1138:
891:
62:. 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
636:
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
89:
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
98:. 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).
361:
627:
1133:{\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}}}
1253:
463:
116:
781:
896:
1170:
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404:
838:
814:
1165:
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862:
1345:
702:
356:{\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})}}}
1792:
2018:
2008:
1280:
659:
of an index or other benchmark, and thus reflect tracking errors relative to short positions in the underlying index or benchmark.
622:{\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}
637:
returns (two standard deviations from the mean) can be expected to fall between +2x and -2x per cent of the mean excess return.
1660:
90:
portfolio managers to control risk. Various types of ex-ante tracking error models exist, from simple equity models which use
1939:
1742:
672:
they are attempting to replicate, and this problem may be solved using standard optimization techniques. To begin, define
457:
2013:
1338:
652:
1625:
1517:
1689:
1777:
1358:
453:
1635:
1600:
1605:
43:; it indicates how closely a portfolio follows the index to which it is benchmarked. The best measure is the
1944:
1665:
1650:
1592:
1331:
817:
54:, are expected to replicate, before trading and other costs, the returns of an index exactly, while others '
1640:
1318:
1308:
1982:
1884:
1857:
1842:
1610:
1465:
1386:
1248:{\displaystyle y_{j}={\begin{cases}1,\quad &w_{j}>0\\0,\quad &{\text{otherwise}}\end{cases}}}
885:
456:
problem of maximizing the return, subject to tracking error and linear constraints, may be solved using
406:
is the active return, i.e., the difference between the portfolio return and the benchmark return and
1987:
1919:
1909:
1864:
1797:
1630:
1381:
1167:
is the logical condition of whether or not an asset is included in the index fund, and is defined as:
32:
1192:
1977:
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1401:
675:
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107:
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55:
40:
36:
823:
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investable assets in the index, it is sometimes better practice to only invest in a subset
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Many portfolios are managed to a benchmark, typically an index. Some portfolios, notably
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for the assets in the index. While creating an index fund could involve holding all
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1275:. Mathematics, Finance and Risk. Cambridge University Press. pp. 178–180.
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Index funds are expected to minimize the tracking error with respect to the
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is the vector of active portfolio weights relative to the benchmark. The
1807:
1303:
77:
Dividing portfolio active return by portfolio tracking error gives the
20:
1874:
58:' the portfolio by deviating from the index in order to generate
1889:
1327:
816:
is the vector of active weights for each asset relative to the
1241:
47:
of the difference between the portfolio and index returns.
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of the assets. These considerations lead to the following
776:{\displaystyle \omega ^{2}=(w-w_{b})^{T}\Sigma (w-w_{b})}
66:(return) from specific stock selection or industry and
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1309:Tracking error: A hidden cost of passive investing
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81:, which is a risk adjusted performance measure.
70:"betas", it can also include risk (return) from
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649:are expected to have minimal tracking errors.
94:as a primary determinant to more complicated
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1271:Cornuejols, Gerard; Tütüncü, Reha (2007).
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106:The ex-post tracking error formula is the
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110:of the active returns, given by:
1621:Conditional Value-at-Risk (CVaR)
96:multi-factor fixed income models
1273:Optimization Methods in Finance
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655:are designed to perform as the
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31:is a measure of the risk in an
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1743:Asset and liability management
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458:second-order cone programming
445:{\displaystyle (w_{p}-w_{b})}
1518:Operational risk management
1319:What is the Tracking Error?
692:{\displaystyle \omega ^{2}}
399:{\displaystyle r_{p}-r_{b}}
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2019:Investment fund indicators
1690:Proportional hazards model
1641:Interest rate immunization
16:Measure of investment risk
2009:Financial risk management
1973:
1359:financial risk management
1636:First-hitting-time model
1601:Arbitrage pricing theory
1945:Stress test (financial)
1651:Modern portfolio theory
833:{\displaystyle \Sigma }
809:{\displaystyle w-w_{b}}
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1160:{\displaystyle y_{j}}
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2014:Convex optimization
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1446:Holding period risk
664:Index fund creation
1955:Structured product
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1717:Cash flow at risk
1713:Liquidity at risk
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1407:Credit derivative
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877:{\displaystyle K}
857:{\displaystyle N}
842:covariance matrix
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79:information ratio
41:portfolio manager
37:active management
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1870:Growth investing
1788:Enterprise value
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1703:and extensions (
1646:Market portfolio
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1905:Systematic risk
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1783:Economic bubble
1778:Diversification
1773:Cost of capital
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1481:Inflation risk
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1471:Margining risk
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1463:
1461:Valuation risk
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1453:
1430:Commodity risk
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1412:Securitization
1409:
1404:
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1367:
1363:
1362:
1355:Financial risk
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1314:Tracking error
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1306:
1300:Tracking Error
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1294:External links
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1681:Sortino ratio
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1574:Systemic risk
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1408:
1405:
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1569:Profit risk
1456:Equity risk
1434:Volume risk
1422:Market risk
1374:Credit risk
647:Index funds
74:decisions.
52:index funds
29:active risk
2003:Categories
1548:Legal risk
1528:Model risk
1442:Shape risk
1438:Basis risk
1366:Categories
1258:References
820:index and
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