1132:
453:(including both principal and interest), while WAL is the weighted-average time to receive simply the principal payments (not including interest, and not discounting). For an amortizing loan with equal payments, the WAL will be higher than the duration, as the early payments are weighted towards interest, while the later payments are weighted towards principal, and further, taking present value (in duration) discounts the later payments.
816:
1127:{\displaystyle {\begin{aligned}{\text{WAL}}&=\sum _{i=1}^{n}{\frac {P_{i}}{P}}t_{i}\\{\text{WAL}}\times P&=\sum _{i=1}^{n}P_{i}t_{i}&&=\sum _{i=1}^{n}P_{i}{\frac {i}{12}}\\{\text{WAL}}\times P\times r&=\sum _{i=1}^{n}iP_{i}{\frac {r}{12}}&&={\frac {r}{12}}\sum _{i=1}^{n}iP_{i}\end{aligned}}}
505:
WAL is a measure that can be useful in credit risk analysis on fixed income securities, bearing in mind that the main credit risk of a loan is the risk of loss of principal. All else equal, a bond with principal outstanding longer (i.e., longer WAL) has greater credit risk than a bond with shorter
417:, the WAL cannot be computed from the amortization schedule alone; one must also make assumptions about the prepayment and default behavior, and the quoted WAL will be an estimate. The WAL is usually computed from a single cash-flow sequence. Occasionally, a
1243:
616:
Note that as the interest rate increases, WAL increases, since the principal payments become increasingly back-loaded. WAL is independent of the principal balance, though payments and total interest are proportional to principal.
536:
On a 30-year amortizing loan, paying equal amounts monthly, one has the following WALs, for the given annual interest rates (and corresponding monthly payments per $ 100,000 principal balance, calculated via an
120:
1370:
1970:
513:
WAL should not be used to estimate a bond's price-sensitivity to interest-rate fluctuations, as WAL includes only the principal cash flows, omitting the interest payments. Instead, one should use
821:
1537:
1884:
714:
379:
479:
than half the principal will have been paid off at the WAL. Intuitively, this is because most of the principal repayment happens at the end. Formally, the distribution of repayments has
1614:
766:
More rigorously, one can derive the result as follows. To ease exposition, assume that payments are monthly, so periodic interest rate is annual interest rate divided by 12, and time
1915:
756:
666:
231:
1652:
805:
399:
1709:
1400:
1679:
1435:
1273:
331:
280:
175:
1830:
1801:
300:
251:
195:
146:
1143:
620:
For a coupon of 0%, where the principal amortizes linearly, the WAL is exactly half the tenor plus half a payment period, because principal is repaid
1654:. This is demonstrated in the following table, which shows the amortization schedule, broken up into principal repayments, where each column is a
1372:. Both of these quantities are the time-weighted total principal of the bond (in periods), and they are simply different ways of slicing it: the
730:
This can be understood intuitively as: "The average dollar of principal is outstanding for the WAL, hence the interest on the average dollar is
49:
1566:
For instance, if the principal amortized as $ 100, $ 80, $ 50 (with paydowns of $ 20, $ 30, $ 50), then the sum would on the one hand be
2070:
2032:
2015:
1290:
2110:
1923:
2049:
1451:
1838:
682:
336:
1569:
489:
2115:
2085:
538:
483:: the small principal repayments at the beginning drag down the WAL (mean) more than they reduce the median.
1892:
733:
631:
422:
2090:
202:
1619:
1563: − 1 periods, and so forth. Using this, the sums can be re-arranged to be equal.
414:
769:
2066:
384:
1780:, one can compute the WAL without knowing the amortization schedule. The total payments are
1684:
1375:
472:
33:
1657:
1413:
1251:
309:
258:
153:
2095:
1806:
1776:
The above can be reversed: given the terms (principal, tenor, rate) and amortized payment
402:
25:
533:(non-amortizing) is exactly the tenor, as the principal is repaid precisely at maturity.
1999:
1783:
1238:{\displaystyle \sum _{i=1}^{n}Q_{i}{\frac {r}{12}}={\frac {r}{12}}\sum _{i=1}^{n}Q_{i},}
1988:
285:
236:
180:
131:
2104:
514:
480:
445:
440:
530:
621:
541:
and the formulas below relating amortized payments, total interest, and WAL):
506:
WAL. In particular, WAL is often used as the basis for yield comparisons in
758:, and now one multiplies by the principal to get total interest payments."
507:
421:
may be computed from multiple cash-flow scenarios, such as those from an
676:
WAL allows one to easily compute the total interest payments, given by:
17:
1889:
Similarly, the total interest as percentage of principal is given by
628:
of the period). So for a 30-year 0% loan, paying monthly, the WAL is
468:
464:
115:{\displaystyle {\text{WAL}}=\sum _{i=1}^{n}{\frac {P_{i}}{P}}t_{i},}
233:
is the fraction of the total principal that is included in payment
1287:
interest payment is based). The statement reduces to showing that
433:
WAL should not be confused with the following distinct concepts:
460:
40:: it's the average time until a dollar of principal is repaid.
1559: − 1)th) principal payment is outstanding for
1555:
periods, while the principal paid off by the second to last ((
282:
is the time (in years) from the calculation date to payment
1365:{\displaystyle \sum _{i=1}^{n}iP_{i}=\sum _{i=1}^{n}Q_{i}}
496:
of multiple loans, not an average of principal repayments.
1547:
principal payments. The principal paid off by the last (
443:
is the weighted-average time to receive the discounted
177:
is the principal repayment that is included in payment
1965:{\displaystyle {\text{WAL}}\times r={\frac {An-P}{P}}}
1926:
1895:
1841:
1809:
1786:
1687:
1660:
1622:
1572:
1454:
1416:
1378:
1293:
1254:
1146:
819:
807:(time in years is period number in months, over 12).
772:
736:
685:
634:
387:
339:
312:
288:
261:
239:
205:
183:
156:
134:
52:
1406:each dollar of principal is outstanding (it slices
1964:
1909:
1878:
1824:
1795:
1703:
1673:
1646:
1608:
1531:
1429:
1394:
1364:
1267:
1237:
1126:
799:
750:
708:
660:
393:
373:
325:
294:
274:
245:
225:
189:
169:
140:
114:
1532:{\displaystyle Q_{n}=P_{n},Q_{n-1}=P_{n}+P_{n-1}}
1879:{\displaystyle {\text{WAL}}={\frac {An-P}{Pr}}}
1539:, and so forth: the principal outstanding when
456:Time until 50% of the principal has been repaid
1543:periods remain is exactly the sum of the next
475:(of time) for a flat payment amortizing loan,
1551:th) principal payment is outstanding for all
709:{\displaystyle {\text{WAL}}\times r\times P,}
8:
374:{\displaystyle {\frac {1}{12}}(i+\alpha -1)}
463:, while "50% of the principal repaid" is a
1609:{\displaystyle 20+2\cdot 30+3\cdot 50=230}
1941:
1927:
1925:
1896:
1894:
1850:
1842:
1840:
1808:
1785:
1695:
1686:
1665:
1659:
1621:
1571:
1517:
1504:
1485:
1472:
1459:
1453:
1437:counts how much principal is outstanding
1421:
1415:
1386:
1377:
1356:
1346:
1335:
1322:
1309:
1298:
1292:
1259:
1253:
1226:
1216:
1205:
1191:
1178:
1172:
1162:
1151:
1145:
1114:
1101:
1090:
1076:
1058:
1052:
1039:
1028:
1000:
986:
980:
970:
959:
941:
931:
921:
910:
888:
878:
863:
857:
851:
840:
824:
820:
818:
789:
777:
771:
737:
735:
686:
684:
644:
633:
386:
340:
338:
317:
311:
287:
266:
260:
238:
212:
206:
204:
182:
161:
155:
133:
103:
88:
82:
76:
65:
53:
51:
1713:
543:
2063:The handbook of fixed income securities
2045:
2028:
2011:
1981:
7:
1910:{\displaystyle {\text{WAL}}\times r}
1803:and the total interest payments are
1772:Computing WAL from amortized payment
1275:is the principal outstanding at the
751:{\displaystyle {\text{WAL}}\times r}
661:{\displaystyle 15+1/24\approx 15.04}
471:. Since principal outstanding is a
401:is the fraction of a month between
469:difference between mean and median
226:{\displaystyle {\frac {P_{i}}{P}}}
14:
1616:, and on the other hand would be
1283:(it's the principal on which the
723:is the annual interest rate and
28:or amortizing bond, also called
486:Weighted-average maturity (WAM)
368:
350:
1:
1647:{\displaystyle 100+80+50=230}
607:$ 270,229.60/($ 100,000*12%)
596:
590:$ 164,153.60/($ 100,000*8%)
579:
562:
22:weighted-average life (WAL)
2132:
2061:Fabozzi, Frank J. (2000),
800:{\displaystyle t_{i}=i/12}
727:is the initial principal.
573:$ 71,871.20/($ 100,000*4%)
405:and first cash flow date.
381:for a monthly bond, where
148:is the (total) principal,
2086:Amortization calculator
539:amortization calculator
409:WAL of classes of loans
394:{\displaystyle \alpha }
1966:
1911:
1880:
1826:
1797:
1705:
1704:{\displaystyle iP_{i}}
1675:
1648:
1610:
1533:
1431:
1396:
1395:{\displaystyle iP_{i}}
1366:
1351:
1314:
1269:
1239:
1221:
1167:
1128:
1106:
1044:
975:
926:
856:
801:
752:
710:
662:
423:option-adjusted spread
419:simulated average life
395:
375:
327:
296:
276:
247:
227:
191:
171:
142:
116:
81:
2111:Fixed income analysis
2091:Amortization schedule
1967:
1912:
1881:
1827:
1798:
1706:
1676:
1674:{\displaystyle Q_{i}}
1649:
1611:
1534:
1439:at each point in time
1432:
1430:{\displaystyle Q_{i}}
1397:
1367:
1331:
1294:
1270:
1268:{\displaystyle Q_{i}}
1240:
1201:
1147:
1129:
1086:
1024:
955:
906:
836:
802:
753:
711:
663:
517:, which incorporates
492:is an average of the
396:
376:
328:
326:{\displaystyle t_{i}}
297:
277:
275:{\displaystyle t_{i}}
248:
228:
192:
172:
170:{\displaystyle P_{i}}
143:
117:
61:
1976:Notes and references
1924:
1893:
1839:
1825:{\displaystyle An-P}
1807:
1784:
1685:
1658:
1620:
1570:
1452:
1414:
1376:
1291:
1252:
1144:
817:
770:
734:
683:
632:
413:In loans that allow
385:
337:
310:
286:
259:
237:
203:
181:
154:
132:
50:
38:principal repayments
36:of the times of the
1448:Working backwards,
333:can be expanded as
2000:Bloomberg Glossary
1962:
1907:
1876:
1822:
1796:{\displaystyle An}
1793:
1701:
1681:, and each row is
1671:
1644:
1606:
1529:
1427:
1392:
1362:
1265:
1235:
1137:Total interest is
1124:
1122:
797:
748:
706:
658:
391:
371:
323:
292:
272:
243:
223:
187:
167:
138:
112:
1960:
1930:
1899:
1874:
1845:
1832:, so the WAL is:
1769:
1768:
1199:
1186:
1084:
1066:
1003:
994:
891:
872:
827:
740:
689:
614:
613:
348:
295:{\displaystyle i}
246:{\displaystyle i}
221:
190:{\displaystyle i}
141:{\displaystyle P}
97:
56:
2123:
2075:
2053:
2042:
2036:
2025:
2019:
2008:
2002:
1997:
1991:
1986:
1971:
1969:
1968:
1963:
1961:
1956:
1942:
1931:
1928:
1916:
1914:
1913:
1908:
1900:
1897:
1885:
1883:
1882:
1877:
1875:
1873:
1865:
1851:
1846:
1843:
1831:
1829:
1828:
1823:
1802:
1800:
1799:
1794:
1714:
1710:
1708:
1707:
1702:
1700:
1699:
1680:
1678:
1677:
1672:
1670:
1669:
1653:
1651:
1650:
1645:
1615:
1613:
1612:
1607:
1538:
1536:
1535:
1530:
1528:
1527:
1509:
1508:
1496:
1495:
1477:
1476:
1464:
1463:
1436:
1434:
1433:
1428:
1426:
1425:
1401:
1399:
1398:
1393:
1391:
1390:
1371:
1369:
1368:
1363:
1361:
1360:
1350:
1345:
1327:
1326:
1313:
1308:
1274:
1272:
1271:
1266:
1264:
1263:
1244:
1242:
1241:
1236:
1231:
1230:
1220:
1215:
1200:
1192:
1187:
1179:
1177:
1176:
1166:
1161:
1133:
1131:
1130:
1125:
1123:
1119:
1118:
1105:
1100:
1085:
1077:
1069:
1067:
1059:
1057:
1056:
1043:
1038:
1004:
1001:
995:
987:
985:
984:
974:
969:
948:
946:
945:
936:
935:
925:
920:
892:
889:
883:
882:
873:
868:
867:
858:
855:
850:
828:
825:
806:
804:
803:
798:
793:
782:
781:
757:
755:
754:
749:
741:
738:
715:
713:
712:
707:
690:
687:
667:
665:
664:
659:
648:
544:
521:the cash flows.
473:concave function
429:Related concepts
400:
398:
397:
392:
380:
378:
377:
372:
349:
341:
332:
330:
329:
324:
322:
321:
301:
299:
298:
293:
281:
279:
278:
273:
271:
270:
252:
250:
249:
244:
232:
230:
229:
224:
222:
217:
216:
207:
196:
194:
193:
188:
176:
174:
173:
168:
166:
165:
147:
145:
144:
139:
121:
119:
118:
113:
108:
107:
98:
93:
92:
83:
80:
75:
57:
54:
34:weighted average
2131:
2130:
2126:
2125:
2124:
2122:
2121:
2120:
2101:
2100:
2096:Amortizing loan
2082:
2073:
2060:
2057:
2056:
2043:
2039:
2026:
2022:
2009:
2005:
1998:
1994:
1987:
1983:
1978:
1943:
1922:
1921:
1891:
1890:
1866:
1852:
1837:
1836:
1805:
1804:
1782:
1781:
1774:
1691:
1683:
1682:
1661:
1656:
1655:
1618:
1617:
1568:
1567:
1513:
1500:
1481:
1468:
1455:
1450:
1449:
1417:
1412:
1411:
1402:sum counts how
1382:
1374:
1373:
1352:
1318:
1289:
1288:
1255:
1250:
1249:
1222:
1168:
1142:
1141:
1121:
1120:
1110:
1068:
1048:
1017:
997:
996:
976:
947:
937:
927:
899:
885:
884:
874:
859:
829:
815:
814:
773:
768:
767:
764:
732:
731:
681:
680:
674:
630:
629:
556:WAL Calculation
527:
503:
431:
411:
403:settlement date
383:
382:
335:
334:
313:
308:
307:
284:
283:
262:
257:
256:
235:
234:
208:
201:
200:
179:
178:
157:
152:
151:
130:
129:
99:
84:
48:
47:
26:amortizing loan
12:
11:
5:
2129:
2127:
2119:
2118:
2116:Bond valuation
2113:
2103:
2102:
2099:
2098:
2093:
2088:
2081:
2078:
2077:
2076:
2071:
2055:
2054:
2037:
2020:
2003:
1992:
1989:PIMCO glossary
1980:
1979:
1977:
1974:
1973:
1972:
1959:
1955:
1952:
1949:
1946:
1940:
1937:
1934:
1906:
1903:
1887:
1886:
1872:
1869:
1864:
1861:
1858:
1855:
1849:
1821:
1818:
1815:
1812:
1792:
1789:
1773:
1770:
1767:
1766:
1763:
1760:
1757:
1753:
1752:
1750:
1747:
1744:
1740:
1739:
1737:
1735:
1732:
1728:
1727:
1724:
1721:
1718:
1698:
1694:
1690:
1668:
1664:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1526:
1523:
1520:
1516:
1512:
1507:
1503:
1499:
1494:
1491:
1488:
1484:
1480:
1475:
1471:
1467:
1462:
1458:
1424:
1420:
1389:
1385:
1381:
1359:
1355:
1349:
1344:
1341:
1338:
1334:
1330:
1325:
1321:
1317:
1312:
1307:
1304:
1301:
1297:
1262:
1258:
1246:
1245:
1234:
1229:
1225:
1219:
1214:
1211:
1208:
1204:
1198:
1195:
1190:
1185:
1182:
1175:
1171:
1165:
1160:
1157:
1154:
1150:
1135:
1134:
1117:
1113:
1109:
1104:
1099:
1096:
1093:
1089:
1083:
1080:
1075:
1072:
1070:
1065:
1062:
1055:
1051:
1047:
1042:
1037:
1034:
1031:
1027:
1023:
1020:
1018:
1016:
1013:
1010:
1007:
999:
998:
993:
990:
983:
979:
973:
968:
965:
962:
958:
954:
951:
949:
944:
940:
934:
930:
924:
919:
916:
913:
909:
905:
902:
900:
898:
895:
887:
886:
881:
877:
871:
866:
862:
854:
849:
846:
843:
839:
835:
832:
830:
823:
822:
796:
792:
788:
785:
780:
776:
763:
760:
747:
744:
717:
716:
705:
702:
699:
696:
693:
673:
672:Total Interest
670:
657:
654:
651:
647:
643:
640:
637:
612:
611:
608:
605:
602:
599:
595:
594:
591:
588:
585:
582:
578:
577:
574:
571:
568:
565:
561:
560:
557:
554:
553:Total Interest
551:
548:
526:
523:
510:calculations.
502:
499:
498:
497:
494:maturity dates
487:
484:
457:
454:
446:present values
438:
430:
427:
410:
407:
390:
370:
367:
364:
361:
358:
355:
352:
347:
344:
320:
316:
304:
303:
291:
269:
265:
254:
242:
220:
215:
211:
198:
186:
164:
160:
149:
137:
123:
122:
111:
106:
102:
96:
91:
87:
79:
74:
71:
68:
64:
60:
43:In a formula,
13:
10:
9:
6:
4:
3:
2:
2128:
2117:
2114:
2112:
2109:
2108:
2106:
2097:
2094:
2092:
2089:
2087:
2084:
2083:
2079:
2074:
2072:0-87094-985-3
2068:
2064:
2059:
2058:
2051:
2047:
2041:
2038:
2034:
2030:
2024:
2021:
2017:
2013:
2007:
2004:
2001:
1996:
1993:
1990:
1985:
1982:
1975:
1957:
1953:
1950:
1947:
1944:
1938:
1935:
1932:
1920:
1919:
1918:
1904:
1901:
1870:
1867:
1862:
1859:
1856:
1853:
1847:
1835:
1834:
1833:
1819:
1816:
1813:
1810:
1790:
1787:
1779:
1771:
1764:
1761:
1758:
1755:
1754:
1751:
1748:
1745:
1742:
1741:
1738:
1736:
1733:
1730:
1729:
1725:
1722:
1719:
1716:
1715:
1712:
1696:
1692:
1688:
1666:
1662:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1564:
1562:
1558:
1554:
1550:
1546:
1542:
1524:
1521:
1518:
1514:
1510:
1505:
1501:
1497:
1492:
1489:
1486:
1482:
1478:
1473:
1469:
1465:
1460:
1456:
1446:
1444:
1440:
1422:
1418:
1410:), while the
1409:
1405:
1387:
1383:
1379:
1357:
1353:
1347:
1342:
1339:
1336:
1332:
1328:
1323:
1319:
1315:
1310:
1305:
1302:
1299:
1295:
1286:
1282:
1278:
1260:
1256:
1232:
1227:
1223:
1217:
1212:
1209:
1206:
1202:
1196:
1193:
1188:
1183:
1180:
1173:
1169:
1163:
1158:
1155:
1152:
1148:
1140:
1139:
1138:
1115:
1111:
1107:
1102:
1097:
1094:
1091:
1087:
1081:
1078:
1073:
1071:
1063:
1060:
1053:
1049:
1045:
1040:
1035:
1032:
1029:
1025:
1021:
1019:
1014:
1011:
1008:
1005:
991:
988:
981:
977:
971:
966:
963:
960:
956:
952:
950:
942:
938:
932:
928:
922:
917:
914:
911:
907:
903:
901:
896:
893:
879:
875:
869:
864:
860:
852:
847:
844:
841:
837:
833:
831:
813:
812:
811:
808:
794:
790:
786:
783:
778:
774:
761:
759:
745:
742:
728:
726:
722:
703:
700:
697:
694:
691:
679:
678:
677:
671:
669:
655:
652:
649:
645:
641:
638:
635:
627:
623:
618:
609:
606:
603:
600:
597:
592:
589:
586:
583:
580:
575:
572:
569:
566:
563:
558:
555:
552:
549:
546:
545:
542:
540:
534:
532:
529:The WAL of a
524:
522:
520:
516:
515:bond duration
511:
509:
500:
495:
491:
488:
485:
482:
481:negative skew
478:
474:
470:
466:
462:
458:
455:
452:
448:
447:
442:
441:Bond duration
439:
437:Bond duration
436:
435:
434:
428:
426:
424:
420:
416:
408:
406:
404:
388:
365:
362:
359:
356:
353:
345:
342:
318:
314:
289:
267:
263:
255:
240:
218:
213:
209:
199:
184:
162:
158:
150:
135:
128:
127:
126:
109:
104:
100:
94:
89:
85:
77:
72:
69:
66:
62:
58:
46:
45:
44:
41:
39:
35:
31:
27:
23:
19:
2062:
2046:Fabozzi 2000
2040:
2029:Fabozzi 2000
2023:
2012:Fabozzi 2000
2006:
1995:
1984:
1888:
1777:
1775:
1756:3 × 50
1743:2 × 30
1731:1 × 20
1565:
1560:
1556:
1552:
1548:
1544:
1540:
1447:
1442:
1438:
1408:horizontally
1407:
1403:
1284:
1280:
1276:
1247:
1136:
809:
765:
729:
724:
720:
718:
675:
625:
619:
615:
604:$ 270,299.60
587:$ 164,153.60
535:
528:
518:
512:
504:
501:Applications
493:
476:
450:
444:
432:
418:
412:
306:If desired,
305:
124:
42:
37:
30:average life
29:
21:
15:
2033:pp. 616–617
2016:pp. 588–589
1441:(it slices
570:$ 71,871.20
531:bullet loan
449:of all the
2105:Categories
1443:vertically
1279:of period
622:in arrears
601:$ 1,028.61
451:cash flows
415:prepayment
1951:−
1933:×
1902:×
1860:−
1817:−
1595:⋅
1583:⋅
1522:−
1490:−
1333:∑
1296:∑
1277:beginning
1203:∑
1149:∑
1088:∑
1026:∑
1012:×
1006:×
957:∑
908:∑
894:×
838:∑
743:×
698:×
692:×
653:≈
459:WAL is a
389:α
363:−
360:α
63:∑
32:, is the
2080:See also
624:(at the
584:$ 733.76
567:$ 477.42
525:Examples
508:I-spread
668:years.
550:Payment
425:model.
197:, hence
125:where:
18:finance
2069:
2050:p. 805
1248:where
810:Then:
719:where
610:22.52
593:20.52
576:17.97
467:; see
465:median
24:of an
20:, the
762:Proof
656:15.04
253:, and
2067:ISBN
1404:long
559:WAL
547:Rate
477:less
461:mean
1929:WAL
1898:WAL
1844:WAL
1765:50
1726:50
1720:100
1717:230
1642:230
1624:100
1604:230
1445:).
1002:WAL
890:WAL
826:WAL
739:WAL
688:WAL
626:end
598:12%
519:all
490:WAM
55:WAL
16:In
2107::
2065:,
2048:,
2031:,
2014:,
1917::
1762:50
1759:50
1749:30
1746:30
1734:20
1723:80
1711::
1636:50
1630:80
1598:50
1586:30
1574:20
1197:12
1184:12
1082:12
1064:12
992:12
795:12
650:24
636:15
581:8%
564:4%
346:12
2052:)
2044:(
2035:)
2027:(
2018:)
2010:(
1958:P
1954:P
1948:n
1945:A
1939:=
1936:r
1905:r
1871:r
1868:P
1863:P
1857:n
1854:A
1848:=
1820:P
1814:n
1811:A
1791:n
1788:A
1778:A
1697:i
1693:P
1689:i
1667:i
1663:Q
1639:=
1633:+
1627:+
1601:=
1592:3
1589:+
1580:2
1577:+
1561:n
1557:n
1553:n
1549:n
1545:k
1541:k
1525:1
1519:n
1515:P
1511:+
1506:n
1502:P
1498:=
1493:1
1487:n
1483:Q
1479:,
1474:n
1470:P
1466:=
1461:n
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1423:i
1419:Q
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1354:Q
1348:n
1343:1
1340:=
1337:i
1329:=
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1303:=
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1285:i
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1218:n
1213:1
1210:=
1207:i
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1153:i
1116:i
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1108:i
1103:n
1098:1
1095:=
1092:i
1079:r
1074:=
1061:r
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1036:1
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1030:i
1022:=
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989:i
982:i
978:P
972:n
967:1
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961:i
953:=
943:i
939:t
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915:=
912:i
904:=
897:P
880:i
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861:P
853:n
848:1
845:=
842:i
834:=
791:/
787:i
784:=
779:i
775:t
746:r
725:P
721:r
704:,
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695:r
646:/
642:1
639:+
369:)
366:1
357:+
354:i
351:(
343:1
319:i
315:t
302:.
290:i
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219:P
214:i
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185:i
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159:P
136:P
110:,
105:i
101:t
95:P
90:i
86:P
78:n
73:1
70:=
67:i
59:=
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