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題名 利用Quantopian交易平台設計演算法交易策略
Design algorithmic trading strategy by Quantopian trading platform
作者 吳雅岩
Wu, Ya Yen
貢獻者 謝明華
吳雅岩
Wu, Ya Yen
關鍵詞 演算法交易
金融科技
因子投資
績效指標
權重策略
投資組合
Algorithmic trading
FinTech
Factor investing
Performance index
Weight
Portfolio
日期 2017
上傳時間 31-Aug-2017 12:05:55 (UTC+8)
摘要 本文以全球第一個演算法交易雲端平台-Quantopian進行研究,藉由平台社群討論區內公開之演算法交易策略,透過交易策略篩選和初步優化,以演算法交易策略為投資標的,搭配不同權重策略建構投資組合。權重策略部分,本文提出適用於組合式交易策略的績效指標加權 (Performance Index Weighted) 法,應用因子投資的觀念,融合排序相關性較低、不同面向之績效指標作為報酬率驅動因子,並參考Asness et al. (2013) 以因子排序作為權重計算依據,提供了簡單直覺、非最適化求解而且穩健的加權方式,更直接地將交易策略各面向績效的優劣反應在權重上。
根據數值分析,發現組合式交易策略長期而言,整體績效表現平均優於個別演算法交易策略,最小變異、績效指標加權和均等權重投資組合的風險亦明顯低於個別交易策略,且最小變異、績效指標加權和均等權重投資組合在降低投資組合風險的同時,並未犧牲過多報酬,風險調整後績效表現優於個別交易策略。而績效指標加權投資組合之年化報酬率、風險衡量和風險調整後績效表現皆優於最小變異、平均數-變異數、均等權重的加權投資組合,此種權重策略可使投資組合之夏普比率 (Sharpe ratio) 顯著提升,且投資組合的風險大幅降低,最大跌幅 (Max drawdown) 在四年半的實驗區間內降至10%以下的水準,風險調整後績效優異。
透過Quantopian社群演算法交易平台,個人投資者也能站在巨人的肩膀上學習,集合眾人的力量,憑藉量化交易創造出和機構法人一樣具有競爭力的投資組合。如Chan (2009) 所言,個人投資者也能憑藉量化交易,設計一套演算法交易策略。
Quantopian is a crowd-sourced hedge fund which allows members on the platform to develop their own algorithmic strategies and even get capital allocations from Quantopian. In this paper, we constructed portfolios by Quantopian trading platform and proposed Performance Index Weighted method which generate consistently profit in our study. First, we filtered algorithmic trading strategies shared on the Quantopian community and improved the performance slightly. Second, we combined multiple algorithmic strategies with varied portfolio weight method, such as minimize-variance, performance index weighted, mean-variance, and equal weighed method to construct a portfolio.
To elaborate, Performance Index Weighted portfolio is actually an application of factor investing, in which the portfolio weight depends on the ranking of performance index (factors), and these index measure returns, risk, and also risk-adjusted returns, which truly reflects how well the algorithmic strategy is. As a result, we used the performance index as a return driver and invested more in well-ranked strategies directly. Performance index weighted is a simple, robust, and fully intuitively way to construct a portfolio.
In numerical analysis, we found that using multiple strategies to construct a portfolio could generate better performance than a single algorithm strategy on average. Moreover, the annual returns, risk measure, and risk-adjusted returns of Performance Index Weighted portfolio turn out to be better than minimize-variance portfolio, mean-variance portfolio, and equal weighted portfolio. As a result, Performance Index Weighted portfolio has significantly higher Sharpe ratio and lower Max Drawdown (lower than 10% in our out-of-sample test) than other portfolios, which shows excellent risk-adjusted performance.
Most important of all, retail traders could learn more precisely by standing on the shoulders of giants through Quantopian trading platform. Also, by collecting wisdom from the crowd, we create an opportunity for retail traders to construct competitive portfolios just as institutional investors do.
參考文獻 Acerbi, C., & Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487-1503.
Ang, A. (2014). Asset management: A systematic approach to factor investing: Oxford University Press.
Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2006). The cross‐section of volatility and expected returns. The journal of finance, 61(1), 259-299.
Asness, C. S., Frazzini, A., & Pedersen, L. H. (2014). Quality minus junk.
Asness, C. S., Moskowitz, T. J., & Pedersen, L. H. (2013). Value and momentum everywhere. The journal of finance, 68(3), 929-985.
Aumann, R. J., & Serrano, R. (2008). An economic index of riskiness. Journal of Political Economy, 116(5), 810-836.
Bali, T. G., Cakici, N., & Chabi-Yo, F. (2011). A generalized measure of riskiness. Management science, 57(8), 1406-1423.
Ballings, M., Van den Poel, D., Hespeels, N., & Gryp, R. (2015). Evaluating multiple classifiers for stock price direction prediction. Expert Systems with Applications, 42(20), 7046-7056.
Banz, R. W. (1981). The relationship between return and market value of common stocks. Journal of financial economics, 9(1), 3-18.
Bernard, V. L., & Thomas, J. K. (1989). Post-earnings-announcement drift: delayed price response or risk premium? Journal of Accounting research, 1-36.
Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. The review of financial studies, 4(2), 315-342.
Biglova, A., Ortobelli, S., Rachev, S. T., & Stoyanov, S. (2004). Different approaches to risk estimation in portfolio theory. The journal of portfolio management, 31(1), 103-112.
Black, F. (1993). Beta and return. The journal of portfolio management, 20(1), 8-18.
Breiman, L. (2001). Random forests. Machine learning, 45(1), 5-32.
Breiman, L., Friedman, J., Stone, C. J., & Olshen, R. A. (1984). Classification and regression trees: CRC press.
Carhart, M. M. (1997). On persistence in mutual fund performance. The journal of finance, 52(1), 57-82.
Chan, E. (2009). Quantitative trading: how to build your own algorithmic trading business (Vol. 430): John Wiley & Sons.
Chen, Y.-T., Huang, R. J., Shih, P.-T., & Tzeng, L. Y. (2015). Capital Asset Pricing Model Based on a Generalized Economic Index of Riskiness.
Choueifaty, Y., & Coignard, Y. (2008). Toward maximum diversification. The journal of portfolio management, 35(1), 40-51.
Clenow, A. (2016). Stocks on the Move: So schlagen Sie den Markt mit den Momentum-Strategien der Hedgefonds: Börsenbuchverlag.
Conrad, J., & Kaul, G. (1998). An anatomy of trading strategies. The review of financial studies, 11(3), 489-519.
Da, Z., Engelberg, J., & Gao, P. (2011). In search of attention. The journal of finance, 66(5), 1461-1499.
Daniel, K., Grinblatt, M., Titman, S., & Wermers, R. (1997). Measuring mutual fund performance with characteristic‐based benchmarks. The journal of finance, 52(3), 1035-1058.
De Bondt, W. F., & Thaler, R. H. (1987). Further evidence on investor overreaction and stock market seasonality. Journal of finance, 557-581.
DeMiguel, V., Garlappi, L., Nogales, F. J., & Uppal, R. (2009). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management science, 55(5), 798-812.
Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of financial economics, 33(1), 3-56.
Fama, E. F., & French, K. R. (2017). International tests of a five-factor asset pricing model. Journal of financial economics, 123(3), 441-463.
Foster, D. P., & Hart, S. (2009). An operational measure of riskiness. Journal of Political Economy, 117(5), 785-814.
Goodwin, T. H. (1998). The information ratio. Financial Analysts Journal, 54(4), 34-43.
Gultekin, M. N., & Gultekin, N. B. (1983). Stock market seasonality: International evidence. Journal of financial economics, 12(4), 469-481.
James, G., Witten, D., & Hastie, T. (2014). An Introduction to Statistical Learning: With Applications in R.
Jegadeesh, N. (1990). Evidence of predictable behavior of security returns. The journal of finance, 45(3), 881-898.
Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. The journal of finance, 48(1), 65-91.
Jorion, P. (1997). Value at risk: the new benchmark for controlling market risk: Irwin Professional Pub.
Keating, C., & Shadwick, W. F. (2002). A universal performance measure. Journal of performance measurement, 6(3), 59-84.
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.
Levy, R. A. (1967). Relative strength as a criterion for investment selection. The journal of finance, 22(4), 595-610.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The review of economics and statistics, 13-37.
Maillard, S., Roncalli, T., & Teïletche, J. (2010). The properties of equally weighted risk contribution portfolios. The journal of portfolio management, 36(4), 60-70.
Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.
Moat, H. S., Curme, C., Avakian, A., Kenett, D. Y., Stanley, H. E., & Preis, T. (2013). Quantifying Wikipedia usage patterns before stock market moves. Scientific reports, 3, 1801.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica: Journal of the econometric society, 768-783.
Mundial, F. E. (2015). The Future of Financial Services: How disruptive innovations are reshaping the way financial services are structured, provisioned and consumed. Ginebra: FEM. Consultado en: http://www3. weforum. org/docs/WEF_The_future__of_financial_services. pdf.
Preis, T., Moat, H. S., & Stanley, H. E. (2013). Quantifying trading behavior in financial markets using Google Trends. Scientific reports, 3, srep01684.
Reinganum, M. R. (1981). Misspecification of capital asset pricing: Empirical anomalies based on earnings` yields and market values. Journal of financial economics, 9(1), 19-46.
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.
Schnytzer, A., & Westreich, S. (2013). A global index of riskiness. Economics Letters, 118(3), 493-496.
Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance, 19(3), 425-442.
Sharpe, W. F. (1994). The sharpe ratio. The journal of portfolio management, 21(1), 49-58.
Sortino, F. A., & Price, L. N. (1994). Performance measurement in a downside risk framework. the Journal of Investing, 3(3), 59-64.
Thenmozhi, M., & Kumar, M. (2009). Dynamic interaction among mutual fund flows, stock market return and volatility. NSE Research Papers.
Treynor, J. L. (1961). Toward a theory of market value of risky assets. Unpublished manuscript, 6.
Young, T. W. (1991). Calmar ratio: A smoother tool. Futures, 20(1), 40.
Yu, J.-R., Lee, W.-Y., & Chiou, W.-J. P. (2014). Diversified portfolios with different entropy measures. Applied Mathematics and Computation, 241, 47-63.
Zivot, E. (2013). Portfolio Theory with Matrix Algebra. viewed on May, 5, 2012.
描述 碩士
國立政治大學
風險管理與保險學系
104358016
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0104358016
資料類型 thesis
dc.contributor.advisor 謝明華zh_TW
dc.contributor.author (Authors) 吳雅岩zh_TW
dc.contributor.author (Authors) Wu, Ya Yenen_US
dc.creator (作者) 吳雅岩zh_TW
dc.creator (作者) Wu, Ya Yenen_US
dc.date (日期) 2017en_US
dc.date.accessioned 31-Aug-2017 12:05:55 (UTC+8)-
dc.date.available 31-Aug-2017 12:05:55 (UTC+8)-
dc.date.issued (上傳時間) 31-Aug-2017 12:05:55 (UTC+8)-
dc.identifier (Other Identifiers) G0104358016en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/112360-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 風險管理與保險學系zh_TW
dc.description (描述) 104358016zh_TW
dc.description.abstract (摘要) 本文以全球第一個演算法交易雲端平台-Quantopian進行研究,藉由平台社群討論區內公開之演算法交易策略,透過交易策略篩選和初步優化,以演算法交易策略為投資標的,搭配不同權重策略建構投資組合。權重策略部分,本文提出適用於組合式交易策略的績效指標加權 (Performance Index Weighted) 法,應用因子投資的觀念,融合排序相關性較低、不同面向之績效指標作為報酬率驅動因子,並參考Asness et al. (2013) 以因子排序作為權重計算依據,提供了簡單直覺、非最適化求解而且穩健的加權方式,更直接地將交易策略各面向績效的優劣反應在權重上。
根據數值分析,發現組合式交易策略長期而言,整體績效表現平均優於個別演算法交易策略,最小變異、績效指標加權和均等權重投資組合的風險亦明顯低於個別交易策略,且最小變異、績效指標加權和均等權重投資組合在降低投資組合風險的同時,並未犧牲過多報酬,風險調整後績效表現優於個別交易策略。而績效指標加權投資組合之年化報酬率、風險衡量和風險調整後績效表現皆優於最小變異、平均數-變異數、均等權重的加權投資組合,此種權重策略可使投資組合之夏普比率 (Sharpe ratio) 顯著提升,且投資組合的風險大幅降低,最大跌幅 (Max drawdown) 在四年半的實驗區間內降至10%以下的水準,風險調整後績效優異。
透過Quantopian社群演算法交易平台,個人投資者也能站在巨人的肩膀上學習,集合眾人的力量,憑藉量化交易創造出和機構法人一樣具有競爭力的投資組合。如Chan (2009) 所言,個人投資者也能憑藉量化交易,設計一套演算法交易策略。
zh_TW
dc.description.abstract (摘要) Quantopian is a crowd-sourced hedge fund which allows members on the platform to develop their own algorithmic strategies and even get capital allocations from Quantopian. In this paper, we constructed portfolios by Quantopian trading platform and proposed Performance Index Weighted method which generate consistently profit in our study. First, we filtered algorithmic trading strategies shared on the Quantopian community and improved the performance slightly. Second, we combined multiple algorithmic strategies with varied portfolio weight method, such as minimize-variance, performance index weighted, mean-variance, and equal weighed method to construct a portfolio.
To elaborate, Performance Index Weighted portfolio is actually an application of factor investing, in which the portfolio weight depends on the ranking of performance index (factors), and these index measure returns, risk, and also risk-adjusted returns, which truly reflects how well the algorithmic strategy is. As a result, we used the performance index as a return driver and invested more in well-ranked strategies directly. Performance index weighted is a simple, robust, and fully intuitively way to construct a portfolio.
In numerical analysis, we found that using multiple strategies to construct a portfolio could generate better performance than a single algorithm strategy on average. Moreover, the annual returns, risk measure, and risk-adjusted returns of Performance Index Weighted portfolio turn out to be better than minimize-variance portfolio, mean-variance portfolio, and equal weighted portfolio. As a result, Performance Index Weighted portfolio has significantly higher Sharpe ratio and lower Max Drawdown (lower than 10% in our out-of-sample test) than other portfolios, which shows excellent risk-adjusted performance.
Most important of all, retail traders could learn more precisely by standing on the shoulders of giants through Quantopian trading platform. Also, by collecting wisdom from the crowd, we create an opportunity for retail traders to construct competitive portfolios just as institutional investors do.
en_US
dc.description.tableofcontents 第一章 研究緒論 8
1.1 研究動機與研究目的 8
1.2 QUANTOPIAN交易平台 9
1.3 研究架構 10
第二章 文獻探討 11
2.1 交易策略 11
2.2 權重策略 15
2.3 績效指標 17
第三章 研究方法 24
3.1 交易策略篩選 24
3.2 績效指標加權法 29
3.3 投資組合權重設計 32
第四章 數值分析 34
4.1 建構組合式交易策略 34
4.2 投資組合之績效與風險衡量 37
4.3 投資組合績效分析 41
第五章 結論與建議 48
5.1 結論 48
5.2 研究限制與研究建議 49
參考文獻 50
zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0104358016en_US
dc.subject (關鍵詞) 演算法交易zh_TW
dc.subject (關鍵詞) 金融科技zh_TW
dc.subject (關鍵詞) 因子投資zh_TW
dc.subject (關鍵詞) 績效指標zh_TW
dc.subject (關鍵詞) 權重策略zh_TW
dc.subject (關鍵詞) 投資組合zh_TW
dc.subject (關鍵詞) Algorithmic tradingen_US
dc.subject (關鍵詞) FinTechen_US
dc.subject (關鍵詞) Factor investingen_US
dc.subject (關鍵詞) Performance indexen_US
dc.subject (關鍵詞) Weighten_US
dc.subject (關鍵詞) Portfolioen_US
dc.title (題名) 利用Quantopian交易平台設計演算法交易策略zh_TW
dc.title (題名) Design algorithmic trading strategy by Quantopian trading platformen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Acerbi, C., & Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487-1503.
Ang, A. (2014). Asset management: A systematic approach to factor investing: Oxford University Press.
Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X. (2006). The cross‐section of volatility and expected returns. The journal of finance, 61(1), 259-299.
Asness, C. S., Frazzini, A., & Pedersen, L. H. (2014). Quality minus junk.
Asness, C. S., Moskowitz, T. J., & Pedersen, L. H. (2013). Value and momentum everywhere. The journal of finance, 68(3), 929-985.
Aumann, R. J., & Serrano, R. (2008). An economic index of riskiness. Journal of Political Economy, 116(5), 810-836.
Bali, T. G., Cakici, N., & Chabi-Yo, F. (2011). A generalized measure of riskiness. Management science, 57(8), 1406-1423.
Ballings, M., Van den Poel, D., Hespeels, N., & Gryp, R. (2015). Evaluating multiple classifiers for stock price direction prediction. Expert Systems with Applications, 42(20), 7046-7056.
Banz, R. W. (1981). The relationship between return and market value of common stocks. Journal of financial economics, 9(1), 3-18.
Bernard, V. L., & Thomas, J. K. (1989). Post-earnings-announcement drift: delayed price response or risk premium? Journal of Accounting research, 1-36.
Best, M. J., & Grauer, R. R. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. The review of financial studies, 4(2), 315-342.
Biglova, A., Ortobelli, S., Rachev, S. T., & Stoyanov, S. (2004). Different approaches to risk estimation in portfolio theory. The journal of portfolio management, 31(1), 103-112.
Black, F. (1993). Beta and return. The journal of portfolio management, 20(1), 8-18.
Breiman, L. (2001). Random forests. Machine learning, 45(1), 5-32.
Breiman, L., Friedman, J., Stone, C. J., & Olshen, R. A. (1984). Classification and regression trees: CRC press.
Carhart, M. M. (1997). On persistence in mutual fund performance. The journal of finance, 52(1), 57-82.
Chan, E. (2009). Quantitative trading: how to build your own algorithmic trading business (Vol. 430): John Wiley & Sons.
Chen, Y.-T., Huang, R. J., Shih, P.-T., & Tzeng, L. Y. (2015). Capital Asset Pricing Model Based on a Generalized Economic Index of Riskiness.
Choueifaty, Y., & Coignard, Y. (2008). Toward maximum diversification. The journal of portfolio management, 35(1), 40-51.
Clenow, A. (2016). Stocks on the Move: So schlagen Sie den Markt mit den Momentum-Strategien der Hedgefonds: Börsenbuchverlag.
Conrad, J., & Kaul, G. (1998). An anatomy of trading strategies. The review of financial studies, 11(3), 489-519.
Da, Z., Engelberg, J., & Gao, P. (2011). In search of attention. The journal of finance, 66(5), 1461-1499.
Daniel, K., Grinblatt, M., Titman, S., & Wermers, R. (1997). Measuring mutual fund performance with characteristic‐based benchmarks. The journal of finance, 52(3), 1035-1058.
De Bondt, W. F., & Thaler, R. H. (1987). Further evidence on investor overreaction and stock market seasonality. Journal of finance, 557-581.
DeMiguel, V., Garlappi, L., Nogales, F. J., & Uppal, R. (2009). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management science, 55(5), 798-812.
Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of financial economics, 33(1), 3-56.
Fama, E. F., & French, K. R. (2017). International tests of a five-factor asset pricing model. Journal of financial economics, 123(3), 441-463.
Foster, D. P., & Hart, S. (2009). An operational measure of riskiness. Journal of Political Economy, 117(5), 785-814.
Goodwin, T. H. (1998). The information ratio. Financial Analysts Journal, 54(4), 34-43.
Gultekin, M. N., & Gultekin, N. B. (1983). Stock market seasonality: International evidence. Journal of financial economics, 12(4), 469-481.
James, G., Witten, D., & Hastie, T. (2014). An Introduction to Statistical Learning: With Applications in R.
Jegadeesh, N. (1990). Evidence of predictable behavior of security returns. The journal of finance, 45(3), 881-898.
Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency. The journal of finance, 48(1), 65-91.
Jorion, P. (1997). Value at risk: the new benchmark for controlling market risk: Irwin Professional Pub.
Keating, C., & Shadwick, W. F. (2002). A universal performance measure. Journal of performance measurement, 6(3), 59-84.
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management science, 37(5), 519-531.
Levy, R. A. (1967). Relative strength as a criterion for investment selection. The journal of finance, 22(4), 595-610.
Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. The review of economics and statistics, 13-37.
Maillard, S., Roncalli, T., & Teïletche, J. (2010). The properties of equally weighted risk contribution portfolios. The journal of portfolio management, 36(4), 60-70.
Markowitz, H. (1952). Portfolio selection. The journal of finance, 7(1), 77-91.
Moat, H. S., Curme, C., Avakian, A., Kenett, D. Y., Stanley, H. E., & Preis, T. (2013). Quantifying Wikipedia usage patterns before stock market moves. Scientific reports, 3, 1801.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica: Journal of the econometric society, 768-783.
Mundial, F. E. (2015). The Future of Financial Services: How disruptive innovations are reshaping the way financial services are structured, provisioned and consumed. Ginebra: FEM. Consultado en: http://www3. weforum. org/docs/WEF_The_future__of_financial_services. pdf.
Preis, T., Moat, H. S., & Stanley, H. E. (2013). Quantifying trading behavior in financial markets using Google Trends. Scientific reports, 3, srep01684.
Reinganum, M. R. (1981). Misspecification of capital asset pricing: Empirical anomalies based on earnings` yields and market values. Journal of financial economics, 9(1), 19-46.
Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.
Schnytzer, A., & Westreich, S. (2013). A global index of riskiness. Economics Letters, 118(3), 493-496.
Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance, 19(3), 425-442.
Sharpe, W. F. (1994). The sharpe ratio. The journal of portfolio management, 21(1), 49-58.
Sortino, F. A., & Price, L. N. (1994). Performance measurement in a downside risk framework. the Journal of Investing, 3(3), 59-64.
Thenmozhi, M., & Kumar, M. (2009). Dynamic interaction among mutual fund flows, stock market return and volatility. NSE Research Papers.
Treynor, J. L. (1961). Toward a theory of market value of risky assets. Unpublished manuscript, 6.
Young, T. W. (1991). Calmar ratio: A smoother tool. Futures, 20(1), 40.
Yu, J.-R., Lee, W.-Y., & Chiou, W.-J. P. (2014). Diversified portfolios with different entropy measures. Applied Mathematics and Computation, 241, 47-63.
Zivot, E. (2013). Portfolio Theory with Matrix Algebra. viewed on May, 5, 2012.
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