1. Introduction

The Holy Grail of finance is an empirical asset pricing model that explains stock returns. Most models fall under the risk/return umbrella where risk is positively related to return. There are two basic models in empirical asset pricing, the standard Capital Asset Pricing Model, CAPM [15, 17, 21] and the Fama/French three-factor model, FF3 [5]. The basic idea behind the CAPM is that market movements matter a lot for capturing the relationship between risk and return. The systematic risk measure, beta, is an estimate of the sensitivity of a security or portfolio's

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

returns to market movements. In the risk/return world, the CAPM is considered a one-factor model in that a single factor, the market return, does all the heavy lifting. The model specification is as follows:

$$r\_{\rm it} - r f\_{\rm t} = \alpha\_{\rm i} + \beta\_{\rm i}^\* \left( rm\_{\rm t} - rf\_{\rm t} \right) + \varepsilon\_{\rm t} \tag{1}$$

The idea of basically finding other factors besides the market that explain equity returns has generated many different versions of factor models. One that has gained widespread acceptance is the Fama and French three-factor model (FF3). The general consensus is that the FF3 has greater explanatory power than the CAPM. The Fama-French model adds to the explanatory power of the standard CAPM by including two additional factors, firm size and the bookto-market ratio. Both factors were found in previous research to matter for explaining equity returns. That small firms outperform large cap firms is found in Banz [1], while Barr Rosenberg, Kenneth Reid, and Ronald Lanstein [19] find a positive relationship between average stock returns and book-to-market ratio. Low B/M firms are considered "value stocks" while high B/M are "growth stocks." There is strong consensus around the idea that smaller cap firms are riskier and therefore, generating greater returns beyond what would be expected from simple market beta exposure is a widely accepted explanation for the size factor. There is less agreement for an explanation of the value premium, but one is rooted in behavior where basically relatively cheap stocks outperform relatively expensive stocks because optimism and pessimism persist among investors. Investors bid up growth stocks leading to future under performance, and keep down value stocks leading to future over-performance. Both size and B/M factors are added to their model as factors that account for returns, along with the market

An Application of Wavelets to Finance: The Three-Factor Fama/French Model

http://dx.doi.org/10.5772/intechopen.74165

83

factor as found in the CAPM. The FF3 model is specified as follows:

sign of the size and book-to-market factors varying across scale.

Book-to-market is defined and total assets less total liabilities.

4

<sup>∗</sup> rmt � rf <sup>t</sup> beta2<sup>i</sup>

∗

where SMBt and HMLt are the size and book-to-market factors, respectively. The book-tomarket ratio is intended to capture the difference between value and growth stocks in the sense that the book-to-market ratio is high for value stocks and low for growth stocks.4

Several studies have examined the Fama-French 3-factor model at the scale level. Kim and In [10–14] apply wavelets to the Fama-French 3-factor model using monthly data from 1964 to 2004 for 12 industry portfolios. They find that the market variable plays an important role in explaining stock returns across all scales. In addition, they find that the estimated coefficients for the SMB and the HML are significant in specific time scales, depending on the industry. Trimtech et al. [22] apply wavelet analysis to the Fama-French model to study monthly returns for the French stock market for the period 1985. They find that the r-square of the medium and high scale versions of the Fama-French model exceed that of the standard model. They also find that the risk sensitivity of the factors depends on the time scale with the magnitude and

We use multi-scale analysis and a rolling 250-day window to estimate the Fama-French 3-factor model of stock returns for 49 industry stocks of US industry portfolios. The data set, which consists of daily observations, covers the period from July 1, 1969 to September 29, 2017. We find through risk-adjusting the portfolios using the FF3 model that there are distinct risk dynamics during recessions. The rolling window estimation approach allows us to capture the behavior of an investor who periodically reallocates his portfolio. Using periodic estimates of expected return we implement a set of out-of-sample long/short investment strategies based on

SMBt þ β3<sup>i</sup>

<sup>∗</sup>HMLt <sup>þ</sup> et (2)

rit � rf <sup>t</sup> ¼ α<sup>i</sup> þ β<sup>i</sup>

where rit = return of firm i at time t, rmt = market return at time t, and rft = risk free rate at time t. The slope term, βit, estimates systematic risk. The intercept, αi, measures abnormal returns, or returns not explained by market exposure of the security or portfolio. In the context of the CAPM, α<sup>i</sup> is expected to be zero since only non-diversifiable, also referred to as systematic or market risk, represents the risk that matters for explaining returns.

While the CAPM remains a cornerstone of financial theory, numerous empirical studies have called into question the ability of the CAPM to explain the cross-section of expected stock returns (see for instance, [3]). Several studies have used wavelets to examine the CAPM across scale. Gencay [7] first proposed the use of wavelets to estimate systematic risk in the Capital Asset Pricing Model. They estimate the beta of each stock annually for 6 wavelet scales using daily returns for the period January 1973 to November 2000 for stocks that were in the S&P 500. They find a positive relationship between portfolio returns and beta. Gencay et al. [8] extend their 2003 study by including stocks from the Germany and UK. They find that scale matters in other markets in that the relationship between portfolio returns and beta becomes stronger at high scales. Fernandez [6] applies wavelet analysis to a model of the international CAPM using a data set that consists of daily aggregate equity returns for seven emerging markets for the period 1990–2004.<sup>1</sup> The ICAPM<sup>2</sup> was estimated at 6 scales (2–128 day dynamics). Fernandez finds that market sensitivities are generally greatest at the higher scales of 5 and 6. In addition, the R<sup>2</sup> peaked at scales 5 and 6. She concludes that the ICAPM does its best at capturing the relationship between risk and return at the medium scale or long-term scale that for their data set is 32–128 days. An important takeaway from research employing wavelet measures of beta is that when the environment is distinguished by slowly changing features, or low frequency events the CAPMs' applicability in terms of providing a measure of systematic risk improves when using wavelets. This is consistent with the findings of Rua and Nunes [20] that employs wavelet methodology and provides evidence that market risk varies across time and over frequencies.<sup>3</sup>

The adage the proof of the pudding is in the eating is of particular relevance for empirical asset pricing models. Practitioners want to know if they employ a specific empirical asset pricing model will their investors benefit? The fierce competition to develop a winning model continues among various market players, especially hedge funds [2]. The prescription to basically accept that markets are efficient and form a portfolio that passively tracks the market has contributed to the growth of index investing, but has not slowed the search for a better model.

<sup>1</sup> Brazil, Chile, Mexico, Indonesia, South Korea, Malaysia, and Thailand.

<sup>2</sup> ICAPM for two countries E rð Þ¼ <sup>i</sup> � r β1cov ri ð Þþ ;rw β2cov ri ð Þ ;s , where ri = returns for domestic asset, rw = returns for world portfolio, s is the percent change in the exchange rate for domestic and foreign currency.

<sup>3</sup> Their application is to Emerging Markets.

The idea of basically finding other factors besides the market that explain equity returns has generated many different versions of factor models. One that has gained widespread acceptance is the Fama and French three-factor model (FF3). The general consensus is that the FF3 has greater explanatory power than the CAPM. The Fama-French model adds to the explanatory power of the standard CAPM by including two additional factors, firm size and the bookto-market ratio. Both factors were found in previous research to matter for explaining equity returns. That small firms outperform large cap firms is found in Banz [1], while Barr Rosenberg, Kenneth Reid, and Ronald Lanstein [19] find a positive relationship between average stock returns and book-to-market ratio. Low B/M firms are considered "value stocks" while high B/M are "growth stocks." There is strong consensus around the idea that smaller cap firms are riskier and therefore, generating greater returns beyond what would be expected from simple market beta exposure is a widely accepted explanation for the size factor. There is less agreement for an explanation of the value premium, but one is rooted in behavior where basically relatively cheap stocks outperform relatively expensive stocks because optimism and pessimism persist among investors. Investors bid up growth stocks leading to future under performance, and keep down value stocks leading to future over-performance. Both size and B/M factors are added to their model as factors that account for returns, along with the market factor as found in the CAPM. The FF3 model is specified as follows:

$$r\_{it} - r f\_{\;t} = \alpha\_i + \beta\_i^\* \left( rm\_t - rf\_{\;t} \right) \\\\beta\_{2i} + \beta\_{3i}^\* \text{HML}\_{\;t} + e\_t \tag{2}$$

where SMBt and HMLt are the size and book-to-market factors, respectively. The book-tomarket ratio is intended to capture the difference between value and growth stocks in the sense that the book-to-market ratio is high for value stocks and low for growth stocks.4

Several studies have examined the Fama-French 3-factor model at the scale level. Kim and In [10–14] apply wavelets to the Fama-French 3-factor model using monthly data from 1964 to 2004 for 12 industry portfolios. They find that the market variable plays an important role in explaining stock returns across all scales. In addition, they find that the estimated coefficients for the SMB and the HML are significant in specific time scales, depending on the industry. Trimtech et al. [22] apply wavelet analysis to the Fama-French model to study monthly returns for the French stock market for the period 1985. They find that the r-square of the medium and high scale versions of the Fama-French model exceed that of the standard model. They also find that the risk sensitivity of the factors depends on the time scale with the magnitude and sign of the size and book-to-market factors varying across scale.

We use multi-scale analysis and a rolling 250-day window to estimate the Fama-French 3-factor model of stock returns for 49 industry stocks of US industry portfolios. The data set, which consists of daily observations, covers the period from July 1, 1969 to September 29, 2017. We find through risk-adjusting the portfolios using the FF3 model that there are distinct risk dynamics during recessions. The rolling window estimation approach allows us to capture the behavior of an investor who periodically reallocates his portfolio. Using periodic estimates of expected return we implement a set of out-of-sample long/short investment strategies based on

returns to market movements. In the risk/return world, the CAPM is considered a one-factor model in that a single factor, the market return, does all the heavy lifting. The model specifica-

where rit = return of firm i at time t, rmt = market return at time t, and rft = risk free rate at time t. The slope term, βit, estimates systematic risk. The intercept, αi, measures abnormal returns, or returns not explained by market exposure of the security or portfolio. In the context of the CAPM, α<sup>i</sup> is expected to be zero since only non-diversifiable, also referred to as systematic or

While the CAPM remains a cornerstone of financial theory, numerous empirical studies have called into question the ability of the CAPM to explain the cross-section of expected stock returns (see for instance, [3]). Several studies have used wavelets to examine the CAPM across scale. Gencay [7] first proposed the use of wavelets to estimate systematic risk in the Capital Asset Pricing Model. They estimate the beta of each stock annually for 6 wavelet scales using daily returns for the period January 1973 to November 2000 for stocks that were in the S&P 500. They find a positive relationship between portfolio returns and beta. Gencay et al. [8] extend their 2003 study by including stocks from the Germany and UK. They find that scale matters in other markets in that the relationship between portfolio returns and beta becomes stronger at high scales. Fernandez [6] applies wavelet analysis to a model of the international CAPM using a data set that consists of daily aggregate equity returns for seven emerging markets for the period 1990–2004.<sup>1</sup> The ICAPM<sup>2</sup> was estimated at 6 scales (2–128 day dynamics). Fernandez finds that market sensitivities are generally greatest at the higher scales of 5 and 6. In addition, the R<sup>2</sup> peaked at scales 5 and 6. She concludes that the ICAPM does its best at capturing the relationship between risk and return at the medium scale or long-term scale that for their data set is 32–128 days. An important takeaway from research employing wavelet measures of beta is that when the environment is distinguished by slowly changing features, or low frequency events the CAPMs' applicability in terms of providing a measure of systematic risk improves when using wavelets. This is consistent with the findings of Rua and Nunes [20] that employs wavelet methodology and provides evidence that market risk varies

The adage the proof of the pudding is in the eating is of particular relevance for empirical asset pricing models. Practitioners want to know if they employ a specific empirical asset pricing model will their investors benefit? The fierce competition to develop a winning model continues among various market players, especially hedge funds [2]. The prescription to basically accept that markets are efficient and form a portfolio that passively tracks the market has contributed to the growth of index investing, but has not slowed the search for a better model.

ICAPM for two countries E rð Þ¼ <sup>i</sup> � r β1cov ri ð Þþ ;rw β2cov ri ð Þ ;s , where ri = returns for domestic asset, rw = returns for

<sup>∗</sup> rmt � rf <sup>t</sup>

<sup>þ</sup> et (1)

rit � rf <sup>t</sup> ¼ α<sup>i</sup> þ β<sup>i</sup>

market risk, represents the risk that matters for explaining returns.

tion is as follows:

82 Wavelet Theory and Its Applications

across time and over frequencies.<sup>3</sup>

Their application is to Emerging Markets.

Brazil, Chile, Mexico, Indonesia, South Korea, Malaysia, and Thailand.

world portfolio, s is the percent change in the exchange rate for domestic and foreign currency.

1

2

3

<sup>4</sup> Book-to-market is defined and total assets less total liabilities.

the standard Fama-French model, and also the scale versions of the model. We find that for the sample as a whole the strategy based on the standard model outperforms each of the scale based strategies. In other words, frequency-based information does not appear to matter for portfolio performance when spanning the entire time period. However, during the majority of recessions, the higher scale long/short strategies tend to outperform the standard approach. The frequency content of information does appear to matter during recessions. We conclude that most recessions reflect a time-varying market regime where scale dynamics matter for portfolio performance. In terms of practioners the results suggest that an avenue for potential improvement in portfolio performance is found by taking scale into consideration when faced with potential recessionary periods.

Sector Name Industry Mean Std.Dev Skewness Kurtosis Business Equipment Chips Electronic equipment 0.0302 1.6051 0.415 6.48 Business Equipment Hardw Computers 0.0247 1.6601 0.048 11.34 Business Equipment Softw Computer software 0.0271 2.2342 0.023 7.55 Chemicals Chems Chemicals 0.0310 1.2503 �0.142 8.77 Consumer Durables Hshld Consumer goods 0.0227 1.0970 �0.236 9.76 Consumer Non-Durables Agric Agriculture 0.0305 1.4128 0.390 14.57 Consumer Non-Durables Beer Beer & liquor 0.0359 1.1521 �0.401 14.72 Consumer Non-Durables Books Printing and publishing 0.0219 1.2104 �0.284 10.71 Consumer Non-Durables Clths Apparel 0.0266 1.2706 �0.057 6.84 Consumer Non-Durables Food Food products 0.0334 0.9178 �0.044 9.93 Consumer Non-Durables Smoke Tobacco products 0.0523 1.4031 �0.366 7.11 Consumer Non-Durables Soda Cand & soda 0.0350 1.4357 �0.284 10.67 Consumer Non-Durables Toys Recreation 0.0163 1.4800 �0.035 18.54 Consumer Non-Durables Txtls Textiles 0.0268 1.3612 �0.843 22.88 Energy Coal Coal 0.0318 2.4043 �0.187 8.55 Energy Mines Non-metallic and metal 0.0274 1.6273 �0.355 9.47 Energy Oil Petroleum and natural gas 0.0306 1.3598 �0.126 3.94 Health Drugs Pharmaceutical products 0.0341 1.1545 0.205 10.03 Health Hlth Healthcare 0.0243 1.5270 0.471 12.22 Health MedEq Medical equipment 0.0309 1.1857 0.117 7.20 Manufacturing Aero Aircraft 0.0368 1.3506 �0.197 9.90 Manufacturing Autos Automobiles and trucks 0.0213 1.4643 �0.269 6.62 Manufacturing Boxes Shipping containers 0.0294 1.2786 �0.319 10.36 Manufacturing ElcEq Electrical equipment 0.0355 1.3878 �0.380 10.49 Manufacturing FabPr Fabricated products 0.0150 1.5073 �0.441 9.15 Manufacturing Guns Defense 0.0424 1.3798 0.246 16.64 Manufacturing LabEq Measuring and control equip. 0.0290 1.4337 �0.307 10.14 Manufacturing Mach Machinery 0.0272 1.3123 �0.122 7.25 Manufacturing Paper Business supplies 0.0276 1.1143 �0.291 14.22 Manufacturing rubbr Rubber and plastic products 0.0279 1.1525 �0.131 6.20 Manufacturing Ships Shipbuilding, railroad equip. 0.0322 1.5089 �0.296 10.80 Manufacturing Steel Steel works, etc. 0.0165 1.6334 �0.236 9.17 Money Banks Banking 0.0295 1.4384 �0.184 6.51 Money Fin Trading 0.0351 1.4694 �0.564 14.32

An Application of Wavelets to Finance: The Three-Factor Fama/French Model

http://dx.doi.org/10.5772/intechopen.74165

85

The remainder of this chapter is organized as follows: Section 2 presents the data and basic statistics. Section 3 describes the methodology. Section 4 presents the empirical findings, and Section 5 follows with our concluding comments.
