本研究首先以1991年至2000年為取樣期間，運用選股策略，每個投組選取五十支股票，檢視投組是否可以顯著超越大盤報酬率。選股變數分為價值型、成長型、規模、組合型等四大類變數。本研究第二階段再以1998年至2000年為取樣期間，檢驗加入指數期貨的操作策略，是否存在超越前述投組的超額報酬。三種衍生性策略分別為現金增值策略（一成資金做為期貨保證金，另外九成投入固定收益商品）、期貨現貨互轉套利策略（以一定程度價差比為標準，互相轉換期貨與現貨）、避險策略（以九成資金持有現貨部位，另外一成資金做為避險用途）。相關實證結果如下： 1.選股策略中，單一變數以價值型變數的超額報酬率較佳，其中又以低本益比（L-P/E）、低本銷比（L-P/S）二個投組績效最佳，平均年超額報酬率分別達到9.5%與8.2%。 2.成長型變數的投組只有高淨值報酬率（H-ROE）、高營收成長率（H-Sales G%）二個投組，在一個風險調整後的指標具有顯著的超額報酬，其餘變數皆不顯著。 3.台灣股市大型股的報酬率比小型股突出，但未達統計上顯著的程度。 4.組合型策略透過價值型與成長型變數雙重篩選，除兼顧成長性外，更透過價值型變數過濾掉價格過高者，出現較單一變數更佳的績效。其中又以高營收成長率與低本銷比兩個變數的組合最佳，平均年超額報酬率可達到11.5%。只要彼此互相篩選，價值型篩選掉低成長性的股票，成長型篩選掉價格過高的股票，價值型與成長型的投資哲學是可以兼顧的。組合型策略是選股策略中最佳的操作選擇。 5.現金增值策略的超額報酬不顯著，但可以取代選股策略，成為單純指數型基金的操作策略。 6.期貨現貨互轉套利策略，加計交易成本後，相較於投組，仍具有顯著的超額報酬。較佳轉換的組合，其價差百分比分別是（-1,1）、（0,0）、（-1,-1）。例如第一個組合是期貨相對於現貨出現-1%的價差比時，將現貨轉為期貨，待出現1%價差比時再轉回現貨，其餘類推。此策略是為了多賺取期貨低估的報酬率。這三個組合相較於投組的平均年超額報酬率高達39.8%、33.2%、26.0%，各項檢定均呈現顯著。不過限於指數期貨交易歷史不長，取樣期間只有三十六個月為其缺憾。 7.逆差避險策略無法超越投組績效，須另外尋找價格發現的指標。 以上的研究結果，說明台灣股市的確可以利用組合型選股的策略，達到超越大盤的報酬，本研究更進一步加上搭配期貨現貨互轉套利策略，使報酬率比單純的組合型選股策略更佳。 台灣股市的效率性的確不若歐美先進國家。本研究所採用的策略，可以顯著擊敗大盤。市場特性的不同，積極型的操作手法在台灣也許可以獲致更佳的報酬率。

指數基金係提供投資者追隨市場指數成長的投資工具，且投資者僅需考量市場風險即可，其建構方式有完全複製法、分層法、抽樣法、及最佳化法。本論文使用目標規劃模型建構指數基金，此法可歸類為最佳化法。由於模型中每種股票的投資數量設為整數變數，加上控制股票種類數量的0-1變數，因此所建構的目標規劃模型為混合型整數線性規劃問題。此問題在大尺度模型時往往無法求得其最佳解，我們研究此模型的結構提出一組縮小解集合空間的合理不等式，應用切面法加入必需的不等式後再根據本模型的對偶性質發展出有效率的啟發式演算法，最後將此模型及演算法應用在模擬台灣發行量加權股價指數。 / Index fund is an investment tool which tracks a stock-market index and thus is associated with market risk only. Its attraction to investors is low investment risk and low administrative expenses. Four different approaches to index fund construction can be classified as full replication, stratification, sampling, and optimizing respectively. In this thesis, we construct an index fund via the goal programming model with the optimizing approach. The model can be formulated as a mixed integer linear programming. The exact optimal solution can not be obtained when the model becomes large. We then develop a valid inequality and use this valid inequality to develop a cutting plane method. We also propose an efficient heuristic by adopting the dual property. Finally, an empirical study applying to the Taiwan Stock Exchange Capitalization Weighted Stock Index is given to show the efficiency of the algorithm.

通常已建立的指數基金，經過一段時間後其追蹤指數的效能已經無法滿足初期建購時的要求，此時管理者便面臨指數基金投資組合的調整問題。本論文融合建構指數基金的方法及最小化交易成本的概念，提出一個新的混合整數線性規劃模型以調整指數基金投資組合。模型亦考慮實務中交易成本、最小交易單位及批量、固定交易費用比率、以及資產總類數等限制。因此，模型包含整數變數及二元變數，求解也較為困難許多。本論文以啟發式演算法增進求解的效率，並以台灣50指數的相關資料做為實證研究的對象。 / The efficiency of index-tracking in index fund, which has been built, has usually been incapable to meet the needs after a period of time. In this moment, the managers have to face with the problems of the adjusting for index fund portfolio. In this paper, we integrate the methods of constructing index fund and the concepts of minimum transaction cost with it, and propose a new mixed integer linear program model to adjust the index fund portfolio. Moreover, the model also considers some limitations, such as the transaction costs, minimum transaction units and lots, fixed proportional transaction rates, and cardinality constraint in practical operating. For this reason, a set of integer variables and binary variables are introduced. However, they increase the computational complexity in model solution. Due to the difficulty of the MILP problem, a heuristic algorithm has been developed for the solution. The computational results are presented by applying the model to the Taiwan 50 index.

自從槓桿型指數基金於各股票市場發行後，各國主管機關皆紛紛發出聲明表示，該商品並不適合長期持有；因此，該類型投資商品的公開說明書皆會註明不宜長期投資。然而，本研究實證結果發現，持有期間長短並非主要風險來源。雖然，如大家所知，槓桿型指數基金多是以「日」為單位追蹤指數，而導致複利效果 (Compounding Effect) 使基金長期報酬與槓桿倍數不同。 根據算出上述的報酬差異(Return Difference)可以發現不論是正向2倍或是反向1倍皆與台灣50報酬率的標準差有統計上顯著關係。反向1倍皆與台灣50報酬率的標準差有顯著負相關；反之，正向2倍與台灣50報酬率的標準差有顯著正向相關。然而，從已實現乘數(Realized Multiple)的分佈中可發現，不合理值並不隨投資期間越長而越多。意即儘管投資期間越長，並不一定會導致複利效果越大，而與目標槓桿倍數脫節。再者，隨著投資期間越長，波動度(volatility)的對於報酬差異的解釋力越強；因此，若想長期投資槓桿型指數資金，預測標的波動度的能力更顯為重要。 / When we browse the reports about the inverse and leveraged ETF, most of them emphasize that the LETF is not appropriate to long-term investors. However, in this research, we attempt to demonstrate the main factor of the performance of the leveraged and inverse ETF is not how long the LETF we hold, but the volatility of the underlying index or ETF. Observing the empirical test, no matter how long the investment horizon is, the coefficient of the variance of the Taiwan 50 is statistically significant both in the Taiwan 50 Bear -1X and the Taiwan 50 Bull. However, its effect on the Bear -1X is opposite to that on the Bull 2X. First, the relationship between the volatility and the return difference of Taiwan 50 and the Bear -1X is negative. In contrast, the relationship between the volatility and the return difference of Taiwan 50 and the Bull 2X is positive. However, in accordance with the distribution of the realized multiples, the frequency of either the Bear -1X or Bull 2X was not more and more when the holding period is longer. As a result, our research show the variance has a significant effect on both, no matter how long investors hold. If the volatility is moderate, the return difference may be close to zero; then the LETFs would be a convenient way to investors who desire to magnify the market return. Moreover, due to the increasing explanatory power of the volatility, we may make a further inference that whether the compounding effect is positive or negative depends on the volatility, especially within longer holding period. Therefore, without the great ability to forecast the variance, the LETFs are not recommended to the long-term investors.

投資者面對龐大的股票市場，希望選取少量的股票使如指數基金般達到追蹤市場的效果，傳統的作法是使用指數追蹤的技術，建立一組投資組合使得報酬率與市場報酬率的績效相同。本論文除了最小化指數追蹤的下方追蹤誤差，還加入beta值的限制式，利用不同的beta值建立一組與市場成長趨勢相當或可能超越市場績效的投資組合。論文中使用提出之模型針對不同範圍的beta值進行研究，分析比較標的指數與建立的投資組合之績效表現。最後以台灣股票市場作為實證研究對象，實證結果顯示本論文模型所建立之投資組合在三個月內與標的指數表現相當，並在三個月後超越標的指數。 關鍵字：beta值、指數追蹤、下方追蹤風險、指數基金

建立指數基金時，通常是利用追蹤指數的技巧，選取少量的股票建構指數基金使得報酬率與標的指數(benchmark index)報酬率同步的投資組合。如果能建立包含少量股票的投資組合，就可達到指數追蹤的效果，那麼也能利用少量的股票建立績效可以超越指數基金的投資組合。本論文利用建構指數基金的方法以及大中取小的概念，挑選出一個績效可以超越標的指數的投資組合。本論文提出的模型亦考慮實務上交易所需的各項成本、整數交易單位與資產總類數等限制。因此，模型包含整數變數與二元變數。最後以台灣加權股價指數的相關資料做為實證研究的對象，實證結果顯示本論文提出的模型所建立的投資組合超越標的指數的績效平均年化報酬率25%。 / Setting up an index fund usually uses techniques of index-tracking that choosing few stocks forming a portfolio to obtain the same return rate as the benchmark index. Similarly we can use the same concept to set up a portfolio such that the performance is better than index’s. In this thesis we use index-tracking methods and minimax rule to obtain a portfolio which outperforms the benchmark index. In the proposed mathematical model we will consider the transaction costs, integer trading unit volume, and the total number of assets in the portfolio. Therefore the resulting model is a mixed integer nonlinear programming including integer variables and binary variables. Finally, the empirical study will be performed by using the data from the Taiwan stock market to verify the performance of our model. The empirical study shows that the portfolios created by our models outperform the benchmark index up to 25% in average.

從投資組合理論與效率市場假說來看，影響投資績效之主要因素為資產配置而非選股或擇時能力，相關之實証亦發現主動式投資策略並無法獲得持續優於大盤的報酬，因此被動式投資策略將是一較佳的選擇。自先鋒集團在1976年率先推出指數基金之後，此類指數化投資商品即呈現指數化成長。而我國亦在2003年6月30日正式推出第一檔指數股票型基金「臺灣50指數股票型基金，簡稱TTT」。 本論文以投資人的角度來分析TTT之報酬率及成交量，從追蹤誤差及溢折價來觀察基金之報酬率，研究造成追蹤誤差之主要影響因子為何與實物創造或贖回機制是否能有效發揮功用，使TTT之市價可貼進基金淨值，以免如同封閉型基金大都折價交易，而影響投資人之獲利率。此外，更進一步比較TTT與其他指數化商品之差異性以供投資人選擇投資標的時參考。 本研究發現影響追蹤誤差之因素主要為指數成份股之增刪、公眾流通係數之調整與現金股利的發放。雖然在短期下上述因素會影響基金報酬，但長期而言基金報酬與指數相當，甚或更為略高，故在衡量長期績效時臺灣50指數可做為一良好的報酬指標。而從溢折價來分析時，由於市場交易機制，使得TTT自上市以來之流動性不足，市場效率性不高，溢折價幅度較大。此點與成交量之分析結果一致，研究發現TTT之成交量並未因出現套利機會而顯著增加，投資人買賣TTT主要在於避險或投機需求。 雖然現階段我國尚未出現指數基金，但若未來有出現類似之商品時，在目前我國停徵證券交易所得稅的環境下，投資金額的大小將非決定選擇指數基金或TTT之關鍵因素，投資期間與「質」的因素才是主要關鍵。當投資期間愈長，TTT之高交易成本的影響程度將降低，其可在盤中隨時買賣及低追蹤誤差等「質」方面的優勢將提高，TTT將相對較具吸引力。而臺灣50指數期貨由於交易成本及流動性風險較高，因而投資人較不偏好操作臺灣50指數期貨。

指數基金的投資策略，已經被愈來愈多的投資者和投資機構所接受。在實務上，指數基金的建構方法大多都採取簡化的方法或是最佳化的方法，簡化的方法可以快速求得解答，但答案未必是最佳，而一般的最佳化方法又過於耗時；為了在效率和最佳化之間求得平衡，這篇論文中提出了目標規劃的模型以及一套有效率的演算法來計算實際的問題；本文還提出了一個新的測度方法，用來衡量指數追蹤的誤差，衡量的方法主要是依據指數的數值和所建構的投資組合其價值相差的絕對值。本文的實證分析採用了摩根台灣加權指數來測試所建構的模型和演算法，結果顯示所建構的投資組合能準確的追蹤指數，誤差不超過0.8%。 / Creating index-tracking stock baskets has been accepted by more and more investors or institutes as one part of a total investment strategy. In practice, the selection methods widely adopted are some simplified methods (e.g. stratification) combined with some criteria, and some optimization models to minimize the traditional tracking error. Simplified method facilitates for obtaining a feasible answer, optimal in no sense, while the optimization model usually requires larger computational efforts. For bridging the gap between having efficiency and seeking optimality, we propose a goal programming model and develop an efficient solution algorithm. We also suggest a new measure of tracking error basing on the absolute difference between the value of the benchmark and the index computed from the portfolio obtained from our model. Empirical analyses employ the Morgan Stanley Capital International (MSCI) Taiwan Index to assess the tracking efficacy of the model. Computational results show that the constructed portfolio can track the index with error less than 0.8%.

本文係針對臺灣50指數期貨與基金於2003年6月30日上市之後，臺灣期貨及現貨市場報酬率間領先落後關係與波動性的變化來進行探討。研究分為兩部份，第一部份是觀察臺灣50指數期貨與現貨之間的關聯性，並探討臺灣加權股價指數、金融保險類股股價指數及電子類股股價指數期貨與現貨市場間的變化；第二部份是採用可模擬現貨走勢的臺灣50指數基金、國泰金及臺積電的股價來做為現貨的替代變數，觀察其與期貨之間的關連性是否與第一部份的結果類似，若是實證結果極為相同，則相關機構與一般投資人將可運用各期貨與其標的指數中市值最大的股票來進行套利操作。此外，本文在進行模型估計時，首度採用一階段估計法，來聯合估計雙變量GARCH模型中的條件平均數方程式與條件變異數方程式，以避免過去相關文獻將兩條方程式個別估計時所造成的估計誤差。 實證結果所獲得的重要結論如下：首先，臺灣期貨市場的發展仍未趨成熟，並不具有價格發現的功能，在考慮風險溢酬方面，僅有臺灣50指數期貨與現貨的投資人會在報酬率之外，額外要求用以補償的風險溢酬，再者，臺灣50指數期貨與基金的上市，並沒有對臺灣現有的期貨與現貨市場造成顯著的影響，然而，替代變數並不能完全取代現貨指數，但相較之下，國泰金在臺灣50指數期貨與基金上市之後的那段期間模擬成效最好。 / This paper investigates the change of lead-lag relationship in returns and volatilities in Taiwan futures and spot markets after the introduction of Taiwan 50 Futures and Taiwan Top50 Tracker Fund (TTT) on June 30, 2003. The study divides into two parts. The first part examines the relationship between Taiwan 50 Futures and spot markets, and also discusses the change of Taiwan Stock Exchange Capitalization Weighted Stock Index, Taiwan Stock Exchange Banking and Insurance Sector Index, and Taiwan Stock Exchange Electronic Sector Index in futures and spot markets. Another part uses the stock price of TTT, Cathay Financial Holding Company and Taiwan Semiconductor Manufacturing Company as the substitutive variables of spot index and goes a step further to examine the relationships between them and futures individually. Additionally, this research used One-Pass Method for first time to estimate jointly the conditional mean equation and conditional variance equation of Bivariate GARCH Model to avoid estimating error in previous relative studies with Two-Pass Method. The major empirical results are as follows: first, the development of Taiwan futures market is incomplete. The futures market does not play the price discovery role to the spot market. Second, under the consideration of risk premium, only investors in Taiwan 50 Futures and spot markets would ask for compensated risk premium excepting returns. Third, the opening of Taiwan 50 Futures and TTT does not influence significantly Taiwan futures and spot markets. Last but not least, these substitutive variables can not replace spot index perfectly. However, comparing with others, the stock price of Cathay Financial Holding Company is the very model of Taiwan Stock Exchange Banking and Insurance Sector Index after the introduction of Taiwan 50 Futures and TTT.