1710:. In dynamic programming, the last period decision rule, contingent on available wealth and the realizations of all previous periods' asset returns, is devised in advance; then the next-to-last period's decision rule is devised, taking into account how the results of this period will influence the final period's decisions; and so forth backward in time. This procedure becomes complex very quickly if there are more than a few time periods or more than a few assets.
1747:
is young that turn out badly can be reacted to by supplying more labor than anticipated in subsequent time periods to at least partially offset the lost wealth; since an older person with fewer subsequent time periods is less able to offset bad investment returns in this way, it is optimal for an investor to take on less investment risk at an older age.
86:
from the portfolio of the period before that, etc. However, under certain circumstances the optimal portfolio decisions can be arrived at in a way that is separated in time, so that the shares of wealth placed in particular assets depend only on the stochastic asset return distributions of that particular period.
1746:
Intertemporal portfolio models in which portfolio choice is conducted jointly with intertemporal labor supply decisions can lead to the age effect of conservatism increasing with age as advocated by many investment advisors. This result follows from the fact that risky investments when the investor
1451:
for choice under risk, including the log and power utility functions dealt with above. Mossin showed that under HARA utility, optimal portfolio choice involves partial time-independence of decisions if there is a risk-free asset and there is serial independence of asset returns: to find the optimal
85:
after the first will depend on the amount of wealth that results from the previous period's portfolio, which depends on the asset returns that occurred in the previous period as well as that period's portfolio size and allocation, the latter having depended in turn on the amount of wealth resulting
1572:
If there is not serial independence of returns through time, then the expectations operator cannot be applied separately to the various multiplicative terms. Thus the optimal portfolio for any period will depend on the probability distribution of returns for the various assets contingent on their
953:
for intertemporal portfolio choice states that, when asset return distributions are identical in all periods, a particular portfolio replicated each period will outperform all other portfolio sequences in the long run. Here the long run is an arbitrarily large number of time periods such that the
1435:
Hence under this condition we again have intertemporal independence of portfolio decisions. Note that the log utility function, unlike the power utility function, did not require the assumption of intertemporal independence of returns to obtain intertemporal independence of portfolio decisions.
1576:
Moreover, the optimal actions in a particular period will have to be chosen based on knowledge of how decisions will be made in future periods, because the realizations in the present period for the asset returns affect not just the portfolio outcome for the present period, but also the
1721:
is gradual entry into risky assets; it is frequently advocated by investment advisors. As indicated above, it is not confirmed by models with log utility. However, it can emerge from an intertemporal mean-variance model with negative serial correlation of returns.
954:
distributions of observed outcomes for all assets match their ex ante probability distributions. The Kelly criterion gives rise to the same portfolio decisions as does the maximization of the expected value of the log utility function as described above.
1382:
1730:
With HARA utility, asset returns that are independently and identically distributed through time, and a risk-free asset, risky asset proportions are independent of the investor's remaining lifetime. Under certain assumptions including
1689:
550:
1256:
1271:—that is, if the realization of the return on any asset in any period is not related to the realization of the return on any asset in any other period—then this expected utility expression becomes
1739:(1,1) process, a necessary but not sufficient condition for increasing conservatism (decreasing holding of the risky asset) over time (which is often advocated by investment advisors) is negative first-order
1567:
1130:
890:
421:
1014:
172:
1433:
1277:
1776:
through time, and with a risk-free asset, one can obtain an explicit solution for the demand for the unique optimal portfolio, and that demand is linear in initial wealth.
329:
923:
621:
583:
296:
240:
126:
714:
683:
656:
266:
202:
1584:
These considerations apply to utility functions in general with the exceptions noted previously. In general the expected utility expression to be maximized is
1590:
1023:< 1. With this utility function instead of the log one, the above analysis leads to the following expected utility expression to be maximized:
970:, a property that tends to cause decisions to scale up proportionately without change as initial wealth increases. The power utility function is
208:
portfolio return in any period (the imperfectly predictable amount that the average dollar in the portfolio grows or shrinks to in a given period
941:
for the optimal shares in a particular period do not contain the stochastic return information or the decision information for any other period.
1452:
current-period portfolio, one needs to know no future distributional information about the asset returns except the future risk-free returns.
432:
1141:
1795:
1765:
1911:
1448:
82:
1736:
71:
1743:, while non-negative first-order serial correlation gives the opposite result of increased risk-taking at later points in time.
1706:
The mathematical method of dealing with this need for current decision-making to take into account future decision-making is
1578:
177:
then decisions are intertemporally separate. Let initial wealth (the amount that is investable in the initial period) be
1466:
1029:
722:
353:
1444:
1924:
Balvers, Ronald J., and
Mitchell, Douglas W., "Autocorrelated returns and optimal intertemporal portfolio choice",
937:
involves taking the derivatives of one additively separate expression with respect to the various shares, and the
1790:
976:
967:
950:
133:
1854:
585:
refers to the stochastic return (the imperfectly predictable amount that the average dollar grows to) of asset
1756:
1377:{\displaystyle a\cdot W_{0}^{a}\cdot {\text{E}}R_{1}^{a}\cdot {\text{E}}R_{2}^{a}\cdots {\text{E}}R_{T}^{a};}
1993:
1398:
1998:
1945:
1815:
1909:
Balvers, Ronald J., and
Mitchell, Douglas W., "Efficient gradualism in intertemporal portfolios",
1810:
59:
963:
39:. Since the returns on almost all assets are not fully predictable, the criterion has to take
1838:
1718:
938:
301:
1387:
maximization of this expected utility expression is equivalent to separate maximization (if
898:
596:
558:
271:
215:
101:
1800:
1768:
with hyperbolic absolute risk aversion, with asset returns whose evolution is described by
692:
661:
634:
244:
180:
8:
1875:
1805:
1732:
1707:
36:
1460:
As per the above, the expected utility of final wealth with a power utility function is
51:
of the value of the portfolio after a certain number of time periods—that is, the
1926:
1892:
1843:
1740:
62:
that are attained by withdrawing some funds from the portfolio after each time period.
56:
1958:
1850:
1962:
1954:
1940:
1884:
1761:
52:
48:
20:
1785:
1769:
205:
28:
44:
40:
1943:(1971). "Optimum Consumption and Portfolio Rules in a Continuous-Time Model".
331:
inherited from the previous period that are allocated at the start of period
55:
of final wealth. Alternatively, it may be a function of the various levels of
1987:
1684:{\displaystyle {\text{E}}U(W_{T})={\text{E}}U(W_{0}R_{1}R_{2}\cdots R_{T}),}
32:
95:
1967:
1896:
1870:
1573:
previous-period realizations, and so cannot be determined in advance.
545:{\displaystyle R_{t}=w_{1t}r_{1t}+w_{2t}r_{2t}+\cdots +w_{nt}r_{nt},}
1251:{\displaystyle R_{t}=w_{1t}r_{1t}+w_{2t}r_{2t}+\cdots +w_{nt}r_{nt}}
1888:
35:
some criterion. The set of asset proportions at any time defines a
658:
above to express outcome-contingent utility, substituting in for
72:
Stochastic programming § Multistage portfolio optimization
81:
In a general context the optimal portfolio allocation in any
24:
1773:
268:
depends on the portfolio allocation—the fractions
1973:(Chapter I of his Ph.D. dissertation; Chapter 5 in his
716:
gives the expected utility expression to be maximized:
1581:
for future asset returns and hence future decisions.
929:
are additively separate, giving rise to the result of
94:
If the investor's utility function is the risk averse
1593:
1469:
1401:
1280:
1144:
1032:
979:
901:
725:
695:
664:
637:
599:
561:
435:
356:
304:
274:
247:
218:
183:
136:
104:
1269:
If there is serial independence of the asset returns
1019:with positive or negative, but non-zero, parameter
1873:(1968). "Optimal multiperiod portfolio policies".
1842:
1683:
1561:
1427:
1376:
1250:
1124:
1008:
917:
884:
708:
677:
650:
615:
577:
544:
415:
323:
290:
260:
234:
196:
166:
120:
1562:{\displaystyle a\cdot W_{0}^{a}\cdot {\text{E}}.}
1125:{\displaystyle a\cdot W_{0}^{a}\cdot {\text{E}},}
885:{\displaystyle \ln+\sum _{t=1}^{T}{\text{E}}\ln.}
631:) are constrained to sum to 1. Taking the log of
416:{\displaystyle W_{T}=W_{0}R_{1}R_{2}\cdots R_{T}}
1985:
933:: optimizing for any particular decision period
931:intertemporal independence of optimal decisions
966:for any value of the power parameter exhibits
689:, and taking the expected value of the log of
1735:and a single asset with returns following an
76:
43:into account. Typically the criterion is the
31:, repeatedly over time, in such a way as to
1833:
1831:
1455:
1009:{\displaystyle {\text{Utility}}=aW_{T}^{a}}
167:{\displaystyle {\text{Utility}}=\ln W_{T},}
1966:
1837:
1796:Intertemporal capital asset pricing model
1774:independently and identically distributed
1449:von Neumann-Morgenstern utility functions
1912:Journal of Economic Dynamics and Control
1849:. Totowa, NJ: Rowman & Littlefield.
1828:
1713:
1447:(HARA) is a feature of a broad class of
895:The terms containing the choice shares
1986:
1939:
1869:
1701:
1930:43(11), November 1997, pp. 1537-1551.
1579:conditional probability distributions
1428:{\displaystyle {\text{E}}R_{t}^{a}.}
1845:Theory of Financial Decision Making
962:Like the log utility function, the
19:is the process of allocating one's
13:
1750:
944:
14:
2010:
1903:
1445:Hyperbolic absolute risk aversion
957:
65:
1791:Intertemporal budget constraint
1439:
968:constant relative risk aversion
1933:
1918:
1863:
1725:
1675:
1629:
1615:
1602:
1553:
1499:
1116:
1062:
876:
783:
745:
732:
89:
17:Intertemporal portfolio choice
1:
1821:
1959:10.1016/0022-0531(71)90038-X
1395:<0) of each of the terms
7:
1779:
1391:>0) or minimization (if
10:
2015:
1946:Journal of Economic Theory
1757:Merton's portfolio problem
1754:
77:Time-independent decisions
69:
1816:Two-moment decision model
1698:is the utility function.
98:function of final wealth
1456:Time-dependent decisions
1975:Continuous-Time Finance
1811:Modern portfolio theory
593:, and where the shares
324:{\displaystyle W_{t-1}}
1839:Ingersoll, Jonathan E.
1685:
1563:
1429:
1378:
1252:
1126:
1010:
964:power utility function
939:first-order conditions
919:
918:{\displaystyle w_{it}}
886:
771:
710:
679:
652:
617:
616:{\displaystyle w_{it}}
579:
578:{\displaystyle r_{it}}
546:
417:
325:
292:
291:{\displaystyle w_{it}}
262:
236:
235:{\displaystyle R_{t}.}
198:
168:
122:
121:{\displaystyle W_{T},}
1719:Dollar cost averaging
1714:Dollar cost averaging
1686:
1564:
1430:
1379:
1261:for each time period
1253:
1127:
1011:
920:
887:
751:
711:
709:{\displaystyle W_{T}}
680:
678:{\displaystyle R_{t}}
653:
651:{\displaystyle W_{T}}
618:
580:
547:
418:
326:
293:
263:
261:{\displaystyle R_{t}}
237:
199:
197:{\displaystyle W_{0}}
169:
123:
1801:Intertemporal choice
1591:
1467:
1399:
1278:
1142:
1030:
977:
899:
723:
693:
662:
635:
597:
559:
433:
354:
302:
272:
245:
216:
181:
134:
102:
1994:Financial economics
1876:Journal of Business
1806:Investment strategy
1733:exponential utility
1708:dynamic programming
1702:Dynamic programming
1552:
1534:
1516:
1490:
1421:
1370:
1347:
1324:
1301:
1115:
1097:
1079:
1053:
1005:
1999:Portfolio theories
1927:Management Science
1741:serial correlation
1681:
1559:
1538:
1520:
1502:
1476:
1425:
1407:
1374:
1356:
1333:
1310:
1287:
1248:
1122:
1101:
1083:
1065:
1039:
1006:
991:
915:
882:
706:
675:
648:
613:
575:
542:
413:
321:
298:of current wealth
288:
258:
232:
194:
164:
118:
57:goods and services
23:wealth to various
1941:Merton, Robert C.
1624:
1597:
1497:
1405:
1354:
1331:
1308:
1060:
983:
775:
140:
2006:
1978:
1972:
1970:
1937:
1931:
1922:
1916:
1915:24, 2000, 21-38.
1907:
1901:
1900:
1867:
1861:
1860:
1848:
1835:
1762:Robert C. Merton
1690:
1688:
1687:
1682:
1674:
1673:
1661:
1660:
1651:
1650:
1641:
1640:
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1614:
1613:
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1568:
1566:
1565:
1560:
1551:
1546:
1533:
1528:
1515:
1510:
1498:
1495:
1489:
1484:
1434:
1432:
1431:
1426:
1420:
1415:
1406:
1403:
1383:
1381:
1380:
1375:
1369:
1364:
1355:
1352:
1346:
1341:
1332:
1329:
1323:
1318:
1309:
1306:
1300:
1295:
1257:
1255:
1254:
1249:
1247:
1246:
1234:
1233:
1212:
1211:
1199:
1198:
1183:
1182:
1170:
1169:
1154:
1153:
1135:where as before
1131:
1129:
1128:
1123:
1114:
1109:
1096:
1091:
1078:
1073:
1061:
1058:
1052:
1047:
1015:
1013:
1012:
1007:
1004:
999:
984:
981:
924:
922:
921:
916:
914:
913:
891:
889:
888:
883:
875:
874:
862:
861:
840:
839:
827:
826:
811:
810:
798:
797:
776:
773:
770:
765:
744:
743:
715:
713:
712:
707:
705:
704:
684:
682:
681:
676:
674:
673:
657:
655:
654:
649:
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646:
622:
620:
619:
614:
612:
611:
584:
582:
581:
576:
574:
573:
551:
549:
548:
543:
538:
537:
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524:
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502:
490:
489:
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461:
460:
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414:
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411:
399:
398:
389:
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379:
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366:
365:
330:
328:
327:
322:
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297:
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289:
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286:
267:
265:
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259:
257:
256:
241:
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228:
227:
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192:
173:
171:
170:
165:
160:
159:
141:
138:
127:
125:
124:
119:
114:
113:
53:expected utility
49:concave function
29:financial assets
2014:
2013:
2009:
2008:
2007:
2005:
2004:
2003:
1984:
1983:
1982:
1981:
1938:
1934:
1923:
1919:
1908:
1904:
1868:
1864:
1857:
1836:
1829:
1824:
1786:Decision theory
1782:
1770:Brownian motion
1766:continuous time
1764:showed that in
1759:
1753:
1751:Continuous time
1728:
1716:
1704:
1669:
1665:
1656:
1652:
1646:
1642:
1636:
1632:
1621:
1609:
1605:
1594:
1592:
1589:
1588:
1547:
1542:
1529:
1524:
1511:
1506:
1494:
1485:
1480:
1468:
1465:
1464:
1458:
1442:
1416:
1411:
1402:
1400:
1397:
1396:
1365:
1360:
1351:
1342:
1337:
1328:
1319:
1314:
1305:
1296:
1291:
1279:
1276:
1275:
1239:
1235:
1226:
1222:
1204:
1200:
1191:
1187:
1175:
1171:
1162:
1158:
1149:
1145:
1143:
1140:
1139:
1110:
1105:
1092:
1087:
1074:
1069:
1057:
1048:
1043:
1031:
1028:
1027:
1000:
995:
980:
978:
975:
974:
960:
951:Kelly criterion
947:
945:Kelly criterion
906:
902:
900:
897:
896:
867:
863:
854:
850:
832:
828:
819:
815:
803:
799:
790:
786:
772:
766:
755:
739:
735:
724:
721:
720:
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694:
691:
690:
669:
665:
663:
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659:
642:
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632:
604:
600:
598:
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594:
566:
562:
560:
557:
556:
530:
526:
517:
513:
495:
491:
482:
478:
466:
462:
453:
449:
440:
436:
434:
431:
430:
407:
403:
394:
390:
384:
380:
374:
370:
361:
357:
355:
352:
351:
309:
305:
303:
300:
299:
279:
275:
273:
270:
269:
252:
248:
246:
243:
242:
223:
219:
217:
214:
213:
188:
184:
182:
179:
178:
155:
151:
137:
135:
132:
131:
109:
105:
103:
100:
99:
92:
79:
74:
68:
12:
11:
5:
2012:
2002:
2001:
1996:
1980:
1979:
1953:(4): 373–413.
1932:
1917:
1902:
1889:10.1086/295078
1883:(2): 215–229.
1862:
1855:
1826:
1825:
1823:
1820:
1819:
1818:
1813:
1808:
1803:
1798:
1793:
1788:
1781:
1778:
1772:and which are
1755:Main article:
1752:
1749:
1727:
1724:
1715:
1712:
1703:
1700:
1692:
1691:
1680:
1677:
1672:
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1631:
1628:
1620:
1617:
1612:
1608:
1604:
1601:
1570:
1569:
1558:
1555:
1550:
1545:
1541:
1537:
1532:
1527:
1523:
1519:
1514:
1509:
1505:
1501:
1493:
1488:
1483:
1479:
1475:
1472:
1457:
1454:
1441:
1438:
1424:
1419:
1414:
1410:
1385:
1384:
1373:
1368:
1363:
1359:
1350:
1345:
1340:
1336:
1327:
1322:
1317:
1313:
1304:
1299:
1294:
1290:
1286:
1283:
1259:
1258:
1245:
1242:
1238:
1232:
1229:
1225:
1221:
1218:
1215:
1210:
1207:
1203:
1197:
1194:
1190:
1186:
1181:
1178:
1174:
1168:
1165:
1161:
1157:
1152:
1148:
1133:
1132:
1121:
1118:
1113:
1108:
1104:
1100:
1095:
1090:
1086:
1082:
1077:
1072:
1068:
1064:
1056:
1051:
1046:
1042:
1038:
1035:
1017:
1016:
1003:
998:
994:
990:
987:
959:
956:
946:
943:
925:for differing
912:
909:
905:
893:
892:
881:
878:
873:
870:
866:
860:
857:
853:
849:
846:
843:
838:
835:
831:
825:
822:
818:
814:
809:
806:
802:
796:
793:
789:
785:
782:
779:
769:
764:
761:
758:
754:
750:
747:
742:
738:
734:
731:
728:
703:
699:
672:
668:
645:
641:
610:
607:
603:
572:
569:
565:
553:
552:
541:
536:
533:
529:
523:
520:
516:
512:
509:
506:
501:
498:
494:
488:
485:
481:
477:
472:
469:
465:
459:
456:
452:
448:
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439:
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423:
410:
406:
402:
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393:
387:
383:
377:
373:
369:
364:
360:
318:
315:
312:
308:
285:
282:
278:
255:
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231:
226:
222:
191:
187:
175:
174:
163:
158:
154:
150:
147:
144:
117:
112:
108:
91:
88:
78:
75:
67:
64:
45:expected value
41:financial risk
9:
6:
4:
3:
2:
2011:
2000:
1997:
1995:
1992:
1991:
1989:
1976:
1969:
1964:
1960:
1956:
1952:
1948:
1947:
1942:
1936:
1929:
1928:
1921:
1914:
1913:
1906:
1898:
1894:
1890:
1886:
1882:
1878:
1877:
1872:
1866:
1858:
1852:
1847:
1846:
1840:
1834:
1832:
1827:
1817:
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1797:
1794:
1792:
1789:
1787:
1784:
1783:
1777:
1775:
1771:
1767:
1763:
1758:
1748:
1744:
1742:
1738:
1734:
1723:
1720:
1711:
1709:
1699:
1697:
1678:
1670:
1666:
1662:
1657:
1653:
1647:
1643:
1637:
1633:
1626:
1618:
1610:
1606:
1599:
1587:
1586:
1585:
1582:
1580:
1574:
1556:
1548:
1543:
1539:
1535:
1530:
1525:
1521:
1517:
1512:
1507:
1503:
1491:
1486:
1481:
1477:
1473:
1470:
1463:
1462:
1461:
1453:
1450:
1446:
1437:
1422:
1417:
1412:
1408:
1394:
1390:
1371:
1366:
1361:
1357:
1348:
1343:
1338:
1334:
1325:
1320:
1315:
1311:
1302:
1297:
1292:
1288:
1284:
1281:
1274:
1273:
1272:
1270:
1266:
1264:
1243:
1240:
1236:
1230:
1227:
1223:
1219:
1216:
1213:
1208:
1205:
1201:
1195:
1192:
1188:
1184:
1179:
1176:
1172:
1166:
1163:
1159:
1155:
1150:
1146:
1138:
1137:
1136:
1119:
1111:
1106:
1102:
1098:
1093:
1088:
1084:
1080:
1075:
1070:
1066:
1054:
1049:
1044:
1040:
1036:
1033:
1026:
1025:
1024:
1022:
1001:
996:
992:
988:
985:
973:
972:
971:
969:
965:
958:Power utility
955:
952:
942:
940:
936:
932:
928:
910:
907:
903:
879:
871:
868:
864:
858:
855:
851:
847:
844:
841:
836:
833:
829:
823:
820:
816:
812:
807:
804:
800:
794:
791:
787:
780:
777:
767:
762:
759:
756:
752:
748:
740:
736:
729:
726:
719:
718:
717:
701:
697:
688:
670:
666:
643:
639:
630:
626:
608:
605:
601:
592:
588:
570:
567:
563:
539:
534:
531:
527:
521:
518:
514:
510:
507:
504:
499:
496:
492:
486:
483:
479:
475:
470:
467:
463:
457:
454:
450:
446:
441:
437:
429:
428:
427:
408:
404:
400:
395:
391:
385:
381:
375:
371:
367:
362:
358:
350:
349:
348:
346:
342:
338:
334:
316:
313:
310:
306:
283:
280:
276:
253:
249:
229:
224:
220:
211:
207:
189:
185:
161:
156:
152:
148:
145:
142:
130:
129:
128:
115:
110:
106:
97:
87:
84:
73:
66:Discrete time
63:
61:
58:
54:
50:
46:
42:
38:
34:
30:
27:, especially
26:
22:
18:
1974:
1968:1721.1/63980
1950:
1944:
1935:
1925:
1920:
1910:
1905:
1880:
1874:
1865:
1844:
1760:
1745:
1729:
1717:
1705:
1695:
1693:
1583:
1575:
1571:
1459:
1443:
1440:HARA utility
1392:
1388:
1386:
1268:
1267:
1262:
1260:
1134:
1020:
1018:
961:
948:
934:
930:
926:
894:
686:
628:
624:
590:
586:
554:
425:
344:
340:
336:
332:
209:
204:and let the
176:
93:
80:
16:
15:
1871:Mossin, Jan
1726:Age effects
589:for period
96:log utility
90:Log utility
83:time period
60:consumption
1988:Categories
1856:0847673596
1822:References
335:to assets
206:stochastic
70:See also:
21:investable
1663:⋯
1536:⋯
1518:⋅
1492:⋅
1474:⋅
1349:⋯
1326:⋅
1303:⋅
1285:⋅
1217:⋯
1099:⋯
1081:⋅
1055:⋅
1037:⋅
845:⋯
781:
753:∑
730:
685:for each
627:=1, ...,
508:⋯
401:⋯
343:=1, ...,
314:−
149:
37:portfolio
1841:(1987).
1780:See also
47:of some
33:optimize
1897:2351447
982:Utility
347:). So:
139:Utility
1895:
1853:
1694:where
555:where
426:where
25:assets
1893:JSTOR
212:) be
1851:ISBN
1737:ARMA
949:The
1963:hdl
1955:doi
1885:doi
1265:.
1990::
1977:).
1961:.
1949:.
1891:.
1881:41
1879:.
1830:^
778:ln
727:ln
146:ln
1971:.
1965::
1957::
1951:3
1899:.
1887::
1859:.
1696:U
1679:,
1676:)
1671:T
1667:R
1658:2
1654:R
1648:1
1644:R
1638:0
1634:W
1630:(
1627:U
1623:E
1619:=
1616:)
1611:T
1607:W
1603:(
1600:U
1596:E
1557:.
1554:]
1549:a
1544:T
1540:R
1531:a
1526:2
1522:R
1513:a
1508:1
1504:R
1500:[
1496:E
1487:a
1482:0
1478:W
1471:a
1423:.
1418:a
1413:t
1409:R
1404:E
1393:a
1389:a
1372:;
1367:a
1362:T
1358:R
1353:E
1344:a
1339:2
1335:R
1330:E
1321:a
1316:1
1312:R
1307:E
1298:a
1293:0
1289:W
1282:a
1263:t
1244:t
1241:n
1237:r
1231:t
1228:n
1224:w
1220:+
1214:+
1209:t
1206:2
1202:r
1196:t
1193:2
1189:w
1185:+
1180:t
1177:1
1173:r
1167:t
1164:1
1160:w
1156:=
1151:t
1147:R
1120:,
1117:]
1112:a
1107:T
1103:R
1094:a
1089:2
1085:R
1076:a
1071:1
1067:R
1063:[
1059:E
1050:a
1045:0
1041:W
1034:a
1021:a
1002:a
997:T
993:W
989:a
986:=
935:t
927:t
911:t
908:i
904:w
880:.
877:]
872:t
869:n
865:r
859:t
856:n
852:w
848:+
842:+
837:t
834:2
830:r
824:t
821:2
817:w
813:+
808:t
805:1
801:r
795:t
792:1
788:w
784:[
774:E
768:T
763:1
760:=
757:t
749:+
746:]
741:0
737:W
733:[
702:T
698:W
687:t
671:t
667:R
644:T
640:W
629:n
625:i
623:(
609:t
606:i
602:w
591:t
587:i
571:t
568:i
564:r
540:,
535:t
532:n
528:r
522:t
519:n
515:w
511:+
505:+
500:t
497:2
493:r
487:t
484:2
480:w
476:+
471:t
468:1
464:r
458:t
455:1
451:w
447:=
442:t
438:R
409:T
405:R
396:2
392:R
386:1
382:R
376:0
372:W
368:=
363:T
359:W
345:n
341:i
339:(
337:i
333:t
317:1
311:t
307:W
284:t
281:i
277:w
254:t
250:R
230:.
225:t
221:R
210:t
190:0
186:W
162:,
157:T
153:W
143:=
116:,
111:T
107:W
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