Jegadeesh and Titman (1993) Journal of Finance Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency Jegadeesh and Titman (1993) Journal of Finance

I. Introduction Reversal and contrarian strategies: De Bondt and Thaler (1985,1987). The early literature on market efficiency focused on relative strength strategies that buy past winners and sell past losers. Besides, a number of practitioners use relative strength as one of their stock selection criteria. Reconcile: One possibility is that the abnormal returns realized by these practitioners are either spurious or are unrelated to their tendencies to buy past winners. A second possibility is that the discrepancy is due to the

I. Introduction difference between the time horizons used in the trading rules examined in the recent academic papers and those used in practice. 本文研究 1. An analysis of relative strength trading strategies over 3- to 12-month horizons. The analysis documents significant profits in the 1965 to 1989 sample period. 2. Provide a decomposition of the momentum profits into different sources and develop tests that allow us to evaluate their relative importance. The evidence is consistent with delayed price reaction to firm-specific information.

I. Introduction 3. Further tests suggest that part of the predictable price changes that occur during these 3- to 12-month holding periods may not be permanent.

II. Trading Strategies and Returns 形成期(formation periods)與持有期(holding returns):在樣本期間的每一個月,根據每檔股票過去1~4季的累積報酬將股票分組形成投組,並考慮形成投組後持有1~4季的持有期報酬,因此共有16種投資策略. 持有期報酬率的計算:採用overlapping holding periods 在計算t月的holding returns時,為當月所形成之投組與之前K-1個月(K為持有期)所形成的投組, 共K個投組之報酬率的平均(個別投組的報酬率是以等權方式計算).

II. Trading Strategies and Returns The holding returns of relative strength portfolios(買進贏家組合賣出輸家組合) 表1: The returns of all the zero-cost portfolios are positive. All these returns are statistically significant except for the (3,3) strategy that does not skip a week. 換言之,相對強勢策略的確能賺取所謂的動能利潤,接著作者將以(6,6)為例,檢驗該動能利潤的來源:是系統風險、股價對共同因子的延遲反應、或股價對個別公司因子的延遲反應.

III. Sources of Relative Strength Profits 本節提出兩個簡單的報酬產生過程來分解之前所發現之相對強勢組合的異常報酬,並進而解釋其來源. 1. Case1:allows for factor-mimicking portfolio returns to be serially correlated but requires individual stocks to react instantaneously to factor realization (假設共同因子的報酬率有自我相關,但個別股票對共同因子的反應無時間落差). 2. Case 2:允許個別股票對共同因子的反應有時間落差 (lead-lag relationship).

III. Sources of Relative Strength Profits A simple one-factor model(p.71) 1. 考慮一個p.71 eq1的單因子模型: 為unconditional expected return. 為該共同因子的非預期報酬率(unexpected或surprise的部分,預期到的部分已經反應在 ), 為公司特有因子對報酬率的影響. 2.相對強勢策略有異常報酬顯示 , 我們可以將該式想成為: 以個股前期報酬扣除前期等權平均報酬之差作為本期個股權重的zero-cost trading strategy ( ).

III. Sources of Relative Strength Profits 3.以eq2方式所做出的相對強勢投組的超額報酬為 4.5% (semiannually), 與本文前節以等權權重所計算出之超額報酬的相關程度為0.95.但以WRSS的方式提供我們分解異常報酬的來源: eq3 (p.72). 第一項:cross-sectional variance of expected returns (期望報酬率的橫斷面分散程度):真實報酬率中含有期望報酬率的部份,故本期報酬率相對高的個股預期未來的報酬率也會相對來的高(即反映高期望報酬的部份).

III. Sources of Relative Strength Profits 第二項: the potential to time the factor, cross-sectional variances of factor sensitivities.指共同因子本身有自我相關,而相對強勢策略是否能利用此相關性來創造異常報酬,則決定於個股對該因子敏感度的差異(比如若因子有正相關,則在本期該因子倘使出現高報酬,則高敏感度的股票相對容易成為相對強勢股票,而投資這樣的股票在下一期由於因子有正的自我相關,故又產生相對高的報酬). 第三項:average serial covariance of the idiosyncratic components of security returns(指非系統因子的自我相關) 若異常報酬來自前兩項:並非市場無效率;若異常報酬來自第三項: market inefficiency.

III. Sources of Relative Strength Profits 前述三項報酬來源的檢驗 1.第一項: 報酬的來源是因為選了相對風險高的個股? 由表2發現loser比winner的beta還高(兩者皆高於其他組群),市值相對也小(兩者皆小於其他組群). 2.第二項: 由(4)式,為檢驗共同因子是否有正的自我相關,須計算serial covariance of the equally weighted index returns,結果發現該值為負(-0.0028),顯示第二項不太可能是報酬的來源. 3.第三項:根據market model 計算個股的residual returns,然後計算該residual return的serial covariance並平均, 發現其值為正,顯示報酬的來源可能是:stocks under-reacting to firm-specific information(但也無法排除可能是某些股票對共同因子的反應有時間落差).

III. Sources of Relative Strength Profits 模型二的檢測:Lead-lag effect 1.第二個模型假設共同因子並無自我相關,但個股對共同因子的反應卻可能出現時間落差:eq5(p.74). 代表反應時間落差的部份, 係數為正代表underreaction,為負代表overreaction,由eq5可推導出eq6,7. 若時間落差是決定報酬的重要因素,則每期動能報酬的高低必定與共同因子前期所實現的報酬率平方成正相關(positively related to the squared factor portfolio return in the previous period, see eq8).反之

III. Sources of Relative Strength Profits 若時間落差非影響因素,回到model 1,則根據eq8下面的式子,動能利潤應與共同因子前期所實現的報酬率平方呈負相關,究竟何者得到支持呢? 作者由迴歸分析來剖析: 1. 被解釋變數為relative strength portfolio(根據過去6個月的累積報酬所形成)在t期的報酬,而解釋變數為加權市場投組過去6個月(t-6 through t-1)的demeaned return取平方. 2. 結果 估計值為負,顯示支持model 1.故相對強勢策略的利潤來源並非lead-lag effect所造成,而是stocks under-reacting to firm-specific information.

IV. Size- and Beta-Based Sub-samples 根據規模與ex ante estimates of beta分成子樣本後, 在子樣本下討論relative strength profits. 以觀察相對強勢策略的利潤來源是否僅限於某種類型的股票? 由於許多文獻論及規模以及beta值與報酬、風險有密切關係,因此我們可以再度測試動能利潤的來源:若利潤來自期望報酬(系統風險),則在同一子樣本下的個股,因其風險特性較為接近,故在此子樣本下所計算的relative strength利潤應該會變小;反之若利潤來自serial covariance in idiosyncratic returns,則子樣本下的利潤不見得小於總樣本下的利潤,而且在

IV. Size- and Beta-Based Subsamples 小規模公司樣本所產生的利潤傾向大於大規模公司. 1. 表3: (6,6)策略下每一個子樣本的平均報酬,panel A顯示,各個子樣本下的動能利潤與總樣本下的動能利潤差異不大,不過,此動能利潤與規模,beta有相關性: largest firms generate lower abnormal returns and the returns in the sub-samples segmented by beta are monotonically increasing in beta.

IV. Size- and Beta-Based Subsamples 2. These findings indicate that the relative strength profits are not primarily due to cross-sectional differences in the systematic risk of the stocks in the sample. 3. 表3 panel B: 透過eq9調整beta風險的部份來計算risk-adjusted returns(即eq9估計出之截距項), 此risk-adjusted returns在各子樣本群組下結果與panel A的raw returns一致.