Mean–variance vs trend–risk portfolio selection

Abstract

In this paper, we provide an alternative trend (time)-dependent risk measure to Ruttiens’ accrued returns variability (Ruttiens in Comput Econ 41:407–424, 2013). We propose to adjust the calculation procedure to achieve an alternative risk measure. Our modification eliminates static mean component and it is based on the deviation of squared dispersions, which reflects the trend (time factor) precisely. Moreover, we also present a new perspective on dependency measures and we apply a PCA to a new correlation matrix in order to determine a parametric and nonparametric return approximation. In addition, the two-phase portfolio selection strategy is considered, where the mean–variance portfolio selection strategies represent the first optimization. The second one is the minimization of deviations from their trend leading to identical mean and final wealth. Finally, an empirical analysis verify the property and benefit of portfolio selection strategies based on these trend-dependent measures. In particular, the ex-post results show that applying the modified measure allows us to reduce the risk with respect to the trend of several portfolio strategies.

1 Introduction

In a portfolio optimization theory, a rationally minded investor focuses on maximizing potential profit with respect to possible portfolio losses (risk incurred). In general, the question of how to adequately estimate and evaluate the risk of a large-scale portfolio or a single asset remains an important problem in financial risk management (Hallerbach and Spronk 2002). In the existing literature, return time series are generally identified as specifically distributed random variables with fat tails and higher peaks (Fama 1965; Mandelbrot 1963; Ortobelli et al. 2017). Decades ago, Mandelbrot (1963) analysed an empirical distribution of equity returns, proved that the assumption of normality made in the then existing financial theories (e.g. Markowitz 1952; Tobin 1958) was a misconception, and explained the distribution of returns by a stable distribution. Recently, this finding was pursued by Rachev and Mittnik (2000), Rachev et al. (2008), and Ortobelli and Tichý (2015).

Since the beginning of the modern portfolio theory presented by Markowitz, portfolio selection has depended on a mean and a risk represented by the variance of the historical joint distribution of all asset returns (Markowitz 1952). However, the variance examines and penalizes both-side deviations and so it does not correspond to rational thinking about financial exposure. In the measurement of investment risk, it is more relevant to use semivariance or semideviation, which takes into account only downside deviations (losses) (Markowitz 1959). Even during the last dozen years, numerous variants of portfolio risk measures have been proposed in the literature (Artzner et al. 1999; Rockafellar and Uryasev 2002; Szegö 2002; Rachev et al. 2008).

A specific group of risk measures evaluates possible losses based on the quantiles of the return distribution, such as Value-at-Risk (VaR) or Conditional Value-at-Risk (CVaR) (Duffie and Pan 1997; Altman et al. 1998; Jorion 2000; Rockafellar and Uryasev 2002). Furthermore, the coherent risk measures presented in Artzner et al. (1999) and their subsequent adjustments (Delbaen et al. 1998; Miller and Ruszczyński 2008, and references therein) were designed to detect the amount of capital requirement by expressing the riskiness of the financial portfolio numerically. The above-mentioned risk measurement approaches are established on the basis of historical returns, which essentially eliminates any time impact.

Therefore, a significant problem related to these risk measures is the incorporation of the time factor, which plays an important role in the financial area. In this context, Ruttiens (2013) proved that for the statistical measures used on resorted historical data, the time factor is not negligible, and proposed alternative dynamic risk measures. Recently, Ortobelli et al. (2017) provided an empirical application of such a time-dependent risk measure in the portfolio selection framework. Even though Ruttiens provides a suitable time-compromising approach to measure risk, we modify its original formulation into the trend analysis framework.

The major contribution of this paper is to provide a time-dependent risk (dispersion) measure which is an alternative to the “accrued returns variability” (ARV) introduced by Ruttiens (see Ruttiens 2013). Moreover, we enhance the existing literature focused on portfolio theory and trend-dependent risk measures (Markowitz 1952; Rockafellar and Uryasev 2002; Szegö 2002; Rachev et al. 2008; Ortobelli et al. 2019). In the original work of Ruttiens, the ARV is calculated as the standard deviation of spreads between cumulative returns and a trend risk free variation of these returns at each time leading to the same final cumulative returns (or the final value of a portfolio). Rather, the definition of the non-volatile linear alternative formulation could correspond to the analysis of trend dispersion. The mathematical formulation of the presented dynamic risk measure can be more precise when we consider the mean of squared spreads of cumulative return series with respect to the non-volatile benchmark. Therefore, our “modified accrued returns variability” (modARV) is more accurate in the portfolio optimization framework while minimizing distortion with respect to the predefined zero-risk linear trend. Additionally, we define dependency measures, such as covariance and correlation, derived from the proposed modifications that can be used in the principal component analysis (PCA) framework.

Furthermore, we apply a newly proposed trend–risk measures and related dependency matrices in various mean–variance portfolio selection strategies to demonstrate their properties in the portfolio theory. In particular, to evaluate the effect of trend-dependent risk measures, we compare the mean–variance optimization strategy with a compounded double optimization strategy. This double optimization strategy consists of two steps. In the first step, we fit optimal portfolios of the mean–variance efficient frontier. In the second step, we determine the minimum modified Ruttiens risk measure fixing the expected mean and the final wealth of the optimal mean–variance portfolios in order to minimize the deviations with respect to the trend. To reduce the dimensionality of the portfolio, we use parametric and nonparametric returns approximation techniques with PCA applied to linear or trend-dependent correlation matrices (Ruppert and Wand 1994; Ortobelli et al. 2019; Kouaissah and Hocine 2021). According to this empirical analysis, the newly proposed approach leads to the mitigation of shortcomings and improves the ex-post portfolio statistics compared to the mean–variance scenarios.

This paper is structured as follows. In Sect. 2, we discuss the trend–risk and trend-dependency measures based on ARV. In Sect. 3, we discuss parametric and nonparametric return approximation techniques based on the PCA. Section 4 presents the empirical analysis. Conclusions are summarized in Sect. 5.

2 Theoretical aspects of trend–risk measures

The purpose of this section is to discuss the trend–risk measure called “accrued returns variability” (ARV) presented by Ruttiens and to provide our comprehensive modification that leads to a more precise and accurate computation in the portfolio optimization process.¹

First, let us begin with an example that helps to understand the main reason why measuring the risk using cumulative return series alongside the classical statistics of returns (variance, standard deviation, or semi-deviation). Basically, classical statistics measure the one-sided or two-sided risk associated with each return series as the mean of the deviations from the average. An important issue in financial time series arises from considering static types of statistics in the computation to determine the individual deviations with the time-dependent variable. Due to the constancy of the mean value over time, alternative penalty methods have been proposed to emphasize the time dependence of the returns (Fastrich et al. 2015).

To illustrate the idea of a time-dependent (trend-dependent) risk measure, consider two identical portfolio return series, but ordered differently. Both series of returns lead to the same final cumulative value as captured in Fig. 1. The commonly analysed statistics, e.g. mean, variance, standard deviation, skewness, and kurtosis, are identical for both series. However, from the investor’s perspective, the ways to obtain the final cumulative return are not equal. The cumulative return of portfolio2 looks more volatile. Rationally, we should compare the cumulative returns with the equally accrued return leading to the same cumulative return, which incorporates the impact of time. The non-volatile alternative investment (zero-risk linear trend) can also be seen as a benchmark.

Fig. 1

Evolution of cumulative returns of two portfolios compared to the linear trend line

Fig. 2

Impact of ARV and ARV$_{\text {mod}}$ on the portfolio optimization task

It is apparent that Ruttiens defined ARV as the standard deviation of the positive and negative distances between the cumulative returns at a specific time and the analogous ones with a zero volatility linear trend.

Based on this concept, let us assume the portfolio returns are $x^{\prime }r$, where the vector of portfolio weights is denoted by $x=[x_1,\dots , x_z]$ and the vector of returns is $r=[r_{1},r_{2},\dots ,r_{z}]$ with $x_i$, respectively, $r_i$, the weight and return of the ith asset for $i=1,2,\dots ,z$. Suppose that we have T observations of the portfolio returns. Then we denote the tth observation of the portfolio by $x^{\prime }r_{(t)}$. Assume the cumulative portfolio return is $c_{{x^{\prime }r}}$. Then its tth observation is given by $c_{{x^{\prime }r},t}=c_{{x^{\prime }r},t-1} (1+{x^{\prime }r}_{(t)})$, for $t=1,\dots , T$. At the same time, the tth observation of the equally accrued return $e_{x^{\prime }r}$ is $e_{{x^{\prime }r},t}=c_{{x^{\prime }r},0}+\frac{t}{T} (c_{{x^{\prime }r},T}-c_{{x^{\prime }r},0})$ and $c_{{x^{\prime }r},0}$ represents the initial investment. Recall that the slope of the linear line depends on the original series of the returns and may have an increasing or a decreasing tendency. In addition, if at least one value in the original series is changed, the linear trend changes as well. Furthermore, the mean of the differences $m_{x^{\prime }r}=E(c_{x^{\prime }r}-e_{x^{\prime }r})$ is approximated by its empirical mean $\frac{1}{T} \sum _{t=1}^{T} (c_{x^{\prime }r,t}-e_{x^{\prime }r,t})$. According to Ruttiens (2013), $ARV(x^{\prime }r)$ is the standard deviation of $c_{x^{\prime }r}-e_{x^{\prime }r}$ and is approximated by its empirical standard deviation:

$$\begin{aligned} ARV(x^{\prime }r)\cong \sqrt{ \frac{1}{T} \sum _{t=1}^T [(c_{x^{\prime }r,t}-e_{x^{\prime }r,t})-m_{x^{\prime }r}]^2}. \end{aligned}$$

(1)

Observe that the ARV is a dynamic measure that is sensitive to the order of the return observations over time. Thus, the ARV measure provides better perspectives for measuring the risk of portfolios of financial series than do the classical static risk measures (such as the variance). Clearly, in portfolio theory, a rational investor generally prefers to minimize the so called trend risk, which is formally defined as follows.

Definition 1

We call trend risk the variability around a defined trend.

The basic idea of Ruttiens’ ARV measure is to control the trend risk while maintaining the cumulative returns close to the desired trend throughout the investment period. Unfortunately, Ruttiens’ ARV may lead to an inaccurate valuation of the trend risk, because ARV measures the distance between the cumulative returns and the trend line $e_{x^{\prime }r}$ summed with the average of the deviations $m_{x^{\prime }r}$, as illustrated in Fig. 2.

To mitigate this drawback, we present a modified risk measurement with a refinement of the computation and greater usefulness in the portfolio optimization framework. Improving on Ruttiens (2013), we suggest using the second moment of the deviations $y_{x^{\prime }r}=c_{x^{\prime }r}-e_{x^{\prime }r}$, instead of its standard deviation. Thus, we define a modification of the original ARV measure given by $ARV_{\text {mod}}(x^{\prime }r)=E(y_{x^{\prime }r}^2)$, which is approximated by the empirical second moment of the deviations:

$$\begin{aligned} ARV_{\text {mod}}(x^{\prime }r)\cong \frac{1}{T} \sum _{t=1}^T y_{x^{\prime }r,t}^2, \end{aligned}$$

(2)

where $ARV_{\text {mod}}(x^{\prime }r) \ge 0$ similarly to $ARV(x^{\prime }r)$. It is worth noting that these two trend-dependent measures are highly correlated.

Proposition 1

When $ARV_{\text {mod}}$ equals zero, then the cumulative volatility equals the zero volatility trend line (that is, $ARV_{\text {mod}}(x^{\prime }r)=0 \Leftrightarrow c_{x^{\prime }r}= e_{x^{\prime }r}$), which is required for the elimination of the volatility around the trend line. This is in contrast to the original formulation of Ruttiens, for which ARV equals zero implies the cumulative returns essentially equal the zero volatility trend line plus the average of the deviations (that is, $ARV({x^{\prime }r})=0$ if and only if $c_{x^{\prime }r}-e_{x^{\prime }r}=E(y_{x^{\prime }r})$, where generally $E(y_{x^{\prime }r})\ne 0$).

Remark 1

Observe that the expected deviation constant function is different from the linear trend function or the equally accrued return $e_{x^{\prime }r}$ in Ruttiens’ work, resulting in an inaccuracy in the computation. Thus, when we minimize $ARV_{\text {mod}}(x^{\prime }r)=E(y_{x^{\prime }r}^2)$, the cumulative return portfolio tends to the zero volatility trend line (the equally accrued return $e_{x^{\prime }r}$ is fixed as an optimal wealth path and, therefore, $c_{x^{\prime }r}\rightarrow e_{x^{\prime }r}$). In contrast, considering the minimization of ARV causes a shift of the optimal path by the value of the mean deviation $m_{x^{\prime }r}$, which can lead to a distortion, since $c_{x^{\prime }r}\rightarrow (e_{x^{\prime }r} +m_{x^{\prime }r})$. This situation is also presented in Fig. 2.

Ruttiens’ ARV still offers a usable alternative to measuring risk based on the transformation of the original series to the spread series. Recall that our modification is concentrated on the elimination of trend deviations and mitigating the imperfection mentioned in the computation (optimization) framework. Due to the modification presented, the results of a decision-making process that includes ARV or ARV$_{\text {mod}}$ are not considerably different but, within the theoretical concept examined, more accurate. This measure can also be assigned to the category of trend-deviation measures. In general, we work with the linear trend, though in the financial sphere either an exponential or non-linear trend that replicates a specified index could be desirable.

2.1 Dependency measures

As one of the objectives of this study is to evaluate the effect in the classical Markowitz mean–variance portfolio analysis, the portfolio risk is logically measured through a variance (Markowitz 1952). To calculate this, a dependency structure, expressed by a variance–covariance matrix $\Sigma =[\sigma _{i,j}]$, between all combinations of assets, is essential. Mathematically, the covariance between the ith and the jth return is given by

$$\begin{aligned} \sigma _{i,j}=E[(r_{i,t} - \mu _i)(r_{j,t} - \mu _j)], \end{aligned}$$

(3)

where $\mu _i$ is the mean of the ith returns. Of course, variance and covariance are static types of measures.

Therefore, to evaluate a time dependent measure, we can consider the deviations between the cumulative returns and the zero volatile trend line. In particular, the corresponding Ruttiens covariance structure of the spread series of $(c_i - e_i)$ associated with the ith asset return is formulated as follows:

$$\begin{aligned} \sigma _{i,j}^{Rutt}=E[(c_{i} - e_{i})- m_i ][(c_{j} - e_{j})-m_j], \end{aligned}$$

(4)

where $m_i$ is the expected value of $(c_{i} - e_{i})$. According to Ruttiens (2013), $\sigma _{i,j}^{Rutt}$ is generally higher than the static covariance $\sigma _{i,j}$.

Similarly to the Ruttiens covariance in (4), we propose the modified covariance $\sigma _{i,j}^{mod\,Rutt}$ between the ith and the jth return series based on the spreads. Again, we eliminate the mean component $m_i$ in the calculation so as to take into account only the trend. From the previous assumption, we obtain $\sigma _{i,j}^{mod\,Rutt}$ as follows:

$$\begin{aligned} \sigma _{i,j}^{mod\,Rutt}=E[(c_{i} - e_{i})(c_{j} - e_{j})], \end{aligned}$$

(5)

where the properties of this dependency measure are similar to those of $\sigma _{i,j}^{Rutt}$.

According to this concept, the path-dependent risk measure of the portfolio $ARV(x^{\prime }r)= x^{\prime }\Sigma ^{Rutt} x$ and $ARV_{\text {mod}}(x^{\prime }r)=x^{\prime }\Sigma ^{mod\,Rutt} x$, where $\Sigma ^{Rutt}=[\sigma _{i,j}^{Rutt}]$ and $\Sigma ^{mod\,Rutt}=[\sigma _{i,j}^{mod\,Rutt}]$. To express the dependency structure, linear correlation coefficients, such as those of Pearson or Kendall, are widely used in financial modelling. Many types of correlation measures assess the dependency structure between financial variables, but only some of them are appropriate for reducing the dimensionality with a PCA type of composition or to evaluate the dispersion statistic of portfolios (Ortobelli and Tichý 2015).

The general one is the linear Pearson correlation (PC) formulated as

$$\begin{aligned} \rho _{r_i,r_j}=\frac{E[(r_{i}-\mu _{i})(r_{j}-\mu _{j})]}{\sigma _{r_i} \sigma _{r_j}}, \end{aligned}$$

(6)

where $\sigma _{r_i}=\sqrt{E (r_{i}-\mu _i )^2 }$.

An alternative correlation measure can be used when we assume that the returns are in the domain of attraction of an $\alpha$-stable sub-Gaussian distributed $S_{\alpha } (\gamma ,\beta ,\mu )$ random variable ( Rachev and Mittnik 2000), characterized by its index of stability $\alpha \in (0,2]$, its asymmetry parameter $\beta \in [-1,1]$, its dispersion parameter $\gamma >0$, and its location parameter $\mu$. Thus, we can estimate the stable correlation measure (SC) as

$$\begin{aligned} \rho _{r_{i}, r_{j}}^{Stable} =\frac{E[(r_{i}-\mu _{i}) {\text {sign}}(r_{j}-\mu _{j})]}{2 E(\mid r_{i}-\mu _{i} \mid )} + \frac{E[(r_{j}-\mu _{j}) {\text {sign}}(r_{i}-\mu _{i})]}{2 E( \mid r_{j}-\mu _{j} \mid )}, \end{aligned}$$

(7)

for all $\alpha$-stable sub-Gaussian distributed returns with a finite mean (i.e. $\alpha >1$) (Ortobelli and Tichý 2015).

Since we use the covariance between random variables with respect to their trend in formulas (4) and (5), we can define the following alternative Ruttiens correlation (RC) under the same assumptions:

$$\begin{aligned} \rho _{r_{i}, r_{j}}^{Rutt}=\frac{E\left\{ \left[ \left( c_{i}-e_{i}\right) -m_{i}\right] \left[ \left( c_{j}-e_{j}\right) -m_{j}\right] \right\} }{\sigma _{\left( c_{i}-e_{i}\right) } \sigma _{\left( c_{j}-e_{j}\right) }} , \end{aligned}$$

(8)

where the standard deviation $\sigma _{\left( c_{i}-e_{i}\right) }=\sqrt{E[(c_{i}-e_{i} )-m_{i} ]^2}$. Thus, our proposed alternative, called the modified Ruttiens correlation (MRC), is formulated as

$$\begin{aligned} \rho _{r_{i}, r_{j}}^{mod\,Rutt}=\frac{E[(c_{i}-e_{i} )(c_{j}-e_{j} )]}{\nu _{(c_i-e_i)} \nu _{(c_j-e_j)} }, \end{aligned}$$

(9)

where the standard deviation of $(c_{i}-e_{i})$ is given by $\nu _{\left( c_{i}-e_{i}\right) }=\sqrt{E (y_{i}^2) }$.

3 Approximation of returns with parametric and nonparametric regressions

In this section, we present essential aspects of return approximation techniques using parametric and nonparametric regressions and principal component analysis (PCA). In particular, we reduce a large-scale portfolio complexity using a multifactor model that incorporates an adequate number of factors (not too large) such that the explained variability is significantly non-zero (Ortobelli et al. 2019). For this purpose, we consider the PCA of the correlation matrix of asset returns to determine the main factors. Similarly to the previous literature (e.g. Ortobelli and Tichý 2015; Kouaissah and Hocine 2021), to identify the main s factors (principal components), we apply a PCA to both the Pearson correlation and our proposed trend-dependent correlation matrix, respectively. Moreover, we consider the following linear multifactor model:

$$\begin{aligned} r_i= b_{i,0}+\sum _{j=1}^s b_{i,j} f_j +\varepsilon _i, ~~\text {for}~~ i=1,\dots ,z, \end{aligned}$$

(10)

where $b_{i,0}$ is the fixed constant for asset $i=1,\dots ,z$, $b_{i,j}$ is the coefficient related to factor $f_j$, s is the number of factors, and $\varepsilon _i$ is the error part. Generally, an ordinary least squares (OLS) estimator is usually used to estimate the coefficients of the parametric linear regression problem (Fan et al. 2008). Using parametric regression, we replace the original z correlated return series $\{r_i\}_{i=1}^z$ with z uncorrelated time series $\{g_i\}_{i=1}^z$ obtained from the PCA, where each $r_i$ follows a linear function of $g_i$. Dimensionality is reduced by using only a few factors that express a large part of the total variability, while the remaining factors form the error part. Thus, the linear multifactor model of returns series that is estimated by the OLS method is formulated as

$$\begin{aligned} \hat{r_{i}}=b_{i, 0}+\sum _{j=1}^{s} b_{i, j} f_{j}+\sum _{j=s+1}^{z} b_{i, j} f_{j}=b_{i, 0}+\sum _{j=1}^{s} b_{i, j} f_{j}+\epsilon _{i}, \end{aligned}$$

(11)

where $({\hat{r}}_i)$ is the approximate gross return of asset i, $b_{i,0}$ is the fixed constant for the ith asset, $b_{i,j}$ is the coefficient of the factor $f_j$, s is the number of factors, and $\epsilon _{i}$ is the error part of asset i.

The OLS estimator works well if the initial series is normally distributed, but this is not obvious for a return series (Rachev and Mittnik 2000). This strong assumption may lead to a misleading final estimate when applied to financial data. Recently, Ortobelli et al. (2019) proposed a nonparametric approximation to reduce the dimensionality of a large-scale portfolio, where the factors are again determined applying a PCA to the linear or another type of correlation structure and the $s+1$th factor $B_{s+1}$, that is,

$$\begin{aligned} r_i= E(r_i \mid f_1,\dots ,f_s,B_{s+1})+\varepsilon = \tau (f)+\varepsilon , \end{aligned}$$

(12)

where $(f_1,\dots ,f_s)$ represents s uncorrelated factors and $B_{s+1}$ is a benchmark index. In this paper, we use the market upper stochastic bound defined as $B_{s+1}=\max _i (r_{i})$ that gives the maximum possible return among the available assets at any time. The nonparametric regression model is consistent even if the returns commonly follow the stable distribution (Rachev and Mittnik 2000; Nolan and Ojeda-Revah 2013). The recently preferred tool for the estimation of the function $\tau (f)$ is the estimator based on locally weighted least squares (hereinafter only RW) proposed by Ruppert and Wand (1994). To estimate the regression function $\tau (f)$ by RW, we have to estimate the parameter a by optimizing the following problem:

$$\begin{aligned} \min _{a, b} \sum _{t=1}^{T}\left\{ r_{i}-a-b^{T}\left( f_{(t)}-f\right) \right\} ^{2} K_{H}\left( f_{(t)}-f\right) , \end{aligned}$$

(13)

where $f_{(t)}$ is the tth observation of the vector of factor f, $K_H$ is a multivariate kernel estimator of an $s\times s$ symmetric positive definite matrix H that depends on the sample size T.

For the multivariate kernel estimator $K_H$, Scott (2015) suggests employing the s-dimensional multivariate Gaussian density with variance–covariance bandwidth $H =diag(h_1,\dots ,h_s)$:

$$\begin{aligned} \text { Scott's rule in } {\mathbb {R}}^{s}: {\hat{h}}_{i}={\hat{\sigma }}_{i} T^{-1 /(s+4)}, i=1, \ldots , s, \end{aligned}$$

(14)

where ${\hat{\sigma }}_{i}$ means the estimated standard deviation of variable $f_i$, and T is the number of observations. More details and a discussion of non-parametric regression are provided in Ruppert and Wand (1994); Bowman and Azzalini (1997); Scott (2015); Ortobelli et al. (2019). The empirical analysis of this paper uses the normal kernel function and the bandwidth selection method suggested by Scott (2015).

4 Empirical analysis of the portfolio selection strategy

In this empirical section, we analyse ex-post and ex-ante portfolio statistics of various portfolio selection strategies using approximated returns from factors by parametric and nonparametric regression models. In particular, we use a new framework that integrates both the trend–risk measures and trend-dependency structure introduced in Sect. 2 into several variations of portfolio optimization scenarios. These scenarios are basically derived from the mean–variance portfolio model, where we assume 40 various risk-averse strategies that ultimately form an efficient frontier (Markowitz 1952). Furthermore, we also propose a new portfolio selection framework with a double optimization process and a trend-correlation PCA. The advantage of the double optimization consists of setting a benchmark portfolio statistics in the first part and then finding the effective (less trend-risky) way of how to achieve at least the same level of profitability. According to the results obtained, we show the effect of different strategies on the wealth paths, return statistics, and diversification measures of particular portfolios.

In this analysis, we consider a dataset consisting of daily observations of 182 U.S. stocks, which were included as components of the S &P 100 index during the period from 2 January 2002 to 31 December 2021, which means 5036 daily observations in total. In addition, we monitored changes in the composition of the index that occurred regularly every three months and created a composition matrix. It follows that we are able to capture all the changes in the data-set to make it dynamic in time and eliminate the impact of survivorship bias. This fact makes the analysis more realistic, accurate, and credible. All data were downloaded from Thomson Reuters Datastream.

4.1 Ex-post comparison according to nonparametric regression among different portfolio strategies

To illustrate the effect using an ex-post empirical analysis, we assume that each portfolio is re-optimized at monthly intervals (every 21 days) based on a one-year moving window of historical observations (252 days). Furthermore, we assume that short sales are not allowed and the initial wealth of the portfolio $W_0=1$ is first invested on 2 January 2003.² In order to investigate the effect on portfolios on the efficient frontier, we decided to examine 40 portfolio strategies, which differ in risk attitudes and required minimum value of expected return while maintaining equidistant intervals between them.³ For all 40 mean–variance strategies, we find the composition of the optimal portfolio, where the following steps are performed:

Step 1 In the first scenario, apply the PCA to the classical Pearson correlation matrix (i.e. PC), select 17 principal components (which explain at least 85% of the variability) and then use the OLS estimator or RW estimator to approximate the returns (i.e. OLS-Pearson or RW-Pearson). In the second scenario, apply the PCA to the stable conditional correlation matrix of returns (i.e. SC) and simultaneously to one of the Ruttiens (i.e. RC) or modRuttiens (i.e. MRC) correlation matrices. Then approximate the returns using the nonparametric RW estimator (i.e. RW-Ruttiens, respectively RW-modRuttiens). In this scenario, we select 3 factors of a particular PCA on trend-dependent correlations (explaining at least 90% of the particular variability), 13 factors from a stable PCA (explaining at least 60% of the stable variability), and factor $B_{s+1}$ as a max benchmark. Thus, both strategies consist of 17 factors to approximate returns (see, e.g. Ortobelli et al. 2019).⁴

Step 2 Determine the optimal weights for all 40 portfolio strategies and create the efficient frontier. In particular, the simple mean–variance (mean–variability) model consists of determining the optimal vector of asset weights x for the first strategy while minimizing the global variance of the portfolio (GMV) as the following quadratic optimization problem:

$$\begin{aligned} \begin{aligned} \min (x^{\prime } \Sigma x) \\ x^{\prime } \varphi =1 \\ 0 \le x_i \le 1;\; i=1,\dots ,z, \end{aligned} \end{aligned}$$

(15)

where $\Sigma$ is the classical variance–covariance matrix and $\varphi ^{\prime }= (1,1,\dots ,1)$ is a vector of ones. Then, determine the vector x for the last strategy (i.e. 40th) solving the optimization problem in order to maximize the mean expected return of the portfolio $\max E(x^{\prime }r)$, while the conditions of the model remain the same as in (15). Furthermore, compute the individual vectors of weights x for the inner portfolios of the efficient frontier. To do so, minimize again the variance of the portfolio formulated as $\min (x^{\prime } \Sigma x)$, while achieving a predetermined lower bound of expected return M. Thus, add a condition $E(x^{\prime }r) \ge M$ to model (15).

In contrast, the double optimization strategy is further composed of the second optimization process, which considers the minimization of the modified Ruttiens risk measure ARV$_{\text {mod}}$ formulated in Equation (2). To define the zero volatility trend line for each portfolio $e_{x^{\prime }r}$, we use the first mean–variance (MV) optimization (15). Employing this assumption, compute $e_{x^{\prime }r,t}=W_{0}+\frac{t}{T} (W_{x^{\prime }r,T}^{MV}-W_{0})$, where $W_0=1$. As previously assumed, the condition in the optimization model is to maintain the required expected return of the portfolio $S_M$ in each step computed from the MV model. Therefore, based on the formulation in Sect. 2.1, we have to solve the following quadratic optimization problem:

$$\begin{aligned} \begin{aligned} \min x^{\prime }\Sigma ^{mod\,Rutt} x \\ x^{\prime }r_{[0,T]} = S_M \\ x^{\prime } \varphi =1 \\ 0 \le x_i \le 1;\; i=1,\dots ,z. \end{aligned} \end{aligned}$$

(16)

where $r_{[0,T]}$ is the gross return vector observed during the last window of observations [0, T] (which is, in our analysis, one year, i.e. 252 daily returns). By adding this part into the algorithm, we should reduce another type of variability of the mean–variance selection strategy, achieving at least the same expected return and final wealth with a different path that reduces the deviation between cumulative portfolio returns and the zero volatile trend line.

Step 3 Compute the ex-post final wealth for each kth re-calibration interval as follows:

$$\begin{aligned} W_{t_{k+1}}=W_{t_k} (x^{\prime }r_{t_{k+1}}^{ex-post}), \end{aligned}$$

(17)

where $r_{t_{k+1}}^{ex-post}$ is the vector of gross returns between time $t_k$ and $t_{k+1}$, meaning $t_{k+1}=t_k+\varrho$ where $\varrho =21$.

The entire algorithm (Steps 1 to 3) is repeated until daily return observations are available.

The results of our analysis are reported in Tables 1, 2, 3 and Figs. 3, 4, 5, and 6. Tables 1, 2, and 3 report important portfolio statistics of ex-post returns, i.e. mean (%), standard deviation (%), skewness, kurtosis, VaR5% (%), CVaR5% (%), selected performance measures,⁵ i.e. SR (%), TDR1, and TDR2, and final wealth for the scenarios proposed and all portfolio strategies (Ortobelli et al. 2017; Malavasi et al. 2020).

Table 1 Statistics computed for daily ex-post returns obtained for particular mean–variance strategies with parametric OLS-Pearson and nonparametric RW-Pearson ex-ante return approximations

Table 1 illustrates the results of classical mean–variance portfolio selection strategies on ex-post approximated returns using PCA on the Pearson correlation matrix with parametric OLS and nonparametric RW regression models. It is evident that for the strategies with minimal risk and maximal expected returns located at the beginning and at the end, the OLS slightly outperforms the RW regression model in the profitability statistics, i.e. mean and final wealth. In contrast, strategies in the middle with RW approximation of returns lead to higher ex-post mean and final wealth. However, by analysing the risk measures of particular strategies, we observe that a significantly lower variability (standard deviation, VaR5%, or CVaR5%) of ex-post portfolio returns is achieved when including the RW regression model. As expected, the significance decreases with increasing strategies that maximize the expected return. In relation to these facts, it can be noted that the use of RW regression shows generally higher SR results. The most striking peculiarity shown in Table 1 is that TDR1 and TDR2 often contradict the results observed for the other indicators due to the noticeably higher values for most of RW strategies, except for those around 30. This is mainly caused by the lower rate of the trend-dependent risk measure. In addition, the ex-post returns of all strategies with OLS and especially with the RW approximation are negatively skewed and strongly leptokurtic.

However, to obtain deeper insights into these findings, Figs. 3 and 4 illustrate the Log-Wealth paths of all strategies considering both the parametric OLS-Pearson regression approach and the nonparametric RW-Pearson correlation approach, respectively.

Fig. 3

Ex-post Log-Wealth of mean–variability strategies with parametric OLS return approximation scenario

Fig. 4

Ex-post Log-Wealth of mean–variability strategies with nonparametric RW-Pearson return approximation scenario

It is clearly observable that the application of the nonparametric regression technique generates a similar final Log-Wealth for the less risky strategy and outperforms the OLS in the middle strategies, as concluded from Table 1. However, from the surface plot of wealth paths for all strategies, we can better see the difference in riskiness between these regressions. Especially in the part of the figures monitoring riskier strategies, the impact of the financial crisis in 2008–2009 can be seen, as well as the ability of the strategy to adapt. These figures confirm the findings that using the RW approximation smooths the wealth paths compared to their corresponding strategies with OLS, which concurrently means a reduction of the portfolio risk. Recall that these findings are based on a strategy with a mean–variance optimization.

According to the previous discussion and the existing literature that evaluates the accuracy of the OLS and RW approximation by testing the concave dominance between the estimated error parts Ortobelli et al. (2019), the RW method is preferred for further analysis.

Furthermore, in Tables 2 and 3, we illustrate the same ex-post statistics of all portfolio strategies, while applying nonparametric RW-Ruttiens and RW-modRuttiens return approximation scenario simultaneously with a double optimization strategy for a monthly re-calibration.

Table 2 Statistics computed for daily ex-post returns obtained for particular mean–trend–risk strategies with nonparametric RW-Ruttiens ex-ante return approximation

Table 3 Statistics computed for daily ex-post returns obtained for particular mean–trend–risk strategies with nonparametric RW-modRuttiens ex-ante return approximation

The results give a broad overview of the profitability and benefits of incorporating the proposed trend measures into the double optimization portfolio strategy. In particular, we observe that in most strategies, the proposed approach using RW approximated returns with Ruttiens and modRuttiens correlations outperforms the RW-Pearson approximation with a mean–variance optimization in terms of profitability, except for the first risk-minimizing strategies that achieve higher wealth using the RW-Pearson approach. Furthermore, using the RW-Pearson scenario leads to a lower level of riskiness (standard deviation, VaR5%, CVaR5%) than using the other two approaches, where the high risk-averse strategies show approximately one-half the RW-Ruttiens results, but with higher risk tolerance, the differences narrow almost to zero. Probably the reason for this is that the second optimization focussing on minimizes the deviations from the trend allow smoothing the wealth path, but the weights vary significantly from the optimum of the global minimum variance strategy. However, we can observe a trend that as the rate of return increases, portfolio performance expressed by SR increases. It is also evident that more risky mean–trend–risk strategies (above the average) generally outperform the mean–variance ones. Nevertheless, RW-modRuttiens generates higher trend-dependent performance ratios only for the least risky and the most profitable strategies. Furthermore, it is evident that the ex-post return series of all RW-Pearson strategies are more asymmetric and leptokurtic than RW-Ruttiens or RW-modRuttiens. Generally, according to the findings above, the RW-modRuttiens approach is more appropriate for less risk-averse investors than RW-Ruttiens.

Fig. 5

Ex-post Log-Wealth of mean–variability strategies with nonparametric RW-Ruttiens return approximation scenario

Fig. 6

Ex-post Log-Wealth of mean–variability strategies with nonparametric RW-modRuttiens return approximation scenario

In Figs. 5 and 6, we show the Log-Wealth paths of all strategies based on the nonparametric RW-Ruttiens and RW-modRuttiens scenarios. They essentially confirm that trend–risk strategies have a fundamental impact on ex-post performance in portfolio selection strategies even if they have a slightly higher variability in their ex-post wealth paths.

Fig. 7

Comparison of portfolio Final Wealth and selected daily statistics for particular strategies

To show and easily compare the results of the different strategies, Fig. 7 with selected portfolio indicators is presented. In particular, we also present the results of trend–risk measures and performance ratios. For all partial panels a–h, the x axis represents an individual portfolio strategy (1–40), where 1 means the GMV strategy and 40 is the MER strategy. On the y axes, we show Final Wealth (panel a), Mean (panel b), Standard Deviation (panel c), Sharpe Ratio (panel d), ARV$_{\text {mod}}$ (panel e), ARV (panel f), TDR1 (panel g), and TDR2 (panel h). The discussion of the classical portfolio statistics presented in Tables 1, 2, and 3 was provided above. However, when we focus on both trend-dependent risk measures ARV$_{\text {mod}}$ and ARV, it can be surprising that this kind of risk of whole portfolio paths with the double optimization strategy is habitually higher than the simple mean–variance model. Only for the less and relatively high risky portfolios are the obtained results of these scenarios lower than the other ones. We can observe that the results of ARV$_{\text {mod}}$ for the most risky strategies are not depicted in the figure due to the scale used (around a value of 20) in order to better show the differences.

4.2 Ex-ante analysis of diversification impact

Having shown the comparison of portfolios between the three strategies in terms of ex-post portfolio performance, we proceed to analyse the diversification effect of individual strategies. To do so, we select four basic diversification measures.

The first simple indicator is the number of assets (DM1) to which the investment funds are allocated, i.e. the number of assets with the non-zero weight after the kth re-calibration. The results of DM1 are in the interval [0, z], where 0 indicates that the investor interrupted the investment at the re-calibration time k and z means that all funds are split into all available assets.

The second diversification measure (DM2) is the simple concentration index, also well known as the Herfindahl–Hirschman index, which is generally used to measure market concentration (Hirschman 1964). However, because of its explanatory power, it can be applied to portfolio analysis. Its formulation is as follows:

$$\begin{aligned} DM2=\sum _{i=1}^z x_i^2, \end{aligned}$$

(18)

where $DM2 \in [\frac{1}{z},1]$ due to the cases where naive and single asset portfolios are considered. If the value of DM2 approaches zero (one), then the investment is divided into a huge (small) number of assets and vice-versa.

The last measure of diversification DM3 is more significant for evaluating portfolio diversification from a risk perspective. For this purpose, DM3 is the ratio between the risk of the portfolio and the average variability of all z assets for the kth re-calibration. It is formulated as

$$\begin{aligned} DM3=\frac{x^{k \prime } Q^{k} x^{k}}{\frac{1}{z} \sum _{i=1}^{z}\left\{ \frac{1}{T} \sum _{t=1}^{T}\left[ r_{i, t}^{k}-\textrm{E}\left( r_{i}^{k}\right) \right] ^{2}\right\} }, \end{aligned}$$

(19)

where $DM3 \ge 0$ and $T=(1,\dots ,252)$. If $DM3=1$ then the risk of the portfolio is equal to the mean risk of the assets, which can substitute for the average market risk. If $DM3<1$ ($DM3>1$) the portfolio has a lower (higher) risk than the market average.

The last indicator examined is turnover (Moorman 2014). According to Biglova et al. (2014), turnover is defined as the change in the composition of the optimal portfolio after the kth re-calibration. Therefore, it can be expressed as follows:

$$\begin{aligned} \varrho _{k}=\sum _{i=1}^{z} \mid x_{i}^{k}-x_{i}^{k-1}\mid , \end{aligned}$$

(20)

where $x_{i}^{k}$ is the proportion of funds invested in the ith portfolio component at the re-calibration time k. Moreover, based on this formulation, the indicator value $\varrho _{k}$ belongs to [0, 2] interval, where $\varrho _{k}=0$ means the unchanged composition of the portfolio after the kth re-calibration, while $\varrho _{k}=2$ indicates the situation in which the portfolio composition has completely changed.

Table 4 reports the average results of all selected diversification measures (i.e. DM1, DM2, and DM3) and the turnover indicator $\varrho$ of all optimal portfolio strategies considering different RW approximation scenarios.

Table 4 Diversification statistics obtained for all mean–trend–risk strategies with nonparametric RW returns approximation

From the results in Table 4 we observe that according to the number of non-zero weight assets (i.e. DM1) optimal portfolios are always divided among more than 2 assets and less than 17 assets on average for RW-Pearson and for RW with trend-correlation measures less than 5 assets on average. As might be expected, risk-averse strategies are more diversified than the riskiest strategies. This fact has been confirmed in research on portfolio diversification ( Egozcue et al. 2011; Woerheide and Persson 1992, and the references therein). As a result, the values of DM2 behave in inverse relationship depending on the selected strategy, meaning that a higher value of DM1 corresponds to a lower value of DM2 and vice versa. However, the differences between risk-averse strategies are not very pronounced. In addition, when we compare the more sophisticated diversification measure DM3, we can observe that if the RW-Pearson selection strategy is applied, the interval is deeper compared to the RW-Ruttiens and RW-modRuttiens strategies. Furthermore, due to increased diversification, risk-averse strategies have drastically lower variability (approximately 10%) than the average of assets in the market, and risk-seeking strategies are more than 1.5 times riskier than the market average. In contrast, when considering the RW-Ruttiens and RW-modRuttiens strategies, the variability is almost always below the market average, even for the riskiest portfolio strategies (around 40). In summary, these approaches stabilize the portfolio variability.

In addition, an examination comparing types of investor reveals that for classical mean–variance optimization in the RW-Pearson scenario, risk-averse investors exhibit lower portfolio turnover than risk-takers. However, using the proposed double optimisation in RW-Ruttiens or RW-modRuttiens, lower turnover is present in the strategies around the middle. Considering the average values of the turnover, which are around 1.4, we can deduce that RW-Ruttiens or RW-modRuttiens (for almost all strategies) reflect a significant change in portfolio composition. Recall that proportional transaction costs are strongly influenced by turnover.

We have also tested the second order stochastic dominance among the ex-post log-returns of the different portfolio selection strategies. We observed that about 23 per cent of trend–risk portfolio strategies second order stochastically dominates the corresponding mean–variance strategies. However, among the trend–risk type strategies, we do not observed the second order stochastic dominance relationship.

4.3 Discussion

According to the previous ex-post and ex-ante empirical analysis, we have demonstrated that:

• The integration of time-dependent or trend–risk measures as an alternative in the optimization process expands our insight into the issue of portfolio selection strategy;

• A trend–risk double optimization portfolio strategy outperforms the profitability of the simple mean–variance selection strategy;

• In general, these risk measures appear to be more useful in the finance sense as well as attractive to risk-avoiding investors.

In addition, we also used exponential and other variations of the trend line as a substitute for the equally accrued return (required trend) in risk measurement equations, which generate similar results and benefits. Nevertheless, we did not cover them in this analysis, but this adjustment could merit further research. In other words, we can integrate different types of function (e.g. the exponential) to capture trend preferences or replace it with the market trend. Furthermore, we could consider replacing the mean–variance model for the first optimization with a max-ratio model.

Recall that in this paper we expanded the possibility of using trend analysis to measure risk and dependence in the portfolio selection problem. The results should not only compare the modified indicators with the original ones provided by Ruttiens, but rather present the advantage of the proposed portfolio selection strategies.

5 Conclusion

In this paper, we have proposed a new modification of the trend-dependent risk measure based on Ruttiens’ ARV, which evaluates the mean of the squared deviations between the returns and their linear trend. In this context, we are able to include the trend in the optimization process more accurately than Ruttiens supposed. Moreover, we have also defined new trend-dependency measures, i.e. covariance and correlation, that are appropriate to use for the PCA.

To empirically apply the proposed trend-dependent measures in the portfolio selection problem, we have considered the mean–variance portfolio model. In this context, we applied a PCA to several correlation matrices (linear or trend-dependent) to reduce the dimensionality of the portfolio and approximate the returns using both parametric and nonparametric regression models. Additionally, we proposed a new double optimization portfolio selection strategy, which consists of the classical Markowitz mean–variance model followed by a minimization of the deviations from the trend alternative obtained from the previous optimization generating at least identical mean return and final wealth. We evaluated the impact on the ex-post portfolio statistics for 40 strategies with different preferences and risk attitudes of investors. The empirical results showed that using the nonparametric RW approximation, the wealth is smoother during the investment and return series are less variable, with insignificant differences in the profitability. Furthermore, double optimization strategies with trend correlation PCA and RW regression model outperforms the final wealth of strategies based on the Pearson correlation PCA. Finally, the average results of ex-ante diversification analysis show that the suggested strategies reduce the number of assets in the portfolio and their allocation compared to the RW-Pearson strategy.

Note that this type of trend-dependent risk measure is only one alternative, which can subsequently be adapted to the requirements of analysts or researchers.