夏普指標廣泛地運用在投資組合的績效評量上，但是夏普指標為單位風險可以得到多少超額報酬的概念，此概念只考慮到一階，二階動差，因此當目標資料的報酬率不符合常態分配時，會造成夏普指標與實際報酬率互相矛盾的現象。近年來已有許多學者，透過實證方法，提出金融資產報酬率有偏態、厚尾高峰等三階四階動差的特徵，這是常態分配下無法捕捉到的情形，換言之，夏普指標已不合時宜。
本研究目的為推導出一種新的夏普指標，並與傳統上的夏普指標比較優劣。樣本為各國重要指數，包括法國巴黎指數(CAC40)、德國法蘭克福指數(DAX)、英國倫敦金融時報指數(FTSE100)、香港恆生指數(Hang Seng index)、那斯達克指數(NASDAQ)、日本日經指數(NIKKEI225)、美國標準普爾指數(S&P500)和台灣加權指數(TAIEX)共８種全球重要指數，研究期間為2001/01/01~2010/12/31。我們首先推導出JD、VG、NIG、Hyperbolic、GH此五種夏普指標，透過赤池資訊準則、以及貝氏資訊準則找出最適的分配過程。
分析比較傳統夏普指標和改良過後的一般化夏普指標彼此間的差異，研究的方法包括不同夏普指標的相關係數和個自的自我相關係數比較，探討基金經理人可以透過賣選擇權的方式，操弄夏普指數，以及不同的夏普指數下投資組合累積報酬的預測程度。
根據上述的方法，得到的結果如下：第一、GH在配適指數資料上，是一個較好的指標，第二、GH分配下的夏普指標有傳統的夏普指標呈現低度相關，顯示兩者有不小的差異，第三、經理人雖然可以操弄投資組合的夏普指數，但修正過後的夏普指標可以反應出實際的報酬優劣。第四、在投資組合的預測程度上，修正後的夏普指標績效表現較好。

關鍵詞：夏普指標、Levy過程、GH分配、投資組合、效用函數

Sharpe ratio is extensively used in performance of portfolio. However, it is based on assumption that return follows normal distribution. In other words, when return in asset is not normal distribution, the Sharpe ratio is not meaningful.
This research focuses on Generalized Sharpe ratio with different distribution in eight indexes from 2001/12/31 to 2010/12/31. We try to find a suitable levy process to fit our data. Instead of Normal distribution assumption, we use Jump diffusion, Variance Gamma, Normal Inverse Gaussian, Hyperbolic, Generalized Hyperbolic, as our distribution to solve stylized fact like skewness and kurtosis.
Compared the difference between standard Sharpe ratio and Generalized Sharpe ratio, we come to these conclusions: first of all, Generalized Hyperbolic is better levy process to fit our eight indexes. Second, Sharpe ratio under GH levy process has low autocorrelation, and it present that modified Sharpe ratio is more elastic. Third, Generalized Sharpe ratio can uncover the strategy that fund manager manipulate Sharpe ratio. At last, Generalized Sharpe ratio have better predict than standard Sharpe ratio.


Keywords: Sharpe ratio, Levy process, GH distribution, portfolio, utility function

1. Introduction 1
1.1 Background and motivation 1
1.2 Purpose 2
2. Literature review 4
2.1 Standard Sharpe ratio and Generalized Sharpe ratio 4
2.2 Non-normality distribution of stock’s return 4
2.3 Generalized Sharpe ratio under Variance Gamma process 5
2.4 Generalized Sharpe ratio under Normal Inverse Gaussian process 6
2.5 Generalized Sharpe ratio under four levy process 6
3. Model derivative and method of empirical 8
3.1 Frame of research 8
3.2 Levy process 8
3.3 model derivation 14
3.4 Method of empirical 18
4. Empirical results 22
4.1 Descriptive statistics 22
4.2 AIC and BIC 24
4.3 Correlation and autocorrelation analysis 25
4.4 Estimating performances of portfolios with manipulated Sharpe ratios 31
4.5 Performance comparison between SR and GSR in assets allocation 32
4.6 Quartile analysis 36
5 Conclusion and Suggestion 38
5.1 Conclusion 38
5.2 Suggestion 39
Reference 41

Bernardo, Antonio E and Oliver Ledoit. (2000). “Gain, loss and asset pricing,” Journal of Political Economy, 108(1), 144–172.
Cass, David and Joseph E. Stiglitz. (1970). “The structure of investor preferences and asset returns, and separability in portfolio allocation: A contribution to the pure theory of mutual funds,” Journal of Economic Theory, 2(2), 122-160.
Carr, Peter and Hélyette Geman. (2002). “The Fine Structure of Asset Returns: An Empirical Investigation,” The Journal of Business, 75(2), 305-332.
Carr Peter, Hélyette Geman, Dilip B. Madan and Marc Yor. (2003). “Stochastic Volatility for Levy Processes,” Mathematical Finance, 13(3), 345-382.
Dowd, Kevin. (2004). “Adjusting for risk: An improved Sharpe ratio,” International Review of Economics and Finance, 9, 209–222.
Favre, Laurent and José-Antonio Galeano. (2002). “Mean-modified value-at-risk optimization with hedge funds,” Journal of Alternative Investments, 5(2), 21-25.
Simone, Farinelli S., Ferreira Manuel, Rossello Damiano, Thoeny Markus, Tibiletti Luisa. (2008). “Beyond Sharpe ratio: optimal asset allocation using different performance ratios,” Journal of Banking & Finance, 32(10), 2057-2063.
Hodges, S.. (1998). “A generalization of the Sharpe ratio and its applications to valuation bounds and risk measures,” Working Paper, Financial Options Research Centre, University of Warwick.
McCulloch, J. Huston. (1978). “Continuous time processes with stable increments,” Journal of Business, 51(4), 601-619.
Madan, Dilip B. and Gavin S. McPhail. (2000). “Investing in skews,” Journal of Risk Finance, 2(1), 10-18.
Madan Dilip B., Peter P. Carr and Eric C. Chang. (1998). “The variance Gamma process and option pricing,” European Finance Review, 2(1), 79-105.
Rachev, Svetlozar., Jasic Teo, Stoyanov Stoyan and Frank J. Fabozzi. (2007). “Momentum strategies based on reward-risk stock selection criteria,” Journal of Banking and Finance, 31(8), 2325-2346.
Ramezani Cyrus A. and Zeng Yong (2007). “Maximum likelihood estimation of the double exponential jump-diffusion process,” Annals of Finance, 3, 487-507.
.Sortino, Frank A. and Lee N. Price. (1994). “Performance measurement in a downside risk framework,” Journal of Investing, 3(3), 59-64.
Zakamouline, Valeri and Steen Koekebakker. (2009). “Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance,” Journal of Banking and Finance, 33(7), 1242-1254.