Median-adaptive portfolios: a minimum criteria approach to asset allocation

Abstract

We propose a new class of adaptive portfolios for asset allocation, based on a one-parameter variation of the equally weighted portfolio and the use of the median-ranked asset. Our methodological contribution offers a simple way of performing, static or optimized, allocation of assets in portfolios of any dimension, thus easily circumventing the “curse of dimensionality”. Our results show that, even for a static selection of the parameter that defines our allocation, we obtain improved performance compared to the equally weighted benchmark in all the standard metrics. For the case of an optimized selection of the parameter we offer results from minimum variance optimization, that do require the estimation of the covariance matrix, but our approach can easily be adapted to other kinds of portfolio objective functions. This new class of portfolios can easily be added to, as a complement or substitute, to any existing portfolio allocation method.

1 Introduction

One of the most important portfolio benchmarks is the equally weighted (EW) portfolio or the 1/N-portfolio, where N signifies the number of assets. There is considerable interest in this kind of simple asset allocation approach, precisely because of its simplicity but also because of its performance. We can, for example, look at DeMiguel et al. (2009) where they examine the distinction between “optimal” and “naive” (i.e., EW) diversification and the extend of the inefficiency of the EW portfolio, but also on Kritzman et al. (2010) where they come to the defense of optimization and the “fallacy” of 1/N. There are many recent references that use the EW portfolio, either as a benchmark of performance or as an input or extension to another method, for example and among others: Mainik et al. (2015) consider the EW portfolio as a benchmark in optimization of heavy-tailed assets; Pae and Sabbaghi (2015) compare the EW portfolio with value-weighted ones; Hanke et al. (2019) discuss equal weighting in the context of managed portfolios while Jiang et al. (2019) combine the minimum-variance and EW portfolio for improving performance; Mynbayeva et al. (2022) discuss the impact of (multi-parameter) estimation on creating problems for Markowitz-type portfolios while Harris and Mazibas (2022) and Harris et al. (2022) use the EW portfolio as a benchmark to other asset allocation approaches.

The continued interest on the potential uses of the EW portfolio is the origin for this paper, coupled with ideas of adaptive learning, see for example Kyriazi et al. (2019), that go back to the use of exponential smoothing. The main research question that we offer to address with the current work is this: what if we take as our starting point the EW portfolio and just “tweak it” just a bit? What will happen if this tweak becomes dynamic via a learning-rule? The novelty of the paper is that we introduce the median-ranked portfolios, a new, one-parameter class derived from the EW portfolios, that can incorporate learning and dynamic updating. The properties of four (4) types of portfolios in this new class are examined, in terms of their ability in generating different sets of portfolio weights; we also examine the impact of the choice of the single parameter that defines this new class of portfolios. Furthermore, we show how this parameter can be easily optimized—if so desired—by any portfolio objective function; however, we illustrate the application of optimization within the context of variance-minimization only. We offer a set of empirical results that support our idea of using this simple class of portfolios in addition to, or mainly as new substitutes for, the EW portfolio which they easily outperform in various metrics. The use of just a single parameter to control the temporal evolution of the portfolio weights, makes this new class very easily, and immediately, adaptable to any dimension of the portfolio, i.e., to any number of assets, thus easily by-passing the “curse of dimensionality”.

Table 1 ETF list used in the construction of portfolios

Table 2 Statistics for portfolio P1, no optimization

Table 3 Statistics for portfolio P1, with optimization

Table 4 Statistics for portfolio P2, no optimization

Table 5 Statistics for portfolio P2, with optimization

Our main research questions can thus be summarized as follows:

• Can we effect performance-enhancements in the EW portfolio without altering its inherent simplicity and speed of computation?

• Is dynamic learning of the portfolio weights useful in our suggested methodology, compared to a static approach?

• Does one have to optimize the single parameter that defines the new portfolio class or ex ante calibration can consistently provide better performance?

• What is the interpetation, if any, of the suggested portfolio weighting schemes?

The rest of the paper is structured as follows: in Sect. 2 we offer a small literature review of papers that related to portfolio construction; in Sect. 3 we present the new methodology of median-adaptive portfolios and the four weighting schemes that we consider; in Sect. 4 we describe our data and offer an extended discussion of empirical performance of the proposed new class of portfolios; finally, Sect. 5 concludes and offers some suggestions for future research.

2 Literature review

There is a plethora of articles and approaches on modern portfolio management, since the seminal work of Markowitz (1952), and see also Markowitz (1999), and is virtually impossible to review everything here. We shall attempt to give a general overview as it relates to our suggested methodology—a relatively recent review article is Kolm et al. (2014). The rest of our literature review is grouped thematically and chronologically, and is focused on other methods not necessarily associated with the EW portfolio class.

Arditti and Levy (1975) considered the use of higher moments for portfolio optimization. Gilli and Kellezi (2001) suggested a portfolio optimization problem which contain nonlinear and non-convex constraints, optimized with a heuristic method which called threshold accepting. Consigli (2002) presented a jump-diffusion model for markets behavior for the measurement of the portfolio value-at-risk (VaR), while Kibzun and Kuznetsov (2006) analyzed and a compared criteria for VaR and CVaR (conditional VaR) in a portfolio. Cesari and Cremonini (2003) provided a wide comparison between various portfolio management techniques using an extensive simulation comparison of popular dynamic strategies of asset allocation, while Ferson and Khang (2002) introduced a new performance measure based on conditional weight which combines the use of portfolio holdings data and other conditioning information. Clarke et al. (2006) constructed a minimum-variance portfolios and they proved that the empirical research was independent of any assumption about expected returns. Metaxiotis and Liagkouras (2012) considered multiobjective evolutionary algorithms for the selection of the objective function of the portfolio; Fu et al. (2013) used genetic algorithms (GA) to obtain the distribution of weights in a portfolio, while Mittal and Mehlawat (2014) proposed a model including transaction costs also optimized using GA. Xidonas et al. (2017) developed a robust, minimum-variance portfolio optimization model; Plachel (2019) introduced a unified model for regularized and robust large-scale portfolio optimization, which deals with dimension reduction and parameter uncertainty in the covariance matrix estimation, while Bian et al. (2020) proposed a robust estimator for a high dimensional covariance matrix, used in the construction of optimal mean-variance portfolios; recently, Sleire et al. (2022) considered a new method related to return asymmetries where the global covariance matrix can be replaced by a local approximation. Henrique et al. (2019) considered machine-learning methodologies for the prediction of moments of financial variables and Doering et al. (2019) used metaheuristics for achieving portfolio optimization; Pinelis and Ruppert (2022) found economically and statistically significant gains when using machine learning for portfolio allocation between a market index and a risk-free asset.

Bhattacharya and Galpin (2011) discussed the popularity and usage of value-weighting in portfolio construction, while Alper et al. (2017) compared selected portfolio management strategies in the presence of correlation between the assets and transaction fees, and showed that the wealth dynamics of a balanced portfolio is a log-normal process. Gruszka and Szwabiński (2020) continued the work of Alper et al.. Shimizu and Shiohama (2020) introduced an allocation strategy based on inverse factor volatility and Cederburg et al. (2020) showed that neither of these methods suggests a pervasive link between volatility management and improved performance for real-time investors. Hautsch and Voigt (2019) argued that for large-scale portfolios the transaction costs act either as a turnover penalization or as regularization in the covariance matrix. Branger et al. (2017) argued that including jumps we have better portfolio performance, while Zhou et al. (2019) argued that these jumps give rise to substantial model risk. Arouri et al. (2019) investigated the international equity markets affect, and the corresponding international allocation and the diversification benefits. Stoilov et al. (2021) developed an active policy for optimization based on repetitive application of a modified Black-Litterman (BL) portfolio model.

Neumann and Skiadopoulos (2013) investigated if there are predictable patterns in the dynamics of higher order risk neutral moments and Le (2020) explored the benefits of incorporating conditional higher moments in an international portfolio allocation, suggesting that investors tilt towards countries with higher skewness and less kurtosis. Schwaiger and Masood (2021) showed that portfolios consisting of high momentum stocks have exhibited persistent outperformance versus market cap weighted benchmarks and Ehsani and Linnainmaa (2022) showed that momentum is a dynamic portfolio that times other factors. Gao (2021) argued that there are several volatility measures that can be used to manage momentum risk.

Table 6 Statistics for portfolio P3, no optimization

Table 7 Statistics for portfolio P3, with optimization

Table 8 Statistics for portfolio P4, no optimization

Table 9 Statistics for portfolio P4, with optimization

Table 10 Statistics for portfolio P5, no optimization

Table 11 Statistics for portfolio P5, with optimization

As one can easily infer from the above references, there are plenty of advanced methods for portfolio allocation and optimization. Many of these methods use both advanced statistics (e.g., higher moments, expansions, robust covariance estimation etc.) and advanced methods of allocation and optimization. These methods are usually benchmarked and against the EW portfolio and other portfolio methods such as the global-minimum-variance or maximum Sharpe ratio portfolios. However, many of these methods have their explicit or implicit constraints. For example, as the number of assets increases (for any given sample size or data frequency) so do the problems associated with the allocation decision: problems of accurate estimation of moments, problems of covariance singularities, problems of optimization performance and others like them. Clearly there are ways to deal with all of them, but the practical application of asset allocation requires less complexity—for complexity comes at a cost of increased computational times and uncertainty of estimation and inference. Therefore, the contribution we are making in the context of this discussion is that we offer this simplicity without sacrificing a strong theoretical foundation and a prior explanation for our results, within the context of the easily understood EW portfolio. By introducing our new class of portfolios, we offer the practitioner of asset allocation a simple and direct approach to performance-enhancements to the EW portfolio benchmark, thus offering an additional level of comparison with the benchmark and, more importantly, a new method that is fast, explainable and robust in any asset context and any dimension.

3 Methodology of median-adaptive portfolios

3.1 Portfolio structure and properties

We have M assets available, along with their associated returns $r_{ti}$ for $i=1, 2, \dots , M$. We denote the $\displaystyle (M \times 1)$ unit vector by $\displaystyle \varvec{e}\displaystyle {\mathop {=}\limits ^{\text{ def }}}\left[ 1, 1, \dots 1\right] ^{\top }$ and the $\displaystyle (M \times 1)$ vector that has a one in the $i^{th}$ position at time t, and zeros elsewhere, by $\displaystyle \varvec{e}_{t}^{i_{t}} = \displaystyle {\mathop {=}\limits ^{\text{ def }}}\left[ 0, 0, \dots , 1, \dots , 0\right] ^{\top }$. The equal weights are defined as $\displaystyle \varvec{e}/M$ and the dynamic portfolio weights are defined as $\displaystyle \varvec{w}_{t+1|t,j}$, for weights determined at time t and used for evaluating the portfolio returns at time $t+1$; here $j=1, 2, 3, 4$ denotes the corresponding adaptive portfolio—we will examine four (4) weighting schemes.

Let $\gamma \in \left[ 0,1\right]$ be the portfolio tuning parameter. Our first adaptive portfolio and weighting scheme is based on the equal weights as follows: at $t=0$ we start the recursion with $\displaystyle \varvec{w}_{1|0,1}=\displaystyle \varvec{e}/M$, the vector of equal weights. When the first asset returns are observed at $t=1$, say $r_{1i}$, we find the asset that has the median return among all M assets, i.e., we find the index $i_{1}$ that satisfies:

$$\begin{aligned} i_{1}=argmin_{i=1, 2, \dots , M}\left| r_{1i}-m_{1}\right| \end{aligned}$$

(1)

where $m_{1}$ is the (cross-sectional) median return at $t=1$. Then we update the initial, equal, weights by adding more weight to the median asset and less weight to all other assets, as in:

$$\begin{aligned} \displaystyle \varvec{w}_{2|1,1}=(1-\gamma )\frac{\displaystyle \varvec{e}}{M}+\gamma \displaystyle \varvec{e}_{1}^{i_{1}}=\frac{\displaystyle \varvec{e}}{M} +\gamma \left( \displaystyle \varvec{e}_{1}^{i_1}-\frac{\displaystyle \varvec{e}}{M}\right) \end{aligned}$$

(2)

or, in general at time t we have that:

$$\begin{aligned} i_{t}=argmin_{i=1, 2, \dots , M}\left| r_{ti}-m_{t}\right| \end{aligned}$$

(3)

and,

$$\begin{aligned} \displaystyle \varvec{w}_{t+1|t,1}=(1-\gamma )\frac{\displaystyle \varvec{e}}{M}+\gamma \displaystyle \varvec{e}_{t}^{i_{t}}=\frac{\displaystyle \varvec{e}}{M} +\gamma \left( \displaystyle \varvec{e}_{t}^{i_{t}}-\frac{\displaystyle \varvec{e}}{M}\right) \end{aligned}$$

(4)

Note that the new weights continue to be positive and sum to one, with all of equal weights being marginally reduced by $(1-\gamma )$ and the weight at position $i_{t}$ being marginally increased by $\gamma$. That the weights continue to sum-up to one is immediate since we have that:

$$\begin{aligned} \displaystyle \varvec{e}^{\top }\displaystyle \varvec{w}_{t+1|t,1}=(1-\gamma )\frac{\displaystyle \varvec{e}^{\top }\displaystyle \varvec{e}}{M}+\gamma \displaystyle \varvec{e}^{\top }\displaystyle \varvec{e}_{t}^{i_{t}} = (1-\gamma ) + \gamma = 1 \end{aligned}$$

(5)

For the $i_{t}$-asset that its weight is being increased, we note that the following holds:

$$\begin{aligned} w_{t+1|t,1}^{i_{t}}=\frac{1}{M}+\gamma \left( 1-\frac{1}{M}\right) =\gamma + (1-\gamma )\frac{1}{M} \end{aligned}$$

(6)

which shows the “boosting”, over the equal weight, of the $i_{t}$ asset by $\gamma (M-1)/M$. Furthermore, this increase goes to $\gamma$ as M increases, thus making the $\gamma$ parameter the extra weight in progressively larger portfolios. We can see that this first weighting scheme is repetitive on the equal weights every period t, and there is no feedback from past weights in the recursion.

Remark What is the rationale behind our use of the median asset? The easiest answer at this stage of our research is that it works! But there is actually more to it. One can make a twofold argument about the median asset choice. First, if the market exhibits (at least local) positive momentum in some assets and negative momentum in some other assets in the portfolio, then weighting more the tails of the cross-sectional return distribution will skew the return towards our choice; if this choice is correct all will be well, but if it is not correct then this will be detrimental to performance. Thus, the choice of the median-ranked asset makes some a priori sense as being rather “safer” than the tails of the cross-sectional distribution. Second, if the market exhibits mean-reversion, a similar argument about the choice of the assets to assign more weight can be made. If assets change quickly their ranking positions we must have sudden changes in their returns and, again, the median asset should be the one in the center of the cross-sectional distirbution with minimal change in its return. Therefore, either in the case of momentum, or in the case of mean-reversion, the choice of the median-ranked asset should be preferred in terms of these a priori volatility arguments.

The second weighting scheme we consider uses the weights from the first portfolio as input and the same procedure for altering them, like a “double-smoothing” approach. We now write:

$$\begin{aligned} \displaystyle \varvec{w}_{t+1|t,2}=(1-\gamma )\displaystyle \varvec{w}_{t+1|t,1}+\gamma \displaystyle \varvec{e}_t^{i_{t}} = \displaystyle \varvec{w}_{t+1|t,1}+\gamma (\displaystyle \varvec{e}_t^{i_{t}}-\displaystyle \varvec{w}_{t+1|t,1}) \end{aligned}$$

(7)

and we can easily see that it has to obey the same properties as $\displaystyle \varvec{w}_{t+1|t,1}$, in maintaining positivity of the weights and summability to one. Now observe that the $i_{t}$-asset will be further increased in weight so as to have that $w_{t+1|t,2}^{i_{t}} = w_{t+1|t,1}^{i_{t}} + \gamma (1-w_{t+1|t,1}^{i_{t}})$, while all other weights will be further reduced as in $w_{t+1|t,2}^{j} = (1-\gamma )^{2}M^{-1}$, for $j\ne i_{t}$. So, in the second adaptive portfolio the median asset gets an extra “boosting” on its weight and the others are reduced accordingly.

Next, we have the third weighting scheme, where now we have a true recursion, as the current weights have feedback from the previous period. In this portfolio the weights adapt relative to the current and past positioning of the median asset as follows:

$$\begin{aligned} \displaystyle \varvec{w}_{t+1|t,3}= \displaystyle \varvec{e}_t^{i_{t}}+\gamma (\displaystyle \varvec{w}_{t|t-1,3}-\displaystyle \varvec{e}_{t-1}^{i_{t-1}}) \end{aligned}$$

(8)

and we will show that there are some significant differences from the two previous adaptive portfolios. The recursion is also initialized with $w_{1|0,3} = \displaystyle \varvec{e}/M$, the equal weights.

We first observe that the weights will again sum up to one by construction, and is really straightforward to show this. Next, we note that the weights can now become both higher than one and negative and, thus, obtain an “automatic” leveraging and also a short-positioning, if so desired. Observe that $\displaystyle \varvec{w}_{t|t-1,3}$ will always start-off with all its elements being in $\left[ 0, 1\right]$, but if $\displaystyle \varvec{e}_t^{i_{t}}$ is different that $\displaystyle \varvec{e}_{t-1}^{i_{t-1}}$ we will have that:

$$\begin{aligned} \begin{array}{lcl} w_{t+1|t,3}^{i_{t}} &{} = &{} 1+\gamma w_{t|t-1,3}^{i_{t}} > 1\\ w_{t+1|t,3}^{i_{t-1}} &{} = &{} 0+\gamma (w_{t|t-1,3}^{i_{t-1}}-1) < 0 \\ \end{array} \end{aligned}$$

(9)

and, therefore, we have the option to either leave the short-positioning of the past median asset in place, or eliminate that asset altogether, or re-assign to it a fraction of equal weighting. In our applications we follow the simplest approach and eliminate the assets with negative weights and proceed with a “no-short” assumption (and we re-scale all the weights to be less than one by diving them with their sum).

Note that if $\displaystyle \varvec{e}_t^{i_t}=\displaystyle \varvec{e}_{t-1}^{i_{t-1}}$ and the median asset position does not change we get:

$$\begin{aligned} w_{t+1|t,3}^{i_{t}}=1+\gamma (w_{t|t-1,3}^{i_{t}}-1)= \gamma w_{t|t-1,3}^{i_{t}}+(1-\gamma )> 0 \end{aligned}$$

(10)

because of our assumption that $\gamma$ lies in the unit interval.

It is easy to show that, according to this “no-short” assumption for the above weighting scheme, as the median asset’s position changes we will end-up with an asset rotation: that is, as the index of the median asset changes, the old (existing) median asset gets a zero weight and then in all subsequent periods will continue with a zero weight unless it becomes again the median asset—this implies asset elimination over time. This is, of course, a problem if one wants to maintain a multi-asset allocation across time (it is not however a problem if this portfolio is used as a purely quantitative investment strategy). We can, furthermore, show from solving the recursion of Eq. (8) that, as t increases, the weights are those of a pure rotation to the median asset. Note that, by substitution, we obtain:

$$\begin{aligned} \begin{array}{lcl} \displaystyle \varvec{w}_{t+1|t,3} &{} = &{} \displaystyle \varvec{e}_t^{i_{t}}+\gamma \left[ \displaystyle \varvec{e}_{t-1}^{i_{t-1}}+\gamma (\displaystyle \varvec{w}_{t-1|t-2,3}-\displaystyle \varvec{e}_{t-2}^{i_{t-1}})-\displaystyle \varvec{e}_{t-1}^{i_{t-1}}\right] \\ \ {} &{} = &{} \displaystyle \varvec{e}_t^{i_{t}}+\gamma ^{2}(\displaystyle \varvec{w}_{t-1|t-2,3}-\displaystyle \varvec{e}_{t-2}^{i_{t-1}}) \\ \dots &{} = &{} \dots \\ \ {} &{} = &{} \displaystyle \varvec{e}_t^{i_{t}} + \gamma ^{t}\displaystyle \varvec{e}/M \\ \end{array} \end{aligned}$$

(11)

and its easy to see that the last line of the above equation converges to $\displaystyle \varvec{e}_{t}^{i_{t}}$ as t increases. To illustrate the weights resulting from the solution of the recursion, and how one can create a viable portfolio structure, consider the following. Take $M=10$ and $\gamma =0.9$ and assume that first $t=6$. Then, all the weights (re-scaled by their sum) are 68.76% for the median asset and 3.47% for all other assets. For $t=12$ the respective weights become 80.17% for the median asset and 2.20% for all other assets. A calculation like this might be a (very) crude guide as to when one wants to possibly “re-start” the recursion so as to maintain a particular weighting scheme and to avoid pure rotation that leads into asset elimination.

A simple and straightforward approach to convert the above asset rotation, for large t, again into a feasible portfolio is to smooth the weights like the following:

$$\begin{aligned} \displaystyle \varvec{w}_{t+1|t,3}^{S} = \frac{1}{K}\sum _{j=0}^{K-1}\displaystyle \varvec{w}_{t+1-j|t-j,3} \end{aligned}$$

(12)

using a K-order moving average and, in this way, one will maintain all the properties of the resulting smoothed weights (positivity and summability to one) and a multi-asset allocation. We will illustrate these two approaches (rotation and smoothing) for the third adaptive portfolio in our empirical applications section.¹

We next present our fourth weighting scheme by offsetting the $\displaystyle \varvec{w}_{t|t-1,3}$ weights not by $\displaystyle \varvec{e}_{t-1}^{i_{t-1}}$ but by $\displaystyle \varvec{e}_{t}^{i_{t}}$. This results in the following updating formula, initialized with equal weights, that now has adaptive learning:

$$\begin{aligned} \displaystyle \varvec{w}_{t+1|t,4}= \displaystyle \varvec{e}_t^{i_{t}}+\gamma (\displaystyle \varvec{w}_{t|t-1,4}-\displaystyle \varvec{e}_{t}^{i_{t}}) = (1-\gamma )\displaystyle \varvec{e}_{t}^{i_{t}} + \gamma \displaystyle \varvec{w}_{t|t-1,4} \end{aligned}$$

(13)

and note that the weight on the current median asset is that of Eq. (10), i.e., in this fourth recursion we maintain the structure of the third recursion as if the median asset does not change (although it clearly does in practice); furthermore, note that this last recursion will deliver the most smooth evolution of the initial equal weights, as it is a convex combination of existing weights (starting with the equal ones) and a “boosting” of $(1-\gamma )$ on the median asset. As we did for the third adaptive portfolio, we can solve the recursion of Eq. (13) to obtain the solved weights as an exponentially weighted sum of all past median assets, as follows:

$$\begin{aligned} \begin{array}{lcl} \displaystyle \varvec{w}_{t+1|t,4} &{} = &{} (1-\gamma )\displaystyle \varvec{e}_{t}^{i_{t}} + \gamma \left[ (1-\gamma )\displaystyle \varvec{e}_{t-1}^{i_{t-1}} + \gamma \displaystyle \varvec{w}_{t-1|t-2,4}\right] \\ \ {} &{} = &{} (1-\gamma )\displaystyle \varvec{e}_{t}^{i_{t}} + \gamma (1-\gamma )\displaystyle \varvec{e}_{t-1}^{i_{t-1}} + \gamma ^{2}\displaystyle \varvec{w}_{t-1|t-2,4} \\ \dots &{} = &{} \dots \\ \ {} &{} = &{} (1-\gamma )\sum _{j=0}^{t-1}\gamma ^{j}\displaystyle \varvec{e}_{t-j}^{i_{t-j}} + \gamma ^{t}\displaystyle \varvec{e}/M \end{array} \end{aligned}$$

(14)

and we can see both the exponential structure on the past median assets and the relative weighting that we can obtain for each asset.²

Although is not strictly needed, we also consider the application of weight smoothing of Eq. (12) not just to the third adaptive portfolio but to all portfolios that we are presenting. We also note that the smoothed weights are used in the calculation of the corresponding portfolio returns but are not used in the recursions (so smoothing only comes at the end while the recursions continue as described above). Following Eq. (12) we write $\displaystyle \varvec{w}_{t+1|t,m}^{S}$ for the smooth weights of portfolio $m=1, 2, 3, 4$.

We close this section by noting that the methodology just presented offers certain practical advantages over other existing methods. First, there is no need for any optimization; second, it can be applied to large portfolios with no computational burden; third, it does not require the estimation of any moments of the asset returns; fourth, it has a natural benchmark in the equally weighted portfolio; and, finally, there is no reason as to why these adaptive portfolios, that are initialized by using equal weights, cannot be adapted to work with any other weighting scheme from any method of constructing portfolio weights.

3.2 Is there an “optimal” value for $\gamma$?

The results on the previous section hold for any value of the tuning parameter $\gamma$ in the unit interval. However, as the weights are functions of that parameter, it is natural to consider whether an “optimal” value for this parameter can be found. Doing so, however, requires that we move away from the apparent simplicity of the proposed weighting schemes and, at a minimum, require that some assumptions are made about the estimation of the variance-covariance matrix of the asset returns, in the context of a minimum-variance objective function.³ Furthermore, it should be apparent from the nature of the weights in Eqs. (4), (7), (8) and (13), that only the weights of the first adaptive portfolio have an explicitly linear structure; the other three adaptive portfolios have a non-linear structure on the $\gamma$ parameter. We now show that for the first adaptive portfolio we can obtain an explicit value for the variance-minimizing $\gamma$.

Take the weights of the first adaptive portfolio, from Eq. (4). These weights depend only on the scalar parameter $\gamma$, irrespective of the number of assets M. Suppose that we wish to minimize the variance of the portfolio, given an estimate⁴ of the variance-covariance of the assets—say $\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}$. Then, denoting by $\displaystyle \varvec{w}_{t+1|t,1}(\gamma )$ the vector of the weights of Eq. (4), we want to find the variance-minimizing value of $\gamma _{t,1}^{*}$ from:

$$\begin{aligned} \gamma _{t,1}^{*} \displaystyle {\mathop {=}\limits ^{\text{ def }}}{{\,\textrm{argmin}\,}}_{\gamma _{t}}\displaystyle \widehat{\sigma }_{t+1|t,1}(\gamma _{t}) = \displaystyle \varvec{w}_{t+1|t,1}(\gamma _{t})^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{w}_{t+1|t,1}(\gamma _{t}) \end{aligned}$$

(15)

where we have made explicit the time-varying nature of the parameter. This is a natural consequence of the adaptive nature of the weights, for they depend on $\displaystyle \varvec{e}_{t}^{i_{t}}$ which changes with t; therefore, the variance-minimizing value of $\gamma$ must be time-varying in this context.

This is a standard problem that has an explicit solution via linear algebra, which we now illustrate. Taking the first order conditions we obtain:

$$\begin{aligned} \frac{\partial \displaystyle \widehat{\sigma }_{t+1|t,1}(\gamma _{t})}{\partial \displaystyle \varvec{w}_{t+1|t,1}(\gamma _{t})} = 2\displaystyle \varvec{w}_{t+1|t,1}(\gamma _{t})^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\frac{\partial \displaystyle \varvec{w}_{t+1|t,1}(\gamma _{t})}{\partial \gamma _{t}} = 0 \end{aligned}$$

(16)

and is immediate to see from Eq. (4) that:

$$\begin{aligned} \frac{\partial \displaystyle \varvec{w}_{t+1|t,1}(\gamma _{t})}{\partial \gamma _{t}} = \displaystyle \varvec{e}_{t}^{i_{t}} - \frac{\displaystyle \varvec{e}}{M} \end{aligned}$$

(17)

We can combine terms and then solve for $\gamma _{t,1}^{*}$ from the following equation:

$$\begin{aligned} \left[ (1-\gamma _{t,1}^{*})\frac{\displaystyle \varvec{e}}{M}+\gamma _{t,1}^{*} \displaystyle \varvec{e}_{t}^{i_{t}}\right] ^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\left( \displaystyle \varvec{e}_{t}^{i_{t}} - \frac{\displaystyle \varvec{e}}{M}\right) = 0 \end{aligned}$$

(18)

whereas, upon completing the algebra, we can find the following variance-minimizing value of $\gamma _{t,1}{*}$ to be:

$$\begin{aligned} \gamma _{t,1}^{*} \displaystyle {\mathop {=}\limits ^{\text{ def }}}\frac{\left( \displaystyle \varvec{e}^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}_{t}^{i_{t}}/M - \displaystyle \varvec{e}^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}/M^{2}\right) }{\left( \displaystyle \varvec{e}^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}_{t}^{i_{t}}/M - \displaystyle \varvec{e}^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}/M^{2} - \displaystyle \varvec{e}_{t}^{i_{t},\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}_{t}^{i_{t}} - \displaystyle \varvec{e}_{t}^{i_{t},\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}/M\right) } \end{aligned}$$

(19)

We note that, in the above equation, the components of the variance-minimizing value of $\gamma$ are as follows:

• The (scaled by M) variance and sum of all covariances for the $i_{t}$ asset, i.e., $\displaystyle \varvec{e}^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}_{t}^{i_{t}}/M$.

• The (scaled by $M^{2}$) sum of all variances and covariances, i.e., $\displaystyle \varvec{e}^{\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}/M^{2}$.

• The variance of the $i_{t}$ asset, i.e., $\displaystyle \varvec{e}_{t}^{i_{t},\top }\displaystyle \widehat{\displaystyle \varvec{\Sigma }}_{t+1|t}\displaystyle \varvec{e}_{t}^{i_{t}}$.

Although we have this explicit, algebraic, solution, the optimization problem is also trivial to solve numerically. We thus opt to go with the numerical optimization route in finding the variance-minimizing value of $\gamma$ for the A1, A2 and A4 portfolios—and report results accordingly.

3.3 The probabilistic structure of the portfolio weights

The portfolio allocation problem, as presented above, has the advantage of having a decision set of minimum dimension—a problem with the minimum required number of criteria for decision making. The decision variables are just M, the number of assets and $\gamma$, the parameter that determines the weights—and this is so in any of the four weighting schemes that we consider in the paper. This is a distinct advantage over any other method, as it requires minimum computational time and minimum structure to be imposed for solving the portfolio allocation problem. There is no need, in the fixed $\gamma$ case, for computing any moments of the returns and, furthermore, the cross-sectional information is contained in the median ranked asset that determines the weight rebalancing.

The minimum dimension of the decision set is aided by the, naturally occurring, time-variation of the weights through the use of the median-ranked asset and the associated learning that takes place over time. In other words, this weight adaptability works in our favor for we do not have make choices over time. This time-variation of the weights has a probabilistic structure and therefore an a priori interpretation. To explain this let us consider the first weighting scheme of Eq. (4).

We start by noting that the vector with 1 in the position of the median ranked asset follows a one-draw, multinomial distribution, i.e., $\displaystyle \varvec{e}_{t}^{i_{t}} \sim \mathcal{M}(n=1, \displaystyle \varvec{p}_{t})$, where $\displaystyle \varvec{p}_{t} \displaystyle {\mathop {=}\limits ^{\text{ def }}}\left[ p_{t1}, p_{t2}, \dots p_{tM}\right] ^{\top }$ is the $\displaystyle (M \times 1)$ vector of probabilities of each asset being the median-ranked asset in period t. In this set-up, and without additional assumptions about the temporal characteristics and properties of these probabilities, it is immediate to find the expected weights for our four weighting schemes. If we look at Eqs. (4), (11) and (14) we can see that the expected weights for weighting schemes 1, 3 and 4 are respectively given by (for large t):

$$\begin{aligned} \begin{array}{lcl} \displaystyle \textsf{E}\left[ \displaystyle \varvec{w}_{t+1|t,1}\right] &{} = &{} (1-\gamma )\displaystyle \varvec{e}/M + \gamma \displaystyle \varvec{p}_{t} \\ \displaystyle \textsf{E}\left[ \displaystyle \varvec{w}_{t+1|t,3}\right] &{} = &{} \displaystyle \varvec{p}_{t}\\ \displaystyle \textsf{E}\left[ \displaystyle \varvec{w}_{t+1|t,4}\right] &{} = &{} (1-\gamma )\sum _{j=0}^{t-1}\gamma ^{j}\displaystyle \varvec{p}_{t-j}\\ \end{array} \end{aligned}$$

(20)

which shows the underlying data generating process of the weights based on these probabilities. The above equation explicitly shows that our approach is a minimum criteria approach, for the only parameters of choice are M and $\gamma$, and that once these parameters are selected then the updating of the weights occurs under the probabilistic law of $\displaystyle \varvec{p}_{t}$. Finally, note that one might actually opt to use the expected weights in place of the actual weights of the method, and this is done quite easily: the associated probabilities can be found by computing proportions of the number of times an asset becomes the median asset. This can be done in standard fashion, using a rolling or a recursive window of observations, but do note that it will add an additional layer of complexity in the suggested methodology. If one is willing to accept this additional dimension for making a final decision, then the expected weights can be used in applications.

3.4 Portfolio allocation as an MCDA problem

An interesting aspect of the proposed methodology is that it can be viewed as, or interpreted, a minimum-criteria decision analysis problem. Within the larger, generic, context of multi-criteria decision analysis we can observe that in our methodology one is faced with a decision problem (the allocation of weights in the portfolio), given the number of assets or choosing the number of assets, and then only one other choice must really be made: that of the single parameter $\gamma$. If the number of assets M is assumed to be exogenously given by other considerations (e.g. via an expert or via fundamental measurements on company performance), then the decision problem is essentially a one-parameter one. Therefore, we can and do say that our approach is a minimum-criteria decision analysis problem. Furthermore, we should note that even the choice of $\gamma$ can be further automated if an objective function (such as variance minimization) is selected as a target.

However, we rarely face the case where the decision variables are predetermined. What would thus be the implications for our methodology if we were to allow for more flexibility in the decision space? Let us explore this a bit further. The total number of variables that we have are these: M, $\gamma$, K the smoothing parameter and the choice of a rolling or recursive estimate of the covariance matrix (as discussed in the empirical section that follows). If all these are free to vary then one has to consider in what order of importance might one choose them. Let’s start with M the number of assets in the portfolio. Here we have two decision branches: if we do not involve a target function for optimization the number of assets becomes largely irrelevant, for even if M is in the thousands one still needs to choose the value of $\gamma$ only, for any given K. If one decides on having a target function for optimization then the number of assets clearly plays a role: if we have more observations that assets then standard methods of estimation might be employed (using either a rolling or recursive approach to estimation); if we have more assets than observations then one has to consider dimension-reduction methods or more complicated approaches to covariance estimation.

What needs to be clearly seen here is that if M is in the hundreds of assets, then there is going to be a more even distribution of the allocation in each rebalancing period and the portfolio will behave more closely to the equally weighted one. While our proposed method can, as explained above, handle this situation we conjecture that one might be better off choosing not just the median asset but a number of assets around the median asset; this way one can induce additional variability in the weights and potentially improve performance. This is something that we consider in our ongoing research.

A final note on the choice of $\gamma$. In the current implementation this is the parameter of interest and we have shown that it can either be selected in the unit interval or optimized via a target function. There is an alternative way of looking at this parameter in the context of an MCDA analysis: that $\gamma$ becomes a random variable that has some distribution (which clearly cannot be the uniform!). In this case, the nature of the problem changes completely for we have to introduce additional structure to handle it—and this will take away the simplicity of the proposed method. This is also something that we consider in our ongoing research.

4 Empirical applications

4.1 Data and portfolios

For our empirical applications we consider a number of portfolios that consist of Exchange Traded Funds (ETFs). The use of ETFs is widespread due to their breadth of asset representation and relatively low cost on their transactions. We consider monthly returns for the computation of the position of the median asset and, correspondingly, we have monthly-rebalanced portfolios. Table 1 has the ETF list, from which we construct our various portfolios. The data start from the earliest common date in each portfolio (varies) and end in August 2022.

Our portfolios are as follows:

• P1. This is the portfolio for the extended example that follows; it consists of SPY (S &P500), TLT (long-term fixed income), SHY (short-term fixed income, cash equivalent), SH (short positioning on SPY), OIH (oil & oil services) and GLD (gold shares).

• P2. This is a modification of the extended example portfolio above; it consists of SPY (S &P500), OIH (oil and oil services), GLD (gold shares), EEM (emerging markets), TLT (long-term fixed income), IWF (Russell 1000 Growth), SHY (short-term fixed incomes, cash equivalent) and SH (short positioning on SPY).

• P3. This is a portfolio for the sectoral ETFs of the S &P500; it consists of SPY (S &P500), XLK (technology sector), XLU (utilities sector), XLP (consumer staples sector), XLF (financial sector), XLE (energy sector), XLI (industrial sector), XLB (materials sector), XLV (health care sector) and XLY (consumer discretionary sector).

• P4. This is a smaller, subset, portfolio of P1 and P2; it consists of SPY (S &P500), TLT (long-term fixed income), SHY (short-term fixed income, cash equivalent), SH (short positioning on SPY), i.e., it does not have the commodities.

• P5. Our final portfolio is the largest of all four above and nests them all; it consists of SPY (S &P500), TLT (long-term fixed income), SHY (short-term fixed income, cash equivalent), SH (short positioning on SPY), OIH (oil & oil services), IWF (Russell 1000 Growth), IWD (Russell 1000 Growth), GLD (gold shares), SLV (silver shares) EEM (emerging markets) XLK (technology sector), XLU (utilities sector), XLP (consumer staples sector), XLF (financial sector), XLE (energy sector), XLI (industrial sector), XLB (materials sector), XLV (health care sector), XLY (consumer discretionary sector), KRE (regional banking), VNQ (real estate), XHB (homebuilders), IYT (transportation), IBB (biotechnology), SMH (semi-conductors), QQQ (NASDAQ), FDN (DJ Internet index), XME (metals and mining), EZU (Eurozone), EWJ (Japan), EWU (United Kindgom), USO (United States Oil), DBC (Commodity Index track), LQD (investment grade corporate bonds), HYG (high yield corporate bonds), SSO (2x-leveraged S &P500) and SH (short positioning on SPY).

Our results are evaluated using standard measures as follows: we compute and report the (annualized) mean return and the associated volatility, their (Sharpe) ratio, the total return (TR) of the portfolio performance since inception and the maximum drawdown. These measures appear—in this order—in the tables, for each of the adaptive portfolios and the equally weighted benchmark.

4.2 Comparative results and discussion

We start our discussion by considering a comparison on the performance of the fixed $\gamma$ portfolios, with the values of $\gamma =0.5$ and $\gamma =0.75$. Our baseline case is that of no weight smoothing, which appears as the top panel in Tables 2, 4, 6, 8 and 10. If we look at total return performance we can see that almost all the new weighting schemes A1 to A4 lead to higher total returns with the higher value of $\gamma =0.75$. This would have been a possibly anticipated result should we have chosen the asset with maximum return, i.e., if we have followed momentum. However, in our case we are choosing to “boost” the weight of the median-ranked asset and the good performance might be attributed to mean-reversion considerations rather than to momentum. If we look at the risk-return reward of the Sharpe ratio, and also the maximum drawdown, as our metrics of performance evaluation, then many of the weighting schemes perform better with the lower value of $\gamma =0.5$, which is consistent with the result on total returns—it would nice to have the same value of $\gamma$ leading to higher returns and lower risk at the same time, but it does not occur everywhere. But when we look at the A4 weighting scheme we do see that it exhibits a higher Sharpe ratio consistently with a higher value of $\gamma =0.75$; this is a suggestive and practical result because the A4 weighting scheme, which is based on adaptive learning and is dynamic, while maintaining weights that are “spread” out and not collapsing into one asset as in the A3 weighting scheme. We can see, therefore, that the use of adaptive learning combined with the median asset update in A4 leads to overall performance enhancements. Finally, we note that in terms of the lowest maximum drawdown its the A1 weighting scheme that performs better in most cases; however, the results are more mixed and data dependent. There are two main winners here, the A1 and the A4 weighting schemes, and we note that the use of the smaller value of $\gamma =0.5$ links more closely with A1 while the larger value of $\gamma =0.75$ links more closely with A4 (Tables 3, 4, 5, 6).

Turning now to the case where we apply weight smoothing, we can see among the relevant tables in the second and third panel of each one, that, in terms of Sharpe ratio and total return performance, the results are rather more mixed than in the case of no-smoothing—but there are some important differences.⁵ We can observe that we obtain better results for the smaller value of smoothing (K = 3), and the two top performing weighting approaches are A3 and A4, as before. However, we can also observe that now it is the higher value of $\gamma =0.75$ that links with higher Sharpe ratio and the lower value of $\gamma =0.5$ that links with higher total return—in contrast to the no-smoothing. More specifically, note that in portfolios P3 and P5 we have that the A3 weighting scheme is an overall winner in terms of Sharpe ratio and total return, with the higher value of $\gamma =0.75$ and both values of smoothing, (K = 3 and K = 12). Now, in this case of smoothing this result on the outperformance of the A3 weighting scheme is neither unexpected nor difficult to explain. The A3 weighting scheme tends over time to concentrate weight on one asset only an bud becomes asset rotation, rather than a portfolio, and as long as the median-ranked asset offers a boost the rotation will outperform the portfolio. The smoothing induces more dispersion on the weights and therefore it is natural that with smoothing the performance of the A3 weighting scheme will be good both in terms of total returns and in terms of the Sharpe ratio. We continue to see that in this case the A1 weighting scheme offers the best performance overall for lower maximum drawdowns.

Let us turn now to the case where the value of $\gamma$ is selected by variance minimization. The results are collected in Tables 3, 5, 7, 9 and 11. We present results for weighting schemes A1, A2 and A4, with and without smoothing and with recursive and rolling estimation of the covariance matrix.⁶ The left panel of each table has the results with the recursive estimation and the right panel has the results of the rolling estimation of the covariance matrix. As anticipated, when we have a particular objective function to optimize, there are significant performance enhancements with respect to risk measures. With the exception of only one portfolio P3, for all other four portfolios that we examine we can see that the optimization leads to very significant drops in the maximum drawdown and increases in the Sharpe ratio (in many instances also of considerable magnitude). Although these enhancements come at the expense of lower total returns vis-a-vis the fixed $\gamma$ case, we do note that the total returns are higher than those of the equally weighted portfolio in almost all instances. Furthermore, and to counter an argument that the equally weighted portfolio is not the appropriate benchmark here, we note that we are optimizing a single parameter at almost nil computational cost. Comparing our results with a global minimum variance portfolio is certainly appropriate in terms of using optimization; but we would not be comparing the same computational complexity and certainly not the same weight structure that we use here. Our main benchmark is the equally weighted portfolio and that we can easily beat, be that with the A1 weighting scheme that is static or with the A4 weighting scheme that is dynamic (Tables 8, 9, 10, 11).

5 Concluding remarks

We have introduced a new class of one-parameter portfolios and several approaches for obtaining the portfolio weights. This class derives from the equally weighted (EW) class, one of the most fundamental and robust benchmarks in the theory and practice of asset allocation. Our main contribution is that we offer a new methodology of moment-free asset allocation that leads to considerable performance enhancements to the EW benchmark, without having to sacrifice much in terms of simplicity and speed of application and, importantly, without any constraints in terms of the dimensionality of the problem. Our results depend on our second main contribution, the introduction of the position of the median asset of the portfolio into the weighting schemes we considered. This new class of portfolios can be implemented in any portfolio size and it requires no optimization (unless so desired).

Our new methodology offers considerable simplifications to the asset allocation decision because of its following characteristics: no need to compute any moments for the assets (expected returns and/or covariance matrices) thus avoiding completely problems of estimation; adaptation and dynamic learning for the weights over time; no need for optimization as the tuning parameter can be set with ex-ante considerations in mind based on how far one wants to deviate from the equally weighted portfolio; if one wants to optimize the value of the single parameter in this new class of portfolios, it can easily be accomplished with any objective function that defines portfolio performance.

Furthermore, we note that there is also a number of additional practical considerations that suggest the use of proposed methodology. The methodology is easy to understand and implement, without complications of set-up or optimization constraints because of the size of the portfolio; it is, correspondingly, a low-maintenance methodology in terms of resources needed; it is equally suitable to individual or institutional investors, particularly for the latter when simplicity and transparency of operation is required by the regulatory authorities.

Our empirical results suggest that with our new approach we can obtain significant performance enhancements over the equally weighted benchmark, in terms of higher total returns and Sharpe ratios, but also lower maximum drawdowns. Thus, we consider that this new class of portfolios can easily be used as a compliment of, or even a competitive substitute, to the equally weighted portfolio. Given our empirical results, we do note that we have not performed comparisons with other literature benchmarks such as the global minimum variance portfolio or the maximum Sharpe ratio portfolio; although these portfolios are not directly comparable with either the EW portfolio or our new ones, it will greatly enhance the scope and range of application of the median-adaptive portfolios if we can show that they outperform other industry benchmarks as well. Finally, we are currently working for a better understanding of the reasons for which the median-ranked asset works well in our suggested approach, and not an asset in some other position of the cross-sectional distribution of the portfolio returns. We are planning to address these and related questions in our future research.