoid this shortcoming. Next, Estrada (2008) developed a theory to evaluate the downside risk which is derived from the concept of the semivariance. Bawa and Lindenberg (1977) developed a theory to evaluate the downside risk model named "Mean Lower Partial Moment" (MLPM) model which is derived from the concept of the Lower Partial Moment. Then, Konno and Yamazaki (1991) proposed the mean mean absolute deviation (MMAD) as alternative to the mean variance (MV) model. Finally, Rockafellar and Uryasev (2000) developed a theory to evaluate the downside risk model named "Mean Conditional Value-at-Risk" (MCVaR) model which is derived from the concept of the Conditional Value-at-Risk (CVaR). They claim it retains all the positive features of the MV model, saves the investor computing time, and dose not required the covariance matrix. The main subject of this paper is to make some comparisons and analyses among these portfolio risk models whose risks measured by variance, SV, LPM, MAD and CVaR under normal and nonnormal real data, respectively.