13.4 Enhanced index tracking
Enhanced index tracking refers to variatiations on the basic index tracking formulation (13.9). They generally attempt to increase returns by building portfolios around index-like positions but then adding tactical tilts toward specific styles or individual stocks. This has been used by professional portfolio managers for decades (Xu et al., 2022) and some examples will be explored next.
13.4.1 Alternative tracking error measures
The tracking error used in Section 13.3 is based on the \(\ell_2\)-norm between the achieved returns \(\bm{X}\w\) and the benchmark returns \(\bm{r}^\textm{b}\): \[\begin{equation} \textm{TE}(\w) = \frac{1}{T}\left\|\bm{r}^\textm{b} - \bm{X}\w\right\|^2_2. \tag{13.11} \end{equation}\] This is a very simple way to measure the size of the tracking error and one could consider many other alternatives by either changing the norm (e.g., using the \(\ell_1\)-norm or the \(\ell_p\)-norm (Beasley et al., 2003)) or changing the error measure altogether, such as the excess return \(\frac{1}{T} \bm{1}^\T\left(\bm{X}\w - \bm{r}^\textm{b}\right)\) (Beasley et al., 2003; Dose and Cincotti, 2005), the downside risk, value at risk, or conditional value at risk (as defined in Chapter 6 and explored in detail in Chapter 10 for portfolio design). We now consider the downside risk and the \(\ell_1\)-norm tracking error for illustration purposes.
Downside risk error measure
A particularly interesting alternative is the downside risk that only takes into account when the achieved returns are worse than those of the benchmark: \[\begin{equation} \textm{DR}(\w) = \frac{1}{T}\left\|\left(\bm{r}^\textm{b} - \bm{X}\w\right)^+\right\|^2_2, \tag{13.12} \end{equation}\] where the operator \((\cdot)^+\triangleq\textm{max}(0,\cdot)\) precisely only considers returns smaller than the benchmark. This falls into the realm of enhanced index tracking.
It is worth pointing out that while using this tracking error measure may have the benefit of possibly beating the index, it is not appropriate for hedging purposes since that requires tracking the index in both directions.
Two key observations arise regarding the portfolio optimization under the downside risk \(\textm{DR}(\w)\). First, \(\textm{DR}(\w)\) is a convex function of \(\w\) and, therefore, it can be effectively optimized in practice with an appropriate solver. Second, it can be easily majorized for the purpose of using the MM method; in particular, we can choose a majorizer with the form of an \(\ell_2\)-norm and then the methods presented in Section 13.3 can be readily employed.
Lemma 13.2 (Majorizer of the downside risk) The downside risk function \(\textm{DR}(\w)\) in (13.12) is majorized at \(\w = \w_0\) by \(\textm{TE}(\w)\) in (13.11) with shifted benchmark returns \(\tilde{\bm{r}}^\textm{b}\): \[ \textm{DR}(\w) \le \frac{1}{T}\left\|\tilde{\bm{r}}^\textm{b} - \bm{X}\w\right\|^2_2, \] where \(\tilde{\bm{r}}^\textm{b} = \bm{r}^\textm{b} + \left(\bm{X}\w_0 - \bm{r}^\textm{b}\right)^+\) (Benidis et al., 2018b, 2018a).
The shifted benchmark returns \(\tilde{\bm{r}}^\textm{b}\) in Lemma 13.2 have an interesting interpretation: they are an improvement of the original returns \(\bm{r}^\textm{b}\) for those returns that were outperformed by the nominal portfolio \(\w_0\).
According to Lemma 13.2, index tracking under the downside risk can still be accomplished with Algorithm 13.1 under a slight modification: at each iteration \(k\), use the shifted benchmark returns \[\big(\tilde{\bm{r}}^\textm{b}\big)^k = \bm{r}^\textm{b} + \left(\bm{X}\w^k - \bm{r}^\textm{b}\right)^+.\]
\(\ell_1\)-norm tracking error
The TE in (13.11) is based on the \(\ell_2\)-norm, but this is a rather arbitrary choice. In fact, one may argue that the \(\ell_1\)-norm would make more sense: \[\begin{equation} \textm{TE}_1(\w) = \frac{1}{T}\left\|\bm{r}^\textm{b} - \bm{X}\w\right\|_1. \tag{13.13} \end{equation}\] This error measure is a convex function and, furthermore, it can be conveniently majorized in the form of an \(\ell_2\)-norm as in the case of the tracking error considered in Section 13.3.
Lemma 13.3 (Majorizer of the \(\ell_1\)-norm TE) The \(\ell_1\)-norm TE function \(\textm{TE}_1(\w)\) in (13.13) is majorized at \(\w = \w_0\) by a weighted version of the \(\textm{TE}(\w)\) in (13.11): \[ \textm{TE}_1(\w) \le \frac{1}{T}\left\|\bm{r}^\textm{b} - \bm{X}\w\right\|_{2,\bm{\alpha}}^2, \] where \(\|\bm{x}\|_{2,\bm{\alpha}}^2 \triangleq \sum_{i=1}^T \alpha_i x_i^2\) is the squared weighted \(\ell_2\)-norm with weights \(\bm{\alpha} = 1/(2|\bm{r}^\textm{b} - \bm{X}\w_0|)\). See Section B.7 in Appendix B for details.
The weights in the weighted \(\ell_2\)-norm TE in Lemma 13.3 have a natural interpretation: their role is to down-weight the errors so that they grow approximately in a linear fashion like in the \(\ell_1\)-norm, as expected.
According to Lemma 13.3, index tracking under the \(\ell_1\)-norm TE can still be accomplished with Algorithm 13.1 under a slight modification: at each iteration \(k\), use the weighted \(\ell_2\)-norm with weights \[\bm{\alpha}^k = 1/(2|\bm{r}^\textm{b} - \bm{X}\w^k|).\]
13.4.2 Robust tracking error measures
Robustness against outliers in the data is paramount in order to mitigate the effects of data contamination and avoid being sensitive or even break down (Section 3.5 in Chapter 3 covers the topic of robust estimators).
Any tracking error measure can be robustified to make it less sensitive to outliers. For illustration purposes, take the tracking error in (13.11) used in Section 13.3. This measure is based on the \(\ell_2\)-norm between the achieved returns \(\bm{X}\w\) and the benchmark returns \(\bm{r}^\textm{b}\), but precisely the \(\ell_2\)-norm is not a robust measure because it is sensitive to (anomalous) large errors due the the squaring operation. Of course one alternative is to use the \(\ell_1\)-norm tracking error in (13.13), which is naturally robust.
Alternatively, we can robustify the \(\ell_2\)-norm with the Huber penalty function (Huber, 2011): \[\phi^{\textm{hub}}(x)=\begin{cases} x^{2}, & \quad|x|\leq M\\ M\left(2|x| - M\right), & \quad|x|>M, \end{cases}\] which essentially behaves as the square function \(x^2\) for arguments up to a magnitude of \(M\) and as a linear function otherwise (this way outliers are not amplified). We can then define a Huberized version of the tracking error: \[\begin{equation} \textm{Hub-TE}(\w) = \frac{1}{T}\sum_{t=1}^T \phi^{\textm{hub}}(r_t^\textm{b} - \bm{X}_{t,:}\w), \tag{13.14} \end{equation}\] which is a convex function. In addition, it can be conveniently majorized in the form of an \(\ell_2\)-norm as in the case of the tracking error considered in Section 13.3.
Lemma 13.4 (Majorizer of the Huberized TE) The Huberized TE function \(\textm{Hub-TE}(\w)\) in (13.14) is majorized at \(\w = \w_0\) by a weighted version of the \(\textm{TE}(\w)\) in (13.11): \[ \textm{Hub-TE}(\w) \le \frac{1}{T}\left\|\bm{r}^\textm{b} - \bm{X}\w\right\|_{2,\bm{\alpha}}^2 + \textm{const}, \] where \(\|\bm{x}\|_{2,\bm{\alpha}}^2 \triangleq \sum_{i=1}^T \alpha_i x_i^2\) is the squared weighted \(\ell_2\)-norm with weights \[ \bm{\alpha} = \textm{min}\left(\bm{1}, \frac{M}{|\bm{r}^\textm{b} - \bm{X}\w_0|}\right), \]
and the term \(\textm{const}\) refers to an irrelevant constant term (Benidis et al., 2018b, 2018a).
The weights in the Huberized TE have a very natural interpretation: their role is to down-weight the errors that are larger than \(M\) in magnitude so that the squared values grow linearly instead.
According to Lemma 13.4, index tracking under the Huberized TE can still be accomplished with Algorithm 13.1 under a slight modification: at each iteration \(k\), use the weighted \(\ell_2\)-norm with weights \[ \bm{\alpha}^k = \textm{min}\left(\bm{1}, \frac{M}{|\bm{r}^\textm{b} - \bm{X}\w^k|}\right). \]
13.4.3 Holding constraints
In practice, portfolios typically have holding constraints:
upper bounds (\(\w \le \bm{u}\)): this is to avoid risk from allocating too much budget to a single asset and to promote diversification; and
lower bounds (\(\w \ge \bm{l}\)): this is to avoid very small positions that are irrelevant to the overall portfolio and just complicate the logistics.
These bound constraints are trivial linear inequality constraints. However, when dealing with sparsity, the lower bounds become incompatible with sparse solutions. In this case, the lower bounds should only become active for active assets: \[ w_i \; \begin{cases} \ge l_i, & \textm{if }w_i > 0\\ = 0, & \textm{otherwise}. \end{cases} \] These complicated nonconvex constraints can be easily incorporated into evolutionary algorithms (Beasley et al., 2003; Maringer and Oyewumi, 2007). Interestingly, they can also be approximated and solved via the MM framework in a convenient way (Benidis et al., 2018b, 2018a).
13.4.4 Group sparsity
Stocks and other financial assets are classified and grouped together into sectors and industries. This organization is convenient for investors in order to easily diversify their investment across different sectors (which presumably are less correlated than stocks within each sector).
All the index tracking formulations considered thus far are able to control the sparsity of the portfolio via the term \(\|\w\|_0\). Nevertheless, it is possible to have a more refined control by taking account of stock industry profiles. One way is by replacing the overall sparsity term \(\|\w\|_0\) by a “group sparsity” term that is able to construct a portfolio composed of a specific number of stocks concentrating on a few industries, and ensures both industry-wise and within-industry sparsity (Xu et al., 2022).
13.4.5 Numerical experiments
We now compare the tracking of the S&P 500 index based on different tracking error measures of the cumulative returns, namely:
Figure 13.11 shows the tracking over time with approximately \(K=20\) active assets. The tracking portfolios are computed on a rolling window basis with a lookback period of two years and recomputed every six months. As expected, one can observe that the design based on the downside risk beats the market (suitable for investment purposes) while the other measures generally track the index in both directions (appropriate for hedging purposes).
Figure 13.11: Tracking over time of the S&P 500 index under different tracking error measures.