Return to Answer

Below, I have provided some examples of statistical measures, that compares the extreme re-balancing of different portfolios. They do not show how the concentration is allocated, only if the allocation is extreme. Many of the measures can be found in the empirical portfolio section of Patton et al. (2018), where they compare different covariance forecasts under one portfolio setup. However, the measures are still applicable under different portfolio setups, incorporating the same covariance forecasts.

Let $w_t = \left[w_{1t},\ldots,w_{dt}\right]$ be a $d$-dimensional vector of weights at time $t$, found from one of your portfolio allocation schemes. Then, turning to the paper of DeMiguel et al. (2014), one of the important features of stable asset allocation schemes is, that they produce less portfolio turnover. From the paper, we can define the portfolio turnover rate as:

Portfolio Concentration:

Below, I have provided some examples of statistical measures, that compare the extreme re-balancing of different portfolios. They do not show how the concentration is allocated, only if the allocation is extreme. Many of the measures can be found in the empirical portfolio section of Patton et al. (2018), where they compare different covariance forecasts under one portfolio setup. However, the measures are still applicable under different portfolio setups, incorporating the same covariance forecasts.

Let $w_t = \left[w_{1t},\ldots,w_{dt}\right]$ be a $d$-dimensional vector of weights at time $t$, found from one of your portfolio allocation schemes. Then, turning to the paper of DeMiguel et al. (2014), one of the important features of stable asset allocation schemes comes from the fact, that they produce less portfolio turnover. From the paper, we can define the portfolio turnover rate as:

Portfolio Concentration:

Then, $HHI_t \in [\frac{1}{d}, 1]$ under the "full investment" principle.

\begin{equation} cHHI_t = 1 - \frac{1-HHI_t}{1-\frac{1}{d}}. \end{equation} Now, the $cHHI_t \in [0,1]$. The notation follows from Chammas (2017) (see pp. 71 - 76), who also give examples of other portfolio concentration measures. In Patton et al. (2018) they use an alternative measure of portfolio concentration, which utilizes the Euclidean norm to measure the distance between the weights.

It might be ideal to measure the total portfolio short positions, since less extreme and fewer short positions are likely to facilitate the practical implementation of the portfolios, and help further mitigate higher transaction fees (related to shorting). One way of doing so, can be formulated as:

None of the above measures shows you how your portfolio is actually concentrated. If you have a large asset space, one idea would be to group them into sectors and then show how the weights might change from sector to sector via a time-series graph. Here, you could also look at one particular sector and plot the time-series of the weights, to show the changes within.

then, $HHI_t \in [\frac{1}{d}, 1]$ under the "full investment" principle.

\begin{equation} cHHI_t = 1 - \frac{1-HHI_t}{1-\frac{1}{d}}. \end{equation}

Now, the $cHHI_t \in [0,1]$. The notation follows from Chammas (2017) (see pp. 71 - 76), who also give examples of other portfolio concentration measures. In Patton et al. (2018) they use an alternative measure of portfolio concentration, which utilizes the Euclidean norm to measure the distance between the weights.

It might be ideal to measure the total portfolio short positions, since less extreme and fewer short positions are likely to facilitate the practical implementation of the portfolios, and further help to mitigate higher transaction fees (related to shorting). One way of doing so, can be formulated as:

None of the above measures shows how your portfolio is actually concentrated. If you have a large asset space, one idea would be to group them into sectors and then show how the weights might change from sector to sector via a time-series graph. Here, you could also look at one particular sector and plot the time-series of the weights, to show the changes within.

Below, I have provided some examples of statistical measures, that – in some way or another – compares the extreme re-balancing of different portfolios. They do not show how the concentration is allocated, only if the allocation is extreme. Many of the measures can be found in the empirical portfolio section of Patton et al. (2018), where they compare different covariance forecasts under one portfolio setup. However, the measures are still applicable under different portfolio setups, incorporating the same covariance forecasts.

Below, I have provided some examples of statistical measures, that compares the extreme re-balancing of different portfolios. They do not show how the concentration is allocated, only if the allocation is extreme. Many of the measures can be found in the empirical portfolio section of Patton et al. (2018), where they compare different covariance forecasts under one portfolio setup. However, the measures are still applicable under different portfolio setups, incorporating the same covariance forecasts.