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

portfolios”

AUTH ORS

Francesco Cesarone https://orcid.org/0000-0003-2326-4204

Fabiomassimo Mango https://orcid.org/0000-0002-3092-6731

Gabriele Sabato

ARTICLE INFO

Francesco Cesarone, Fabiomassimo Mango and Gabriele Sabato (2020). Z-

score vs minimum variance preselection methods for constructing small

portfolios. Investment Management and Financial Innovations, 17(1), 64-76.

doi:10.21511/imfi.17(1).2020.06

DOI http://dx.doi.org/10.21511/imfi.17(1).2020.06

RELEASED ON Friday, 14 February 2020

RECE IVED ON Wednesday, 25 September 2019

ACCEPTED ON Tuesday, 24 December 2019

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This work is licensed under a Creative Commons Attribution 4.0 International

License

JOURNAL "Investment Management and Financial Innovations"

ISSN PRINT 1810-4967

ISSN ONLINE 1812-9358

PUBLISHER LLC “Consulting Publishing Company “Business Perspectives”

FOUNDER LLC “Consulting Publishing Company “Business Perspectives”

NUMBER OF REFERENCES

52

NUMBER OF FIGURES

5

NUMBER OF TABLES

5

© The author(s) 2020. This publication is an open access article.

businessperspectives.org

64

Investment Management and Financial Innovations, Volume 17, Issue 1, 2020

http://dx.doi.org/10.21511/im.17(1).2020.06

Abstract

Several contributions in the literature argue that a signicant 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. is 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 eectiveness of a minimum vari-

ance approach and that of an innovative index (the new Altman Z-score) based on the

creditworthiness of the companies. Dierent 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.

Francesco Cesarone (Italy), Fabiomassimo Mango (Italy), Gabriele Sabato (UK)

Z-score vs minimum

variance preselection

methods for constructing

small portfolios

Received on: 25 of September, 2019

Accepted on: 24 of December, 2019

INTRODUCTION

e issue of constructing small portfolios is a well-known problem in

the nancial industry, particularly in the case of small investor who

should stem costs due to the complexity of management. However, al-

so big investors could take advantage of this practice if small portfoli-

os can achieve better performance than large portfolios. is analysis

provides further evidence that an appropriate preselection of the as-

sets in a market can lead to a signicant improvement in portfolio per-

formance. More precisely, this paper investigates the eectiveness of

the Z-score index for preselecting the assets of an investment universe

compared with that achieved by the minimum variance approach.

e 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. Dierent

classes of portfolio models are examined on real-world data by apply-

ing both the minimum variance and the Z-score preselection meth-

ods. 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.

e 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 precise-

ly, subsection 2.1 describes the portfolio selection strategies analyzed.

© Francesco Cesarone, Fabiomassimo

Mango, Gabriele Sabato, 2020

Francesco Cesarone, Assistant

Professor, Department of Business

Studies, Roma Tre University, Italy.

Fabiomassimo Mango, Associate

Professor of Banking and Finance,

Department of Management,

Sapienza University of Rome, Italy.

Gabriele Sabato, Ph.D. in Banking

and Finance, Sapienza University

of Rome; Co-founder and CEO of

Wiserfunding Limited, UK.

is is an Open Access article,

distributed under the terms of the

Creative Commons Attribution 4.0

International license, which permits

unrestricted re-use, distribution,

and reproduction in any medium,

provided the original work is properly

cited.

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LLC “P “Business Perspectives”

Hryhorii Skovoroda lane, 10,

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BUSINESS PERSPECTIVES

JEL Classification C61, C63, G11

Keywords asset allocation, risk diversication, risk parity, portfolio

optimization, credit scoring

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Subsection 2.2 is devoted to discussing the new Altman Z-score model, while, subsection 2.3 explains

the preselection procedures applied to an investment universe, and describes the method used to evalu-

ate the performance. e computational results based on real-world data are presented in section 3,

where the main empirical ndings are also discussed. Finally, the last section contains some concluding

remarks.

1. LITERATURE REVIEW

e rst empirical evidence showing that small

portfolios tend to achieve a drastic elimination of

the diversiable risk in a market is probably due

to a work by Evans and Archer (1968) who dis-

covered that the average standard deviation de-

creases quickly when the portfolio size increases.

ey concluded that no more than about ten as-

sets 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 sucient to obtain

a signicant 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, 2018, and references therein).

Aer the global nancial crisis started in 2008,

the weakness of some classical portfolio selec-

tion approaches based on risk-gain analysis

(Markowitz, 1952, 1959) has given rise to a new re-

search stream that is based on capital (DeMiguel,

Garlappi, & Uppal, 2009; Tu & Zhou, 2011; Pug,

Pichler, & Wozabal, 2012) and risk diversica-

tion (see Cesarone & Tardella, 2017; Cesarone

& Colucci, 2018; Cesarone, Scozzari, & Tardella,

2019; Lhabitant, 2017; Roncalli, 2014, and ref-

erences therein). Furthermore, in the last few

decades, several scholars have proposed portfo-

lio selection models based on stochastic domi-

nance criteria (see, e.g., Fábián, Mitra, Roman, &

Zverovich, 2011; Roman, Mitra, & Zverovich, 2013;

Bruni, Cesarone, Scozzari, & Tardella, 2017; Valle,

Roman, & Mitra, 2017, and references therein).

is study considers several of these approaches for

portfolio selection purposes and investigates the

eectiveness of the Z-score index for preselecting

the assets of an investment universe compared with

that achieved by the minimum variance approach.

e original Z-score index was introduced by

Altman (1968) for evaluating the default proba-

bility of a company. However, following several

ndings that show the relation between market

prices and credit ratings (Hand, Holthausen, &

Lewich, 1992; Hsueh & Liu, 1992; Kliger & Sarig,

2000; Gonzalez, Haas, Persson, Toledo, Violi,

Wieland, & Zins, 2004; Hull, Predescu, & White,

2004; Norden & Weber, 2004; Micu, Remolona,

& Wooldridge, 2006; Grothe, 2013), a new ver-

sion of the Altman credit-scoring model (Altman,

2002; Altman & Hotchkiss, 2006; Altman, 2013)

is used here to classify the quality of a compa-

ny and its out-of-sample performance in terms of

market price.

2. METHODS

2.1. Portfolio selection models

is subsection gives a brief review of the portfolio

selection models used for this analysis. Specically,

three dierent classes of models for selecting a

portfolio are considered:

1. risk minimization;

2. capital or risk diversication;

3. second-order stochastic dominance.

Hereaer, the linear return of the

-thk

asset at

time

t

is denoted by

, 1,

,

1,

,

tk t k

tk

tk

pp

rp

−

−

−

=

where

,tk

p

represents its price at time

.t

e portfolio return at time

t

is

( )

,

1

,

n

t i it

i

R x xr

=

=∑

where

i

x

is the percentage of capital invested in

the asset

,i

and

n

indicates the number of trada-

ble assets belonging to an investment universe.

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2.1.1. Minimum risk portfolios

is 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 rst portfolio

selection model considered aims at minimizing var-

iance, namely a special case of the Mean-Variance

model (Markowitz, 1952, 1959). In the case of

long-only portfolios, it can be formulated as follows:

11

1

min

.. 1 ,

0 1, ,

nn

ij i j

xij

n

i

i

i

xx

st x

xi n

σ

= =

=

=

≥=

∑∑

∑

(1)

where

ij

σ

is the covariance of the returns of asset

i

and asset

.j

As for asymmetric risk measures, the second port-

folio selection model analyzed consists in mini-

mizing the Conditional Value-at-Risk at a spec-

ied condence level ε (CV aRε), i.e., the mean

of losses in the worst 100epsilon% of the cas-

es (Acerbi & Tasche, 2002), where losses are de-

ned as negative outcomes. A formal denition of

CVaR

is as follows:

( )

( )

( )

0

1,

px

R

CVaR x Q d

ε

ε

αα

ε

= − ∫

(2)

where

( )

( )

px

R

Q

α

is the

α

-quantile function of

the portfolio return

( )

.

px

R

anks to its theoret-

ical 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

satises the properties of mono-

tonicity, sub-additivity, homogeneity, and transla-

tional 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 eciently solved by means of lin-

ear programming (Rockafellar & Uryasev, 2000).

e long-only portfolio that minimizes

CVaR

ε

can be found by solving the following problem:

( )

1

.. 1 .

0 1,

m

,

in

n

i

i

x

i

st x

xi

CV x

n

aR

∈

=

=

≥=

∑

(3)

In these experiments, the condence level

ε

is

xed equal to 10%.

2.1.2. Capital and risk diversification strategies

e concept of diversication can be qualitatively

related to the portfolio risk reduction due to the

process of compensation caused by the co-move-

ment among assets that leads to a potential at-

tenuation of the exposure to risk determined by

individual asset shocks. However, the question

of which measure of diversication is most ap-

propriate is still open (see, e.g., Meucci, 2009;

Lhabitant, 2017).

e oldest and most intuitive way to force diversi-

cation in a portfolio is to equally share the capi-

tal among all securities in an investment universe

(Tu & Zhou, 2011). Formally, the Equally Weighted

(EW) portfolio is dened as

1/ .

EW

xn=

is strat-

egy does not entail the use of any past or future

information, nor needs the resolution of complex

models. From a theoretical point of view, Pug et

al. (2012) prove that when increasing the uncer-

tainty 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 dierent

classical and recent portfolio selection models.

Two recent portfolio selection approaches focused

on risk diversication are described below and

tested in the empirical analysis.

e Risk Parity (RP) strategy, introduced by

Maillard, Roncalli, and Teiletche (2010), requires

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that each asset equally contributes to the total risk

of the portfolio, which is measured by volatility.

e standard approach used for decomposing the

portfolio volatility is the Euler allocation, namely

( ) ( )

1

,

n

i

i

x RC x

σ

=

=

∑

where

( ) ( )

( )

1

1n

i i ik i k

k

i

x

RC x x x x

xx

δσ σ

δσ

=

= =

∑

is the contribution of the

-thi

asset. us, the RP

portfolio can be obtained by imposing the follow-

ing conditions:

( ) ( )

1

1

,.

n

i j ik i k

k

n

jk j k

k

RC x RC x x x

xx i j

σ

σ

=

=

=⇔=

= ∀

∑

∑

Hence, a direct method for nding an RP portfo-

lio is to solve the following system of linear and

quadratic equations and inequalities:

1

1

1, ,

1

0

1, ,

n

ik i k

k

n

i

i

i

xx i n

x

xi n

σλ

=

=

= =

=

≥=

∑

∑

(4)

that has a unique solution, due to the positive

semi-deniteness of the covariance matrix

Σ

(Cesarone et al., 2019).

An alternative approach to diversifying the risk,

introduced by Choueifaty and Coignard (2008),

consists in maximizing the so-called diversica-

tion ratio:

( )

11

,

n

ii

i

nn

ij i j

ij

x

DR x

xx

σ

σ

= =

=∑

∑∑

(5)

where

i

σ

is the volatility of asset

.i

Note that,

thanks to the subadditivity property of volatility,

( )

1.DR x ≥

As described by Choueifaty, Froidure,

and Reynier (2013), the Most Diversied (MD)

portfolio, namely the optimal portfolio that max-

imizes the diversication ratio (5), can be found

by solving the following (convex) quadratic pro-

gramming problem:

11

1

min

.. 1 .

0 1, ,

nn

ij i j

yij

n

ii

i

i

yy

st y

yi n

σ

σ

= =

=

=

≥=

∑∑

∑

(6)

Clearly, the normalized portfolio weights are

*

*

MD i

in

k

k

y

x

y

=

∑

with

1, , ,in=

where

*

y

is the optimal solution

of Problem (6).

2.1.3. Portfolio selection based on SSD

Second-order Stochastic Dominance (SSD) is a ra-

tional principle of decision making under uncer-

tainty, widely studied and investigated in the liter-

ature (see, e.g., Bruni et al., 2017; Valle et al., 2017,

and references therein). is subsection discusses

the portfolio optimization method for Enhanced

Indexation (EI), provided by Fábián et al. (2011),

Roman et al. (2013) who select a portfolio whose

return distribution SSD dominates that of a given

benchmark. For nding an SSD ecient portfo-

lio w.r.t. a specic benchmark RB, in the case of T

equally likely scenarios, the authors propose the

following multi-objective optimization problem:

( )

( )

( )

1

1

max min

.. 1 ,

0 1, ,

tp tB

tT

xTT

n

i

i

i

Tail R x Tail R

st x

xi n

≤≤

=

−

=

≥=

∑

(7)

where

( )

/tT p

Tail R is the unconditional expecta-

tion of the worst (t/T)100% outcomes of

.

p

R

is

problem can be expressed as an LP problem, us-

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Investment Management and Financial Innovations, Volume 17, Issue 1, 2020

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ing the

CVaR

reformulation of Rockafellar and

Uryasev (2000, 2002). However, due to the high

number of variables and constraints (more than

2

T

), Problem (7) is solved by implementing cut-

ting planes techniques, as explained in Roman et

al. (2013).

e complete list of portfolio models analyzed in

this study is reported in Table 1.

Table 1. List of porolio strategies

Model Abbreviaon

Minimum risk strategy

Minimum variance porolio MinV

Minimum condional value-at-risk porolio

with ε = 0,10 MinCVaR

Capital diversicaon strategy

Equally weighted porolio EW

Risk diversicaon strategy

Risk parity porolio RP

Most diversied por olio MD

Porolio selecon based

on Second-order Stochasc Dominance

SSD por olio SSD

2.2. Z-score: from default

to price prediction

e Z-score index was introduced by Altman (1968)

to predict the default probability of a rm. is in-

dex was originally built as multiple linear regres-

sion of ve explanatory variables represented by

common business ratios. Given its high accuracy

and eectiveness in predicting a rm bankruptcy

(see, e.g., Altman, Haldeman, & Narayanan, 1977;

Altman, 2002; Altman & Hotchkiss, 2006; Altman,

2013), the Altman Z-score model has become one

of the state-of-the-art approaches for assessing

the credit risk of a company. is version of the

Altman credit-scoring model, also called SME

Z-score, is calibrated by the country and indus-

trial sector to maximize its prediction power. e

SME Z-score index is obtained by multiple linear

regression with forty explanatory variables, which

can be divided into three main groups:

• nancial variables such as those belonging

to the following accounting ratio categories:

leverage, liquidity, protability, coverage,

activity;

• corporate governance and managerial varia-

bles such as size and age of the company, in-

dustry sector, age/experience of managers,

location, market position, number of board

members, etc.;

• macroeconomic variables such as industry de-

fault rate, GDP growth rate, consumer con-

dence index, consumer price index, unem-

ployment rate, interest rate, etc.

As mentioned in the introduction, following sev-

eral ndings that highlight the connection be-

tween market prices and credit ratings, in this

paper, a variant of the SME Z-score model is ap-

plied on a set of large corporates belonging to

the Eurostoxx market, assessing the eectiveness

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.

2.3. Preselection process

and methodology

is paper aims to study and compare the ability

of the new Altman Z-score index and that of the

minimum variance criterion for preselecting as-

sets. e 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-sam-

ple data of 1 year (250 nancial days), and ten

preselected assets are those with the highest

weights in such MinV portfolio (minimum

variance preselection).

e empirical analysis is based on a rolling time win-

dows approach. As already mentioned, an in-sample

time window of 1 year is used. e portfolio perfor-

mance is then assessed in the following month (20

nancial days, called out-of-sample window). Next,

the in-sample window is shied by one month, thus

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covering the previous out-of-sample period; again,

the optimal portfolio w.r.t. the new in-sample win-

dow is computed, and this procedure is repeated up

to the end of the data. us, for each monthly port-

folio rebalancing, ten assets are preselected through

both the Z-score preselection and minimum vari-

ance preselection strategies. en, all the portfolio

selection approaches, listed in Table 1, are applied on

these ten preselected assets.

e out-of-sample performance of each portfo-

lio strategy is examined using as a benchmark the

Equally Weighted (EW) portfolio throughout the

investment universe (Bench). More specically, the

following performance measures (where the con-

stant risk-free rate of return is set equal to 0) are con-

sidered: mean (Mean); volatility (Vol), Sharpe ratio

(Sharpe) (Sharpe, 1966, 1994), maximum drawdown

(MDD) (see, e.g., Chekhlov, Uryasev, & Zabarankin,

2005, and references therein), Ulcer index (Ulcer)

(MacCann, 1989), Sortino ratio (Sortino & Satchell,

2001) (Sortino), Rachev ratio (with a condence level

equal to 5% and 10%, named Rachev5 and Rachev10,

respectively) (Rachev, Biglova, Ortobelli, & Stoyanov,

2004), Jensen’s Alpha (JensenA) (Jensen, 1968), and

Information ratio (Info) (Goodwin, 1998).

3. RESULTS AND DISCUSSION

is section provides the empirical results obtained

by all the strategies listed in Table 1 with and without

preselection on the Eurostoxx market. Specically, a

subset of this investment universe, containing 31 as-

sets, is considered, where companies belonging to

the banking, insurance, and nancial sector are ex-

cluded. e 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 possi-

ble bankruptcy of non-nancial companies, which

can be traded, or not, in a market. As described in

subsection 2.2, the SME Z-score model uses sever-

al categories of budget indicators to forecast the de-

fault probability of a company, but these variables

are explanatory only for a specic sector. Indeed, -

nancial companies are based on completely dierent

rules and dynamics w.r.t. non-nancial ones. For in-

stance, some budget indicators are representative of

the degree of solvency only for non-nancial com-

panies, and therefore cannot be used for the same

purpose in the case where the debt is part of the pro-

duction 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 se-

curities market. Furthermore, at least in theory, -

nancial companies can borrow indenitely. Indeed,

except for specic regulatory constraints, they can

cover all (or almost all) costs of production factors

if they are able to generate a signicant and positive

spread between the lending and borrowing rates. On

the other hand, non-nancial companies should not

directly allow debts to produce goods and/or servic-

es, but they should use debts only to meet the needs

of the circulating and xed capital. In addition, they

can borrow up to a specic threshold, beyond which

the cost of the debt is too expensive for any prota-

ble 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 nancial

and non-nancial 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-nancial and non-insurance compa-

nies, 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

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Table 2. List of 31 assets belonging to the investment universe considered

No. Company name Ticker symbol ISIN number Ticker Bloomberg

1 DAIMLER AG DAI DE0007100000 DAI GY Equity

2TOTA L S.A . FP FR0000120271 FP FP Equity

3BAYERISCHE MOTOREN WERKE AKTIENGESELL SCHAFT BMW DE0005190003 BMW GY Equity

4SIEMENS AG SIE DE0007236101 SIE GY Equity

5EN I S .P. A. ENI IT0 003132476 ENI IM Equity

6ENEL SPA ENEL IT0003128367 ENEL IM Equity

7BASF SE BAS DE00 0BASF111 BAS GY Equit y

8KONINKLIJKE AHOLD DELHAIZE N.V. AD NL0 0117 940 37 AD NA Equity

9TELEFONICA SA TEF ES0178430E18 TEF SQ Equit y

10 LVMH MOET HENNESSY – LOUIS VUIT TON SE MC FR0000121014 MC FP Equity

11 VINCI DG FR0000125486 DG FP Equity

12 ORANGE ORA FR0 00013330 8 ORA FP Equit y

13 BAYER AG BAYN DE000BAY0017 BAYN GY Equity

14 SANOFI SAN FR0000120578 SAN FP Equity

15 IBERDROL A, S.A . IBE ES0 14458 0Y14 IBE SQ Equity

16 FRESENIUS SE & CO. KGAA FRE DE0005785604 FRE GY Equity

17 L’O RE AL S A OR FR0000120321 OR FP Equity

18 SCHNEIDER ELECTRIC SE SU FR0000121972 SU FP Equity

19 DANONE S.A . BN FR0000120644 BN FP Equity

20 SAP SE SAP DE0007164600 SAP GY Equity

21 NOKIA OYJ NOKIA FI0009000681 NOKIA FH Equity

22 SAFRAN S.A. SAF FR0000073272 SAF FP Equity

23 ADIDAS AG ADS DE000A1EWWW0 ADS GY Equity

24 L’AIR LIQUIDE AI FR0000120073 AI FP Equity

25 KONINKLIJKE PHILIPS N.V. PHIA NL0000009538 PHIA NA Equity

26 KERING KER FR0000121485 KER FP Equity

27 VIVENDI VIV FR0000127771 VIV FP Equity

28 ASML HOLDING N.V. ASML NL0010273215 ASML NA Equity

29 ESSILORLUXOTTI CA EL FR0000121667 EL FP Equity

30 UNILEVER NV UNA NL0000009355 UNAT NA Equity

31 CRH PUBLIC LIMITED COMPANY CRG IE0001827041 CRH ID Equity

Figure 1. Ten preselected assets for each rebalancing date using Z-score preselecon method

18/01/2010

04/07/2011

17/12/2012

02/06/2014

16/11/2015

01/05/2017

15/10/2018

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

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25

26

27

28

29

30

31

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minimum variance preselection methods, respec-

tively. According to the rolling time windows ap-

proach discussed above, Figures 1 and 2 are heat-

maps 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.

3.1. Out-of-sample performance

results without preselection

Computational results are presented here for all

the portfolio models listed in Table 1 without us-

ing any preselection procedure. Table 3 reports

the out-of-sample performance for each portfo-

lio strategy, where the rank of the performance is

shown in dierent 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 (cor-

responding to a row). Note that the best perfor-

mances are generally obtained from SSD and MD

portfolios. is behavior is also conrmed 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

is 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 prese-

lection 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 mini-

mum variance methods, respectively.

Table 4 reports the out-of-sample performance for

each portfolio model when the minimum vari-

ance preselection is used. Again, the rank of the

performance is indicated with dierent colors, as

in subsection 3.1. Note that the minimum vari-

ance preselection tends to be ineective compared

to the results obtained without assets preselection.

is is also highlighted in Figure 4, where the

cumulative out-of-sample portfolio returns are

Figure 2. Ten preselected assets for each rebalancing date using

the minimum variance preselecon method

18/01/2010

04/07/2011

17/12/2012

02/06/2014

16/11/2015

01/05/2017

15/10/2018

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

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shown for all the portfolio strategies analyzed.

Conversely, using the new Altman Z-score prese-

lection, general improvement in the performance

of all the models analyzed can be observed, ex-

cept for the SSD model. is phenomenon is easily

veriable 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).

Table 3. Out-of-sample results without preselecon

Approach Mean Vol Sharpe MDD Ulcer Sortino Rachev5 Rachev10 JensenA Info

MinVaR 4.34E-04 8.76E-03 4.95E-02 –0.144 0.043 7.09E-02 0.986 1.008 1.94E-04 1.29E-02

MinCVaR 4.48E-04 8.95E-03 5.01E-02 –0.156 0.046 7.15E-02 0.984 1.005 2.05E-04 1.50E-02

EW 3.58E-04 1.10E-02 3.24E-02 –0.230 0.071 4.65E-02 0.991 1.009 0.00E+00 –

RP 3.82E-04 1.04E-02 3.66E-02 –0.209 0.061 5.26E-02 0.994 1.005 4.50E-05 2.31E-02

MD 5.71E-04 9.78E-03 5.84E-02 –0.185 0.046 8.52E-02 1.019 1.039 2.83E-04 4.64E-02

SSD 7.49E-04 9.88E-03 7.58E-02 –0.244 0.059 1.13E-01 1.056 1.082 4.96E-04 5.73E-02

Bench 3.58E-04 1.10E-02 3.24E-02 –0.230 0.071 4.65E-02 0.991 1.009 – –

Figure 3. Out-of-sample compounded return for all models without preselecon

2010 20 11 2012 2 013 201 4 2015 2016 2 017 2018 20 19

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5 MinV

MinCVaR

EW

RP

MD

SSD

Bench

Table 4. Out-of-sample results using the minimum variance preselecon

Approach Mean Vol Sharpe MDD Ulcer Sortino Rachev5 Rachev10 JensenA Info

MinVaR 4.27E-04 8.77E-03 4.87E-02 –0.144 0.043 6.97E-02 0.988 1.009 1.88E-04 1.18E-02

MinCVaR 4.30E-04 8.96E-03 4.80E-02 –0.153 0.047 6.85E-02 0.995 1.011 1.88E-04 1.18E-02

EW 3.87E-04 9.34E-03 4.14E-02 –0.179 0.048 5.98E-02 1.008 1.029 1.03E-04 7.43E-03

RP 4.08E-04 9.10E-03 4.49E-02 –0.166 0.044 6.49E-02 1.009 1.029 1.34E-04 1.22E-02

MD 4.74E-04 9.13E-03 5.19E-02 –0.173 0.046 7.54E-02 1.030 1.043 2.14E-04 2.19E-02

SSD 6.66E-04 1.00E-02 6.66E-02 –0.207 0.067 9.95E-02 1.065 1.086 4.22E-04 4.15E-02

Bench 3.58E-04 1.10E-02 3.24E-02 –0.230 0.071 4.65E-02 0.991 1.009 –

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Figure 4. Out-of-sample compounded return for all models using minimum variance preselecon

201 0 201 1 201 2 201 3 201 4 201 5 201 6 20 17 20 18 20 19

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5 MinV

MinCVaR

EW

RP

MD

SSD

Bench

Table 5. Out-of-sample results using Z-score preselecon

Approach Mean Vol Sharpe MDD Ulcer Sortino Rachev5 Rachev10 JensenA Info

MinVaR 5.85E-04 9.92E-03 5.90E-02 –0.174 0.048 8.55E-02 0.993 1.021 3.16E-04 3.77E-02

MinCVaR 5.93E-04 1.01E-02 5.89E-02 –0.199 0.055 8.50E-02 0.984 1.008 3.23E-04 3.77E-02

EW 4.87E-04 1.11E-02 4.37E-02 –0.211 0.062 6.33E-02 0.999 1.020 1.42E-04 3.92E-02

RP 5.21E-04 1.07E-02 4.88E-02 –0.203 0.056 7.07E-02 1.001 1.017 1.94E-04 4.58E-02

MD 6.49E-04 1.06E-02 6.12E-02 –0.183 0.047 8.96E-02 1.017 1.037 3.41E-04 5.85E-02

SSD 5.31E-04 1.06E-02 4.98E-02 –0.177 0.061 7.16E-02 0.979 1.010 2.53E-04 2.56E-02

Bench 3.58E-04 1.10E-02 3.24E-02 –0.230 0.071 4.65E-02 0.991 1.009 – –

Figure 5. Out-of-sample compounded return for all models using Z-score preselecon

20 10 2011 2012 2013 201 4 20 15 2 01 6 20 17 2 01 8 20 19

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

MinV

MinCVaR

EW

RP

MD

SSD

Bench

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CONCLUSION

e qualitative goal of portfolio diversication is to avoid over-concentrating the capital in very few se-

curities. However, an important strand of research has shown that a signicant in-sample risk reduction

and good out-of-sample performances can be obtained by constructing small portfolios.

is paper examines for the rst time the eectiveness of a new credit risk index (the new Altman

Z-score) to preselect the assets from an investment universe and compares this with a minimum vari-

ance preselection approach. e eects of these two preselection methods on dierent classes of port-

folio models have been investigated using real-world data. e ndings demonstrate that the Z-score

preselection method tends to generate better out-of-sample performances with respect to those ob-

tained from the minimum variance criterion. Further tests are underway to examine the preselection

eectiveness of the Z-score index compared to that achieved by other strategies on dierent markets,

also including nancial companies.

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