# Statistical methodology for proving the stability in time of asset allocation weights

I am comparing the set of weights obtained by the classical Markowitz allocation process with those of another asset allocation technique I have devised.

Markowitz's weights are unstable, as the algorithm leads to concentrated portfolios where wealth is allocated to only few securities. Moreover, this changes over time leading to extreme re-balancing.

A way to show this, is simply by plotting the time series of weights for each security. On the other hand, I was wondering if there is a statistical methodology that can be used to prove the degree of time-stability of the weights?

Many thanks.

• Markowitz portfolio weights are asymptotically normal around the population analogue. If that population analogue were unchanged in two periods, you could compute an approximate Mahalanobis distance between the two portfolios, which could be rescaled to a chi-square under the null. This would give a time series of Mahalanobis distances. May 4 at 16:09

## Examples of statistical measures to compare extreme rebalancing:

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.

### Portfolio Turnover Rates:

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:

$$\begin{equation} TO_t = \sum_{i=1}^{d} \bigg\vert w^{i}_{t+1} - w^{i}_{t} \frac{1+r_t^{i}}{1+w_t^\intercal r_t} \bigg\vert, \end{equation}$$

where $$w_t^{i}$$ and $$r_t^i$$ corresponds respectively to the allocation on $$i$$'th risky asset and the log-returns for asset $$i$$.

In essence, at the end of trading day $$t$$, the investor rebalances his portfolio using the ex-post covariance matrix to calculate the weights $$w_{t+1}^i$$ for the next day. The moment before implementing the new weights, the notional value of the portfolio for each asset has changed the value of asset $$i$$ to $$w_t^i (1+r_t^{i})$$ and the corresponding realized weight of stock $$i$$ becomes $$w_t^i \frac{(1+r_t^{i})}{1+w_t^\mathsf{T} r_t}$$. The turnover produces a decimal value, which can be interpreted as the fraction of the portfolio that has been bought or sold at time $$t$$.

Under the assumption that you are using the same covariance matrix for all of your portfolios, then the lower portfolio turnover implies a less extreme re-balancing from day $$t$$ to $$t+1$$.

### Portfolio Concentration:

Another way to compare the portfolio allocation scheme is the usage of a corrected Hirschman-Herfindahl Index (HHI) approach. We define the concentration of a portfolio as the measure of inequality relative to the weight allocation of the available wealth among the available stocks. Define the HHI index as the sum of squared weights

$$\begin{equation} HHI_t = \sum_{i=1}^{d} \left(w_t^i \right)^2, \end{equation}$$

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

The intuition for the simple portfolio concentration stems from the fact that an equally weighted portfolio would have the smallest concentration possible. Moreover, if the portfolio manager chooses to invest his entire wealth into one single asset, then that would be the maximum concentration. We can bound the portfolio concentration from 0 to 1 by correcting the HHI estimator above:

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

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.

### Portfolio Short Positions:

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:

$$\begin{equation} SP_t = \sum_{i=1}^d w_t^{i} 1_{\{w_t^{i}\: < \: 0\}}. \end{equation}$$

While this does not directly tell you anything about the extreme rebalancing of your portfolios, it is still nice to know.

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.

Nevertheless, I hope this provide some help.

Letting the Markowitz portfolio over period $$i$$ be $$\hat{w}_i = \hat{\Sigma}_i^{-1} \hat{\mu}_i$$, the distributions of these are asymptotically normal around the population value: $$\hat{w}_i \sim \mathcal{N}\left(w_i, \Omega_i\right),$$ where $$w_i$$ is the population value and $$\Omega_i$$ is the covariance of the sample Markowitz portfolio. If the underlying has not changed, all the $$w_i$$ are equal to some value. Then we have $$\hat{w}_i - \hat{w}_j \sim \mathcal{N}\left(0, \Omega_i + \Omega_j\right).$$ One can estimate the $$\Omega$$ to compute $$d^2 = \left(\hat{w}_i - \hat{w}_j\right)^{\top}\left(\hat{\Omega}_i + \hat{\Omega}_j\right)^{-1} \left(\hat{w}_i - \hat{w}_j\right).$$ This will be approximately $$\chi^2$$ with $$k$$ degrees of freedom for $$k$$ assets.

You can estimate the $$\Omega_i$$ assuming normal returns, or use the delta method as outlined here and implemented in the MarkowitzR R package.