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Tracking error
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<h2 id="definition">Definition</h2> <p>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 portfolio managers to control risk. Various types of ex-ante tracking error models exist, from simple equity models which use <a href="/facts/Beta_(finance)/0u6NPHLF">beta</a> as a primary determinant to more complicated <a href="/facts/Factor_analysis/LT6B9I7D">multi-factor fixed income models</a>. 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). </p> <h3>Formulas</h3> <p>The ex-post tracking error formula is the <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> of the active returns, given by: </p> T E = ω = Var ( r p − r b ) = E [ ( r p − r b ) 2 ] − ( E [ r p − r b ] ) 2 = ( w p − w b ) T Σ ( w p − w b ) {\displaystyle TE=\omega ={\sqrt {\operatorname {Var} (r_{p}-r_{b})}}={\sqrt {{E}[(r_{p}-r_{b})^{2}]-({E}[r_{p}-r_{b}])^{2}}}={\sqrt {(w_{p}-w_{b})^{T}\Sigma (w_{p}-w_{b})}}} <p>where r p − r b {\displaystyle r_{p}-r_{b}} is the active return, i.e., the difference between the portfolio return and the benchmark return<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> and ( w p − w b ) {\displaystyle (w_{p}-w_{b})} is the vector of active portfolio weights relative to the benchmark. The <a href="/facts/Mathematical_optimization/oRn8Iv5I">optimization</a> problem of maximizing the return, subject to tracking error and linear constraints, may be solved using <a href="/facts/Second-order_cone_programming/TNXLKvK7">second-order cone programming</a>: argmax w μ T ( w − w b ) , s.t. ( w − w b ) T Σ ( w − w b ) ≤ ω 2 , A x ≤ b , C x = d {\displaystyle {\underset {w}{\operatorname {argmax} }}\;\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} </p> <h3>Interpretation</h3> <p>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 returns (two standard deviations from the mean) can be expected to fall between +2x and -2x per cent of the mean excess return. </p> <h2 id="examples">Examples</h2> <ul><li><a href="/facts/Index_fund/ULAIr8j0">Index funds</a> are expected to have minimal tracking errors.</li> <li><a href="/facts/Inverse_exchange-traded_fund/y9wuHaMT">Inverse exchange-traded funds</a> are designed to perform as the <i>inverse</i> of an index or other benchmark, and thus reflect tracking errors relative to short positions in the underlying index or benchmark.</li></ul> <h3>Index fund creation</h3> <p>Index funds are expected to minimize the tracking error with respect to the <a href="/facts/Index_(economics)/cMnG0tgd">index</a> they are attempting to replicate, and this problem may be solved using standard optimization techniques. To begin, define ω 2 {\displaystyle \omega ^{2}} to be: ω 2 = ( w − w b ) T Σ ( w − w b ) {\displaystyle \omega ^{2}=(w-w_{b})^{T}\Sigma (w-w_{b})} where w − w b {\displaystyle w-w_{b}} is the vector of active weights for each asset relative to the <a href="/facts/Performance_attribution/36R3usXP">benchmark</a> index and Σ {\displaystyle \Sigma } is the <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> for the assets in the index. While creating an index fund could involve holding all N {\displaystyle N} investable assets in the index, it is sometimes better practice to only invest in a subset K {\displaystyle K} of the assets. These considerations lead to the following <a href="/facts/Quadratic_programming/ct4RIUgs">mixed-integer quadratic programming (MIQP)</a> problem: argmin w ω 2 s.t. w j ≤ y j , ∑ j = 1 N y j ≤ K ℓ j y j ≤ w j ≤ u j y j , y j ∈ { 0 , 1 } , ℓ j , u j ≥ 0 {\displaystyle {\begin{aligned}{\underset {w}{\operatorname {argmin} }}&\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}}} where y j {\displaystyle y_{j}} is the logical condition of whether or not an asset is included in the index fund, and is defined as: y j = { 1 , w j > 0 0 , otherwise {\displaystyle y_{j}={\begin{cases}1,\quad &w_{j}>0\\0,\quad &{\text{otherwise}}\end{cases}}} </p> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.youtube.com/watch?v=A1sB2ynlNrw">Tracking Error</a> - <a href="/facts/YouTube/db276jst">YouTube</a></li> <li><a href="http://monevator.com/2011/01/18/tracking-error-%E2%80%93-a-hidden-cost/">Tracking error: A hidden cost of passive investing</a></li> <li><a href="http://moneyterms.co.uk/tracking-error/">Tracking error</a></li> <li><a href="https://www.trackinsight.com/education/education-what-is-tracking-error">What is the Tracking Error?</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Cornuejols, Gerard; Tütüncü, Reha (2007). Optimization Methods in Finance. Mathematics, Finance and Risk. Cambridge University Press. pp. 178–180. ISBN 978-0521861700. <a href="978-0521861700" target="_blank">978-0521861700</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>