Programmazione Amibroker Tutorial: matrice di covarianza ibrida usando le volatilità implicite delle opzioni

[Cren]

Se hai pazienza fino a lunedì, quando arrivo in ufficio allego direttamente il .xls con colonne e date allineate, così non dovrebbe esserci problemi.

Allegato, buon divertimento

Con calma se ne hai voglia, allega l'algoritmo.

Ricordo che tutti i package utilizzati sono reperibili gratuitamente presso CRAN Packages By Name.

[COLOR=Teal]# ***********************************************************************
# Back test portafoglio a varianza minima
# by Cren
# ***********************************************************************

# Carico i package[/COLOR]

require(gogarch)
require(PerformanceAnalytics)
require(Rsolnp)
require(timeSeries)

[COLOR=Teal]# Calcolo i rendimenti aritmetici degli indici[/COLOR]

dX.0 <- returns(MV_datas, method = 'discrete') ; dX <- dX.0[-1,]

[COLOR=Teal]# Per aiutarmi nel back test, costruisco una funzione per applicare in
# expanding window l'algoritmo: prende in input i dati, la funzione da
# applicare e l'ampiezza iniziale della finestra; mi restituisce l'output
# nel medesimo formato dei dati in ingresso
[/COLOR]
roll.app <- function(data, FUN, width) {
  temp <- matrix(NA, nrow = nrow(data), ncol = ncol(data))
  for(i in 1:(nrow(data) - (width - 1))) {
    temp[i + width - 1,] <- FUN(data[i:(i + width - 1),])
  }
  return(temp)
}

[COLOR=Teal]# Costruisco una funzione per ottimizzare i pesi a minima varianza: la
# funzione prende in input i dati, calcola la matrice di covarianza con
# un modello GARCH(1,1) ortogonale a residui condizionatamente normali e
# infine usa un algoritmo risolutore per estrarre il vettore dei
# parametri che minimizzano la varianza del portafoglio[/COLOR]

cov.pars <- function(data) {
  V <- gogarch(data = data, formula = ~ garch(1,1))@V
  
  fun <- function(w) {
    t(w) %*% V %*% w
  }
  
  eqfun <- function(w) {
    sum(w)
  }
  
  pars <- solnp(pars = rep(1/ncol(V),ncol(V)), fun = fun, eqfun = eqfun,
                eqB = 1, LB = rep(0,ncol(V)), UB = rep(1,ncol(V)))$pars
  return(pars)
}

[COLOR=Teal]# Applico in expanding window la funzione per estrarre i pesi ottimali
# del portafoglio a varianza minima, quindi li trasformo in serie storica[/COLOR]

w.t <- roll.app(data = dX, FUN = cov.pars, width = 200)
w.t <- timeSeries(w.t, charvec = rownames(as.timeSeries(dX)))
colnames(w.t) <- colnames(MV_datas)

[COLOR=Teal]# Visualizzo l'evoluzione dei pesi ottimali nel tempo[/COLOR]

plot(w.t, format = '%y', main = 'Optimal Weights')

[COLOR=Teal]# Calcolo i rendimenti della strategia; notare la funzione 'lag':
# applichiamo al periodo successivo i pesi calcolati nel periodo
# precedente, quindi c'è un ritardo di un periodo nell'adeguamento della
# allocazione ottimale[/COLOR] [COLOR=Teal](come faremmo nella realtà, ovviamente)[/COLOR]

ret <- lag(w.t) * dX
tot.ret <- as.timeSeries(rowSums(ret))

[COLOR=Teal]# Visualizzo alcune misure di performance[/COLOR]

charts.PerformanceSummary(R = tot.ret)

table.DownsideRisk(tot.ret, p = .99)
table.Returns(tot.ret)
table.Drawdowns(tot.ret)

Per applicare il codice, dovete importare i dati dal foglio Excel che ho allegato; poichè il foglio Excel comprende anche la chiusura di venerdì scorso, potreste ottenere risultati leggerissimamente diversi in funzione della performance degli indici nell'ultima settimana.

Se volete risparmiarvi di importare i dati dal foglio Excel, è sufficiente che, prima di eseguire il codice, ricopiate in R la seguente (lunghissima) istruzione, che non fa altro che ricreare l'intera matrice delle serie storiche utilizzate e gli dà il nome MV_datas, di modo che possiate usare il codice:

MV_datas <- structure(c(1229.23, 1275.09, 1243.26, 1225.19, 1279.64, 1239.4, 
1230.13, 1239.22, 1238.33, 1275.47, 1294.59, 1299.29, 1282.8, 
1293.72, 1348.35, 1319, 1356.85, 1335.18, 1345, 1337.8, 1330.29, 
1301.84, 1327.75, 1293.64, 1342.84, 1315.31, 1391.22, 1403.28, 
1418.78, 1356.94, 1328.72, 1300.29, 1327.68, 1336.61, 1348.27, 
1357.24, 1351.66, 1335.42, 1277.36, 1282.81, 1336.02, 1247.41, 
1301.65, 1362.93, 1370.23, 1396.06, 1422, 1416.62, 1433.3, 1417.04, 
1421.03, 1458.34, 1469.25, 1441.47, 1465.15, 1441.36, 1360.16, 
1424.37, 1387.12, 1346.09, 1333.36, 1409.17, 1395.07, 1464.47, 
1527.46, 1498.58, 1516.35, 1356.56, 1434.54, 1452.43, 1432.63, 
1420.96, 1406.95, 1378.02, 1477.26, 1456.95, 1464.46, 1441.48, 
1454.6, 1478.9, 1509.98, 1480.19, 1419.89, 1462.93, 1471.84, 
1491.72, 1506.45, 1520.77, 1494.5, 1465.81, 1448.72, 1436.51, 
1408.99, 1374.17, 1396.93, 1379.58, 1426.69, 1365.98, 1367.72, 
1341.77, 1315.23, 1369.89, 1312.15, 1305.97, 1320.28, 1298.35, 
1318.33, 1342.55, 1354.95, 1349.48, 1314.76, 1301.54, 1245.87, 
1234.19, 1233.43, 1150.54, 1139.83, 1160.33, 1128.43, 1183.5, 
1242.98, 1253.06, 1266.61, 1245.67, 1291.96, 1277.9, 1260.68, 
1264.97, 1214.37, 1225.35, 1224.42, 1190.59, 1215.69, 1210.85, 
1205.83, 1214.36, 1190.17, 1161.98, 1184.94, 1133.59, 1085.79, 
1092.54, 965.8, 1040.95, 1071.38, 1091.65, 1073.49, 1104.61, 
1087.2, 1120.31, 1138.66, 1150.34, 1139.46, 1158.31, 1123.09, 
1144.9, 1161.02, 1172.51, 1145.6, 1127.58, 1133.28, 1122.2, 1096.22, 
1104.18, 1089.84, 1131.78, 1164.31, 1166.16, 1148.7, 1147.39, 
1122.73, 1111.01, 1125.17, 1076.32, 1073.43, 1054.99, 1106.59, 
1083.82, 1067.14, 1027.53, 1007.27, 989.14, 989.81, 989.03, 921.39, 
847.76, 852.84, 864.24, 908.64, 928.77, 940.86, 916.07, 893.92, 
889.81, 845.39, 827.37, 800.58, 835.32, 884.39, 897.65, 900.96, 
894.74, 909.83, 930.55, 936.31, 912.23, 889.48, 895.75, 875.4, 
908.59, 927.57, 901.78, 861.4, 855.7, 829.69, 834.89, 848.17, 
841.15, 828.89, 833.27, 895.79, 863.5, 878.85, 868.3, 893.58, 
898.81, 930.08, 933.41, 944.3, 933.22, 963.59, 987.76, 988.61, 
995.69, 976.22, 985.7, 998.14, 993.32, 998.68, 980.15, 977.59, 
990.67, 993.06, 1008.01, 1021.39, 1018.63, 1036.3, 996.85, 1029.85, 
1038.06, 1039.32, 1028.91, 1050.71, 1053.21, 1050.35, 1035.28, 
1058.2, 1061.5, 1074.14, 1088.67, 1095.89, 1108.48, 1121.86, 
1139.83, 1141.55, 1131.13, 1142.76, 1145.81, 1144.11, 1144.94, 
1156.87, 1120.57, 1109.78, 1108.06, 1141.81, 1139.32, 1134.61, 
1140.6, 1107.3, 1098.7, 1095.7, 1093.56, 1120.68, 1122.5, 1136.47, 
1135.02, 1134.43, 1125.38, 1112.81, 1101.39, 1086.2, 1101.72, 
1063.97, 1064.8, 1098.35, 1107.77, 1113.63, 1123.92, 1128.55, 
1110.11, 1131.5, 1122.14, 1108.2, 1095.74, 1130.2, 1166.17, 1184.17, 
1170.34, 1182.65, 1191.17, 1188, 1194.22, 1210.13, 1211.92, 1186.19, 
1184.52, 1167.87, 1171.36, 1203.03, 1205.3, 1201.59, 1211.37, 
1222.12, 1200.08, 1189.65, 1171.42, 1172.92, 1181.2, 1142.62, 
1152.12, 1156.85, 1171.35, 1154.05, 1189.28, 1198.78, 1196.02, 
1198.11, 1216.96, 1191.57, 1194.44, 1211.86, 1227.92, 1233.68, 
1234.18, 1226.42, 1230.39, 1219.71, 1205.1, 1218.02, 1241.48, 
1237.91, 1215.29, 1228.81, 1195.9, 1186.57, 1179.59, 1198.41, 
1220.14, 1234.72, 1248.27, 1268.25, 1265.08, 1259.37, 1267.32, 
1268.66, 1248.29, 1285.45, 1287.61, 1261.49, 1283.72, 1264.03, 
1266.99, 1287.24, 1289.43, 1287.23, 1281.58, 1307.25, 1302.95, 
1294.83, 1295.5, 1289.12, 1311.28, 1310.61, 1325.76, 1291.24, 
1267.03, 1280.16, 1288.22, 1252.3, 1251.54, 1244.5, 1270.2, 1265.48, 
1236.2, 1240.29, 1278.55, 1279.36, 1266.74, 1302.3, 1295.09, 
1311.01, 1298.92, 1319.87, 1314.78, 1335.85, 1349.58, 1365.62, 
1368.6, 1377.34, 1364.3, 1380.9, 1401.2, 1400.95, 1396.71, 1409.84, 
1427.09, 1410.76, 1418.3, 1409.71, 1430.73, 1430.5, 1422.18, 
1448.39, 1438.06, 1455.54, 1451.19, 1387.17, 1402.85, 1386.95, 
1436.11, 1420.86, 1443.76, 1452.85, 1484.35, 1494.07, 1505.62, 
1505.85, 1522.75, 1515.73, 1536.34, 1507.67, 1532.91, 1502.56, 
1503.35, 1530.44, 1552.5, 1534.1, 1458.95, 1433.06, 1453.64, 
1445.94, 1479.37, 1473.99, 1453.55, 1484.25, 1525.75, 1526.75, 
1557.59, 1561.8, 1500.63, 1535.28, 1509.65, 1453.7, 1458.74, 
1440.7, 1481.14, 1504.66, 1467.95, 1484.46, 1478.49, 1411.63, 
1401.02, 1325.19, 1330.61, 1395.42, 1331.29, 1349.99, 1353.11, 
1330.63, 1293.37, 1288.14, 1329.51, 1315.22, 1370.4, 1332.83, 
1390.33, 1397.84, 1413.9, 1388.28, 1425.35, 1375.93, 1400.38, 
1360.68, 1360.03, 1317.93, 1278.38, 1262.9, 1239.49, 1260.68, 
1257.76, 1260.31, 1296.32, 1298.2, 1292.2, 1282.83, 1242.31, 
1251.7, 1255.08, 1213.01, 1099.23, 899.22, 940.55, 876.77, 968.75, 
930.99, 873.29, 800.03, 896.24, 876.07, 879.73, 887.88, 872.8, 
931.8, 890.35, 850.12, 831.95, 825.88, 868.6, 826.84, 770.05, 
735.09, 683.38, 756.55, 768.54, 815.94, 842.5, 856.56, 869.6, 
866.23, 877.52, 929.23, 882.88, 887, 919.14, 940.09, 946.21, 
921.23, 918.9, 896.42, 879.13, 940.38, 979.26, 987.48, 1010.48, 
1004.09, 1026.13, 1028.93, 1016.4, 1042.73, 1068.3, 1044.38, 
1025.21, 1071.49, 1087.68, 1079.6, 1036.19, 1069.3, 1093.48, 
1091.38, 1091.49, 1105.98, 1106.41, 1102.47, 1126.48, 1115.1, 
1144.98, 1136.03, 1091.76, 1073.87, 1066.19, 1075.51, 1109.17, 
1104.49, 1138.7, 1149.99, 1159.9, 1166.59, 1178.1, 1194.37, 1192.13, 
1217.28, 1186.69, 1110.88, 1135.68, 1087.69, 1089.41, 1064.88, 
1091.6, 1117.51, 1076.76, 1022.58, 1077.96, 1064.88, 1102.66, 
1101.6, 1121.64, 1079.25, 1071.69, 1064.59, 1104.51, 1109.55, 
1125.59, 1148.67, 1146.24, 1165.15, 1176.19, 1183.08, 1183.26, 
1225.85, 1199.21, 1199.73, 1189.4, 1224.71, 1240.4, 1243.91, 
1256.77, 1257.64, 1271.5, 1293.24, 1283.35, 1276.34, 1310.87, 
1329.15, 1343.01, 1319.88, 1321.15, 1304.28, 1279.2, 1313.8, 
1332.41, 1328.17, 1319.68, 1337.38, 1363.61, 1340.2, 1337.77, 
1333.27, 1331.1, 1300.16, 1270.98, 1271.5, 1268.45, 1339.67, 
1343.8, 1316.14, 1345.02, 1292.28, 1199.38, 1178.81, 1123.53, 
1176.8, 1173.97, 1154.23, 1216.01, 1136.43, 1131.42, 1155.46, 
1224.58, 1238.25, 1285.09, 1253.23, 1263.85, 1215.65, 1158.67, 
1244.28, 1255.19, 1219.66, 1265.33, 1257.6, 1277.81, 1289.09, 
1315.38, 1316.33, 1344.9, 1342.64, 1361.23, 1365.74, 1369.63, 
1370.87, 1404.17, 1397.11, 1408.47, 1398.08, 1370.26, 1378.53, 
1403.36, 1369.1, 1353.39, 1295.22, 1317.82, 1278.04, 1325.66, 
1342.84, 1335.02, 1362.16, 1354.68, 1356.78, 1362.66, 1385.97, 
1390.99, 1405.87, 180.29, 186.26, 178.87, 178.07, 179.37, 177.62, 
176.14, 173.44, 175.35, 176.16, 182.28, 185.39, 180.01, 184.44, 
189.19, 190.25, 189.84, 192.16, 193.43, 189.41, 186.7, 182.64, 
185.32, 192.65, 193.44, 191.94, 195.36, 195.38, 196.7, 191.33, 
195.38, 190.36, 192.34, 197.49, 197.87, 199.1, 200.91, 197.16, 
194.34, 197.56, 200.93, 197.58, 198.95, 203.75, 205.63, 208.86, 
212.67, 215.02, 215.65, 218.8, 219.02, 226.88, 231.2, 221.83, 
227.81, 222.89, 221.58, 226.47, 228.28, 224.72, 225.69, 230.45, 
230.53, 227.2, 231.85, 230.84, 228.78, 219.91, 217.65, 217.49, 
216.44, 213.75, 206.51, 206.81, 221.03, 220.98, 219.72, 217.64, 
219.48, 221.61, 220.96, 215.86, 208.45, 207.8, 210.65, 212.48, 
213.77, 215.57, 207.48, 202.52, 198.3, 199.9, 199.13, 191.29, 
189.94, 192.45, 198.57, 194.49, 191.21, 185.66, 187.93, 191.1, 
188.88, 186.37, 191.59, 192.78, 190.07, 192.49, 191.56, 192.3, 
186.7, 184.5, 177.81, 177.96, 179.44, 165.61, 163.6, 166.23, 
167.88, 171.08, 175.01, 176.25, 178.25, 175.89, 176.69, 175.21, 
170.7, 171.45, 166.2, 164.75, 164.29, 158.33, 158.65, 157.94, 
157.77, 162.35, 158.59, 159.33, 160.98, 156.3, 148.73, 138.73, 
129.15, 139.39, 143.22, 145.46, 142.42, 146.9, 145.04, 148.35, 
149.82, 149.54, 149.58, 152.72, 147.27, 147.61, 150.61, 154.24, 
149.82, 147.02, 145.89, 144.28, 142.32, 145.91, 142.74, 147.28, 
155.29, 155.03, 153.48, 153.11, 153.24, 150.72, 155.52, 153.92, 
154.61, 153.43, 157.64, 156.48, 154.75, 149.97, 144.03, 143.44, 
147.87, 147.42, 141.43, 137.47, 128.63, 129.89, 133.43, 134.76, 
136.09, 133.01, 129.05, 126.71, 121.56, 121.52, 117.75, 118.49, 
124.22, 123.01, 124.48, 126.41, 127.15, 129.93, 130.71, 126.88, 
124.79, 124.98, 124.27, 128.75, 128.97, 128.01, 124.02, 121.33, 
119.74, 120.2, 121.09, 118.37, 114.88, 115.17, 120, 117.7, 119.3, 
119.57, 123.35, 122.45, 126.92, 129.36, 131.63, 131.12, 133.32, 
138.12, 140.25, 141.13, 137.46, 139.47, 139.41, 137.44, 141.03, 
139.66, 139.19, 143.12, 144.16, 143.91, 147.86, 148.49, 152.11, 
148.55, 154.24, 157.45, 158.29, 155.08, 157.2, 158.55, 159.98, 
157.08, 160.3, 164.02, 164.77, 166.81, 168.95, 174.04, 178.77, 
176.22, 180.46, 175.34, 177.65, 181.35, 180.67, 179.59, 182.5, 
174.4, 177.49, 176.03, 182.15, 182.42, 179.93, 179.1, 174.28, 
171.74, 165.2, 169.12, 173.7, 173.44, 174, 175.6, 177.19, 176.85, 
176.56, 175.68, 170.33, 171.27, 169.89, 168.29, 170.71, 172.6, 
172.91, 175.95, 176.23, 175.62, 179.38, 181.84, 179.2, 180.88, 
182.76, 189.42, 191.46, 193.56, 195.35, 198.49, 193.07, 196.32, 
201.89, 203.92, 198.02, 198.79, 197.15, 198.95, 202.31, 203.62, 
206.64, 208.25, 211.59, 211.64, 208.15, 203.2, 203.43, 204.07, 
200.07, 200.32, 197.21, 200.94, 196.07, 196.63, 198.42, 198.57, 
198.37, 202.07, 200.37, 199.29, 198.39, 202.43, 204.59, 206.85, 
208.66, 214.88, 210.5, 210.64, 216.59, 218.9, 218.16, 216.39, 
221.41, 216.42, 214.1, 209.31, 211.01, 216.16, 217.41, 218.98, 
220.69, 223.88, 225.95, 230, 231.55, 230.71, 242.59, 242.3, 238.21, 
245.33, 242.79, 241.35, 241.76, 245.5, 244.76, 242.5, 251.04, 
250.18, 251.84, 256.41, 253.57, 262.99, 264.04, 271.41, 268.11, 
252.55, 252.68, 254.26, 238.62, 237.32, 237.87, 249.44, 250.33, 
239.34, 241.52, 251.84, 254.49, 251.1, 257.5, 254.65, 259, 254.3, 
254.71, 255.25, 258.22, 259.95, 262.64, 265.92, 269.33, 268.51, 
272.28, 271.62, 274.69, 277.69, 281.57, 284.24, 283.78, 286.92, 
283.39, 284.58, 287.25, 286.49, 292.2, 293.37, 298.24, 299.68, 
285.29, 288.07, 285.46, 297.61, 296.25, 302.27, 305.26, 310.31, 
308.61, 311.9, 310.11, 311.32, 310.59, 316.5, 307.4, 315.92, 
314.98, 316.21, 322.96, 328.81, 327.49, 309.44, 309.29, 300.31, 
287.12, 302.87, 308.95, 305.96, 309.16, 320.15, 328.14, 334.44, 
338.82, 332.91, 339.34, 338.75, 331.14, 324.35, 319.98, 330.75, 
334.94, 322.59, 318.31, 325.05, 318.7, 311.72, 295.83, 293.03, 
297.79, 283.41, 290.42, 293.53, 299.73, 289.23, 287.56, 280.03, 
293.92, 304.16, 301.08, 305.45, 308.39, 311.94, 308.59, 317.85, 
311.88, 310.65, 305.78, 291, 287.7, 282.33, 273.09, 270.67, 272.08, 
272.6, 269.59, 264.72, 258.59, 255.99, 259.73, 238.89, 241.01, 
240.22, 235.49, 216.5, 170.11, 174.52, 155.35, 172.21, 172.2, 
161.58, 144.42, 161.78, 151.01, 164.38, 171.29, 167.97, 175.83, 
174.12, 160.94, 149.52, 155.9, 160.3, 156.26, 142.83, 140.9, 
132.17, 140.57, 150.36, 155.78, 162.19, 163.05, 167.18, 168.16, 
171.43, 183.93, 180.7, 187.84, 192.4, 193.69, 196.78, 189.83, 
190.47, 189.14, 182.92, 193.82, 203.16, 208.61, 211.35, 212.83, 
214.12, 218.29, 214.3, 225.85, 229.34, 225.51, 219.92, 230.03, 
233.67, 234.64, 224.71, 228.25, 233.67, 230.26, 228.54, 236.29, 
231.48, 227.07, 232.64, 234.4, 240.09, 240.79, 231.26, 223.31, 
214.61, 217.59, 220.99, 221.95, 229.54, 233.85, 233.91, 233.65, 
239.3, 240.77, 240.42, 236.4, 232.8, 209.52, 214.38, 203.11, 
206.21, 203.63, 205.72, 214.16, 209.62, 203.79, 213.59, 215.37, 
219.6, 222.6, 229.07, 219.15, 217.43, 216.06, 224.43, 226.48, 
230.49, 236.23, 238.43, 244.12, 247.59, 245.82, 245.03, 254.1, 
247.56, 246.45, 238.34, 246.05, 246.29, 246.14, 249.67, 252.33, 
250.55, 256.27, 254.89, 255, 258.42, 256.26, 261.97, 257.6, 261.44, 
253.02, 247.39, 256.27, 260.01, 264.77, 261.21, 266.53, 271.02, 
264, 258.69, 256.72, 257.8, 257.82, 251.59, 248.33, 247.55, 259.21, 
258.6, 251.78, 259.06, 254, 229.75, 225.13, 218.13, 219.5, 225.82, 
214.91, 217.45, 199.76, 204.76, 208.04, 218.07, 217.74, 233.27, 
221.51, 219.83, 210.92, 198.76, 217.03, 214.03, 205.86, 210.68, 
211.24, 211.21, 213.68, 222.49, 226.37, 232.01, 231.15, 234.96, 
238.09, 237.44, 234.33, 238.64, 234.68, 234.28, 228.72, 226.16, 
228.4, 229.42, 224.85, 218, 204.46, 203.47, 198.74, 202.99, 207.63, 
207.63, 213.66, 213.11, 210.72, 212.02, 214.24, 217.3, 221.79, 
178.584, 176.388, 178.221, 179.284, 179.416, 176.087, 174.783, 
174.818, 172.886, 172.723, 174.395, 174.202, 174.2, 174.647, 
176.412, 174.96, 174.973, 174.358, 172.514, 171.368, 172.429, 
171.124, 169.571, 167.472, 169.884, 168.057, 170.234, 170.491, 
172.267, 170.671, 170.015, 168.648, 169.047, 170.361, 170.946, 
170.724, 170.8, 171.409, 172.704, 170.584, 170.128, 169.921, 
168.881, 171.318, 172.607, 172.921, 171.838, 171.332, 171.176, 
172.547, 170.28, 169.206, 169.074, 168.732, 167.549, 166.97, 
168.109, 168.89, 167.837, 168.745, 170.677, 170.331, 170.435, 
172.631, 172.466, 174.852, 176.592, 176.654, 175.711, 173.743, 
171.129, 171.207, 171.601, 173.741, 175.816, 176.183, 178.057, 
175.974, 177.981, 178.634, 177.652, 178.892, 178.78, 180.414, 
180.303, 180.719, 181.431, 182.166, 181.844, 181.152, 181.49, 
182.262, 182.554, 183.8, 185.067, 184.4, 183.471, 184.326, 185.354, 
186.408, 188.157, 190.559, 192.152, 194.184, 193.175, 195.882, 
192.619, 193.962, 193.039, 194.792, 195.237, 194.499, 194.984, 
196.692, 197.167, 199.246, 199.254, 198.588, 199.321, 196.348, 
195.469, 195.313, 197.015, 194.701, 194.606, 193.726, 195.627, 
195.85, 197.711, 199.366, 195.946, 196.877, 198.682, 200.069, 
200.863, 200.365, 201.921, 203.523, 202.804, 204.124, 205.038, 
208.775, 207.614, 209.01, 210.477, 208.779, 209.603, 211.077, 
213.21, 213.919, 206.994, 205.571, 208.815, 203.606, 203.757, 
205.142, 205.013, 205.007, 208.718, 208.566, 206.5, 207.87, 209.043, 
209.255, 209.893, 208.157, 204.029, 204.153, 203.361, 203.718, 
206.573, 207.498, 207.27, 209.377, 209.719, 209.35, 207.798, 
209.629, 211.238, 211.308, 214.867, 215.452, 215.256, 214.757, 
218.762, 219.843, 222.09, 223.891, 224.979, 223.463, 225.151, 
226.6, 228.635, 230.089, 232.082, 233.945, 234.051, 232.708, 
228.504, 229.03, 230.819, 233.343, 230.747, 228.487, 228.04, 
230.148, 230.819, 232.825, 235.315, 232.581, 231.01, 233.084, 
234.463, 234.081, 234.615, 234.47, 236.14, 239.028, 239.985, 
238.896, 233.562, 236.787, 236.401, 236.44, 236.64, 237.924, 
237.761, 241.733, 245.178, 246.39, 246.424, 246.797, 250.424, 
246.973, 244.61, 243.318, 244.009, 238.212, 235.994, 232.1, 235.252, 
231.384, 232.051, 232.474, 234.92, 236.767, 238.06, 240.914, 
237.961, 237.54, 235.376, 238.296, 237.364, 235.189, 239.191, 
240.121, 237.481, 239.331, 239.598, 241.339, 241.342, 238.093, 
243.112, 243.982, 243.436, 242.432, 243.558, 244.865, 244.195, 
246.377, 248.629, 250.092, 250.237, 249.441, 244.28, 243.626, 
241.374, 239.304, 238.807, 234.619, 235.226, 235.617, 237.209, 
235.625, 235.444, 237.046, 238.382, 241.64, 241.817, 243.622, 
242.384, 242.028, 246.439, 246.524, 246.696, 246.934, 246.043, 
248.126, 249.551, 250.858, 248.43, 249.477, 250.974, 252.426, 
251.976, 249.526, 249.616, 249.354, 248.811, 249.099, 250.906, 
250.302, 250.252, 250.318, 249.537, 250.635, 251.803, 251.93, 
252.919, 252.883, 250.187, 250.212, 249.611, 246.055, 246.893, 
245.556, 248.208, 248.1, 252.135, 252.322, 253.456, 252.813, 
254.933, 255.05, 256.353, 258.026, 256.915, 256.594, 259.425, 
257.266, 256.682, 255.54, 254.99, 254.201, 252.459, 254.67, 255.633, 
256.071, 259.184, 257.952, 255.58, 256.123, 254.795, 254.374, 
252.664, 254.36, 251.567, 250.033, 251.886, 253.055, 254.311, 
253.047, 253.202, 255.018, 256.243, 256.212, 256.728, 257.252, 
257.416, 254.983, 254.878, 253.522, 254.583, 253.999, 252.519, 
252.193, 253.651, 253.915, 251.188, 249.569, 248.796, 249.911, 
249.658, 249.052, 247.528, 250.061, 250.456, 251.608, 252.063, 
249.744, 248.369, 250.155, 250.482, 251.859, 252.622, 253.8, 
255.677, 254.481, 256.812, 257.848, 258.966, 258.461, 258.378, 
262.157, 261.751, 260.789, 259.171, 259.638, 261.862, 261.389, 
263.709, 263.745, 264.913, 267.223, 265.417, 264.835, 264.586, 
263.196, 264.458, 262.431, 262.627, 261.256, 262.339, 263.191, 
264.967, 265.543, 268.98, 267.665, 268.846, 267.949, 267.609, 
265.903, 265.991, 267.965, 267.846, 269.065, 268.635, 266.523, 
265.673, 264.134, 261.811, 261.014, 261.987, 264.083, 261.576, 
263.316, 266.167, 269.862, 271.253, 269.218, 272.273, 273.604, 
275.765, 279.211, 277.882, 275.246, 276.18, 275.387, 274.759, 
280.633, 280.638, 282.505, 284.754, 285.789, 289.091, 290.562, 
287.613, 285.467, 287.224, 289.245, 293.902, 296.052, 299.466, 
301.242, 301.008, 300.557, 298.501, 298.041, 304.422, 304.664, 
307.643, 310.294, 307.75, 306.941, 306.748, 301.869, 298.497, 
298.591, 301.082, 299.365, 299.593, 295.081, 298.849, 291.591, 
293.763, 298.439, 298.753, 299.298, 297.053, 296.96, 301.063, 
302.098, 304.181, 303.78, 305.287, 307.873, 307.447, 305.533, 
304.872, 310.042, 304.011, 301.229, 307.48, 301.916, 305.452, 
309.925, 320.621, 326.533, 330.91, 335.021, 345.806, 345.776, 
339.896, 340.225, 341.619, 334.264, 329.25, 326.48, 329.878, 
332.191, 327.215, 330.621, 331.187, 338.967, 336.271, 333.324, 
332.772, 332.516, 331.501, 328.28, 326.156, 329.612, 322.666, 
322.255, 313.551, 314.905, 315.03, 320.649, 321.949, 326.896, 
319.456, 319.448, 323.508, 315.407, 322.013, 321.973, 324.975, 
325.522, 328.275, 325.584, 329.154, 332.164, 328.676, 328.475, 
326.726, 329.777, 327.85, 329.87, 331.375, 335.715, 329.714, 
328.491, 328.86, 322.919, 322.317, 322.991, 327, 328.922, 329.826, 
330.791, 328.407, 326.509, 330.695, 329.726, 329.517, 329.99, 
326.661, 324.678, 326.44, 329.685, 328.605, 332.706, 337.936, 
337.779, 342.887, 341.986, 344.593, 344.398, 344.998, 348.291, 
351.662, 350.246, 354.393, 353.003, 356.188, 358.991, 363.538, 
364.32, 363.323, 361.871, 359.6, 361.476, 365.25, 368.259, 372.189, 
368.123, 368.719, 367.924, 371.34, 365.745, 362.554, 362.865, 
359.278, 350.288, 350.954, 349.566, 352.537, 352.261, 352.666, 
350.779, 353.855, 346.055, 346.072, 347.459, 351.943, 350.327, 
352.896, 356.681, 352.164, 352.102, 348.969, 353.663, 354.239, 
357.709, 361.684, 362.285, 363.314, 365.954, 369.143, 369.819, 
370.663, 373.406, 364.498, 369.172, 372.795, 371.451, 376.34, 
382.088, 391.412, 397.342, 394.396, 400.162, 401.955, 398.189, 
403.506, 401.269, 396.58, 392.415, 393.762, 391.505, 399.994, 
400.2, 400.947, 402.152, 400.928, 400.736, 407.066, 402.056, 
406.28, 404.623, 407.462, 403.046, 407.648, 406.89, 406.015, 
405.342, 406.243, 406.571, 405.096, 397.301, 399.059, 400.261, 
405.249, 407.512, 408.208, 409.238, 411.071, 413.19, 416.049, 
415.914, 424.617, 419.307, 421.206, 418.097, 419.594, 422.763, 
424.524, 425.662, 422.922, 422.405, 420.741, 91.03, 92.32, 91.1, 
90.61, 88.91, 88.61, 85.95, 85.72, 86.13, 86.32, 85.84, 84.37, 
83.53, 84.24, 83.39, 89.94, 92.82, 92.56, 93.75, 94.57, 95.04, 
94.17, 94.24, 92.92, 92.57, 90.74, 91.75, 91.04, 90.74, 89.27, 
88.22, 85.73, 85.59, 85.77, 85.11, 85.48, 85.18, 83.44, 81.51, 
81.11, 80.87, 78.92, 78.18, 79.11, 79.35, 78.51, 78.12, 76.78, 
76.4, 74.47, 75.74, 76.51, 78.23, 80.06, 79.96, 80.57, 78.95, 
78.89, 77.71, 76.67, 76.09, 76.83, 76.55, 78.25, 78.56, 78.47, 
80.55, 81.16, 81.7, 83.2, 85.38, 84.98, 84.42, 83.1, 84.5, 85.31, 
86.39, 86.79, 85.07, 88.92, 88.46, 89.18, 91.46, 91.28, 91.21, 
90.54, 88.53, 87.83, 90.37, 90.41, 89.58, 89.78, 87.22, 85.47, 
85.61, 85.02, 84.83, 84.37, 85.18, 85.25, 86.31, 88.52, 89.68, 
89.81, 90.68, 89.9, 90.08, 90.35, 91.24, 91.3, 89.7, 89.79, 88.63, 
91.2, 91.38, 89.56, 88.95, 90.18, 88.93, 88.33, 90.04, 92.04, 
91.72, 91.29, 92.96, 94.6, 94.23, 95.23, 96.74, 97.25, 98.65, 
97.95, 96.92, 97.53, 96.79, 97.46, 99.25, 100.66, 101.2, 98.99, 
96.56, 96.3, 89.17, 94.27, 92.93, 94.44, 92.15, 90.52, 91.76, 
92.63, 94.95, 95.46, 95.64, 97.5, 96.26, 96.63, 97.34, 97.61, 
96.91, 96.98, 96.72, 97.11, 96.83, 97.85, 98.53, 99.37, 102.31, 
102.8, 103.65, 103.54, 104.81, 108.74, 104.84, 103.02, 106.02, 
102.48, 103.43, 103.98, 105.35, 104.95, 105.36, 106.81, 107.46, 
104.32, 98.84, 94.23, 95.69, 97.96, 97.37, 99.19, 99.29, 100.96, 
100.39, 100.09, 98.46, 96.51, 92.95, 90.37, 91.44, 92.17, 92.15, 
92.9, 93.47, 94.31, 94.89, 94.82, 94.41, 94.15, 94.4, 96.1, 94.27, 
91.94, 90.75, 91.99, 91.07, 89.99, 92.51, 92.98, 92.22, 91.61, 
96.78, 94.7, 95.66, 95.76, 98.42, 99.48, 99.64, 102.22, 102.98, 
104.82, 103.53, 106.71, 105.56, 104.64, 105.54, 108.21, 108.44, 
108.24, 109.12, 109.65, 108.82, 108.34, 108.85, 110.43, 112.72, 
111.87, 112.68, 111.69, 116.1, 117.02, 116.5, 113.36, 115.14, 
117.89, 119.16, 116.88, 119.78, 120.13, 121.63, 122.1, 123.16, 
122.71, 122.73, 122.85, 124.56, 127.73, 129.69, 128.84, 128.72, 
130.08, 134.5, 132.67, 134.78, 133.81, 135.86, 124.94, 120.11, 
117.21, 115.23, 112.51, 113.21, 115.74, 122.84, 122.64, 122.5, 
124.24, 126.79, 128.13, 126.93, 128.39, 124.35, 125.22, 126.08, 
125.1, 131.43, 132.98, 135.71, 134.95, 134.75, 132.88, 136.21, 
136.42, 138.29, 137.54, 139.6, 139.8, 145.82, 140.96, 144.81, 
148.97, 150.19, 150.06, 150.35, 150.94, 142.76, 142.14, 143.19, 
138.81, 140.72, 143.59, 144.02, 142.45, 145.5, 140.46, 140.62, 
137.3, 137.55, 137.75, 138.2, 139.98, 143.78, 144.15, 143.77, 
150.28, 147.34, 149.66, 152.84, 156.74, 154.49, 155.71, 159.48, 
158.34, 158.95, 164.04, 156.19, 154.19, 152.97, 153.04, 156.84, 
159.52, 157.4, 152.41, 155.11, 148.75, 146.22, 146.73, 149.33, 
147.52, 153.25, 154.3, 158.17, 157.77, 156.78, 156.5, 157.52, 
154.73, 161.58, 161.4, 160.71, 165.3, 162.86, 161.98, 166.5, 
167.43, 166.53, 171.09, 177.28, 173.14, 174.39, 167.86, 163.59, 
169.43, 168.49, 169.48, 164.13, 161.62, 164.6, 167.96, 168, 165.94, 
163.88, 170.42, 171.87, 169.58, 169.99, 175.95, 178.67, 172.02, 
176.31, 177.17, 179.42, 180.41, 184.61, 183.14, 183.11, 186.71, 
190.58, 190.81, 192.62, 186.98, 188.22, 191.63, 202.11, 201.72, 
200.98, 199.15, 194.55, 198.53, 195.41, 203.07, 207.05, 210.89, 
215.33, 218.01, 217.17, 213.32, 201.33, 202.88, 199.68, 205.01, 
200.9, 203.12, 201.5, 203.73, 204.04, 199.38, 200.57, 188.73, 
189.52, 200.3, 193.12, 190.98, 183.12, 181.2, 187.81, 185.79, 
180.05, 164.4, 162.63, 167.91, 168.38, 172.24, 173.27, 168.78, 
173.66, 178.84, 179.69, 190.55, 186.09, 173.29, 177.83, 170.01, 
165.35, 165.41, 161.03, 164.54, 169.01, 156.67, 158.92, 154.83, 
144.57, 145.04, 139.58, 151.35, 161.1, 150.11, 150.07, 150.7, 
148.75, 142.74, 145.52, 157.2, 152.96, 162.21, 155.76, 163.7, 
167.32, 166.1, 160.6, 166.7, 159.27, 163.01, 158.87, 157.13, 
151.82, 143.93, 140.53, 139.02, 144.36, 145.72, 147.33, 152.41, 
151.35, 148.35, 149.57, 149.18, 153.12, 159, 150.76, 129.23, 
117.29, 104.46, 86.81, 102.94, 95.8, 79.34, 65.46, 80.29, 82.52, 
84.11, 90.64, 89.24, 89.08, 83.79, 82.78, 78.98, 76.66, 79.45, 
69.41, 66.47, 61.43, 54.39, 65.27, 59.09, 62.59, 72.39, 75.65, 
79.08, 81.71, 77.7, 84.84, 75.09, 77.14, 82.19, 85.56, 84.02, 
79.32, 78.97, 75.99, 73.48, 78.62, 84.62, 87.35, 100.29, 95.3, 
96.73, 99.06, 93.88, 99.43, 107.72, 102.23, 98.04, 102.88, 101.42, 
101.68, 98.08, 98.2, 103.8, 103.49, 100.6, 108.86, 107.4, 108.8, 
113.51, 110.33, 110.38, 110.18, 105.71, 104.97, 104.72, 103.62, 
108.92, 109.94, 113.98, 117.54, 119.89, 120.99, 119.99, 124.54, 
120.62, 128.74, 127.1, 119.1, 123.19, 117.16, 120, 113.23, 121.37, 
124.16, 120.64, 111.44, 117.33, 115.13, 122.07, 123.82, 125.19, 
120.43, 119.91, 121.83, 128.5, 126.28, 128.38, 128.39, 126.95, 
129.75, 131.49, 134.05, 131.8, 137.94, 131.2, 128.73, 130.01, 
132.36, 130.97, 129.5, 132.47, 134.39, 134.16, 135.88, 135.42, 
137.25, 138.81, 142.32, 142.78, 142.07, 140.33, 140.31, 138.65, 
139.38, 142.26, 139.41, 142.28, 144.53, 148.74, 145.33, 145.13, 
145.45, 147.29, 145.03, 139.48, 141.78, 141.29, 147.14, 150.38, 
146.85, 150.92, 144.91, 128.76, 131.25, 127.12, 131.18, 131.29, 
130.61, 135.59, 124.24, 121.97, 119.27, 126.88, 131.01, 139.42, 
136.31, 135.44, 131.37, 124.33, 131.52, 133.38, 133.61, 138.72, 
137.57, 137.41, 139.34, 142.8, 146.77, 149.81, 146.71, 147.43, 
146.59, 145.54, 145.51, 149.08, 148.42, 150.36, 148.69, 147.17, 
151.12, 154.51, 153.55, 154.42, 144.39, 148.19, 144.19, 150.27, 
150.9, 150.12, 155.42, 157.32, 158.38, 156.78, 158.48, 159.52, 
156.11, 199.16562, 208.99039, 197.5533, 200.61523, 201.07527, 
193.02649, 191.78073, 190.95576, 192.91347, 203.07455, 214.65221, 
223.49802, 231.82405, 234.03735, 232.26312, 238.15038, 241.98492, 
248.74809, 243.8488, 243.19408, 235.68295, 229.34242, 235.00089, 
246.103, 244.02592, 245.93628, 256.13116, 257.64294, 263.42874, 
266.69, 264.64294, 273.7323, 278.33176, 280.52133, 274.41736, 
281.7854, 296.27029, 298.22076, 303.22653, 303.32944, 272.26331, 
289.96283, 295.16827, 280.66373, 290.94901, 306.2294, 317.25534, 
323.86908, 312.25677, 308.2225, 324.05734, 322.68729, 326.45947, 
314.91293, 342.88577, 354.8768, 346.09775, 360.93497, 364.29111, 
363.49387, 372.72815, 381.10358, 378.16205, 374.95599, 366.13785, 
361.4386, 345.28445, 344.11713, 358.65814, 354.60144, 368.08133, 
391.21274, 394.89923, 393.17133, 394.78046, 392.05936, 396.2887, 
414.91772, 417.74277, 396.79755, 404.18292, 389.58173, 386.20001, 
399.86847, 409.60376, 421.02884, 432.4534, 445.63089, 450.41846, 
464.30493, 444.21457, 432.42612, 433.60513, 465.12033, 450.07019, 
443.47147, 444.64532, 461.38715, 481.64178, 487.63394, 456.5835, 
438.27341, 433.10147, 423.6622, 432.7435, 434.09818, 442.54794, 
452.94342, 447.24319, 457.10315, 454.96841, 440.31741, 435.89301, 
433.51587, 436.5799, 418.23129, 423.80972, 413.97885, 423.82648, 
440.82205, 431.26474, 435.09586, 438.06638, 434.40573, 450.04526, 
432.20068, 427.74976, 430.48889, 430.89536, 412.49448, 406.68176, 
422.01163, 414.29175, 409.06122, 418.19711, 418.52625, 426.10156, 
412.85416, 424.86096, 429.37305, 433.15588, 441.67188, 409.56537, 
389.53668, 380.31824, 377.97763, 373.86356, 372.99014, 354.84131, 
373.01016, 341.61508, 345.94824, 352.00662, 342.20932, 344.97818, 
345.69458, 352.55426, 362.78253, 351.89059, 336.57498, 347.88788, 
351.16116, 349.22083, 357.41919, 351.88559, 366.21677, 378.44418, 
385.54831, 389.2984, 398.77707, 398.07999, 370.89648, 394.73199, 
401.07947, 395.6333, 406.13324, 405.15982, 394.69144, 393.11914, 
387.86902, 398.71155, 394.60739, 408.46259, 410.17065, 414.87167, 
420.93799, 407.91302, 411.7406, 413.02737, 437.92068, 436.8515, 
441.94254, 457.32126, 456.52417, 454.0094, 459.80518, 448.65891, 
444.6524, 454.0271, 429.42505, 432.54044, 411.49112, 410.96909, 
431.30838, 430.26956, 427.08002, 444.12604, 459.61743, 479.81494, 
483.90332, 472.33032, 487.45303, 495.63861, 499.9931, 526.93457, 
529.34473, 532.81097, 546.05786, 558.67957, 520.3866, 452.46167, 
478.05176, 464.72284, 465.82858, 481.82751, 459.13638, 453.0849, 
481.45157, 498.92349, 498.74387, 503.06827, 518.66382, 503.79996, 
506.53238, 499.22064, 509.61368, 520.8006, 521.33398, 516.75262, 
541.30878, 543.14587, 536.28412, 545.08405, 545.04749, 515.24652, 
504.58661, 489.62503, 503.55844, 525.59161, 543.73291, 528.38483, 
534.98444, 527.1795, 542.27014, 564.40149, 556.73468, 548.60846, 
556.09601, 579.685, 584.43555, 576.8584, 582.71265, 606.36053, 
607.5661, 617.99048, 596.67505, 588.0614, 616.89478, 612.19336, 
639.54767, 641.00488, 627.58289, 659.40686, 642.5694, 643.39716, 
671.94696, 668.41333, 650.92102, 663.49634, 687.96582, 688.32843, 
676.12671, 683.72876, 674.85114, 669.52313, 676.2204, 666.29425, 
671.79016, 683.74261, 691.91711, 689.69562, 705.31671, 712.45813, 
732.34967, 747.44867, 702.95227, 710.54712, 703.8233, 734.88831, 
770.31561, 783.76697, 817.74554, 834.80511, 847.16034, 810.60193, 
783.77063, 766.94116, 792.80823, 791.69971, 727.2016, 710.01849, 
767.89514, 749.17151, 736.80939, 740.0224, 759.89288, 765.07324, 
751.03705, 737.48401, 749.79114, 767.24554, 801.7738, 825.71667, 
847.75104, 862.96729, 835.99341, 863.1875, 820.93176, 798.24695, 
831.5368, 786.31329, 786.67249, 758.75769, 755.40979, 790.49231, 
824.34668, 810.9751, 855.37018, 863.54041, 847.81946, 863.30896, 
863.59137, 841.7793, 860.44946, 872.23944, 913.42743, 891.86719, 
891.07269, 921.91058, 885.05945, 866.64307, 889.90167, 920.08289, 
881.70435, 885.21112, 864.16968, 865.21118, 851.61273, 835.16602, 
824.43201, 832.81537, 846.8573, 852.06573, 842.4715, 843.80414, 
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"2007-11-01 23:00:00", "2007-11-08 23:00:00", "2007-11-15 23:00:00", 
"2007-11-22 23:00:00", "2007-11-29 23:00:00", "2007-12-06 23:00:00", 
"2007-12-13 23:00:00", "2007-12-20 23:00:00", "2007-12-27 23:00:00", 
"2008-01-03 23:00:00", "2008-01-10 23:00:00", "2008-01-17 23:00:00", 
"2008-01-24 23:00:00", "2008-01-31 23:00:00", "2008-02-07 23:00:00", 
"2008-02-14 23:00:00", "2008-02-21 23:00:00", "2008-02-28 23:00:00", 
"2008-03-06 23:00:00", "2008-03-13 23:00:00", "2008-03-20 23:00:00", 
"2008-03-27 23:00:00", "2008-04-04 00:00:00", "2008-04-11 00:00:00", 
"2008-04-18 00:00:00", "2008-04-25 00:00:00", "2008-05-02 00:00:00", 
"2008-05-09 00:00:00", "2008-05-16 00:00:00", "2008-05-23 00:00:00", 
"2008-05-30 00:00:00", "2008-06-06 00:00:00", "2008-06-13 00:00:00", 
"2008-06-20 00:00:00", "2008-06-27 00:00:00", "2008-07-04 00:00:00", 
"2008-07-11 00:00:00", "2008-07-18 00:00:00", "2008-07-25 00:00:00", 
"2008-08-01 00:00:00", "2008-08-08 00:00:00", "2008-08-15 00:00:00", 
"2008-08-22 00:00:00", "2008-08-29 00:00:00", "2008-09-05 00:00:00", 
"2008-09-12 00:00:00", "2008-09-19 00:00:00", "2008-09-26 00:00:00", 
"2008-10-03 00:00:00", "2008-10-10 00:00:00", "2008-10-17 00:00:00", 
"2008-10-24 00:00:00", "2008-10-30 23:00:00", "2008-11-06 23:00:00", 
"2008-11-13 23:00:00", "2008-11-20 23:00:00", "2008-11-27 23:00:00", 
"2008-12-04 23:00:00", "2008-12-11 23:00:00", "2008-12-18 23:00:00", 
"2008-12-25 23:00:00", "2009-01-01 23:00:00", "2009-01-08 23:00:00", 
"2009-01-15 23:00:00", "2009-01-22 23:00:00", "2009-01-29 23:00:00", 
"2009-02-05 23:00:00", "2009-02-12 23:00:00", "2009-02-19 23:00:00", 
"2009-02-26 23:00:00", "2009-03-05 23:00:00", "2009-03-12 23:00:00", 
"2009-03-19 23:00:00", "2009-03-26 23:00:00", "2009-04-03 00:00:00", 
"2009-04-10 00:00:00", "2009-04-17 00:00:00", "2009-04-24 00:00:00", 
"2009-05-01 00:00:00", "2009-05-08 00:00:00", "2009-05-15 00:00:00", 
"2009-05-22 00:00:00", "2009-05-29 00:00:00", "2009-06-05 00:00:00", 
"2009-06-12 00:00:00", "2009-06-19 00:00:00", "2009-06-26 00:00:00", 
"2009-07-03 00:00:00", "2009-07-10 00:00:00", "2009-07-17 00:00:00", 
"2009-07-24 00:00:00", "2009-07-31 00:00:00", "2009-08-07 00:00:00", 
"2009-08-14 00:00:00", "2009-08-21 00:00:00", "2009-08-28 00:00:00", 
"2009-09-04 00:00:00", "2009-09-11 00:00:00", "2009-09-18 00:00:00", 
"2009-09-25 00:00:00", "2009-10-02 00:00:00", "2009-10-09 00:00:00", 
"2009-10-16 00:00:00", "2009-10-23 00:00:00", "2009-10-29 23:00:00", 
"2009-11-05 23:00:00", "2009-11-12 23:00:00", "2009-11-19 23:00:00", 
"2009-11-26 23:00:00", "2009-12-03 23:00:00", "2009-12-10 23:00:00", 
"2009-12-17 23:00:00", "2009-12-24 23:00:00", "2009-12-31 23:00:00", 
"2010-01-07 23:00:00", "2010-01-14 23:00:00", "2010-01-21 23:00:00", 
"2010-01-28 23:00:00", "2010-02-04 23:00:00", "2010-02-11 23:00:00", 
"2010-02-18 23:00:00", "2010-02-25 23:00:00", "2010-03-04 23:00:00", 
"2010-03-11 23:00:00", "2010-03-18 23:00:00", "2010-03-25 23:00:00", 
"2010-04-02 00:00:00", "2010-04-09 00:00:00", "2010-04-16 00:00:00", 
"2010-04-23 00:00:00", "2010-04-30 00:00:00", "2010-05-07 00:00:00", 
"2010-05-14 00:00:00", "2010-05-21 00:00:00", "2010-05-28 00:00:00", 
"2010-06-04 00:00:00", "2010-06-11 00:00:00", "2010-06-18 00:00:00", 
"2010-06-25 00:00:00", "2010-07-02 00:00:00", "2010-07-09 00:00:00", 
"2010-07-16 00:00:00", "2010-07-23 00:00:00", "2010-07-30 00:00:00", 
"2010-08-06 00:00:00", "2010-08-13 00:00:00", "2010-08-20 00:00:00", 
"2010-08-27 00:00:00", "2010-09-03 00:00:00", "2010-09-10 00:00:00", 
"2010-09-17 00:00:00", "2010-09-24 00:00:00", "2010-10-01 00:00:00", 
"2010-10-08 00:00:00", "2010-10-15 00:00:00", "2010-10-22 00:00:00", 
"2010-10-29 00:00:00", "2010-11-04 23:00:00", "2010-11-11 23:00:00", 
"2010-11-18 23:00:00", "2010-11-25 23:00:00", "2010-12-02 23:00:00", 
"2010-12-09 23:00:00", "2010-12-16 23:00:00", "2010-12-23 23:00:00", 
"2010-12-30 23:00:00", "2011-01-06 23:00:00", "2011-01-13 23:00:00", 
"2011-01-20 23:00:00", "2011-01-27 23:00:00", "2011-02-03 23:00:00", 
"2011-02-10 23:00:00", "2011-02-17 23:00:00", "2011-02-24 23:00:00", 
"2011-03-03 23:00:00", "2011-03-10 23:00:00", "2011-03-17 23:00:00", 
"2011-03-24 23:00:00", "2011-04-01 00:00:00", "2011-04-08 00:00:00", 
"2011-04-15 00:00:00", "2011-04-22 00:00:00", "2011-04-29 00:00:00", 
"2011-05-06 00:00:00", "2011-05-13 00:00:00", "2011-05-20 00:00:00", 
"2011-05-27 00:00:00", "2011-06-03 00:00:00", "2011-06-10 00:00:00", 
"2011-06-17 00:00:00", "2011-06-24 00:00:00", "2011-07-01 00:00:00", 
"2011-07-08 00:00:00", "2011-07-15 00:00:00", "2011-07-22 00:00:00", 
"2011-07-29 00:00:00", "2011-08-05 00:00:00", "2011-08-12 00:00:00", 
"2011-08-19 00:00:00", "2011-08-26 00:00:00", "2011-09-02 00:00:00", 
"2011-09-09 00:00:00", "2011-09-16 00:00:00", "2011-09-23 00:00:00", 
"2011-09-30 00:00:00", "2011-10-07 00:00:00", "2011-10-14 00:00:00", 
"2011-10-21 00:00:00", "2011-10-28 00:00:00", "2011-11-03 23:00:00", 
"2011-11-10 23:00:00", "2011-11-17 23:00:00", "2011-11-24 23:00:00", 
"2011-12-01 23:00:00", "2011-12-08 23:00:00", "2011-12-15 23:00:00", 
"2011-12-22 23:00:00", "2011-12-29 23:00:00", "2012-01-05 23:00:00", 
"2012-01-12 23:00:00", "2012-01-19 23:00:00", "2012-01-26 23:00:00", 
"2012-02-02 23:00:00", "2012-02-09 23:00:00", "2012-02-16 23:00:00", 
"2012-02-23 23:00:00", "2012-03-01 23:00:00", "2012-03-08 23:00:00", 
"2012-03-15 23:00:00", "2012-03-22 23:00:00", "2012-03-30 00:00:00", 
"2012-04-06 00:00:00", "2012-04-13 00:00:00", "2012-04-20 00:00:00", 
"2012-04-27 00:00:00", "2012-05-04 00:00:00", "2012-05-11 00:00:00", 
"2012-05-18 00:00:00", "2012-05-25 00:00:00", "2012-06-01 00:00:00", 
"2012-06-08 00:00:00", "2012-06-15 00:00:00", "2012-06-22 00:00:00", 
"2012-06-29 00:00:00", "2012-07-06 00:00:00", "2012-07-13 00:00:00", 
"2012-07-20 00:00:00", "2012-07-27 00:00:00", "2012-08-03 00:00:00", 
"2012-08-10 00:00:00"), c("SPX Index", "MXE2 Index", "USG4TR Index", 
"FNAR Index", "DBLCMAVL Index")))

 

[Paolo1956]

Grazie Crengi.

 

[Cren]

Questo me l'ero tenuto sul disco fisso per pubblicarlo più avanti, ma direi che un rapido saltino su IO mi consente il copia & incolla selvaggio

Un back test altrettanto significativo è quello che riguarda la minimizzazione dell'Expected Shortfall (ES): in questo caso non si cerca il vettore dei parametri che minimizza la varianza di portafoglio bensì quello che riduce la perdita attesa nel 1% degli scenari più sfavorevoli.

Anche in questo caso, quindi, non si cerca in alcun modo di prevedere la direzione degli asset, ma ci si limita a minimizzare una misura di rischio e ad applicare i pesi ottimali nel periodo successivo.

I pesi variano nel tempo con maggior schizofrenia rispetto alla minimizzazione della varianza, tuttavia è opportuno osservare che dopo poco tempo oscillano in un intorno ben preciso, consentendo di arrotondare al primo decimale per evitare ribilanciamenti troppo frequenti e conservare nel tempo un valore molto stabile.

Al solito, i rendimenti sono aritmetici: non dovete sommarli nel tempo ma usare una composizione geometrica.

Performance:

  [COLOR=#000000][FONT=Lucida Console] Annualized Sharpe (Rf=0%) 1.4037[/FONT][/COLOR]

Rendimenti mensili (espressi in percentuale), laddove i valori mancanti si riferiscono come al solito al periodo necessario per inizializzare l'algoritmo e alla frazione di 2012 non ancora trascorsa:

[COLOR=#000000][FONT=Lucida Console]       gen  feb  mar  apr  mag  giu  lug  ago  set  ott  nov  dic TS.1
 1999   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA
 2000   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA
 2001   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA
 2002   NA   NA   NA   NA   NA   NA   NA   NA   NA   NA  0.0  1.0  1.0
 2003  0.5  1.0  0.7  0.1  0.2 -1.0 -0.2  0.4  0.5  0.0  0.0  0.3  2.4
 2004 -0.4  0.9 -0.6 -0.9  2.0  0.5  0.3 -0.2  0.2  0.2  0.3  0.2  2.5
 2005 -0.2  0.5 -1.2  0.4  0.5  0.4  0.6  0.0  0.5 -0.4  0.8  0.0  2.0
 2006  0.1  0.2 -0.4 -0.3  0.6  1.4  1.1  0.2  0.7  1.0  0.9 -0.3  5.4
 2007 -0.2  0.4 -0.1  0.4 -0.3  0.6 -0.1  0.7  0.4  0.7  0.7  0.5  3.7
 2008  0.9  1.5 -0.5 -0.8 -1.0  0.7 -0.5  0.0 -0.7  0.0  3.3 -0.3  2.5
 2009 -1.6 -1.1 -0.3 -0.1  0.4  1.4  1.4  0.8  0.3  0.1  1.0 -0.3  2.2
 2010  0.1  1.1 -0.9  1.0  0.1  0.5  1.2 -0.1  1.1 -0.2  0.2  0.8  5.1
 2011  0.8  1.3 -0.7  1.1  1.1  0.3  0.5 -0.1 -0.7  0.3 -0.4  0.8  4.3
 2012  1.3  0.5  0.2  0.5  0.0  1.2 -0.5 -0.3   NA   NA   NA   NA  2.8[/FONT][/COLOR]

Tabella dei cinque draw down più significativi con relativi tempi di recupero (espressi in settimane):

[COLOR=#000000][FONT=Lucida Console]                 From              Trough                  To   Depth Length To Trough 
1 2008-03-27 23:00:00 2008-11-06 23:00:00 2010-04-30 00:00:00 -0.1273    110        33 
2 2004-03-25 23:00:00 2004-05-14 00:00:00 2004-10-08 00:00:00 -0.0707     29         8 
3 2010-11-11 23:00:00 2011-02-03 23:00:00 2011-05-27 00:00:00 -0.0496     29        13 
4 2003-06-20 00:00:00 2003-08-15 00:00:00 2003-10-10 00:00:00 -0.0408     17         9 
5 2005-09-09 00:00:00 2005-10-28 00:00:00 2006-01-05 23:00:00 -0.0285     18         8   
Recovery 
1       77 
2       21 
3       16 
4        8 
5       10[/FONT][/COLOR]

I risultati interessanti dei due back test suggeriscono una direzione di indagine che faccia a meno di indicatori direzionali; intuitivamente si può essere portati a scartare a priori questa strada perchè siamo portati a cercare di comprare quello che vediamo 'salire', ma invito a riflettere sul fatto che i pesi ottimali che variano nel tempo non hanno nulla di differente dai soliti segnali binari a cui i trend follower sono abituati (1 = BUY, -1 = SELL e 0 = FLAT), semplicemente ampliano lo spettro di possibili risultati con tutto ciò che è compreso tra i due valori estremi.

Codice in R (usate i dati allegati al messaggio #72):

[COLOR=Teal]#[/COLOR] [COLOR=Teal]***********************************************************************
# Back test portafoglio a Expected Shortfall minimo
# by Cren
# ***********************************************************************

# Carico i package
[/COLOR]
require(PerformanceAnalytics)
require(Rsolnp)
require(timeSeries)

[COLOR=Teal]# Calcolo i rendimenti aritmetici degli indici
[/COLOR]
dX.0 <- returns(MV_datas, method = 'discrete') ; dX <- dX.0[-1,]

[COLOR=Teal]# Per aiutarmi nel back test, costruisco una funzione per applicare in
# expanding window l'algoritmo: prende in input i dati, la funzione da
# applicare e l'ampiezza iniziale della finestra; mi restituisce l'output
# nel medesimo formato dei dati in ingresso
[/COLOR]
roll.app <- function(data, FUN, width) {
  temp <- matrix(NA, nrow = nrow(data), ncol = ncol(data))
  for(i in 1:(nrow(data) - (width - 1))) {
    temp[i + width - 1,] <- FUN(data[i:(i + width - 1),])
  }
  return(temp)
}

[COLOR=Teal]# Costruisco una funzione per ottimizzare i pesi a minor Expected Shortall: la
# funzione prende in input i dati, calcola l'Expected Shortfall del portafoglio
# e infine usa un algoritmo risolutore per estrarre il vettore dei parametri che
# minimizzano l'EXpected Shortfall del portafoglio[/COLOR]

cov.pars <- function(data) {
  fun <- function(w) {
    wei.m <- function(x) {
      weighted.mean(x, w)
    }
    R <- apply(X = data, MARGIN = 1, FUN = wei.m)
    abs(ES(R = R, p = .99, method = 'modified'))
  }
  eqfun <- function(w) {
    sum(w)
  }
  pars <- solnp(pars = rnorm(ncol(dX), .5, .1), fun = fun, eqfun = eqfun, eqB = 1,
                LB = rep(0, ncol(dX)), UB = rep(1, ncol(dX)), control = list(outer.iter = 1,
                                                                             inner.iter = 10)
  )$pars
  return(pars)
}

[COLOR=Teal]# Applico in expanding window la funzione per estrarre i pesi ottimali
# del portafoglio a varianza minima, quindi li trasformo in serie storica[/COLOR]

w.t <- roll.app(data = dX, FUN = cov.pars, width = 200)
w.t <- timeSeries(w.t, charvec = rownames(as.timeSeries(dX)))
colnames(w.t) <- colnames(MV_datas)

[COLOR=Teal]# Visualizzo l'evoluzione dei pesi ottimali nel tempo[/COLOR]

plot(w.t, format = '%y', main = 'Optimal Weights')

[COLOR=Teal]# Calcolo i rendimenti della strategia; notare la funzione 'lag':
# applichiamo al periodo successivo i pesi calcolati nel periodo
# precedente, quindi c'è un ritardo di un periodo nell'adeguamento della
# allocazione ottimale (come faremmo nella realtà, ovviamente)[/COLOR]

ret <- lag(w.t) * dX
tot.ret <- as.timeSeries(rowSums(ret))

[COLOR=Teal]# Visualizzo alcune misure di performance[/COLOR]

charts.PerformanceSummary(R = tot.ret)

table.DownsideRisk(tot.ret, p = .99)
table.Returns(tot.ret)
table.Drawdowns(tot.ret)

 

[Cren]

Questi TS che tentano, in modi più o meno sofisticati, di estrarre performance interessanti attraverso la minimizzazione di misure di rischio più o meno coerenti, riescono effettivamente a fare meglio del mercato?

Il modo più rapido per verificarlo è di osservare il confronto con il loro benchmark naturale: poichè il valore aggiunto della procedura è dato dal valore che i pesi ottimali assumono nel tempo, il benchmark naturale è rappresentato da un portafoglio equipesato, che evita a priori qualsiasi assunzione sul rischio ed equipara tutte e cinque le asset class.

Con cinque asset class, dobbiamo indagare il comportamento di un portafoglio che suddivide il capitale nel 20% tra ciascuno strumento.

Performance:

[COLOR=#000000][FONT=Lucida Console]
Annualized Sharpe (Rf=0%) 0.4779[/FONT][/COLOR]

Misure di rischio:

[COLOR=#000000][FONT=Lucida Console]Maximum Drawdown                            0.4867 
Historical VaR (99%)                       -0.0596 
Historical ES (99%)                        -0.0784 
Modified VaR (99%)                         -0.0795 
Modified ES (99%)                          -0.0795[/FONT][/COLOR]

Rendimenti mensili (espressi in percentuale):

[COLOR=#000000][FONT=Lucida Console]      Jan  Feb  Mar  Apr  May  Jun  Jul  Aug  Sep  Oct  Nov  Dec TS.1 
1999  0.7  0.3 -0.3  0.4 -1.7 -1.0 -0.5 -0.3 -1.1  1.0  0.2  1.2 -1.3 
2000 -2.0  0.5 -0.5  0.2 -0.5  0.3 -1.2  0.5 -0.4 -0.5 -1.0  1.3 -3.3 
2001 -0.1 -2.0  0.4  0.9 -0.9 -0.4  0.3 -1.5  3.4  0.9  0.5  1.2  2.7 
2002  0.0  2.2  0.4 -0.9 -0.2  1.4 -1.3 -0.3 -0.4  0.6  0.3  0.6  2.5 
2003 -0.1  0.2 -0.1 -0.6  0.9 -1.2  0.4  0.6 -0.6  0.6  0.8  0.3  1.2 
2004 -1.0  1.2 -0.9 -1.1  2.6  0.4  1.0 -0.5  0.4  0.2  0.9  0.0  3.0 
2005 -0.7  1.0 -2.0 -0.7  1.0 -0.5  1.3 -0.2  1.6  0.6  1.3 -0.2  2.5 
2006  1.5  0.9  0.5 -0.5  1.0  3.1  2.2  0.1  1.2  1.3  1.2  0.4 13.9 
2007  0.7  0.3 -0.2  0.4  0.0  0.0 -3.4  1.0  1.0  1.8  0.9  0.3  2.6 
2008  2.5  0.9  0.9  0.4 -0.2 -1.1 -0.6  0.1 -1.7  8.6 10.7 -1.5 19.7 
2009 -0.8 -2.4  2.7  0.6  3.4 -0.1  2.1  1.0 -2.6 -3.1 -0.6 -0.4 -0.6 
2010 -1.7  0.4 -0.2 -0.6  1.3 -1.6  1.3  0.2  1.5 -0.2 -0.3  0.8  0.8 
2011  0.5  0.3  1.6  1.7  0.8 -0.9 -2.1  2.1 -0.5  4.2 -3.6  0.0  4.0 
2012  1.6  1.1  0.0  1.2  0.5  3.2  0.4  0.4   NA   NA   NA   NA  8.7[/FONT][/COLOR]

I cinque peggiori draw down e i corrispondenti tempi di recupero (espressi in settimane):

[COLOR=#000000][FONT=Lucida Console]                 From              Trough         To   Depth Length To Trough 
1 2008-05-23 00:00:00 2009-03-05 23:00:00       <NA> -0.4867    222        42 
2 2000-09-08 00:00:00 2001-09-21 00:00:00 2003-08-29 -0.1576    156        55 
3 2007-11-01 23:00:00 2008-01-17 23:00:00 2008-04-04 -0.0729     23        12 
4 2004-03-11 23:00:00 2004-05-14 00:00:00 2004-08-20 -0.0582     24        10 
5 2007-07-20 00:00:00 2007-08-17 00:00:00 2007-09-21 -0.0568     10         5   
  Recovery 
1       NA 
2      101 
3       11 
4       14 
5        5[/FONT][/COLOR]

L'analisi è abbastanza semplice: in primo luogo notasi che saremmo ancora all'interno di un draw down enorme esploso nel maggio del 2008; in secondo luogo saltano all'occhio le performance deludenti (con alcuni anni da ricordare, però!).

Raffrontando la differente allocazione rispetto ai due TS precedenti, la parte della protagonista l'ha avuta la frazione di capitale allocata al debito statunitense di lungo termine: sia nel TS a minima varianza sia nel TS a minimo ES una parte cospicua del portafoglio è investita sui treasury a 7 - 10 anni di scadenza, ed è questa scelta che ha consentito di salvarsi le chiappe dai momenti più tragici.

Nel portafoglio equipesato, invece, la parte allocata ai treasury è troppo bassa: il 20% si rivela insufficiente perchè, nel momento in cui scoppia il panic selling, le covarianze aumentano in misura eccessiva e l'effetto di diversificazione non è sufficiente.

Credo, Piedi a Terra, che i back test che ho pubblicato dovrebbero esaurire i dubbi che avevi sul comportamento degli algoritmi di minima varianza su portafogli con più di due sottostanti, in particolare portafogli rappresentativi delle più diffuse asset class.

 

[Piedi a Terra]

Una cosa, innanzitutto.
I dati su cui hai testato minima varianza e CVAR sono gli stessi?
Perche' mi sembrava che i dati con cui tu avevi testato la minima varianza erano precedentemente sbagliati riguardo le transizioni in classi diverse del nostro sistema metrico decimale.

 

[Cren]

Una cosa, innanzitutto.
I dati su cui hai testato minima varianza e CVAR sono gli stessi?
Perche' mi sembrava che i dati con cui tu avevi testato la minima varianza erano precedentemente sbagliati riguardo le transizioni in classi diverse del nostro sistema metrico decimale.

Sono esattamente gli stessi per utti e tre i back test.

L'errore di conversione è dipeso dal passaggio al formato .csv, che ha fatto casino a causa del formato dei dati tirati giù da Bloomberg.

Nel formato originale non soffrivano di alcuna anomalia, e ovviamente ho usato quelli.

Ci tengo a precisare che non sto suggerendo di comprare portafogli a minor ES, minor varianza o altro, ma semplicemente che tentativi di minimizzare il rischio hanno evidenziato che in quel periodo storico e su quegli indici la minimizzazione del rischio è andata a coincidere anche con una buona scelta in termini direzionali.

Questo non è un aspetto scontato: un portafoglio con cinque indici può anche lateralizzare sonnecchiando per anni, esibendo ES e varianza relativamente contenuta (non so se è avvenuto, ma sicuramente può avvenire): non è detto a priori che debba necessariamente avere un rendimento medio annuo positivo.

 

[Piedi a Terra]

Questi TS che tentano, in modi più o meno sofisticati, di estrarre performance interessanti attraverso la minimizzazione di misure di rischio più o meno coerenti, riescono effettivamente a fare meglio del mercato?

Eccome !
Con algoritmi tipo Min Var e Expected Shorfall siamo su medie del 6-7% annuo (sottraggo circa 1% per costi di transazione non calcolati) con volatilita' del 5% annue, meno della meta' della volatilita' dell'equipesatura.

Il GTAA di Meban Faber ne esce con la ossa rotte dal confronto, IMHO.

Credo, Piedi a Terra, che i back test che ho pubblicato dovrebbero esaurire i dubbi che avevi sul comportamento degli algoritmi di minima varianza su portafogli con più di due sottostanti, in particolare portafogli rappresentativi delle più diffuse asset class.

I dubbi me li ero gia' chiariti con il comportamento dell'ETF a minima varianza, di cui ho fatto e sto facendo incetta.

L'ETF europeo a minima varianza ha gia' bastonato per bene l'ETF Euroxx, dandogli piu' di 10 punti di distacco da inizio anno (15,2% oggi !!)

L'ETF americano sul Dow Jones a minima varianza batte l'ETF americano, dandogli circa 4 punti di distacco da inizio anno (17,3% oggi )

La mia conclusione e' che il classico momentum in questa fase storica sta mostrando dei limiti (cio' che diviene troppo conosciuto ed esaltato si esaurisce?) mentre il TOM e la minima varianza in questa fase storica stanno vivendo i loro momenti piu' esaltanti.

L'accoppaita Tom & minima varianza dureranno ?
Spero di si, non volermene male se cosi' candidamente te lo confido

 

[Cren]

Eccome !
Con algoritmi tipo Min Var e Expected Shorfall siamo su medie del 6-7% annuo (sottraggo circa 1% per costi di transazione non calcolati) con volatilita' del 5% annue, meno della meta' della volatilita' dell'equipesatura.

Sarei cauto nell'esultare: questi risultati lordi possono essere interessanti in un periodo come quello odierno in cui i tassi d'interesse sono sotto i piedi e le obbligazioni IG a breve termine ormai danno rendimenti nulli o negativi.

Ma in altri periodi storici, in cui si parte da una base risk free del 4 ~ 5%, un TS che offre, nel migliore dei casi, un 2% lordo sul risk free a spese di continui ribilanciamenti mantiene la medesima attrattività?

Ad eccezione forse del periodo odierno, non posso far altro che pensare che un portafoglio di obbligazioni ben giostrato e diversificato, con uscite accorte, possa aver fatto molto meglio di queste strategie per la maggior parte del tempo.

Immagino, tuttavia, che tu abbia scelto di comprare lo spread tra ETF a varianza minima e la versione "classica", non di comprare l'ETF "naked"... vero?

Il GTAA di Meban Faber ne esce con la ossa rotte dal confronto, IMHO.

La GTAA di Faber - sempre ammesso che l'ETF rispetti alla lettera le condizioni originali, cosa di cui dubito - ha un difetto secondo me non trascurabile, che è quello di usare una media mobile di periodo fissato a priori per riversarsi "tutto o niente" in liquidità o a mercato.

Quando il mercato perde la storica sincronia con il periodo prescelto, in questo caso 10 mesi seguendo alla lettera il paper, si spera che almeno un paio delle asset class selezionate (ri)prendano la via del rialzo il prima possibile.

Nel nostro caso, tuttavia, ho il dubbio che questo non sarebbe stato sufficiente: nel periodo considerato, il USG4TR ha espresso una performance strepitosa e con volatilità contenuta; però anche "pescandolo" con la media mobile, il guadagno non sarebbe stato affatto assicurato.

Infatti sarebbero servite altre condizioni:

1. uscire in tempo prima del maggio 2008 da tutte le altre posizioni;
2. evitare che nel mentre la maggior parte delle altre posizioni traforasse la rispettiva media mobile di falsi segnali.

Solo con queste due condizioni si riesce a far sì che un misero 20% di USG4TR possa dare profitti, in tutti gli altri casi bisogna alzare la quota allocata ai treasury per contenere le perdite degli altri asset.

 

[Piedi a Terra]

Questa volta passo.

p.s. sentiamo anche un commento di Paolo su questo tuo ultimo post, cosi' avro' un quadro piu' preciso prima di scrivere altre sciocchezze....

 

[Paolo1956]

No, dunque,

ci sono alcuni aspetti da vedere secondo me:

- sotto quali condizioni un portafoglio di varianza minima è conveniente rispetto ad altri metodi?

- il problema del ribilanciamento va semplificato perché mi pare troppo oneroso in termini di costo e calcoli.

Ci sto ragionando su....