1149:
902:
73:. Tracking error measures the deviation from the benchmark: an index fund has a near-zero tracking error, while an actively managed portfolio would normally have a higher tracking error. Thus the tracking error does not include any risk (return) that is merely a function of the market's movement. In addition to
647:
Under the assumption of normality of returns, an active risk of x per cent would mean that approximately 2/3 of the portfolio's active returns (one standard deviation from the mean) can be expected to fall between +x and -x per cent of the mean excess return and about 95% of the portfolio's active
100:
If tracking error is measured historically, it is called 'realized' or 'ex post' tracking error. If a model is used to predict tracking error, it is called 'ex ante' tracking error. Ex-post tracking error is more useful for reporting performance, whereas ex-ante tracking error is generally used by
109:. In a factor model of a portfolio, the non-systematic risk (i.e., the standard deviation of the residuals) is called "tracking error" in the investment field. The latter way to compute the tracking error complements the formulas below but results can vary (sometimes by a factor of 2).
372:
638:
1144:{\displaystyle {\begin{aligned}\min _{w}&\quad \omega ^{2}\\{\text{s.t.}}&\quad w_{j}\leq y_{j},\quad \sum _{j=1}^{N}y_{j}\leq K\\&\quad \ell _{j}y_{j}\leq w_{j}\leq u_{j}y_{j},\quad y_{j}\in \{0,1\},\quad \ell _{j},\;u_{j}\geq 0\end{aligned}}}
1264:
474:
127:
792:
907:
1181:
461:
708:
415:
849:
825:
1176:
893:
873:
1356:
713:
367:{\displaystyle TE=\omega ={\sqrt {\operatorname {Var} (r_{p}-r_{b})}}={\sqrt {{E}-({E})^{2}}}={\sqrt {(w_{p}-w_{b})^{T}\Sigma (w_{p}-w_{b})}}}
1803:
2029:
2019:
1291:
670:
of an index or other benchmark, and thus reflect tracking errors relative to short positions in the underlying index or benchmark.
633:{\displaystyle \max _{w}\;\mu ^{T}(w-w_{b}),\quad {\text{s.t.}}\;(w-w_{b})^{T}\Sigma (w-w_{b})\leq \omega ^{2},\;Ax\leq b,\;Cx=d}
648:
returns (two standard deviations from the mean) can be expected to fall between +2x and -2x per cent of the mean excess return.
1671:
101:
portfolio managers to control risk. Various types of ex-ante tracking error models exist, from simple equity models which use
1950:
1753:
683:
they are attempting to replicate, and this problem may be solved using standard optimization techniques. To begin, define
468:
2024:
1349:
663:
1636:
1528:
1700:
1788:
1369:
464:
1646:
1611:
1616:
54:; it indicates how closely a portfolio follows the index to which it is benchmarked. The best measure is the
1955:
1676:
1661:
1603:
1342:
828:
65:, are expected to replicate, before trading and other costs, the returns of an index exactly, while others '
1651:
1329:
1319:
1993:
1895:
1868:
1853:
1621:
1476:
1397:
1259:{\displaystyle y_{j}={\begin{cases}1,\quad &w_{j}>0\\0,\quad &{\text{otherwise}}\end{cases}}}
896:
467:
problem of maximizing the return, subject to tracking error and linear constraints, may be solved using
417:
is the active return, i.e., the difference between the portfolio return and the benchmark return and
1998:
1930:
1920:
1875:
1808:
1641:
1392:
1178:
is the logical condition of whether or not an asset is included in the index fund, and is defined as:
43:
1203:
1988:
1945:
1456:
420:
1965:
1960:
1940:
1828:
1631:
1589:
1486:
1412:
686:
380:
118:
55:
1334:
1843:
1833:
1823:
1778:
1773:
1727:
1723:
1696:
1626:
1543:
1417:
1407:
1287:
852:
680:
89:
66:
51:
47:
834:
797:
1880:
1798:
1748:
1731:
1656:
1520:
1505:
1154:
875:
investable assets in the index, it is sometimes better practice to only invest in a subset
1915:
1890:
1848:
1838:
1813:
1793:
1783:
1496:
1460:
1402:
106:
78:
61:
Many portfolios are managed to a benchmark, typically an index. Some portfolios, notably
1863:
1719:
1715:
1574:
1553:
1501:
1491:
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1471:
1440:
1422:
1365:
878:
858:
102:
74:
2013:
1970:
1858:
1768:
1758:
1711:
1691:
1584:
1533:
82:
70:
855:
for the assets in the index. While creating an index fund could involve holding all
1925:
1910:
1686:
1548:
1509:
1975:
1935:
1681:
1666:
1579:
1466:
1444:
1432:
1384:
1286:. Mathematics, Finance and Risk. Cambridge University Press. pp. 178–180.
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1538:
1452:
1448:
1310:
657:
62:
1905:
679:
Index funds are expected to minimize the tracking error with respect to the
17:
1324:
1763:
463:
is the vector of active portfolio weights relative to the benchmark. The
1818:
1314:
88:
Dividing portfolio active return by portfolio tracking error gives the
31:
1885:
69:' the portfolio by deviating from the index in order to generate
1900:
1338:
827:
is the vector of active weights for each asset relative to the
1252:
58:
of the difference between the portfolio and index returns.
895:
of the assets. These considerations lead to the following
787:{\displaystyle \omega ^{2}=(w-w_{b})^{T}\Sigma (w-w_{b})}
77:(return) from specific stock selection or industry and
1184:
1157:
905:
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477:
423:
383:
130:
1741:
1602:
1567:
1519:
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1376:
1320:Tracking error: A hidden cost of passive investing
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1170:
1143:
887:
867:
843:
819:
786:
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632:
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409:
366:
911:
479:
92:, which is a risk adjusted performance measure.
81:"betas", it can also include risk (return) from
1350:
660:are expected to have minimal tracking errors.
105:as a primary determinant to more complicated
8:
1100:
1088:
1380:
1357:
1343:
1335:
1282:Cornuejols, Gerard; Tütüncü, Reha (2007).
1120:
897:mixed-integer quadratic programming (MIQP)
617:
601:
530:
488:
117:The ex-post tracking error formula is the
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1219:
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1189:
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1162:
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1111:
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1055:
1042:
1029:
1019:
997:
987:
976:
962:
949:
937:
927:
914:
906:
904:
880:
860:
836:
811:
799:
775:
753:
743:
721:
715:
694:
688:
592:
576:
554:
544:
525:
512:
493:
482:
476:
444:
431:
422:
401:
388:
382:
353:
340:
324:
314:
301:
292:
281:
268:
255:
243:
228:
218:
205:
190:
188:
174:
161:
146:
129:
1274:
7:
838:
759:
560:
330:
25:
121:of the active returns, given by:
1632:Conditional Value-at-Risk (CVaR)
107:multi-factor fixed income models
1284:Optimization Methods in Finance
1241:
1212:
1106:
1074:
1014:
971:
944:
922:
666:are designed to perform as the
524:
42:is a measure of the risk in an
1951:Strategic financial management
1754:Asset and liability management
781:
762:
750:
730:
582:
563:
551:
531:
518:
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450:
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359:
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321:
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278:
274:
248:
240:
234:
225:
198:
195:
180:
154:
1:
664:Inverse exchange-traded funds
469:second-order cone programming
456:{\displaystyle (w_{p}-w_{b})}
1529:Operational risk management
1330:What is the Tracking Error?
703:{\displaystyle \omega ^{2}}
410:{\displaystyle r_{p}-r_{b}}
2046:
2030:Investment fund indicators
1701:Proportional hazards model
1652:Interest rate immunization
27:Measure of investment risk
2020:Financial risk management
1984:
1370:financial risk management
1647:First-hitting-time model
1612:Arbitrage pricing theory
1956:Stress test (financial)
1662:Modern portfolio theory
844:{\displaystyle \Sigma }
820:{\displaystyle w-w_{b}}
1260:
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1145:
992:
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845:
821:
788:
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457:
411:
368:
50:decisions made by the
1994:Investment management
1896:Investment management
1622:Replicating portfolio
1398:Sovereign credit risk
1261:
1173:
1171:{\displaystyle y_{j}}
1146:
972:
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822:
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369:
1999:Mathematical finance
1931:Risk-return spectrum
1921:Mathematical finance
1876:Fundamental analysis
1809:Exchange traded fund
1393:Consumer credit risk
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1155:
903:
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835:
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381:
128:
44:investment portfolio
2025:Convex optimization
1989:Financial economics
1946:Statistical finance
1712:Value-at-Risk (VaR)
1617:Black–Scholes model
1457:Holding period risk
675:Index fund creation
1966:Structured product
1961:Structured finance
1941:Speculative attack
1627:Cash flow matching
1590:Non-financial risk
1487:Interest rate risk
1413:Concentration risk
1256:
1251:
1168:
1141:
1139:
919:
885:
865:
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817:
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700:
630:
487:
453:
407:
364:
119:standard deviation
56:standard deviation
2007:
2006:
1779:Corporate finance
1774:Capital structure
1728:Cash flow at risk
1724:Liquidity at risk
1697:Survival analysis
1598:
1597:
1544:Reputational risk
1418:Credit derivative
1247:
940:
910:
888:{\displaystyle K}
868:{\displaystyle N}
853:covariance matrix
528:
478:
362:
287:
183:
90:information ratio
52:portfolio manager
48:active management
16:(Redirected from
2037:
1881:Growth investing
1799:Enterprise value
1749:Asset allocation
1732:Earnings at risk
1714:and extensions (
1657:Market portfolio
1521:Operational risk
1506:Refinancing risk
1381:
1359:
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1298:
1297:
1279:
1265:
1263:
1262:
1257:
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1245:
1224:
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1194:
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1177:
1175:
1174:
1169:
1167:
1166:
1150:
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1147:
1142:
1140:
1130:
1129:
1116:
1115:
1084:
1083:
1070:
1069:
1060:
1059:
1047:
1046:
1034:
1033:
1024:
1023:
1012:
1002:
1001:
991:
986:
967:
966:
954:
953:
941:
938:
932:
931:
918:
894:
892:
891:
886:
874:
872:
871:
866:
850:
848:
847:
842:
826:
824:
823:
818:
816:
815:
793:
791:
790:
785:
780:
779:
758:
757:
748:
747:
726:
725:
709:
707:
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699:
698:
639:
637:
636:
631:
597:
596:
581:
580:
559:
558:
549:
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529:
526:
517:
516:
498:
497:
486:
462:
460:
459:
454:
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436:
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416:
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405:
393:
392:
373:
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370:
365:
363:
358:
357:
345:
344:
329:
328:
319:
318:
306:
305:
293:
288:
286:
285:
273:
272:
260:
259:
247:
233:
232:
223:
222:
210:
209:
194:
189:
184:
179:
178:
166:
165:
147:
21:
2045:
2044:
2040:
2039:
2038:
2036:
2035:
2034:
2010:
2009:
2008:
2003:
1980:
1916:Systematic risk
1814:Expected return
1794:Economic bubble
1789:Diversification
1784:Cost of capital
1737:
1594:
1563:
1515:
1497:Volatility risk
1461:Price area risk
1427:
1403:Settlement risk
1372:
1363:
1307:
1302:
1301:
1294:
1281:
1280:
1276:
1271:
1250:
1249:
1242:
1232:
1231:
1215:
1213:
1199:
1185:
1180:
1179:
1158:
1153:
1152:
1138:
1137:
1121:
1107:
1075:
1061:
1051:
1038:
1025:
1015:
1010:
1009:
993:
958:
945:
942:
934:
933:
923:
920:
901:
900:
877:
876:
857:
856:
833:
832:
807:
796:
795:
771:
749:
739:
717:
712:
711:
690:
685:
684:
677:
654:
645:
588:
572:
550:
540:
508:
489:
473:
472:
440:
427:
419:
418:
397:
384:
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378:
349:
336:
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310:
297:
277:
264:
251:
224:
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170:
157:
126:
125:
115:
98:
67:actively manage
46:that is due to
28:
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22:
15:
12:
11:
5:
2043:
2041:
2033:
2032:
2027:
2022:
2012:
2011:
2005:
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2002:
2001:
1996:
1991:
1985:
1982:
1981:
1979:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1908:
1903:
1898:
1893:
1888:
1883:
1878:
1873:
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1866:
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1791:
1786:
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1743:
1742:Basic concepts
1739:
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1720:Margin at risk
1716:Profit at risk
1709:
1707:Tracking error
1704:
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1677:Risk-free rate
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1629:
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1600:
1599:
1596:
1595:
1593:
1592:
1587:
1582:
1577:
1575:Execution risk
1571:
1569:
1565:
1564:
1562:
1561:
1556:
1554:Political risk
1551:
1546:
1541:
1536:
1531:
1525:
1523:
1517:
1516:
1514:
1513:
1502:Liquidity risk
1499:
1494:
1492:Inflation risk
1489:
1484:
1482:Margining risk
1479:
1474:
1472:Valuation risk
1469:
1464:
1441:Commodity risk
1437:
1435:
1429:
1428:
1426:
1425:
1423:Securitization
1420:
1415:
1410:
1405:
1400:
1395:
1389:
1387:
1378:
1374:
1373:
1366:Financial risk
1364:
1362:
1361:
1354:
1347:
1339:
1333:
1332:
1327:
1325:Tracking error
1322:
1317:
1311:Tracking Error
1306:
1305:External links
1303:
1300:
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1293:978-0521861700
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1028:
1022:
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1005:
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996:
990:
985:
982:
979:
975:
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965:
961:
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952:
948:
943:
936:
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926:
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643:Interpretation
641:
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481:
452:
447:
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439:
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430:
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404:
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396:
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387:
375:
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94:
71:active returns
36:tracking error
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2042:
2031:
2028:
2026:
2023:
2021:
2018:
2017:
2015:
2000:
1997:
1995:
1992:
1990:
1987:
1986:
1983:
1977:
1974:
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1971:Systemic risk
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1967:
1964:
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1817:
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1797:
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1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1770:
1769:Capital asset
1767:
1765:
1762:
1760:
1759:Asset pricing
1757:
1755:
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1746:
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1725:
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1710:
1708:
1705:
1702:
1698:
1695:
1693:
1692:Sortino ratio
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1609:
1607:
1605:
1601:
1591:
1588:
1586:
1585:Systemic risk
1583:
1581:
1578:
1576:
1573:
1572:
1570:
1566:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1534:Business risk
1532:
1530:
1527:
1526:
1524:
1522:
1518:
1511:
1507:
1503:
1500:
1498:
1495:
1493:
1490:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1462:
1458:
1454:
1450:
1446:
1442:
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1510:Deposit risk
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1580:Profit risk
1467:Equity risk
1445:Volume risk
1433:Market risk
1385:Credit risk
658:Index funds
85:decisions.
63:index funds
40:active risk
18:Active risk
2014:Categories
1559:Legal risk
1539:Model risk
1453:Shape risk
1449:Basis risk
1377:Categories
1269:References
831:index and
96:Definition
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1819:Financial
1246:otherwise
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