2005. Dissertation Carlos Salas

Multi-Style and Rotation Equity Strategies in European Markets.

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Multi-Style and Rotation Equity Strategies in European Equity Markets

By Carlos Salas, 2005 June,

This study examines the application of different single-style and multi-style equity strategies in European markets taking in consideration a sample of 104 companies during the period 1994-2004. Results from previous research papers, which were mainly focus on the US and the UK Markets, provide evidence that several fundamental ratios have strong influence on stock prices. The main conclusions in this study were the importance of the market capitalization as primary discriminative factor in constructing equity portfolios, whereas PER and PTB showed risk-efficient results as secondary selection factors only for medium and large capitalization stocks. In addition, the last section includes a probabilistic quantitative analysis which sets forth the high degree of accuracy required from an Active Portfolio Manager to top efficiently Passive strategies. This research was completed in 2005 as final dissertation while I was attending lectures in AFI Business School´s “MBA in Finance” during the period 2004-2005. The sources being used for data gathering were Bloomberg, Datastream and JCF/Factset; using Eviews, Excel and VBA as key tools for computational purposes.

Contents

1. Introduction

2. Sample Data.

2.1. Data Description

2.2. Performance by industry and some theoretical concepts

3. Investment Strategies using fundamental Data.

3.1. Portfolio Classification system: Description

3.2. Single-Style Investment Strategies

3.3. Multi-Style Investment Strategies

4. Investment Strategies using Rotation Style.

4.1. Style Rotation strategies as an alternative

4.2. Indicative models for the practice of Style Rotation

5. Conclusions

Notes

Bibliography


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1. Introduction.

This paper discusses different investment strategies created from the use of fundamental

ratios as PTB (price to book value) or PER (price to earnings) during the period 1995-2004.

The use of such a kind of financial indicators as tools in analyzing and creating portfolios has

been very popular over the past 40 years, being a topic very popular in research by several

authors and starting to be used as a basis for different papers by pioneering authors like

Nicholson (1968).

Some articles such as those presented by Chen, Roll and Ross (1986) are focused on the

use of macroeconomic variables in finding additional information not provided by the market

index, while there are other research also aimed at seeking information not priced by the

markets but it can extracted from ratios and data of a fundamental nature. This work is

related to the last view and in line with other financial literature written by relevant authors:

S.Basu (1977) assesses the variable PER as determinant of higher performance, M.Levis

(1985) starts the election debate between small caps or blue chips; as well as Capaul,

Rowley and Sharpe (1993) conduct an influential work on value and growth assets. Of

paramount relevance is the work by Fama and French (1993) highlighting the importance of

market capitalization and book-to-market (inverse of price to book) variables using cross-

section regressions, pointing out the inability inherent in the CAPM model to properly

describe an asset risk-return features. These conclusions were very criticized by CAPM

advocates as Kothari, Shanken and Sloan (1995).

Concerning the sample period on this study, the fact that the first part of the data pertained to

the second half of the 90s might skew the paper conclusions as this period was a booming

one which could be allocated as an outlier whether a longer stock period were to be

appraised. Authors such as Campbell and Schiller (2001) studied the mean-reversion effects

in key ratios like PER and Dividend Yield, as well as their high values during the stock bubble

previously commented, confirming “Stock Price” as the main driver variable within the mean-

reversion phenomenon. Some critics still cast doubted about the former role played by the

stock price in the mean-reversion trend, though a lot of research has backed this conclusion

as aforementioned.


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Different types of research have been based on the former conclusions and using different

data sample as Ahmed, Lockwood and Nanda (2002) for the US markets or Levis and

Liodakis (1999) for the UK markets. This document explores different investment

methodologies concerning portfolio construction for a sample of European stocks using Excel

and VBA. In Section 2 a generic description of the sample data and a resume of portfolio

construction performance by industry sector are displayed. Section 3 is focused on the

analysis of single and multi-style strategies: portfolio creation method, discrimination criteria

and data gathering. Last but not least, results from single and multi-style strategies are

analyzed in-depth.

In Section 4 some brief comments are done regarding to style-rotation investment strategies.

A concise description of the differences between the investment strategies to be developed

in this section and the ones illustrated in section 3 is conducted, afterwards some remarks

related to advantages and requirements in carrying out an active investment strategy

overcoming passive investment strategies are discussed. The second subsection highlights

the importance of different variables in creating and developing investment strategies using

the former methodology, where some econometric models are created for this purpose using

Eviews 3.1 econometrics software.

Finally, in Section 5 are gathered and summarized all the conclusions obtained in each

section, exposing the study biases and how they could be solved in further articles.

2. Sample Data.

2.1. Data description.

The data sample is composed by 104 companies from which different types of information

have been obtained. The purpose of this study is to analyze and evaluate investment

strategies within an increasingly integrated environment such as Europe, using for that

reason a sample comprised only by European companies. This way we can make an

analysis that allows us to both assess the efficiency of European markets and introduce a

sample of data different from the one used by other authors: M.Levis and M.Liodakis (1999)

with the UK; Ahmed, Lockwood and Nanda (2002) with the US.


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The data used in this study is featured for being on a quarterly basis during a ten-year time

horizon (December 31, 1994 to December 31, 2004). The information collected is essentially

of fundamental nature: pricing, market capitalization, price to book ratio (price to stock book

value), price to earnings ratio (price to next year expected earnings), and Dividend Yield ratio

(dividends to price) 1. This data have been chosen since previous works by Fama and

French (1992) and Campbell and Schiller (2001) analyzed the performance of portfolios built

up on criteria related to values of these ratios.

The databases used for data gathering have been JCF version 5.0 and Bloomberg. As it is

shown at Exhibit 1, the Companies used in the study are members in various market indices

representing regional areas or industries: 40 companies in Eurostoxx 50 index, 36

companies in Eurostoxx Mid while 28 companies belong to Small Eurostoxx. Unfortunately

the lack of data available made not possible add to the sample the whole member of each

one of the former indices. Likewise the sample only has been composed with companies

"active" by 31 December 1994, so companies whose start-up dated before were wiped out of

the sample.

Exhibit 1. Company detail

Abertis Casino Guichard Hochtief AG Repsol

Abn Amro Holding Celesio AG Huhtamaki Plc Sacyr Y Vallehermoso

Acerinox CEPSA Iberdrola Safran

Aegon Christian Dior SA Imerys Saint Gobain

Agf Corp Fin Alba Ing Groep Saipem

Ahold Corp Mapfre Inmobiliaria Metrovacesa Sampo Bank

Air Liquide DaimlerChrysler AG Italcementi San Paolo Imi

Alcatel Danone Kesko Sanofi-Aventis

Alleanza Delhaize Group Klepierre Sap AG

Allianz AG Deutsche Bank Kone SBM Offshore NV

Altana AG DSM Lafarge Scor

Amer Sports Corp E.On AG Linde AG Societe Generale

Autostrade Eiffage Loreal Suez

Axa Endesa Lufthansa AG Technip

Banca Fideuram FCC Lvmh Telefonica

Bankinter Fondiaria M Real Television Francaise 1

Basf AG Fortis M6 Metropole Television Thales (Ex Thomson)

Bayer AG Fresenius AG Man AG Total

BBVA Gas Natural Sdg Mondadori Unibail

Beiersdorf AG Gecina Muenchener Rueck Unicredito Italiano

Bic Generali Natexis Unilever Nv

Bnp Paribas Getronics Nokia Union Electrica Fenosa

BSCH Havas Sa Omv Ag Valeo

Buhrmann Heidelbergcement AG Outokumpu Vivendi Universal

Cap Gemini Henkel Philips Wartsila Corp Beff

Carrefour Hermes International Publicis Groupe SA Wolters Kluw er


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2.2 Performance by industry and some theoretical concepts.

Performance sorted by sector is illustrated in Exhibit 2 on a YoY basis (annual returns) as

well as a risk-return proxy ratio:

Exhibit 2. Sector Performance

Market Financials Industrials Energy Technology Construction Retail Media Chemicals Transport

1995-1999 28,47% 21,38% 12,56% 17,27% 38,58% 6,39% 17,69% 33,66% 12,62% 22,98%

2000-2004 -7,13% -12,75% -1,90% 3,81% -25,57% 4,72% -14,25% -24,02% -5,33% 5,14%

1995-2004 9,23% 2,91% 5,08% 10,33% 1,56% 5,56% 0,46% 0,78% 3,26% 13,71%

Mdo Financials Industrials Energy Technology Construction Retail Media Chemicals Transport

1995-1999 1,29 0,81 0,52 0,99 1,34 0,29 1,25 1,28 0,63 1,27

2000-2004 -0,28 -0,22 -0,10 0,21 -0,57 0,33 -0,42 -0,61 -0,20 0,26

1995-2004 0,37 0,10 0,23 0,58 0,04 0,26 0,02 0,02 0,14 0,71

beta 1 1,152 0,740 0,584 1,446 0,577 0,836 1,211 0,761 0,590

YoY Performance / Standard Deviation

YoY Performance

Each industry index has been built using the JCF industry system for classifying only

companies of the sample. Nevertheless there are some as chemical, multimedia sectors,

distribution and transportation which do not reflect the reality of the sector as they only

consist of 3 or 7 companies.

Regarding Exhibit 2 figures, “Transport” is shown as best performer, being only exceeded by

“Technologicals” and “Media” during the second half of the 1990s due to both industries

booming trend. Sectors highlighted by its steady performance were “Energy”, “Construction”

and “Transport”; since at no time any of these sectors have endured losses, being this fact

closely related to low-beta profile shown by them. The outstanding results of the “Transport”

industry are biased by previously commented sample shortage which hides the true level of

risk embedded in the sector.

Main feature to underline is the fact that European markets do not seem to meet efficiency as

the concept proposed by authors as Sharpe or Markowitz. This claim is based on the highest

performance obtained by sectors with low-beta profile such as “Energy” or “Construction” in

relation to high-beta profile sectors as ”Technology” or “Media”.


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3. Investment Strategies using fundamental data.

3.1 Portfolio Classification System: Process description.

Different methodologies to create portfolios/indices will be developed following a given

criteria using Excel XP as a working tool. In a first step some portfolios will be created based

on the value of a single variable (market capitalization, price to book ratio, etc), commonly

called in financial literature as single-style strategies. This type of strategy is based on

classifying sample stocks in three different portfolios each December 31, representing each

portfolio a concrete percentile chosen as a reference to the portfolio. For example in building

three indexes based on market capitalization, those stocks whose market capitalization was

less than or equal to the sample percentile who accumulate 33.33% in terms of capitalization

will be considered members of the “Small” index. In the same way those stocks with market

capitalization higher than percentile 66,66% will be considered within the “Large” index, while

the “Mid” index contains assets with size between both percentiles. Similarly other indexes

will be constructed based on ratios as Price to earnings, Price to book and Dividend Yield.

In Section 3.3 a pair of variables instead of a single one will be used as allocating factors

also known as multi-style strategies. Such strategies consist of 2 breakdown stages that will

take place every December 31. The first stage consists in a classification in terms of its size,

building three groups named “primary”: medium, large and small capitalization stocks. The

second stage begins by choosing a ratio as a criterion for classifying each of the stocks of

each primary group in portfolios of assets according to the value of the ratio of each company

in comparison with the percentile ratio chosen for the primary group to which that company

belongs. By this way each primary group spawns three portfolios based on a ratio chosen as

a secondary criterion: portfolio Low (stocks with a ratio figure equal or lower to 33.33% of

those stocks in the primary group), High (stocks with a ratio greater than or equal to the

percentile 66,66% ratio) and Mid (assets with a value of the ratio between the percentiles

33.33% and 66,66%). In this way, if for example we choose as a secondary criterion the

price-to-book ratio book we can create 9 portfolios: Smallcap-Low, Smallcap-Mid, Smallcap-

High, Midcap-Low, Midcap-Mid, Midcap-High, Largecap-Low, Largecap-Mid and Largecap-

High.


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The reasons for using market capitalization (size) as primary criteria to filter data rely on

conclusions by Fama and French (1992) 2. Also previously named authors demonstrated in

cross-section regressions a relatively higher reliability of ratios as, for example, price to book

to market capitalization.

3.2 Single-style investment strategies.

Filter variables as market capitalization as well as the ratio price-to-book are presented as

those that financial literature has stressed more: Fama and French (1992) or Chan,

Jegadesh and Lakonishok (1995). Other ratios, as the Price-to-earnings and Dividend Yield,

have been studied as a starting point for developing investment strategies: Basu (1977) or

Campbell and Schiller (2001).

Regarding portfolios created by selecting securities according to their size, three different

groups are obtained: Small, Mid and Large. Each one of these indexes is composed by a

33,33% of stocks presented in the sample and are re-allocated every December 31. Results

on a YoY basis as well as in terms of risk-return are presented in Exhibit 3, where small caps

is the asset class with higher profitability (10,58%) and adjusted risk-return (0.48 and 0.137) 3

over the entire period. On the other hand, blue chips (Large index) were the winners during

the first 5 years (23.32% annualized profitability and 1.08 return on deviation), though the

market capitalization-weighted index got the best results.

Exhibit 3. Performance by size.

Mkt Small Mid Large

1995-1999 28,47% 18,46% 16,89% 23,32%

2000-2004 -7,13% 3,21% -9,78% -14,36%

1995-2004 9,23% 10,58% 2,69% 2,77%

beta 1 0,774 0,861 1,085

Mkt Small Mid Large

1995-1999 1,29 0,89 0,89 1,08

2000-2004 -0,28 0,14 -0,40 -0,47

1995-2004 0,37 0,48 0,12 0,10

YoY / beta 0,092 0,137 0,031 0,026

YoY

YoY risk adjusted


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Drawing conclusions from Exhibit 3 according to both absolute and risk-adjusted

performance figures, Small capitalization companies outperformed Large and Medium

capitalization issuers. For which reason it can be confirmed that the European companies in

the sample performed similarly to other international capital markets during the 1995-2004

period. It is remarkable that the amount of negative correlation between profitability and size

is not satisfied in comparing absolute performance between Medium and Large capitalization

companies but it is significant on a risk-return adjusted basis.

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Chart 1. Spread trend: Small-Large vs Market index.

Small vs Large

Market Index

Looking at Chart 1, the performance of large capitalization companies has been positively

and strongly related to the Market index movements (which remind is weighted by

capitalization). This statement is corroborated looking at the higher beta profiles in large

capitalization securities (1.085), by this it can be inferred a pro-cyclical fashion from this sort

of issuers as it was expected and once again points out some caveats in the CAPM model as

a framework to assessing accurately rates of required risk-adjusted return.

As regards the set of indexes created from classifying the stocks using multifarious ratios as

the price-to- book value per share (PTB), the price-to-earnings per share (PER) or Dividend

Yield (DY), 3 types of assets classification are arranged:


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Value assets: stocks whose earning trend evolution is quite stable and characterized

by low PER and PTB ratios as well as moderately high Pay-out ratios (Dividend per

share to earnings per share) that generally are turned into high Dividend yields.

Growth assets: stocks in which investors deposited high hopes for growth in profits,

but with irregular figures, characterized by high PTB and PER ratios while their

Dividend yield is quite scarce mainly caused by low Pay-out ratios.

Blend assets: hybrid stocks whose main features are halfway to those shown by their

value and growth counterparts.

Under the hypothesis of mean-reversion observed in PER PTB or DY ratios, along with the

conclusions by Campbell and Schiller (2001) showing the stock price as the main driver in

bringing these ratios to their long term average 4 variable; different performance paths may

be displayed depending on the inner features of each stocks.

A first sight of the later is shown in Exhibit 4, where three indexes were created similarly to

those built earlier when we analyze the effect size, but using the PTB as filter ratio. This time

the stocks classification comes as it follows: low PTB (Value assets), Medium PTB (Blend

assets) and High PTB (growth assets).

Exhibit 4. Performance by PTB.

Mkt Low Mid High

1995-1999 28,47% 18,51% 17,02% 23,04%

2000-2004 -7,13% 0,91% -7,73% -14,07%

1995-2004 9,23% 9,36% 3,91% 2,82%

beta 1 0,846 0,939 0,937

Mkt Low Mid High

1995-1999 1,29 0,84 0,82 1,20

2000-2004 -0,28 0,04 -0,29 -0,54

1995-2004 0,37 0,40 0,16 0,12

YoY/ beta 0,092 0,111 0,042 0,030

Rentabilidad Anualizada

YoY risk adjusted

Exhibit 4 shows the Low PTB portfolio as the best performer in absolute (9.36%) and risk-

weighted terms, 0.4 and 0,111 respectively. Once again high PTB stocks outperformed Low

PTB during the first sample period, though its performance stood still below the Market

portfolio. This time the nearly 7% performance spread Low PTB- High PTB portfolios is not

possible being explained by the CAPM model as the difference between both portfolios betas

is negative.


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Chart 2. Spread Trend: Market vs Low PTB

Market vs Low PTB

Market iIndex

In Chart 2 it is shown what was commented previously: Low PTB portfolio outperforms the

market in those stages in which market wanes, while in those years when market flourishes

Low PTB is outpaced. It is also remarkable that, as it happened with the market capitalization

when used as selection factor, the PTB ratio generated portfolios yield outcomes within the

mean-reversion hypothesis only concerning Low PTB stocks, being this hypothesis unable to

explain the results obtained between Mid PTB and High PTB indices.

Another ratio used for the construction of indexes has been Price-to-earnings, from which the

conclusions achieved emphasized even more the difference between value and growth style

assets. Comparison between both types of stocks by various authors as Capaul, Rowley and

Sharpe (1993) have led to empirical findings concluding in better value stocks performance

over extended periods of time, findings also present in our sample as it is pointed in Chart 3

(Value index is created by Low PER stocks and Growth by High PER stocks) .


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Chart 3. Accumulated Return Value vs Growth.

Value

Growth

As a reflection of Chart 3, figures in Exhibit 5 show better results for Low PER companies

(Value asset class) than High PER stocks (Growth asset class). Nevertheless the extreme

bull period 1995-1999 resulted in a better performance for Growth and Blend stocks,

although once again the best portfolio turned out to be the Market index.

Exhibit 5. Performance by PER.

Mkt Low Mid High

1995-1999 28,47% 19,41% 20,23% 18,74%

2000-2004 -7,13% 1,22% -6,88% -15,57%

1995-2004 9,23% 9,94% 5,81% 0,13%

beta 1 0,907 0,836 0,976

Mkt Low Mid High

1995-1999 1,29 0,81 1,07 1,00

2000-2004 -0,28 0,05 -0,30 -0,53

1995-2004 0,37 0,39 0,27 0,00

YoY/ beta 0,092 0,110 0,069 0,001

YoY

YoY risk adjusted

Portfolios with blend and growth style as well as the Market index sank from 2000 to 2004,

while Value style stocks outperformed during this sub-period and caught up other investment

styles for the whole period of study. Checking out the former results along with Chart 4 5 one

could say that mean-reversion hypothesis cannot be rejected while CAPM theory fails


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stunningly, since one more time systematic risk, represented by beta, is not a performance

explanatory variable as assets with high beta-profile are not rewarded.

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Chart 4. Average PER and QoQ Average Performance

Average PER

PER Long term Average

QoQ Average Performance (right)

Finally, portfolios created filtering sample data using Dividend Yield ratio are shown in Exhibit

6. The ratio denominator is the variable “Stock price” contrary to what happened in previous

ratios, being the High DY portfolio undoubtedly the best investment vehicle6 (11.39%

annualized profitability and 0,142 return to beta). This High DY portfolio can also be

interpreted as a Value index composed of companies with a high dividend profitability and

generally high payout rates as well as stable profits.

Exhibit 6. Performance by DY.

Mkt Low Mid High

1995-1999 28,47% 21,05% 18,02% 19,46%

2000-2004 -7,13% -18,50% -5,89% 3,86%

1995-2004 9,23% -0,68% 5,39% 11,39%

beta 1 1,051 0,865 0,804

Mkt Low Mid High

1995-1999 1,29 1,03 0,87 0,93

2000-2004 -0,28 -0,61 -0,24 0,16

1995-2004 0,37 -0,02 0,23 0,51

YoY/ beta 0,092 -0,006 0,062 0,142

YoY

YoY risk adjusted


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This slight appreciation becomes even more overwhelming observing the beta figures: High

DY beta (0,804) is below Low DY beta (1,051), jointly with Chart 5 helps to understand the

stronger cyclical bias regarding Low DY stocks. In particular, Low-High DY spread becomes

generally more positive over bull markets while negative spreads were bound to arise during

bear markets. As it happened in earlier sections, CAPM null hypothesis can be rejected as

beta continues showing no signs of significance in explaining relative performance among

portfolios.

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Chart 5. Spread trend: Low DY vs High DY and Market index

Low DY vs High DY

Market Index

To sum up, a common feature among best performer indices in each ratio-class filtering

method is the low sensitivity to market fluctuations or systematic risk. This feature was crucial

especially during the period 2000-2005, when the market slumped violently causing an even

more radical drop to those stocks with high beta features. An easy way to notice the last is by

looking at Chart 6 since this figure highlights quite intuitively how the best single-style

portfolios (Small index along with Low PER, Low PTB and High DY portfolios) share together

the lower market sensitivities, yet this relationship is flawed by the short observational period

chosen.


14

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Chart 6. Market Correlations.

All in all, the best outcomes in absolute and risk-adjusted terms during the sample period of

study came from high dividend securities (Exhibit 7). However, as it was previously

underlined, High DY portfolio does not have a lower beta or lower correlation coefficient in

comparison with the rest of investment strategies. In the last column of the exhibit a “Value”

strategy proxy is depicted, being constructed by averaging PTB Low, Low PER and High DY

outcomes. In short, Small caps strategies still outperformed this “Value” proxy indicator as it

can be reflected from the figures.

Exhibit 7. Optimal Portfolios by Performance.

Mkt I.Small Low PTB Low PER High DY Value

1995-1999 28,47% 18,46% 18,51% 19,41% 19,46% 19,13%

2000-2004 -7,13% 3,21% 0,91% 1,22% 3,86% 2,00%

1995-2004 9,23% 10,58% 9,36% 9,94% 11,39% 10,23%

beta 1 0,774 0,846 0,907 0,804 0,852


1995-1999 22,12% 20,79% 21,93% 24,12% 20,97% 22,34%

2000-2004 25,76% 23,00% 25,26% 26,76% 23,41% 25,14%

1995-2004 25,05% 21,91% 23,68% 25,49% 22,21% 23,79%


1995-1999 8,06% 19,35% 20,58% 15,61% 15,44% 17,21%

2000-2004 45,98% 41,44% 37,83% 47,98% 37,70% 41,17%

1995-2004 44,06% 41,38% 37,49% 47,98% 37,70% 41,06%


1995-1999 128,70% 88,80% 84,39% 80,50% 92,79% 85,90%

2000-2004 -27,67% 13,98% 3,61% 4,57% 16,50% 8,22%

1995-2004 36,85% 48,26% 39,52% 39,01% 51,27% 43,26%

YoY/ beta 0,092 0,137 0,111 0,110 0,142 12,07%

YoY

Stand Deviation annualized

VAR 95%

YoY risk adjusted


15

3.3 Multi-style investment strategies.

Multiple articles as Ahmed, Lockwood and Nanda (2002) have suggested and demonstrated

the fact that applying different criteria to build portfolios may give rise to greater

performances than those obtained through single-style investment strategies. Then the next

step will be to develop a composition of portfolios based on a double-filter methodology: the

first discriminating factor to apply is size (market capitalization), while the second filtering

factor will depend on the chosen (PTB, PER or DY). To begin with PTB as first secondary

discriminatory factor, Exhibit 8 shows nine portfolios built up following the previously

commented methodology.

Exhibit 8. Performance by PTB.

M kt Low M id High M kt Low M id High M kt Low M id High

1995-1999 28,47% 18,75% 7,65% 27,79% 28,47% 19,39% 11,75% 18,30% 28,47% 26,18% 15,84% 27,05%

2000-2004 -7,13% 3,61% 6,71% -1,28% -7,13% -2,71% 11,75% -14,04% -7,13% -14,47% -13,43% -16,20%

1995-2004 9,23% 10,92% 7,18% 12,32% 9,23% 7,78% -1,72% 0,84% 9,23% 3,89% 0,14% 3,19%

beta 1 0,876 0,709 0,731 1 0,798 0,930 0,857 1 1,181 1,065 1,007

M kt Low M id High M kt Low M id High M kt Low M id High

1995-1999 1,29 0,82 0,33 1,37 1,29 0,82 0,64 0,89 1,29 1,11 0,70 1,30

2000-2004 -0,28 0,12 0,31 -0,06 -0,28 -0,12 0,39 -0,54 -0,28 -0,41 -0,42 -0,58

1995-2004 0,37 0,41 0,33 0,58 0,37 0,33 -0,07 0,03 0,37 0,13 0,01 0,12

YoY/ beta 0,092 0,125 0,101 0,169 0,092 0,098 -0,018 0,010 0,092 0,033 0,001 0,032

YoY risk adjusted

YoY YoY

MID LARGESMALL

YoY risk adjusted YoY risk adjusted

YoY

From the last figure some useful insights can be extracted: Smallcaps – High PTB portfolio

got the better results (12.32% annualized return of 12.32% and 0,169 beta-adjusted return).

Results vary depending on the size of enterprises, being medium and large capitalization

enterprises with Low PTB those who better reflect the size premium effect. This figures seem

to unveil some degree of subordination in the PTB criterion to the previous filter by size,

since the results we initially expected according to the indexes created in section 3.2 did

believe that a portfolio consisting of small caps with low ratios PTB would yield superior

performance. However, the results seem to show a greater importance of the size variable

(especially in small caps), being this market capitalization effect offset during the irrational

exuberance fashion which ruled the markets during the first sub-period (1995-1999).

Under the results shown in Exhibit 8, some comments may be made: due to the short time

horizon used (10 years) results could be spurious as the boom and bursts periods in the

sample could have flawed the true role played by both the “Size” and “PTB” variables.


16

In Chart 7, the performance trend of the Smallcaps-High PTB portfolio during the T&IT

Bubble shown the excessive optimism regarding profit growth expectations placed by the

investors in such a kind of companies. As it can be observed, the Small-High PTB portfolio

even outpaced the results obtained by the Small caps index or the low PTB portfolio, which

were the two best single-style investment strategies. So it can be pointed that two elements,

short sample period and irrational exuberance, could have been skewing the final

conclusions about the size and PTB variables effects.

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Chart 7. Accumulated Performance Small caps vs PTB.

Small Index

Small-Low PTB

Low PTB

Small-High PTB

Likewise in Exhibit 9 are shown the results using the PER ratio as secondary selection

variable. In this case the results do not differ from those expected from the single-style

strategies: small capitalization stocks with lower PER ratios performed the best regardless

the size group but in small caps. This phenomenon may have a similar explanation to the

one exhaustively exposed regarding Exhibit 8 data.

Exhibit 9. Performance by PER.

Mkt Low Mid High Mkt Low Mid High Mkt Low Mid High

1995-1999 28,47% 9,35% 27,49% 18,73% 28,47% 21,78% 10,64% 17,46% 28,47% 26,74% 18,77% 23,40%

2000-2004 -7,13% 6,64% 3,26% -1,11% -7,13% -3,34% 10,64% -19,12% -7,13% -11,55% -12,35% -19,98%

1995-2004 9,23% 7,99% 14,74% 8,35% 9,23% 8,50% 1,23% -2,53% 9,23% 5,88% 2,03% -0,63%

beta 1 0,857 0,711 0,748 1 0,822 0,769 0,982 1 1,188 0,951 1,103


1995-1999 1,29 0,37 1,41 0,91 1,29 1,00 0,54 0,89 1,29 1,00 0,92 1,20

2000-2004 -0,28 0,23 0,15 -0,05 -0,28 -0,14 0,49 -0,59 -0,28 -0,34 -0,45 -0,59

1995-2004 0,37 0,30 0,70 0,40 0,37 0,36 0,06 -0,09 0,37 0,19 0,08 -0,02

YoY/ beta 0,092 0,093 0,207 0,112 0,092 0,103 0,016 -0,026 0,092 0,049 0,021 -0,006

YoY

YoY risk adjusted YoY risk adjusted YoY risk adjusted

YoY YoY

MID LARGESMALL


17

Chart 8 (graph below) highlights in accumulated terms how the performance of the

Smallcaps–Mid PER portfolio exceeded sharply that of the other investment strategies,

including passive single-style strategies like Small caps or Low PER indices.

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Chart 8. Accumulated Performance Small caps vs PER.

Small Index

Small-Mid PER

Low PER Index

Small-Low PER

Finally, Exhibit 10 shows the performance for portfolios created using the Dividend Yield as

secondary discriminatory criterion. From this data it can be inferred that market investors

rewarded income generating stocks, regardless the size of the company. A clear evidence of

this income effect is observed on the Small-High DY (14.27% annualized performance and

0,22 beta-adjusted return), being previously anticipated by the aforementioned single-style

results on Small and High DY indices constructed in earlier sections. Once again a graph

showing accumulated returns is present in Chart 9, where the outstanding return of the

Small-High DY portfolio and other related strategies are concisely

depicted.


18

Exhibit 10. Performance by DY.


1995-1999 28,47% 24,54% 13,30% 16,01% 28,47% 14,60% 14,03% 20,88% 28,47% 23,45% 21,02% 24,60%

2000-2004 -7,13% -6,75% 3,91% 12,55% -7,13% -20,24% 14,03% 0,76% -7,13% -21,10% -11,72% -10,44%

1995-2004 9,23% 7,77% 8,51% 14,27% 9,23% -4,39% -0,03% 10,36% 9,23% -1,31% 3,36% 5,64%

beta 1 0,924 0,752 0,643 1 1,073 0,751 0,768 1 1,140 0,990 1,116


1995-1999 1,29 1,00 0,53 0,94 1,29 0,62 0,84 1,01 1,29 1,22 0,94 0,96

2000-2004 -0,28 -0,26 0,17 0,54 -0,28 -0,59 0,58 0,03 -0,28 -0,60 -0,44 -0,33

1995-2004 0,37 0,30 0,36 0,71 0,37 -0,15 -0,00 0,48 0,37 -0,04 0,13 0,19

YoY/ beta 0,092 0,084 0,113 0,222 0,092 -0,041 0,000 0,135 0,092 -0,011 0,034 0,051

YoY risk adjusted YoY risk adjustedYoY risk adjusted

YoY YoY

MID LARGESMALL

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Chart 9. Accumulated Performance Small caps vs DY.

Small Index

Small-High DY

High DY Index

In Exhibit 11 there is a summary of the 6 best strategies both multi-style and single-style.

Overall, the only single-style strategies displayed in the box are those which main investment

themes are high Dividend Yield (High DY) as well as those related to low market

capitalization (Small). The 3 best strategies are multi-style portfolios with small size features:

Small High DY, Small-Mid PER and Small High PTB. An important takeaway from the last

results is the astonishing importance of considering the size factor as primary discriminator.


19

Exhibit 11. Optimal Peformance: Multi-Style and Single-Style.

Small-High DY Small-Mid PER Small-High PTB High DY Small Small-Low PTB

1995-1999 16,01% 27,49% 27,79% 19,46% 18,46% 18,75%

2000-2004 12,55% 3,26% -1,28% 3,86% 3,21% 3,61%

1995-2004 14,27% 14,74% 12,32% 11,39% 10,58% 10,92%

beta 0,643 0,711 0,731 0,804 0,774 0,876


1995-1999 16,99% 19,53% 20,26% 20,97% 20,79% 22,98%

2000-2004 23,13% 21,83% 20,35% 23,41% 23,00% 29,75%

1995-2004 20,03% 21,13% 21,13% 22,21% 21,91% 26,42%


1995-1999 16,81% 14,69% 11,83% 15,44% 19,35% 28,47%

2000-2004 36,39% 40,37% 44,81% 37,70% 41,44% 42,86%

1995-2004 35,23% 38,86% 38,56% 37,70% 41,38% 42,23%


1995-1999 0,94 1,41 1,37 0,93 0,89 0,82

2000-2004 0,54 0,15 -0,06 0,16 0,14 0,12

1995-2004 0,71 0,70 0,58 0,51 0,48 0,41

YoY/ beta 0,222 0,207 0,169 0,142 0,137 0,125

YoY

Stand Dev annualized

VAR 95%

YoY risk adjusted

Something quite remarkable is the fact that Multi-Style strategies offer a Market correlation

coefficient lower than the Single-Style 7 strategies (Chart 10). However, observing the best

strategies for each type there is little distinction among them in sensitivity terms: Single-Style

optimal strategies coefficients are ranging from 0,885 to 0,906 while Multi-Style optimal

strategies coefficient are allocated from 0,804 to 0.96.

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Small Mid Large Small Mid Large Small Mid Large

Chart 10. Market Index Correlations.


20

4. Investment strategies using Style Rotation.

4.1 Style rotation strategies as an alternative.

Multiple investment strategies based on different criteria have been released in the previous

sections. In section 2, Single-Style strategies were built using a single variable; meanwhile in

Section 3 efforts were made to elaborate a more complex portfolio construction process from

using a pair of factors to creating portfolios (primary and secondary discrimination factors)

named Multi-Style strategies.

Single-style and Multi-Style strategies have similar and divergent points, but a great feature

both share is their intrinsic element of passivity. With reference to this inner passivity is the

fact that each 31st December a new portfolio(s) are created based on some parameters and

held until the next allocation window (12 months later), so the name of “Passive Investment

Strategies” is fairly assigned.

However, various authors as Ahmed, Lockwood and Nanda (2001) have tried to analyze the

effect of not betting continuously in a single style strategies over 12-month periods and

experimented with investment style changes, commonly known as Style Rotation, to optimize

portfolio returns. Style rotation strategies may belong to the same group, as changing from

Large Caps to Small Caps, or being a much more sharp change like passing from Large

Caps to Value stocks. The last example is illustrated in Chart 118, where the optimal strategy

is constructed by using the right timing to picking strategies. The quarterly turnover between

styles is useful as the investor may avoid serious drawdowns and manage to get a smooth

returns pattern over the investment horizon. As a result, better absolute and risk-adjusted

returns are achieved in comparison with Passive Investment Strategies.


21

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Chart 11. Spread Trend: Multi-Style and Optimal Strategies .

Small-Large

Value-Growth

Optima

The key in order to achieve a higher performance following active strategies is based on

knowing and anticipate changes in the market cycle. Hence an investor must know a lot of

variables that can explain the change in the leading investment style, besides holding a fairly

outstanding prediction ability whether the investor aims at improving or at least matching

passive strategies performance.

Therefore it is important to find out those expected results linked to a certain level of

prediction under a specific probability of occurrence. To perform the next analysis a set of

previously constructed indexes has been used, choosing the minimum and maximum

performance for each date (accounted for a Rotation fee of 2%) and applying 4 different

levels of prediction. It is assumed that each of the 4 performance generated series are

normally distributed with mean and distribution equal to the computed for each series. Last

step was to implement the inverse normal distribution to find the return linked to each of the

probability values.

For example in Chart 12 some indexes created in Section 2 (Small, Mid and Large) where

used. For each date, the higher and lower returns from these indexes have been gathered;

resulting in 2 data series (minimums data and maximums data) on which we have applied 4

different levels of forecasting to obtain 4 simulated series of expected returns. To put it

another way, the expected returned series of an individual with predictive level of 80% will be


22

simulated as the multiplication of the pair of maximum and minimum series by 80% and 20%

respectively. Thus although by this method the true distribution of expected returns is not

truly forecasted, this rudimentary process is enough to demonstrate some useful insights. To

obtain the true distribution of expected returns for each prediction level, several Style rotation

simulations should be run though is topic up to further research articles.

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Chart 12. Returns Distribution: Size Rotation.

Forecast Level 80% Forecast Level 70%


Small Index

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Chart 13. Returns Distribution: Style Rotation.



Value Index

Both Chart 12 and Chart 13 yielded similar conclusions. Regarding the former one, taking a

level of confidence equal or lower to 55% only an investor with an 80% level prediction would


23

obtained a return equal to the passive strategy (Small caps). In other words, the probability of

outperforming the Small Index for an investor whose forecast accuracy is 80% would be

equal or lower to 45%. An investor with only a 50% forecast ability would only match passive

strategies at a level of confidence equal or lower to 65%, so his chance of getting a higher

profitability would be reduced to 35%.

In Chart 14 a similar process has been followed but using all indices and portfolios created in

sections 2 and 3, in this way we have obtained a series of maximum and minimum for each

date and rotation possibilities include Single-style and Multi-Style strategies. Conclusions to

be drawn from Chart 12, Chart 13 and Chart 14 could be summarize in the difficulty of

outperforming the passive investment strategies without having considerable levels of

prediction.

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Chart 14. Returns Distribution: Total Style Rotation.

Forecast Level80%Forecast Level70%Forecast Level60%Forecast Level50%Small Index

4.2 Indicative models for the practice of Style Rotation.

In Section 4.1 some insights regarding the application of rotation strategies were underlined,

being the most important one the necessity to have certain superior forecasting skills to

anticipate market and style changes. This section will introduce two types of econometric

models whose main objective is to enhance this forecasting ability concerning changes in

cycle and whose explicative variable are 2 sort of spreads: size or style value/growth).


24

The main difference between both models is based on the explicative/independent variables

used to explain the dependent variable 9. The first model studies relationships between

external variables (different to the series of data which make up the spread), while the

second model is based on time series models (use of spread intrinsic information) through

ARMA models and volatility modeling when so required 10.

For the first model, the following variables were used as explanatory regressive factors to

explain the “Small-Large Spread” returns, displaying in parentheses their name as they

appear in Exhibit 12 and Exhibit 13: European Union Inflation (inflation) 11, the 10 years-12

months sovereign slope (slope) 12, the 12 months maturity Spanish Bill quarterly price

change (bill), Dividend yield spread between small and large capitalization companies

(spreadSLDY) and between value and growth companies (spreadVGDY) 13, Equity Market

Risk premium14 (EMRP) and the dollar/euro exchange rate quarterly growth (EURUSD). The

choice of these variables have its basis in research published by M.Levis and M.Liodakis

(1999) for the British market, though further improvement in the regression variables

selection and processing could be done in future research.

Exhibit 12. Small-Large Spread Regression.

Coef t-stat Coef t-stat

intercept 0,003768 1,6362 -0,092772 -2,1147

inflation 1,160985 1,0781 1,807237 2,3341

slope 0.432264 0,1842 1,660048 0,6803

bill 0.053278 0,6062 0,063465 0,8118

spreadSLDY 8,286365 2,2922 1,159402 3,5825

EMRP 0.093910 1,1347 0,226294 3,0546

EURUSD 0.322915 1,4162 0,590844 2,5796

R2 0,412

R2 adjusted 0,301

AIC -2,8345

Univariable Multivariable

In the first two columns of Exhibit 12 can be observed how using a single explanatory

variable yields rather spurious significant results for all the explicative factors (their t statistic

are lower than 1.96 so the null hypothesis “estimated coefficient equal to 0” cannot be

rejected with a 95% level of confidence) saving the variable “spreadSLDY” that is statistically

representative. Nevertheless the situation upturns considerably in considering the last two

columns, in this case all variables gain statistical significance but neither “slope” nor “bill”. To

emphasize the high sensitivity in the dividend regression factor (spreadSLDY), clearly related


25

to the importance previously commented in other sections of the Dividend Yield as key

discriminating factor in the construction of portfolios.

Exhibit 13. Value-Growth Spread Regression.

Coef t-stat Coef t-stat

intercept 0,011322 1,0784 -0,010400 -0,2135

inflation 0,420639 0,5146 0,514377 0,5238

slope -1,094735 -0,6205 -0,422141 -0,2143

bill -0,024461 -0,3674 -0,029563 -0,3930

spreadVGDY 2,161123 1,0074 1,788723 0,6620

EMRP -0,000470 -0,0074 0,026732 0,3329

USDEUR -0,105284 -0,6298 -0,105791 -0,5248

R2 0,048

R2 adjusted -0,125

AIC -2,9228

MultivariableUnivariable

The results presented in table 13 to explain the spread Value-Growth model are much less

encouraging since no variable is certainly significant. A very powerless model is clearly

observable as coefficient of determination low figures corroborates, not even reaching to

explain a 5% of the volatility inherent in the dependent variable.

With reference to the second model, Time-series approach, in Exhibit 14 an ARMA model is

shown using several the fourth, fifth and sixth dependent variable lags as explanatory

variables ((AR (4), AR (5) and AR (6)) as well as the first and fifth regression errors lags (MA

(1) and MA (5)). Despite the fact that after numerous tests this model seems to be the most

successful, the existence of material correlation among the square regression residuals

obliged to modeling also the variance using as independent variable the first lagged squared

residual and an intercept, ARCH (1). 15

Exhibit 14. Small-large Spread: Time-Series Model.

Main Regression

Coef p-val

AR(4) 0,213217 0,0000

AR(5) -0,659728 0,0000

AR(6) 0,359771 0,0046

MA(1) 0,928120 0,0000

MA(5) 1,244422 0,0000

Variance Regression

Coef p-val

intercept 0,001746 0,0002

arch(1) -0,196714 0,0005

R2 0,669

R2 adjusted 0,596

AIC -3,348


26

The model results showed in Exhibit 14 are very encouraging since all variables are

significant (p-value below the 0.05 so it means rejecting null hypothesis of coefficients not

statistically significant). Also sound figures in the coefficient of determination do more to

support the large capacity model to capture the dependent variable volatility.

Main Regression

Coef p-val

intercept 0,004681 0,0000

SPREADVG(-1) 0,606188 0,0001

MA(1) -0,883918 0,0001

MA(2) -0,604951 0,0210

R2 0,491

R2 adjusted 0,447

AIC -3,677

Exhibit 15. Value-Growth Spread: Time-Series model.

Concerning “Value-Growth Spread” dependent variable regression depicted in Exhibit 15, an

ARMA (1,0,2) was built as best alternative. No volatility modeling was required this time as

since regression bore no correlations between square residuals. With regards to the results,

these were similar to those of Exhibit 14, being all independent variables statiscally

significant and with a coefficient of determination around 50%. In this case model time series

for the Spread Value-Growth variable represents a significant leap forward in comparison

with the one presented in Exhibit 13.

Although regarding the “Value-Growth Spread” variable there is no other choice than apply

the Time-Series model, this does not happen so clearly for the “Small-Large Spread”

variable. In the latter case, a better alternative of comparing coefficients of determination is to

use the Akaike information indicator (AIC). The main reason behind this choice is that AIC

copes better with the biases inherent in both the coefficient of determination (artificial

increase by adding independent variables) and the adjusted coefficient of determination (data

set particularities damage this indicator). So the criterion followed compares AIC figures from

both the External factors model and the Time-Series model, and selects the one with smaller

AIC. Fortunately, the Time-Series model offers lower AIC as well as the higher coefficient of

determination, so in this case the choice offers no doubts. 16


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5. Conclusions.

The most significant results obtained in each previous section are highlighted below. Firstly,

the best Single-Style strategies in terms of absolute and risk-adjusted performance are those

portfolios of low market capitalization stocks (Small index) and portfolios containing value

securities (Low PTB, Low PER and High DY portfolios). From the previous portfolios, the

income generating portfolio (High DY) proved to hold a more optimum performance; while the

other side of the coin is observed in the Low PER portfolios, which is displayed as the riskier

alternative since it offers a minimum risk-adjusted return along with high values in standard

deviation and maximum expected loss (VAR95%). In global terms, those indices constructed

with small companies (Size bias) seem to overcome timidly those composed with “Value”

bias, although in terms of VAR the latter submitted lower expected loss.

Regarding Multi-Style strategies, some resemblances with Single-Style strategies were

founded as some portfolios constructed using “Dividend Yield” as secondary discriminator

performed well. In this case the “Small-High DY” portfolio yielded a better performance,

phenomenon that was expected because of the sound results obtained from its Single –Style

"parents": Small index and High DY index. However the results from the rest of Multi-Style

portfolios were not as expected. Given the results obtained in Single-Style tests, it was likely

to have more optimal portfolios with Small-low P/E and Small-Low PTB features, however it

was not the case. Firstly, the PER-class ratio portfolio with best performance was Small-Mid

PER, besides the Small-Low portfolio PER was not even second ranked because it was

surpassed by the Small-High PER portfolio. Secondly, in terms of the PTB as secondary

filtering ratio results also were contrarian to the expected: Small-High PTB portfolio was the

best performer. These inconsistencies are not repeated in portfolios generated from medium

or large capitalization companies, being those portfolios with lower PTB or Low PER clear

winners. These results cast some doubt about the validity of ratios such as the PER and the

PTB, although no definitive conclusions might be made as the small available sample data

along with the irrational exuberant context chosen (T&IT Bubble) tainted the final

conclusions.

Then for the quarterly turnover style investment strategies, two basic conclusions should be

made. The first one refers to the predictive capabilities an investor must own, being advisable

to hold a forecasting accuracy of at least 80% to take advantage of an Active Strategy over a


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Passive Strategy, as the probability to overcome or not a buy-and-hold tactic can oscillate

approximately a 0.2 between an investor with 80% predicting power and one that has only a

50%. The second conclusion focuses on how to achieve or improve this forecasting ability,

being the answer in the optimization of econometric models for the prediction of Style

spread´s sign changes. In this sense, this document shows a certain dominion of Time-

Series regression models against External variables models; yet the variables used as

external independent variables in this research may not be suitable for this sample data

because of certain sample data gathering limitations and the fact that some of this regression

external variables were chosen from the results of other similar articles whose sample data

features were completely different to the one used here.

In conclusion, the major pitfalls of this work are mainly focused in the little sample of

companies gathered (104 stocks), not being a significant number that reflects the European

Union wide range of companies. Another caveat comes in the way in which data have been

studied because it has been collected on a quarterly basis and a ten- year time horizon.

Unfortunately no database was available to collect data more frequently or with a longer

period like 20 years. Finally, it can be said that the document could be updated and improved

including comparisons with the fixed income market and changing the portfolio construction

frequency from an annual window to shorter or longer ones.

Notes

1. Price-to-book ratios as well as Dividend Yield used data from profit and dividends relating to the date in which are located. However,

although the PER its numerator takes the price corresponding to the date of its publication; the denominator reflects the benefit

expected next year rather than current data.

2. Size is used as the primary discriminator while the rest of ratios classify stocks as value, blend or growth. Size as market

capitalization can co-exist in groups of similar size members belonging to the three types previously appointed.

3. Two measures that have been used to assess the risk-adjusted profitability have been the YoY risk-adjusted (annualized return to

standard deviation) and Beta-adjusted returns (annualized return to beta). Although in this way we cannot collect the effect of risk,

premium, both measures are useful for comparing the portfolios created one to each other. Conclusions with both ratios will be

virtually the same as the created portfolios have a high diversification of specific risk.

4. Although mean-reversion studies by Campbell and Schiller are conducted for a sample of companies other than the one used here,

their conclusions may be extrapolated without dramatic consequences.

5. In Figure 4 is calculated the "Average PER " as the arithmetic mean of all ratios PER for each date, while "PER Long Term Average"

is an average for all arithmetic means in each date. Heading "QoQ Average performance" has been built on the average

performance of all stocks on a quarterly basis.

6. Index composed of securities with higher dividend yields is due to the rise in the risk premium priced by the investors. Since the

Bubble burst the majority of investors flew to safety by buying “real” earnings stocks different than the eternal promises made by

Telecommunications and IT companies .

7. Correlations of Single-Style and Multi-Style are in average 0,941's for the former and 0.895 for the later.


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8. For "Value-Growth Spread" series has been necessary to perform three Spread Value-growth series created from different ratios as

main criterion (PTB, PER and DY).

9. Independent variables have been introduced in the model with one lag to the dependent variable. In other words, to explain the

“Small-Large Spread” at t, is necessary to introduce independent variables data at (t-1), being the logic behind this methodology

the aim at analyzing the predictive skills of the model.

10. All regressions have been corrected from heteroskedasticity by White (1980).

11. Inflation in the European Union is not gathered from any number of databases, so it has been calculated approaching as the HICP

annualized variation (Harmonized Price index) of Spain minus 1%. This 1% is the historical spread between the European

inflation and the Spanish inflation.

12. Types have been used for 10 years and 1 year of the Spanish public debt as approximation of European public debt. Could have

chosen the German debt given its most important markets, but it is also true that the process of convergence in recent years has

done that the German debt and Spanish differ by few points Basic.

13. The " Small-Large DY Spread" construction classifies all companies in 2 groups each December 31 depending on its size, being

its average dividend yield calculated on each date as the average between these 2 groups. While for the calculation of "Value-

Growth DY Spread" dividend yield calculation was made each date from the average dividend yield from the groups created using

PTB and PER ratios as allocating variables.

14. Equity Market Risk Premium (EMRP) is obtained through the CAPM model:

Em = Rf + beta * EMRP MERP = Em-Rf

where: Em: Market capitalization-weighted index returns.

Rf: Return of risk-free assets represented by the 10 yr euro sovereign bond.

EMRP: Market risk premium

Beta: sensitivity of the market to movements of the market, which by definition is 1.

15. Model used in theoretical terms is as follows:

Mean Equation: Yt = β * Y t-4 + β * Y t-5 + β * Y t-6 + ut + 1 α * or t-1 + α 2 * or t-2

Variance Equation: V(Yt/It-1) = ht = c + µ * or t-1

where: Yt: Small-Large Spread at t

ut: regression error at t, being on average 0 with a conditioned variance equal to ht and non-serially correlated

(cov(ut;ut-1) = 0) Although not standalone (cov(ut2;_ut-1

2) ≠0).

Box presents the p-value rather than the t statistical t, since in this case variables are distributed by the statistician z whose critical

values not available, although Eviews 3.1 calculates automatically the p-value so that the individual significance test can be safely

done.

16. (AIC) of Akaike information criterion has been used to choose the most optimal model but there is another criterion called Schwartz

Bayesian (SBC). The formal expressions in both methods are:

AIC = T * ln(Squared Residuals Sum) + 2 * n

SBC = T * ln(Squared Residuals Sum) + T * n

n = number of parameters to estimate.

T = number of observations used in the regression.

The use of either criterion in small samples as the current (40 observations) is indifferent, however if the time horizon temporary

had been higher it would have allowed to recommend the SBC.


30

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2005. Dissertation Carlos Salas

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