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                # 9.1 指數跟蹤 · [策略] 指數跟蹤低成本建倉策略 > 來源:https://uqer.io/community/share/56632448f9f06c6c8a91b337 ## 指數跟蹤 指數追蹤通常是指利用某個股票組合復制某一現實指數或者虛擬指數的市場表現,來獲取與指數相近的收益,試圖最小化跟蹤誤差。為什么要跟蹤指數呢?那要從指數廣泛的用途說起:1)很多大型的公募基金都設有專門的指數投資部,通過跟蹤特定的指數標的發行ETF、分級基金等產品,以供投資者使用;2)目前在A股市場上做期限套利者通常需要建倉SH50、HS300、ZZ500等指數的現貨;3)有些投資者熱衷于行業投資,因此其投資標的不是個股而是行業指數;4)近年流行的FOF,是以基金產品作為投資標的,構建基金產品的組合,基金產品也可以認為是代表基金經理投資特定的指數。如果直接購買指數產品有限制或者需要較低成本的建倉指數,那就需要一些策略啦! 通常指數跟蹤分為兩類:完全復制法和部分復制法。顧名思義,完全復制法即為盡可能完全按照指數的成分股和配比權重來調整組合,使之盡可能的跟蹤所選的標的。這種方法對于一般的投資者來講通常成本和操作難度比較大,這種方法通常僅限于大型公募基金指數部門所使用。部分復制法即為選擇某個不同于指數成分股和權重的組合(通常是指數的子集)來盡可能的去跟蹤指數。這就需要采用一些優化策略去做選股和配權重。本文參考了文獻“Carrier Portfolios.--Steven Kusiak”中的思想,意在研究一種部分復制指數的方法。 ## Carrier Portfolios 1)模型: 指數本質上是由其成分股按照特定的權重線性組合而成,其成分股張成一個線性空間,每個成分股的權重即相當于指數在每個維度上的長度。我們希望從這個線性空間中找到一個子空間,并根據子空間的基去配以合適的權重,使得配比后的結果能夠最大程度上的模擬原指數,思想就是主成分分析(PCA)。將其建模成一個優化問題,假設我們構建的資產組合為`P = {w1,w2,...,wN}`,`wi`表示給于第i各成分股的權重。優化的目標便是使得P中權重的1范數的和`∑|wi|,i = 1,2,3,...,N `最小,其含義就是令投資組合的建倉成本達到盡可能小。如果我們只允許做多,那么問題便可以變成更為簡單的使`∑wi`最小。然而如何保證讓構建的P盡可能的跟蹤指數走勢呢?我們可以做這樣一個簡單合理的假設:如果P在過去一段時間窗口`T`日中能夠擬合指數的走勢(體現在收益率的一致性),那么`P`就能下一次調倉前跟蹤指數的走勢。如果令`r(i,t)`表示第`i`個成分股在t日的收益率,`R(t)`表示待跟蹤的標的指數在`t`日的收益率,那么:`∑r(i,t)*wi = R(t)`,`t = 1,2,3,...,T`.那么我們要建模的優化問題便是: ``` min ∑wi, i = 1,2,3,...,N s.t. ∑r(i,t)*wi = R(t), t = 1,2,3,...,T wi >= 0 ``` 2)分析: + ①若 `N > T`: 約束條件為欠定方程組,解空間會有無數的解。我們需要通過迭代運算從解空間中尋找滿足優化目標的最優解,通常在邊界取到; + ②若 `N = T`: 有唯一解,不具備優化的空間; + ③若 `N < T`: 約束條件為超定方程組,解空間無解(假設曲線每天的走勢是不相關的) 3)求解: 通過分析我們知道應該在`N > T`的條件下去做優化,具體的步驟為: + ①:不等式約束`wi >= 0`可通過添加log barrier懲罰項將其約束到優化目標中: `min∑wi-u*log(wi)` + ②:可利用原對偶內點法去求解目標函數,算法原理可參考文獻“Primal-Dual Interior Point algorithms for Linear Programming--George Tzallas-Regas” + ③:依據②中求得的權重去配指數。由于近似最優解一般在邊界上取到,因此必然最優`w`是一個降維后的結果,即有些成分股的權重配比小到無法操作,舍去即可 ## 策略回測 1)策略目標 以上證50為例,我們希望選取部分上證50的成分股并配以一定的優化后的權重構建投資組合 `P = {w1,w2,...,w50}`,讓`P`可以有效的跟蹤上證50的走勢,并且盡可能的選用部分成分股去構建組合。 2)仿真環境 + ①跟蹤標的:上證50 + ②股票池:上證50成分股 + ③調倉:5天(由于只是研究策略,因此假設每次調倉都用新得的股票替換掉原有的,實際做的時候并不需要這樣,只要交易當前組合和原組合的差額即可) ```py from CAL.PyCAL import * import numpy as np import copy as cp start = '2014-05-01' # 回測起始時間 end = '2014-12-01' # 回測結束時間 benchmark = 'SH50' # 策略參考標準 universe = set_universe('SH50') # 證券池,支持股票和基金 capital_base = 100000 # 起始資金 freq = 'd' # 策略類型,'d'表示日間策略使用日線回測,'m'表示日內策略使用分鐘線回測 refresh_rate = 5 # 調倉頻率,表示執行handle_data的時間間隔,若freq = 'd'時間間隔的單位為交易日,若freq = 'm'時間間隔為分鐘 def initialize(account): # 初始化虛擬賬戶狀態 account.portfolioNumList = [] pass def dict2list(dictionary): tmplist = [] for index in dictionary: tmplist.append(dictionary[index]) return tmplist def handle_data(account): # 每個交易日的買入賣出指令 histLength = 15 stockLength = len(account.universe) ####get the return rate of the universe closePrice = account.get_attribute_history('closePrice',histLength+1) uniRetList = [] for index in closePrice: uniRetList.append(((closePrice[index][1:]-closePrice[index][:-1])/closePrice[index][:-1]).tolist()) uniRetMat = np.mat(uniRetList).T ####get the return rate of the benchmark calendar = Calendar('China.SSE') startDate = calendar.advanceDate(account.current_date,'-'+str(histLength+1)+'B').toDateTime() endDate = calendar.advanceDate(account.current_date,'-1B').toDateTime() benchmark = DataAPI.MktIdxdGet(ticker = "000016", field = "closeIndex", beginDate = startDate, endDate = endDate,pandas = '1') bmClose = benchmark['closeIndex'].tolist() bmRet = [] for index in range(len(bmClose)-1): bmRet.append((bmClose[1:][index]-bmClose[:-1][index])/bmClose[:-1][index]) bmRetMat = np.mat(bmRet) ####initialization: constant ##ones: stockLength ones = np.ones(stockLength) ##unitMat: stockLength * stockLength unitMat = np.diag(ones) ##zeros: histLength zeros = np.zeros(histLength) ##zero: stockLength zero = np.zeros(stockLength) ##zeroMatMid: histLength * histLength zeroMatMid = np.diag(zeros) ##zeroMat: stockLength * stockLength zeroMat = np.diag(zero) ##zeroMatRes: histLength * stockLength zeroMatRes = np.zeros((histLength,stockLength)) ##initialization: variables ##w: stockLength # w = np.ones(stockLength)/(stockLength) w = np.ones(stockLength)/(stockLength) ##wmat: stockLength * stockLength wMat = np.diag(w) ##u: histLength u = np.ones(histLength)/histLength ##uMat: histLength * histLength uMat = np.diag(u) ##v: stockLength v = np.ones(stockLength)/(stockLength) ##vMat: stockLength * stockLength vMat = np.diag(v) ##R: histLength * stockLength R = uniRetMat ##Q: histLength * stockLength Q = R ##splMat: (stockLength + histLength + stockLength) * (stockLength + histLength + stockLength) splMatTmp1 = np.hstack([zeroMat,Q.T,unitMat]) splMatTmp2 = np.hstack([Q,zeroMatMid,zeroMatRes]) splMatTmp3 = np.hstack([vMat,zeroMatRes.T,wMat]) splMatTmp = np.vstack([splMatTmp1,splMatTmp2,splMatTmp3]) splMat = splMatTmp ##mulVec: length = stockLength + histLength + stockLength firstCol = np.subtract(ones,np.dot(Q.T,u)) firstCol = np.subtract(firstCol,v) secondCol = np.subtract(bmRetMat,np.dot(Q,w)) thirdCol = np.mat(np.subtract(np.dot(0.1,ones),np.dot(wMat,v))) mulVec = np.hstack([firstCol,secondCol,thirdCol]) ####algorithm iteration part d = 1 mu = 0.1 itera = 0 while itera < 300: # while d > 0.01: ##calculate the dirtaw, dirtau, dirtav temp = np.dot(splMat.I,mulVec.T).tolist() dirtaw = [index[0] for index in temp[:stockLength]] dirtau = [index[0] for index in temp[stockLength:stockLength+histLength]] dirtav = [index[0] for index in temp[stockLength+histLength:]] ##update the vector w, u, v and the matrix wmat, umat, vmat w = np.add(w,dirtaw) u = np.add(u,dirtau) v = np.add(v,dirtav) wMat = np.diag(w) uMat = np.diag(u) vMat = np.diag(v) ##init the matrix splmat: (stockLength + histLength + stockLength) * (stockLength + histLength + stockLength) splMatTmp1 = np.hstack([zeroMat,Q.T,unitMat]) splMatTmp2 = np.hstack([Q,zeroMatMid,zeroMatRes]) splMatTmp3 = np.hstack([vMat,zeroMatRes.T,wMat]) splMatTmp = np.vstack([splMatTmp1,splMatTmp2,splMatTmp3]) splMat = splMatTmp ##init the vector mulvec: length = stockLength + histLength + stockLength firstCol = np.subtract(ones,np.dot(Q.T,u)) firstCol = np.subtract(firstCol,v) secondCol = np.subtract(bmRetMat,np.dot(Q,w)) thirdCol = np.mat(np.subtract(np.dot(mu,ones),np.dot(wMat,v))) ##calculate the iteration condition variable d tmp1 = 0 for index in dirtaw: tmp1 += index**2 tmp2 = 0 for index in dirtau: tmp2 += index**2 d = tmp1 + tmp2 ##update the itera and mu itera += 1 mu = mu*(1-stockLength**(-0.5))**5 ##weight of the components weight = w for index in range(stockLength): if weight[index]<0: weight[index] = 0 weightSum = np.sum(weight) weightReg = [index/weightSum for index in weight] for index in range(stockLength): if weightReg[index]<10*10**(-3): weightReg[index] = 0 count = 0 for index in weightReg: if index != 0: count += 1 account.portfolioNumList.append({account.current_date:count}) ##Sell portfolio for index in account.valid_secpos: order_to(index,0) ####Buy portfolio portfolio = [] for index in range(stockLength): amount = round(100*weightReg[index])*100 if amount != 0: portfolio.append({account.universe[index]:amount}) # amount = account.cash*weightReg[index]/account.referencePrice[account.universe[index]] order(account.universe[index],amount) print 'The portfolio at '+str(account.current_date) + ' has ' + str(len(portfolio)) + ' stocks, which ' print portfolio ``` ![](https://box.kancloud.cn/2016-07-30_579cbdb371e6e.jpg) ``` The portfolio at 2014-05-27 00:00:00 has 15 stocks, which [{'601601.XSHG': 500.0}, {'600111.XSHG': 100.0}, {'600018.XSHG': 500.0}, {'600519.XSHG': 1400.0}, {'601668.XSHG': 100.0}, {'601088.XSHG': 1500.0}, {'601998.XSHG': 500.0}, {'600010.XSHG': 500.0}, {'600637.XSHG': 500.0}, {'600999.XSHG': 300.0}, {'600887.XSHG': 1100.0}, {'601169.XSHG': 1300.0}, {'601988.XSHG': 200.0}, {'601318.XSHG': 900.0}, {'601901.XSHG': 100.0}] The portfolio at 2014-06-04 00:00:00 has 11 stocks, which [{'600583.XSHG': 500.0}, {'600893.XSHG': 300.0}, {'601288.XSHG': 1700.0}, {'600585.XSHG': 700.0}, {'600256.XSHG': 500.0}, {'601800.XSHG': 800.0}, {'601989.XSHG': 100.0}, {'600887.XSHG': 800.0}, {'601398.XSHG': 3400.0}, {'600030.XSHG': 500.0}, {'600150.XSHG': 800.0}] The portfolio at 2014-06-11 00:00:00 has 15 stocks, which [{'601601.XSHG': 600.0}, {'600893.XSHG': 300.0}, {'600111.XSHG': 100.0}, {'600018.XSHG': 400.0}, {'601390.XSHG': 1500.0}, {'601288.XSHG': 800.0}, {'601668.XSHG': 700.0}, {'601818.XSHG': 500.0}, {'600690.XSHG': 1400.0}, {'600010.XSHG': 700.0}, {'600999.XSHG': 500.0}, {'601628.XSHG': 200.0}, {'600887.XSHG': 400.0}, {'601766.XSHG': 400.0}, {'600030.XSHG': 1400.0}] The portfolio at 2014-06-18 00:00:00 has 7 stocks, which [{'600036.XSHG': 100.0}, {'600018.XSHG': 300.0}, {'600256.XSHG': 500.0}, {'600406.XSHG': 7600.0}, {'600048.XSHG': 400.0}, {'600887.XSHG': 200.0}, {'600030.XSHG': 900.0}] The portfolio at 2014-06-25 00:00:00 has 7 stocks, which [{'600050.XSHG': 1200.0}, {'600089.XSHG': 900.0}, {'600256.XSHG': 900.0}, {'600104.XSHG': 1500.0}, {'601989.XSHG': 2000.0}, {'600837.XSHG': 1000.0}, {'601398.XSHG': 2200.0}] The portfolio at 2014-07-02 00:00:00 has 9 stocks, which [{'600036.XSHG': 800.0}, {'601390.XSHG': 1900.0}, {'600016.XSHG': 200.0}, {'601006.XSHG': 600.0}, {'601088.XSHG': 300.0}, {'600585.XSHG': 1300.0}, {'600690.XSHG': 1300.0}, {'601988.XSHG': 2900.0}, {'601186.XSHG': 500.0}] The portfolio at 2014-07-09 00:00:00 has 8 stocks, which [{'601288.XSHG': 2100.0}, {'601668.XSHG': 400.0}, {'601088.XSHG': 2100.0}, {'600089.XSHG': 1400.0}, {'600690.XSHG': 900.0}, {'600010.XSHG': 1400.0}, {'601988.XSHG': 1100.0}, {'601398.XSHG': 400.0}] The portfolio at 2014-07-16 00:00:00 has 6 stocks, which [{'600583.XSHG': 1400.0}, {'601006.XSHG': 1400.0}, {'600256.XSHG': 500.0}, {'601998.XSHG': 1700.0}, {'600048.XSHG': 400.0}, {'600518.XSHG': 4400.0}] The portfolio at 2014-07-23 00:00:00 has 23 stocks, which [{'600583.XSHG': 200.0}, {'601601.XSHG': 700.0}, {'600036.XSHG': 400.0}, {'600018.XSHG': 300.0}, {'600519.XSHG': 100.0}, {'601390.XSHG': 1400.0}, {'601288.XSHG': 200.0}, {'601006.XSHG': 300.0}, {'601088.XSHG': 500.0}, {'600256.XSHG': 500.0}, {'601998.XSHG': 100.0}, {'600015.XSHG': 100.0}, {'600028.XSHG': 500.0}, {'600010.XSHG': 300.0}, {'600999.XSHG': 400.0}, {'600109.XSHG': 100.0}, {'601989.XSHG': 1100.0}, {'600887.XSHG': 200.0}, {'601766.XSHG': 200.0}, {'601169.XSHG': 700.0}, {'601988.XSHG': 1100.0}, {'600030.XSHG': 200.0}, {'601901.XSHG': 300.0}] The portfolio at 2014-07-30 00:00:00 has 8 stocks, which [{'600893.XSHG': 200.0}, {'600104.XSHG': 300.0}, {'600015.XSHG': 100.0}, {'601989.XSHG': 600.0}, {'600518.XSHG': 800.0}, {'601628.XSHG': 2000.0}, {'600887.XSHG': 300.0}, {'601318.XSHG': 5400.0}] The portfolio at 2014-08-06 00:00:00 has 7 stocks, which [{'601328.XSHG': 2800.0}, {'600036.XSHG': 600.0}, {'600111.XSHG': 1200.0}, {'601088.XSHG': 700.0}, {'601800.XSHG': 1300.0}, {'600028.XSHG': 1900.0}, {'600518.XSHG': 1100.0}] The portfolio at 2014-08-13 00:00:00 has 7 stocks, which [{'601601.XSHG': 1400.0}, {'600018.XSHG': 100.0}, {'601818.XSHG': 600.0}, {'601088.XSHG': 2100.0}, {'600406.XSHG': 1600.0}, {'600999.XSHG': 2900.0}, {'600887.XSHG': 1300.0}] The portfolio at 2014-08-20 00:00:00 has 6 stocks, which [{'600000.XSHG': 1900.0}, {'600111.XSHG': 900.0}, {'600016.XSHG': 1600.0}, {'600690.XSHG': 300.0}, {'600150.XSHG': 100.0}, {'601186.XSHG': 5100.0}] The portfolio at 2014-08-27 00:00:00 has 8 stocks, which [{'601328.XSHG': 2000.0}, {'600585.XSHG': 300.0}, {'600048.XSHG': 1300.0}, {'601800.XSHG': 1200.0}, {'600690.XSHG': 700.0}, {'600028.XSHG': 1400.0}, {'601989.XSHG': 1200.0}, {'601988.XSHG': 1800.0}] The portfolio at 2014-09-03 00:00:00 has 8 stocks, which [{'600016.XSHG': 2200.0}, {'601818.XSHG': 1100.0}, {'600585.XSHG': 2500.0}, {'600089.XSHG': 2000.0}, {'600048.XSHG': 400.0}, {'600104.XSHG': 1100.0}, {'600518.XSHG': 500.0}, {'600150.XSHG': 200.0}] The portfolio at 2014-09-11 00:00:00 has 7 stocks, which [{'601857.XSHG': 4300.0}, {'601088.XSHG': 700.0}, {'600048.XSHG': 1000.0}, {'600690.XSHG': 600.0}, {'600518.XSHG': 1000.0}, {'601169.XSHG': 800.0}, {'601901.XSHG': 1400.0}] The portfolio at 2014-09-18 00:00:00 has 8 stocks, which [{'601328.XSHG': 1700.0}, {'600111.XSHG': 500.0}, {'600016.XSHG': 3100.0}, {'600015.XSHG': 300.0}, {'600887.XSHG': 400.0}, {'600030.XSHG': 1800.0}, {'601186.XSHG': 600.0}, {'601318.XSHG': 1700.0}] The portfolio at 2014-09-25 00:00:00 has 11 stocks, which [{'600050.XSHG': 600.0}, {'600583.XSHG': 700.0}, {'600111.XSHG': 3000.0}, {'600048.XSHG': 500.0}, {'600015.XSHG': 300.0}, {'600028.XSHG': 400.0}, {'600837.XSHG': 1400.0}, {'601169.XSHG': 100.0}, {'601988.XSHG': 600.0}, {'601186.XSHG': 100.0}, {'601901.XSHG': 1700.0}] The portfolio at 2014-10-09 00:00:00 has 9 stocks, which [{'600519.XSHG': 1400.0}, {'600585.XSHG': 3000.0}, {'600256.XSHG': 300.0}, {'600048.XSHG': 2500.0}, {'600104.XSHG': 900.0}, {'600015.XSHG': 600.0}, {'600999.XSHG': 300.0}, {'600109.XSHG': 600.0}, {'600887.XSHG': 100.0}] The portfolio at 2014-10-16 00:00:00 has 8 stocks, which [{'600519.XSHG': 800.0}, {'601166.XSHG': 6900.0}, {'600104.XSHG': 700.0}, {'600010.XSHG': 200.0}, {'600999.XSHG': 200.0}, {'601766.XSHG': 700.0}, {'601169.XSHG': 100.0}, {'600030.XSHG': 100.0}] The portfolio at 2014-10-23 00:00:00 has 7 stocks, which [{'601390.XSHG': 400.0}, {'601668.XSHG': 1100.0}, {'600048.XSHG': 1100.0}, {'600104.XSHG': 200.0}, {'600837.XSHG': 4000.0}, {'601398.XSHG': 2200.0}, {'601318.XSHG': 800.0}] The portfolio at 2014-10-30 00:00:00 has 6 stocks, which [{'601857.XSHG': 800.0}, {'600256.XSHG': 600.0}, {'600015.XSHG': 200.0}, {'600887.XSHG': 100.0}, {'601398.XSHG': 7600.0}, {'600030.XSHG': 500.0}] The portfolio at 2014-11-06 00:00:00 has 8 stocks, which [{'601328.XSHG': 4800.0}, {'600519.XSHG': 1000.0}, {'601818.XSHG': 800.0}, {'601088.XSHG': 900.0}, {'600585.XSHG': 600.0}, {'600406.XSHG': 900.0}, {'600104.XSHG': 500.0}, {'601901.XSHG': 400.0}] The portfolio at 2014-11-13 00:00:00 has 7 stocks, which [{'600583.XSHG': 1300.0}, {'601601.XSHG': 2400.0}, {'600893.XSHG': 400.0}, {'601818.XSHG': 1500.0}, {'600104.XSHG': 1700.0}, {'600109.XSHG': 200.0}, {'601318.XSHG': 2300.0}] The portfolio at 2014-11-20 00:00:00 has 9 stocks, which [{'601601.XSHG': 3800.0}, {'600018.XSHG': 300.0}, {'601006.XSHG': 600.0}, {'600089.XSHG': 1300.0}, {'601998.XSHG': 800.0}, {'600015.XSHG': 1300.0}, {'600010.XSHG': 300.0}, {'601989.XSHG': 300.0}, {'601988.XSHG': 1200.0}] The portfolio at 2014-11-27 00:00:00 has 14 stocks, which [{'601601.XSHG': 400.0}, {'600893.XSHG': 1700.0}, {'600111.XSHG': 600.0}, {'600519.XSHG': 300.0}, {'601288.XSHG': 800.0}, {'601088.XSHG': 100.0}, {'600585.XSHG': 1800.0}, {'600089.XSHG': 500.0}, {'600015.XSHG': 500.0}, {'600999.XSHG': 200.0}, {'600109.XSHG': 600.0}, {'600518.XSHG': 300.0}, {'601988.XSHG': 1900.0}, {'601318.XSHG': 100.0}] ``` ## 回測結果 從上面的回測結果中我們可以看到,每個時點的持倉個股數在7-15只的范圍內,然而組合依然可以較好的跟蹤標的上證50指數。跟蹤誤差沒有詳細去計算,應該在400bps(4%)以內,考慮到大大減少了建倉的難度和成本,因此這個跟蹤誤差是可以接受的。 ## 思考 + ① 考慮到我們上面僅僅回測了最簡單的model,事實上我們完全可以在上面的優化問題中加入更多的限制條件,以獲得更為進準的匹配效果。例如,如果我們限制了每只股票的持倉不得高于某個比例,則可以將條件 `wi >= 0` 改為 `xi >= wi >=0`。如果考慮可以融券做空,則優化目標變為 `min ∑|wi|`。如果考慮加入倉位限制,例如倉位控制在70%以上,則可加入 `∑wi > 70%`,等等。 + ② 該模型可以在一定程度上挖掘任意指數、股票型基金曲線的持倉信息。以基金為例:只要是股票型基金,尤其是該基金的投資標的池已知,例如基金嘉實大盤研究精選的標的池很有可能就是HS300,那么便可以用HS300去跟蹤該基金,得到的股票組合很有可能就是嘉實大盤研究精選的持倉。
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