“Z-score vs minimum variance preselection methods for constructing small portfolios”

Several contributions in the literature argue that a significant in-sample risk reduction can be obtained by investing in a relatively small number of assets in an investment universe. Furthermore, selecting small portfolios seems to yield good out-of-sample performances in practice. This analysis provides further evidence that an appropriate preselection of the assets in a market can lead to an improvement in portfolio perfor- mance. For preselection, this paper investigates the eﬀectiveness of a minimum variance approach and that of an innovative index (the new Altman Z-score) based on the creditworthiness of the companies. Diﬀerent classes of portfolio models are examined on real-world data by applying both the minimum variance and the Z-score preselec- tion methods. Preliminary results indicate that the new Altman Z-score preselection provides encouraging out-of-sample performances with respect to those obtained with the minimum variance approach.


INTRODUCTION
The issue of constructing small portfolios is a well-known problem in the financial industry, particularly in the case of small investor who should stem costs due to the complexity of management. However, also big investors could take advantage of this practice if small portfolios can achieve better performance than large portfolios. This analysis provides further evidence that an appropriate preselection of the assets in a market can lead to a significant improvement in portfolio performance. More precisely, this paper investigates the effectiveness of the Z-score index for preselecting the assets of an investment universe compared with that achieved by the minimum variance approach. The Z-score is a predictive index of creditworthiness expressed as a numerical score, which essentially measures the default probability of a company. It is used here to classify the quality of a company and its out-of-sample performance in terms of the market price. Different classes of portfolio models are examined on real-world data by applying both the minimum variance and the Z-score preselection methods. Preliminary results show that the new Altman Z-score method produces encouraging out-of-sample performances with respect to those obtained with the minimum variance approach.
The structure of this paper is as follows. Section 1 presents a survey of the literature on the main research topics covered in this work. Section 2 provides details on the research methodology. More precisely, subsection 2.1 describes the portfolio selection strategies analyzed.

LITERATURE REVIEW
The first empirical evidence showing that small portfolios tend to achieve a drastic elimination of the diversifiable risk in a market is probably due to a work by Evans and Archer (1968) who discovered that the average standard deviation decreases quickly when the portfolio size increases. They concluded that no more than about ten assets are needed to almost completely eliminate the non-systematic risk in the portfolio return. From then on, several contributions in the literature show that investing in a small number of assets from an investment universe is sufficient to obtain a significant in-sample risk reduction in terms of variance and of some other popular risk measures, and good out-of-sample performances in practice (see, e.g., Statman, 1987;Newbould & Poon, 1993;Tang, 2004;Cesarone, Scozzari, & Tardella, 2013, 2016, and references therein).
After the global financial crisis started in 2008, the weakness of some classical portfolio selection approaches based on risk-gain analysis (Markowitz, 1952(Markowitz, , 1959  This study considers several of these approaches for portfolio selection purposes and investigates the effectiveness of the Z-score index for preselecting the assets of an investment universe compared with that achieved by the minimum variance approach.
The original Z-score index was introduced by Altman (1968) for evaluating the default probability of a company. However, following several findings that show the relation between market prices and credit ratings ( is used here to classify the quality of a company and its out-of-sample performance in terms of market price.

Portfolio selection models
This subsection gives a brief review of the portfolio selection models used for this analysis. Specifically, three different classes of models for selecting a portfolio are considered: where i x is the percentage of capital invested in the asset , i and n indicates the number of tradable assets belonging to an investment universe.

Minimum risk portfolios
This subsection describes two portfolio selection models focused on the minimization of portfolio risk, that is measured using both symmetric and asymmetric risk measure.
As for symmetric risk measures, the first portfolio selection model considered aims at minimizing variance, namely a special case of the Mean-Variance model (Markowitz, 1952(Markowitz, , 1959. In the case of long-only portfolios, it can be formulated as follows: where ij σ is the covariance of the returns of asset i and asset . j As for asymmetric risk measures, the second portfolio selection model analyzed consists in minimizing the Conditional Value-at-Risk at a specified confidence level ε (CV aR ε ), i.e., the mean of losses in the worst 100epsilon% of the cases (Acerbi & Tasche, 2002), where losses are defined as negative outcomes. A formal definition of CVaR is as follows: Thanks to its theoretical and computational properties, , CVaR also called expected shortfall or average Value-at-Risk, has become widespread for risk management and asset allocation purposes. From a theoretical point of view, CVaR ε satisfies the properties of monotonicity, sub-additivity, homogeneity, and translational invariance, i.e., the axioms of a coherent risk measure (Artzner, Delbaen, Eber, & Heath, 1999). Furthermore, Ogryczak and Ruszczynski (2002) show that the mean-CVaR model is consistent with second-order stochastic dominance. From a computational point of view, the mean-CVaR portfolio can be efficiently solved by means of linear programming (Rockafellar & Uryasev, 2000).
The long-only portfolio that minimizes CVaR ε can be found by solving the following problem: In these experiments, the confidence level ε is fixed equal to 10%.

Capital and risk diversification strategies
The concept of diversification can be qualitatively related to the portfolio risk reduction due to the process of compensation caused by the co-movement among assets that leads to a potential attenuation of the exposure to risk determined by individual asset shocks. However, the question of which measure of diversification is most appropriate is still open (see, e.g., Meucci, 2009;Lhabitant, 2017).
The oldest and most intuitive way to force diversification in a portfolio is to equally share the capital among all securities in an investment universe (Tu & Zhou, 2011). Formally, the Equally Weighted (EW) portfolio is defined as 1/ .

EW xn =
This strategy does not entail the use of any past or future information, nor needs the resolution of complex models. From a theoretical point of view, Pflug et al. (2012) prove that when increasing the uncertainty of the market, represented by the degree of ambiguity on the distribution of the asset returns, the optimal investment strategy tends to be the EW one. Furthermore, from a practical point of view, DeMiguel et al. (2009) empirically investigate its out-of-sample performance, which seems to be generally better than that obtained from different classical and recent portfolio selection models.
Two recent portfolio selection approaches focused on risk diversification are described below and tested in the empirical analysis.
The Risk Parity (RP) strategy, introduced by Maillard, Roncalli, and Teiletche (2010), requires that each asset equally contributes to the total risk of the portfolio, which is measured by volatility. The standard approach used for decomposing the portfolio volatility is the Euler allocation, namely Thus, the RP portfolio can be obtained by imposing the following conditions: Hence, a direct method for finding an RP portfolio is to solve the following system of linear and quadratic equations and inequalities: portfolio, namely the optimal portfolio that maximizes the diversification ratio (5), can be found by solving the following (convex) quadratic programming problem: Clearly, the normalized portfolio weights are where * y is the optimal solution of Problem (6).

Portfolio selection based on SSD
Second-order Stochastic Dominance (SSD) is a rational principle of decision making under uncertainty, widely studied and investigated in the literature (see, e.g., Bruni 2013) who select a portfolio whose return distribution SSD dominates that of a given benchmark. For finding an SSD efficient portfolio w.r.t. a specific benchmark R B , in the case of T equally likely scenarios, the authors propose the following multi-objective optimization problem: ing the CVaR reformulation of Uryasev (2000, 2002). However, due to the high number of variables and constraints (more than 2 T ), Problem (7) is solved by implementing cutting planes techniques, as explained in Roman et al. (2013).
The complete list of portfolio models analyzed in this study is reported in Table 1. , the Altman Z-score model has become one of the state-of-the-art approaches for assessing the credit risk of a company. This version of the Altman credit-scoring model, also called SME Z-score, is calibrated by the country and industrial sector to maximize its prediction power. The SME Z-score index is obtained by multiple linear regression with forty explanatory variables, which can be divided into three main groups: • financial variables such as those belonging to the following accounting ratio categories: leverage, liquidity, profitability, coverage, activity; • corporate governance and managerial variables such as size and age of the company, industry sector, age/experience of managers, location, market position, number of board members, etc.; • macroeconomic variables such as industry default rate, GDP growth rate, consumer confidence index, consumer price index, unemployment rate, interest rate, etc.
As mentioned in the introduction, following several findings that highlight the connection between market prices and credit ratings, in this paper, a variant of the SME Z-score model is applied on a set of large corporates belonging to the Eurostoxx market, assessing the effectiveness of this approach to classify the quality of a listed company and its future performance in terms of market price. In this variant of the SME Z-score model, the book value of a company is substituted by its market value.

Preselection process and methodology
This paper aims to study and compare the ability of the new Altman Z-score index and that of the minimum variance criterion for preselecting assets. The two preselection strategies are performed as follows: 1) on the date where the assets are preselected, the current monthly values of the new Altman Z-score of all assets in the investment universe are collected, and ten assets with the highest scores are chosen (Z-score preselection); 2) on the same date, the Minimum Variance (MinV) portfolio (1) is computed on in-sample data of 1 year (250 financial days), and ten preselected assets are those with the highest weights in such MinV portfolio (minimum variance preselection).
The empirical analysis is based on a rolling time windows approach. As already mentioned, an in-sample time window of 1 year is used. The portfolio performance is then assessed in the following month (20 financial days, called out-of-sample window). Next, the in-sample window is shifted by one month, thus covering the previous out-of-sample period; again, the optimal portfolio w.r.t. the new in-sample window is computed, and this procedure is repeated up to the end of the data. Thus, for each monthly portfolio rebalancing, ten assets are preselected through both the Z-score preselection and minimum variance preselection strategies. Then, all the portfolio selection approaches, listed in Table 1, are applied on these ten preselected assets.
The out-of-sample performance of each portfolio strategy is examined using as a benchmark the Equally Weighted (EW) portfolio throughout the investment universe (Bench). More specifically, the following performance measures (where the constant risk-free rate of return is set equal to 0) are considered: mean (Mean); volatility (Vol), Sharpe ratio (Sharpe) (Sharpe, 1966(Sharpe, , 1994 (Jensen, 1968), and Information ratio (Info) (Goodwin, 1998).

RESULTS AND DISCUSSION
This section provides the empirical results obtained by all the strategies listed in Table 1 with and without preselection on the Eurostoxx market. Specifically, a subset of this investment universe, containing 31 assets, is considered, where companies belonging to the banking, insurance, and financial sector are excluded. The reasons for this choice are closely linked to the time availability of the new Altman Z-score index, which starts from February 2009, and to the elements on which the Altman Z-score model is based. Indeed, this model aims to assess the possible bankruptcy of non-financial companies, which can be traded, or not, in a market. As described in subsection 2.2, the SME Z-score model uses several categories of budget indicators to forecast the default probability of a company, but these variables are explanatory only for a specific sector. Indeed, financial companies are based on completely different rules and dynamics w.r.t. non-financial ones. For instance, some budget indicators are representative of the degree of solvency only for non-financial com-panies, and therefore cannot be used for the same purpose in the case where the debt is part of the production process. In fact, the banks admit the debt as an element of production as they systematically collect resources for credit activities, mainly aimed at commercial banks and at investments in the securities market. Furthermore, at least in theory, financial companies can borrow indefinitely. Indeed, except for specific regulatory constraints, they can cover all (or almost all) costs of production factors if they are able to generate a significant and positive spread between the lending and borrowing rates. On the other hand, non-financial companies should not directly allow debts to produce goods and/or services, but they should use debts only to meet the needs of the circulating and fixed capital. In addition, they can borrow up to a specific threshold, beyond which the cost of the debt is too expensive for any profitable use. Also, in the case of insurance companies, the new Altman Z-score index cannot be evaluated by the model described in subsection 2.2. Indeed, they have an inverted economic cycle w.r.t. the financial and non-financial companies: revenues occur before production costs due to the collection of insurance premiums.
Since the new Altman Z-score index is available only for non-financial and non-insurance companies, the empirical analysis is performed with the following datasets: • Eurostoxx, containing 31 assets of the Euro Stoxx 50 Market Index (Europe) from February 1, 2009 to January 31, 2019 (daily frequency, source: Bloomberg); • the new Altman Z-score, assessed on the same 31 assets from February 2009 to January 2019 (monthly frequency, source: Wiserfunding Limited). Table 2 reports some details of 31 assets belonging to the analyzed investment universe.
All models have been implemented in Matlab 8.5 on a workstation with Intel Core CPU (i7-6700, 3.4 GHz, 16 Gb RAM) under MS Windows 10. Figures 1 and 2 show the ten preselected assets of the investment universe described in Table 2 for each rebalancing date using the Z-score and the  minimum variance preselection methods, respectively. According to the rolling time windows approach discussed above, Figures 1 and 2 are heatmaps with 31 rows (a row for each asset) and 117 columns (a column for each rebalancing date), where the preselected assets are marked in blue.

Out-of-sample performance results without preselection
Computational results are presented here for all the portfolio models listed in Table 1 without using any preselection procedure. Table 3 reports the out-of-sample performance for each portfolio strategy, where the rank of the performance is shown in different colors. For each column, the colors span from deep-green to deep-red, where deep-green depicts the best performance, while deep-red the worst one. Such a visualization style allows for easier revelation of (possible) persistent behavioral pattern of a portfolio approach (corresponding to a row). Note that the best performances are generally obtained from SSD and MD portfolios. This behavior is also confirmed by the trend of the cumulative out-of-sample portfolio returns reported in Figure 3. Note that there is a clear dominance of the SSD portfolio, followed by the MD portfolio.

3.2.
Out-of-sample performance results using minimum variance and Z-score preselection This subsection provides the empirical results for all the portfolio strategies (see Table 1) applied to a subset of ten assets, which are obtained by means of the minimum variance and the Z-score preselection procedures described in subsection 2.3. As already mentioned, Figures 1 and 2 show, in the rolling time windows scheme of evaluation, ten companies preselected by the Z-score and minimum variance methods, respectively. Table 4 reports the out-of-sample performance for each portfolio model when the minimum variance preselection is used. Again, the rank of the performance is indicated with different colors, as in subsection 3.1. Note that the minimum variance preselection tends to be ineffective compared to the results obtained without assets preselection. This is also highlighted in Figure 4, where the cumulative out-of-sample portfolio returns are  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31 shown for all the portfolio strategies analyzed.
Conversely, using the new Altman Z-score preselection, general improvement in the performance of all the models analyzed can be observed, except for the SSD model. This phenomenon is easily verifiable by comparing Table 5 and Figure 5 with Table 3 and Figure 3, respectively. Furthermore, observe that the empirical tests have also been performed considering a Z-score preselection made on its average over the in-sample period (1 year), but the results tend to remain unchanged w.r.t. the direct use of the monthly Z-score values (see subsection 2.3).