A wavelet approach of investing behaviors and their effects on risk exposures

Abstract

Exposure to market risk is a core objective of the Capital Asset Pricing Model (CAPM) with a focus on systematic risk. However, traditional OLS Beta model estimations (Ordinary Least Squares) are plagued with several statistical issues. Moreover, the CAPM considers only one source of risk and supposes that investors only engage in similar behaviors. In order to analyze short and long exposures to different sources of risk, we developed a Time–Frequency Multi-Betas Model with ARMA-EGARCH errors (Auto Regressive Moving Average Exponential AutoRegressive Conditional Heteroskedasticity). Our model considers gold, oil, and Fama–French factors as supplementary sources of risk and wavelets decompositions. We used 30 French stocks listed on the CAC40 (Cotations Assistées Continues 40) within a daily period from 2005 to 2015. The conjugation of the wavelet decompositions and the parameters estimates constitutes decision-making support for managers by multiplying the interpretive possibilities. In the short-run, (“Noise Trader” and “High-Frequency Trader”) only a few equities are insensitive to Oil and Gold fluctuations, and the estimated Market Betas parameters are scant different compared to the Model without wavelets. Oppositely, in the long-run, (fundamentalists investors), Oil and Gold affect all stocks but their impact varies according to the Beta (sensitivity to the market). We also observed significant differences between parameters estimated with and without wavelets.

Introduction

Markowitz (1952) Modern Portfolio Theory led the development of the Capital Asset Pricing Model (CAPM) which was created by Sharpe (1964), Lintner (1965) and Mossin (1966). Its mathematical equation, the Securities Market Line (SML), is similar to a simple regression model between the asset risk premium and the risks of the Market. This is also known as the Market Model. According to the CAPM hypothesis, the market is the only source of risk and the agents have homogeneous investing behavior. Systematic risk is measured through estimations of the traditional Market Model. Many authors such as Black et al. (1972) and then Fama and MacBeth (1973) highlighted several statistical anomalies in the model, more particularly, the non-robustness of the methods used resulting from the autocorrelation-heteroscedasticity in the residuals of the estimations, and the potential absence of exogenous variables in the model.

Following the first tests of CAPM, Fabozzi and Francis (1978) indicated that the Beta parameter is unstable over time, resulting in confirmation by Bos and Newbold (1984). This characteristic of Beta implies that time-varying parameter estimation is required. Many methods have been tested such as rolling-window, recursive regression, or the Kalman-Bucy filter (Groenewold and Fraser 1997, 1999; Brooks et al. 1998; Faff et al. 2000; Yeo 2001). The multivariate GARCH (Generalised AutoRegressive Conditional Heteroskedasticity) approach is also developed to estimate time variance, according to Bollerslev et al. (1988). Advanced multivariate GARCH models such as the DCC-GARCH (Dynamic Conditional Correlation Generalised AutoRegressive Conditional Heteroskedasticity) or BEKK-GARCH (Baba, Engle, Kraft and Kroner Generalised AutoRegressive Conditional Heteroskedasticity) are also employed as attested by the studies of Choudhry and Wu (2008), Tsay et al. (2014), and Engle (2016).

Another way to improve the CAPM is for it to remain as an addition to explanatory variables in the Market Line equation. This kind of approach appears in Multi-Factors or Multi-Betas Models, originally initiated and theoretically constructed by Merton (1973) and Ross (1976) (with the Arbitrage Pricing Theory or APT). The number and the choice of selected variables vary according to the authors and their analysis. Bantz (1981) and Basu (1983) highlight the importance of considering the effects of accounting variables (specific to each stock) on equity returns, in accordance with their capitalization or size (in annual or quarterly frequencies). In an extension of these works, Fama and French (1992,1996) established three factors that CAPM generally referred to as the Fama–French Model, considering the variables of Price-to-Book and the Company Size (or more precisely, the relative performance of small companies versus big, and of companies with high Price-to-Book levels versus low). Otherwise, Chen et al. (1986) incorporated macro-economic variables as output or interest rates in the Market-Line equation.

Commodity prices also become an important source of risk. Since the oil shocks of the 1980s, several studies specifically highlight the links between financial markets and oil prices. Huang et al. (1996), Jones et al. (2004), Basher and Sadorsky (2006), and Boyer and Filion (2007) show the effects oil price variations have on stock returns. According to these authors, this variable positively affects oil and energy companies within oil producing countries. Lee and Zeng (2011), using Quantile Regressions, yielded similar results for G7 countries. These works establish that there is a relationship between oil and stock markets but there are few analyses about investing results for individual equities. Oil prices provide information about energy demand and it is therefore used as a macroeconomic indicator of economic health. Consequently, oil price risk can be measured by the Multi-Betas Model that provides information about stock’s sensitivity/exposure to this type of risk.

Gold is also an interesting factor because it is generally considered a “safe haven” from counter-cyclical variations to the market as indicated by Baur and Lucey (2010) and Baur and McDermott (2010, 2016). There are many studies about gold-market relationships confirming this fact. Sumner et al. (2010) showed that the market affects gold prices during periods of crisis but the links are weaker in times of expansion. Miyazaki et al. (2012) confirm gold's interest in portfolio management as a counter-cyclical asset with a low correlation in short-term markets. Mirsha et al. (2010) highlighted a bi-causal relationship between gold prices and the Indian stocks market. More recently, Arfaoui and Ben Rejeb (2017) using U.S data and Hussain Shahzad et al. (2017) referring to a panel of European countries (Greece, Ireland, Portugal, Spain, and Italy) confirm that gold prices influence the financial market overall.

These works mainly focus on the analyses of gold-markets relationships, but few studies directly introduce gold into the CAPM to measure equity exposure. Chua et al. (1990) include gold in the CAPM as a dependant variable. They then consider gold as an asset similar to equity and show that it has a weak Beta. These authors don’t address the reverse relation and stock sensitivities to gold price fluctuations. However, Tufano (1998) analyzes the CAPM with gold as an explanatory variable in north-American mining stocks. He concludes that these stocks have greater sensitivity to gold prices compared to market variations because the Beta results related to gold are higher. This author also highlights the effect data frequencies have on the Beta value. Johnson and Lamdin (2015) and He et al. (2018) find similar results with more recent UK-US data (2005–2015). The combination of these different works leads us to introduce oil and gold prices as additional factors to the Market and the Fama–French Factors.

The hypothesis of Asset Pricing Models is discussed with a focus on the behavioral hypothesis of agents engaging in the same horizon investments and making homogenous decisions towards portfolio allocation. Agents are supposed to have homogeneous investing behaviors. A large part of the literature tries to overcome this issue by considering a Behavioral Asset Pricing Model considering cognitive bias, heterogeneity of beliefs, investor attention, among others., as a factor in the model (Tuyon and Ahmad 2018; Wen et al. 2019 ; Gaffeo 2019; Heyman et al. 2019; Nanayakkara et al. 2019; Nasiri et al. 2019).

The objective of this paper is to analyze and compare the effects of short- and long-run exposures to these different sources of risk and discuss the behavioral hypothesis of agents’ homogeneity underlining the CAPM and its extensions using a wavelet approach.

The main interest of this method is that wavelets methodology conserves both time and frequency information of financial time series. This approach is particularly useful for distinguishing short- and long-run co-movements and links between financial variables (Aguiar-Conraria and Soares 2014; Kahraman and Unal 2019).

In practice, investors have heterogeneous behaviors that result in different investing frequencies. In our case, frequency information is translated as the investment horizons of investors. We then use wavelet decompositions that allow for the distinction between short- and long-run sensibilities. We can compare the positions of High-Frequency Traders (HFT) having a short-run vision with those of mutual funds invested in the long-run. These two agents don’t valorize the same market information but they still use the same models and methods for adapting their appetence for creating their own time series. The wavelets related to the time–frequency analysis represent a response to this type of problem. The discreet decompositions or Maximal Overlap Discrete Wavelets Transform MODWT (see Mallat and Meyer works) appear like the easiest and most suitable tool in this case. Gençay et al. (2005) reveal this with U.S data and then Mestre and Terraza (2018) with French data, which show that wavelets can indicate the heterogeneous behavioral hypothesis leading to a Beta differentiation according to various investment horizons in the CAPM framework.

In this paper, we extend the Mestre and Terraza (2018) approach in a multivariate case to appreciate the stock sensitivities to various risks according to investment horizons. We estimate a Time–Frequency Multi-Betas Model considering oil and gold prices and the Fama–French factor with AR-EGARCH errors (AutoRegressive Exponential AutoRegressive Conditional Heteroskedasticity) in order to overcome the previous CAPM’s limits. We use 30 French CAC40 Cotations Assistées Continues 40) equities for which quotations are perennial over a daily period from 2005 to 2015.

In the first part of this study, we estimated parameters of standard Multi-Betas-EGARCH (without wavelets). In the second part, we decompose the wavelets by the variables and we build time–frequency models considering heterogeneous investing behaviors. In the third part, we realize a portfolio application to highlight the usefulness of our model. We discuss the results and the financial perspectives for portfolio managers in the conclusion.

Standard estimation of multi-betas model

Theoretically, the Multi-Beta Model of Merton (1973) or the APT (Arbitrage Pricing Theory) of Ross (1976) are extensions of the CAPM in a multivariate regression framework where more risk factors are considered. The Fama–French Model is considered to be a reference as it includes two additional factors: the difference between the return of portfolios composed by big and small capitalizations called SMB (for Small-Minus-Big) and the difference between the return of portfolios composed by high book-to-market (B/M) Ratios and low B/M ratios called HML (for High-Minus-Low). We also add the oil and gold prices in the CAPM equation because of their specific characteristics: Gold as a “safe haven” asset and oil as a particular variable of global economic vitality.

However, the presence of autocorrelational and heteroscedasticity effects in the CAPM has been observed by many authors such as Diebold et al. (1988) and Giaccoto and Ali (1982). One of the consequences of this observation is that the Beta parameters are inefficient. The (G)ARCH family processes of Engle (1982) and Bollerslev (1986) are currently used to estimate a Beta parameter more efficient than the OLS estimator (Bera et al. 1988; Schwert and Seguin 1990; Corhay and Rad 1996). The consideration of heteroscedasticity seems, however, to affect only periods of the high volatility of the model, as shown by Morelli (2003) by comparing two versions of CAPM (with and without GARCH) for U.K stocks. More recently, Bendod et al. (2017) compared the CAPM and the GARCH-CAPM for the oil sector stocks of Arab and Gulf countries and concluded that the EGARCH model is better adapted to estimate the Beta. Mestre and Terraza (2020), for French stocks, come to similar conclusions. They also specify that the beta differences (between CAPM and EGARCH-CAPM) are not very significant when Betas are less than one whereas a correction is necessary for larger betas (greater than one). For all of these studies, there is a clear improvement of the market line residuals characteristics.

We take into account the statistical limits observed in the model residuals and also agents' heterogeneity by estimating a Multi-Betas Model with AR-EGARCH errors. We estimate 30 French stocks listed on the CAC40 (used as the Market reference) within a daily period between 2005 and 2015. We used the same database as Mestre and Terraza’s (2018) study of time–frequency CAPM and also the WTI oil price/barrel listed on the New York Mercantile Exchange, the gold price per ounce listed on the London Bullion Market, and the SMB and HML of the Fama–French Model to build a Time–Frequency based Multi-Betas Model.

The characteristics of the series in log-form and the results of the Unit-Root Tests (see Table 7a, b) reject the stationary hypothesis. As indicated in the CAPM, the Risk Premium is computed by subtracting the risk-free rate (OAT 10 years rate) from returns (the stationary variables by the first difference filter). Table 7c summarizes the characteristics of the risk premia series. These variables are stationary and zero-mean (see Table 7b).

The Multi-Betas Model, in its standard version (without wavelets), is written as follows:

$$r_{i,t} = { }\beta_{m,i} { }r_{m,t} + { }\beta_{o,i} { }r_{o,t} + { }\beta_{g,i} { }r_{g,t} + { }\beta_{SMB,i} { }SMB_{i;t} + { }\beta_{HML,i} { }HML_{i,t} { } + \varepsilon_{i,t}$$

(1)

where ${ }r_{i,t}$ is the risk premium of asset i, ${ }r_{m,t}$ the Market Premium, ${ }r_{o,t} and { }r_{g,t} { }$ are Oil and Gold Premia, ${ }SMB_{i;t}$ and ${ }HML_{i,t}$ are the Fama–French factors.

Under the OLS Hypothesis, $\varepsilon_{i,t}$ is an i.i.d (0,${{ \sigma }}_{\varepsilon }$) process so in this case beta parameters are consistent estimators. The above studies reject this hypothesis concerning $\varepsilon_{i,t}$. As a substitute, we use the AR(1)-EGARCH(1,1) from Nelson’s (1991) study to characterize it (see Mestre and Terraza , 2020). The parameters of Eq. (1) and those of this process are simultaneously estimated by the Maximum Likelihood methods associated with a non-linear optimization algorithm (see Ye 1997; Ghalanos and Theussl 2011).

The Table 1 summarizes the Model estimations for the 30 equities ranked according to the decreasing value of the $\beta_{m}$.

Table 1 Multi-betas-AR-EGARCH model estimates

All of the Beta parameters are significant and thanks to the determination coefficients (R²), we note that the three variables explain 30–70% of assets total risks. This ranking reveals a relatively significant relationship (R² = 0.33) between the 30 Betas $\beta_{m}$ and their corresponding determination coefficients. Equities with strong (high) $\beta_{m}$ have a globally high R² but this relationship is disrupted by the presence of outliers linked to a few stocks, as exemplified by Alcatel and PSA.

Residuals of the Multi-Betas Model are non-autocorrelated and homoscedastic, but the normality hypothesis is not respected. However, we consider this model as statistically acceptable.

Significance tests of Betas are used to appreciate if the Multi-Betas Model is selected for all stocks. In this context, if $\beta_{o} = { }\beta_{g} = \beta_{SMB} = \beta_{HML} = 0$ the CAPM is selected for the stock. The overall results lead to the following comments.

We observe that the CAPM is retained for only three equities (Vivendi, Carrefour, and Schneider). Thus, the addition of other variables is not relevant because we accept $\beta_{o} = { }\beta_{g} = \beta_{SMB} = \beta_{HML} = 0$. In this case, the CAPM results remain valid for these stocks. For the other 27 equities, there is at least one significant additional variable. For 43.33% of the stocks (13 equities) we notice a significant $\beta_{o}$ while $\beta_{g}$ is significant for 53.33% of stocks (16 equities). Finally, these two additional variables are significant for 30% of the sample (Vivendi, Total, Technip, GDF, LVMH, Alcatel, AXA, BNP, and Crédit Agricole). Concerning the Fama–French factors, $\beta_{SMB}$ is significant for 16 stocks (53.33% of the sample) and $\beta_{HML }$ for 15 (50%). They are both significant for five equities (Essilor, Ricard, L’Oréal, Air Liquide, and Total, Bouygues).

We built an adjustment of OLS Betas in order to quickly appreciate a more consistent $\beta_{m}$ without reestimating the model with AR(1)-EGARCH¹ processes. This adjustment remains valid in the case of the Multi-Betas Model, because we observe no significant differences between the $\beta_{m}$ (and residuals) of Multi-Betas Model in Table 2 and the $\beta_{m}$ of a CAPM with AR(1)-EGARCH(1,1) errors (see Tables 9, 10). The addition of these additional variables has a limited impact on the vast majority of equities because the $\beta_{m}$ are not affected. However, the Multi-Betas Model results have an interest to portfolio managers for analyzing and interpreting the $\beta_{o} {\text{ and }}\beta_{g}$ (their sign and their value) illustrating the sensitivities to oil and gold fluctuations.

For a large part of equities having a significant $\beta_{o}$, we notice that estimators are almost negative, however, sensitivities to oil movements are relatively low varying between − 0.001 and − 0.045%. Considering the classification of stocks by $\beta_{m}$, as in Table 1, we observe that oil affects the different equities profiles in the same way. Technip and Total are a notable exception to this case because their $\beta_{o}$ have higher positive values compared to the other stocks. Thereby, an oil price rise by 1% entails a stock price rise by 0.08% for Total and by 0.14% for Technip. We conclude that stocks of the oil and gas sectors are the most sensitive to oil price fluctuations, which is relevant to their activities.

We note significant and negative $\beta_{g}$ for seven stocks (Alcatel, SG, BNP, CA, AXA, Orange, Vivendi, and GDF) and positive for seven others equities (Publicis, Essilor, Ricard, Air Liquide, Bouygues, Total, and Technip). The Financial sector equities, classified as “risky” due to their $\beta_{m}$ greater than one, are negatively affected by the gold, justifying in part its “safe haven” characteristic. We can extend this result to stocks with strong $\beta_{m}$ (Financial Sector + Alcatel) having a negative and relatively higher sensitivity to gold fluctuations. On the contrary, stocks with $\beta_{m} < 1$ have positive and lower $\beta_{g}$. Once again, Total and Technip are exceptions because they are more sensitive to gold than other stocks.

By comparing these three sensitivity estimators, the market represents a major source of risk because the $\beta_{m}$ are greater than $\beta_{g}$ and $\beta_{o}$ in absolute values. We also note that Gold sensitivity is higher (in absolute values) than oil sensitivity, particularly in financial sectors and for stocks with $\beta_{m} > 1$.

Time–frequency multi-betas model estimates

In the financial markets, the assumption of homogeneity of agents' behavior is difficult to maintain. The investment frequency of a global and a mutual funds portfolio, for example, depends on their buying or selling intentions based on various calculation/financial models. These models don’t differentiate the agents and considers only an aggregation of behaviors (i.e. an “average behavior”) from the financial time series used. The use of wavelet time–frequency decompositions of these time series is justified in the Multi-Betas Model framework because the high-frequencies are related to HFT and the low-frequencies to fundamentalist investors. Wavelets represent a relevant solution to analyzing the behavior of agents that use this kind of financial model. In the rest of the paper, we name Standard Multi-Beta Models, the results of which are summarized in Table 1, to distinguish from its time–frequency versions estimated in this section.

The first paragraph is a brief reminder of wavelets methodology applied to our model before comparing results obtained in the previous part for the Standard Multi-Betas Model with its time–frequency version in the second paragraph. In a third paragraph, we analyse the frequency sensitivities of stock prices to exogenous additional variables.

Wavelets methodology reminder: the maximal overlap discrete wavelets transform (or MODWT):

A Wavelets-mother ${\Psi }\left( {\text{t}} \right)$ with zero-mean and normalised is written as follows²:

$$\mathop \smallint \limits_{ - \infty }^{ + \infty } \psi \left( t \right)dt = 0 \quad {\text{and}}\quad \mathop \smallint \limits_{ - \infty }^{ + \infty } \left| {\psi \left( t \right)} \right|^{2} dt = 1$$

(2)

These properties ensure the Variance/Energy preservation during the decomposition of a series and also guarantee the respect of admissibility condition (Grossman and Morlet 1984).

This wavelet-mother is shifted by the τ parameter and dilated by scale parameters to create “wavelets-daughters’’ regrouping in the wavelets family used as filtering basis:

$$\Psi _{{\uptau ,{\text{s}}}} \left( {\text{t}} \right) = \frac{1}{{\sqrt {\text{s}} }}\Psi \left( {\frac{{{\text{t}} -\uptau }}{{\text{s}}}} \right)$$

(3)

The decomposition of time function x(t) creates/lead to the wavelets coefficients $W\left( {s,\tau } \right)$ as follows:

$$\begin{aligned} & W\left( {s,\tau } \right) = \mathop \smallint \limits_{ - \infty }^{ + \infty } x\left( t \right) \frac{1}{\sqrt s }\psi^{*} \left( {\frac{t - \tau }{s}} \right)dt = <x\left( t \right),\psi_{\tau ,s} \left( t \right) > \\ & \psi^{*} \; is\; the\; complex\;conjuguate\;of\; \psi \\ \end{aligned}$$

(4)

τ and s parameters indicate the time and frequency localization of the coefficient. Thanks to the wavelets, we can represent the temporal localization of the frequency components, hence the name of the time–frequency analysis. These previous equations are a theoretical presentation of wavelet decompositions based on continuous wavelets. A time discreet version is used to decompose time series $x_{t}$ but the principle remains similar because frequencies are still continuous. The practical use of this kind of decomposition implies important computational time and effort, consequently, a frequency discretization is realized for a fast-decomposing time series, as is the MODWT. In this framework, wavelets are defined by a succession repeated J times of high-pass and low-pass filter combinations (Mallat Algorithm 1989, 2009). J is the decomposition order representing the optimal number of repetitions necessary to reconstruct a time series ${\text{ x}}_{{\text{t}}}$ of length N such as ${\text{J}} = \frac{{{\text{Ln}}\left( {\text{N}} \right)}}{{{\text{Ln}}\left( 2 \right)}}$.

Despite this simplified process, the MODWT is still variance/energy preserving. It ensures the perfect reconstruction of the decomposed series, without losses, by adding the high and low-frequencies components:

$${\text{x}}_{{\text{t}}} = {\text{S}}_{{{\text{J}},{\text{t}}}} + { }\mathop \sum \limits_{{{\text{j}} = 1}}^{{{\text{j}} = {\text{J}}}} {\text{D}}_{{{\text{j}},{\text{t}}}}$$

(5)

${\text{S}}_{{{\text{J}},{\text{t}}}}$ is a basic approximation of the series and ${\text{D}}_{{{\text{j}},{\text{t}}}}$ are the details, called also frequency bands, of scale j regrouping the frequencies in the interval $\left[ {\frac{1}{2}^{j + 1} ; \frac{1}{2}^{j} } \right]$.

In Finance, the frequencies interpretation is simplified by translating them in periods that have the same time unit as the original data (for example, days, weeks, etc.). In this case, frequencies represent the different time investment horizons (short-medium-long run). The Table 11 records the frequency bands and the corresponding time horizons in days.

Considering the series length, through wavelets decomposition, we have 11 frequency bands and one approximation. The high-frequencies bands (D1–D2) are related to short-run investments whereas the low-frequencies illustrate long-run horizons. In order to simplify the analysis, we focus on the first six frequency bands: D1 bands are related to 2–4 days investment horizon (high-frequencies) whereas D6 band represents a 3–6 month investment (low-frequencies).

In the Multi-Betas Model framework, we decompose the dependent and the three independent variables by the MODWT. In the Multi-Betas Model, each frequency band of the stock are associated with the corresponding bands of the market, oil, gold, and Fama–French factors. By construction, the frequency bands means are equal to zero.

For an asset i, the Time–Frequency Multi-Betas Model is written as follows:

$$\begin{aligned} D_{j,t}^{asset} & = \beta_{j}^{m} D_{j,t}^{Market} + \beta_{j}^{o} D_{j,t}^{Oil} + \beta_{j}^{g} D_{j,t}^{Gold} + \beta_{j}^{SMB} D_{j,t}^{SMB} + \beta_{j}^{HML} D_{j,t}^{HML} + \varepsilon_{j,t} \\ & \quad \forall j = 1, \ldots , 6 the\; frequency \;band \\ & \quad \varepsilon_{j,t} \sim AR\left( 1 \right) - EGARCH\left( {1,1} \right) \\ \end{aligned}$$

(6)

Betas parameters are estimated in the time–frequency space and represent the asset sensitivities to the five factors considering agents’ investment frequencies and considering stock risk profiling. The different time–frequency regression models are estimated previously by conserving the hypothesis of AR(1)-EGARCH(1,1) errors.

Table 12 summarizes all results³ of Time–Frequency Multi-Betas AR-EGARCH Model estimates.

Time–frequency multi-betas model estimates

$\beta_{m}$ coefficients are highly significant for all equities and frequencies. The coefficients of determination computed on the high-frequencies (D1–D2) are relatively closed to the overall Model of Table 1. But they become more important for medium–low-frequencies (D4–D5) and they are almost equal to 100% on the D6 band. The order of magnitude of the D5–D6 wavelets coefficients are small, therefore, a range of residuals estimates are also small, and then values of R² are high on low-frequencies. However, for all frequency band regressions, we notice a deterioration of residual characteristics. Particularly, the AR-EGARCH process no longer properly captures the heteroscedasticity. By increasing the order of the process, we reduce the autocorrelation and heteroscedasticity without significantly modifying the values of the Betas parameters. Despite these reservations, the Time–Frequency Multi-Betas Model has sufficient statistical properties to analyze its economic results.

Estimates of these parameters play an important role in investor strategies who question the choice of the model according to the significant parameters. Globally, we remark that stocks with a strong $\beta_{m}$ (> 1) have negative $\beta_{o} \;and \;\beta_{g}$ for all frequency bands more particularly in the long-run. The stocks with low $\beta_{m}$ (< 1) have positive and relatively high $\beta_{g}$ whereas $\beta_{o}$ still mainly remain negative. Portfolio managers can thus appreciate the different sources of risk affecting their portfolios when making choices.

Table 2 summarizes the differences between parameters of standard and time–frequency models and represents an additional help to interpret results.

Table 2 Percentages of significantly different Betas between Standard Multi-Betas Model and Time–Frequency Multi-Betas Model (TFMB)

The Betas estimated without wavelets (Standard Multi-Betas) are globally similar to short-run Betas (D1–D2) for the majority of equities whereas the differences are more significant in the long-run. For example, in the long-run (D6), we note that differences between $\beta_{m}$ of the Standard Model and TFMB are significant for 76.67% of equities.

Wavelets provide a differentiated beta estimate according to the investment frequencies, which are useful for identifying and analyzing the effects of investment horizons on systematic risk measures/indicators and on sensitivities to different factors. The intensity of $\beta_{o} { }\;{\text{and }}\;\beta_{g}$ is greater in the long-run than in the short-run. For all equities, the selected variables more strongly affect assets for long-run investments. Therefore, we confirm the results of Gençay et al. (2005) as well as Mestre and Terraza (2018) concerning the interest of wavelets in market models for long-run investments. Both studies indicated that CAPM’s Beta is frequency-varying for long-run investment horizons and the standard estimation of Beta does not hold. Therefore, equities risk profile changes. The Time–Frequency Multi-Betas Model is therefore of strategic interest for long-term investments by estimating its low-frequency exposures to risks in order to adjust their allocation if the initial characteristic is lost (see Part III).

Stocks sensitivities to oil and gold movements

By testing the significance of frequency parameters $\beta_{o} { },{ }\beta_{g} , \beta_{SMB} { }\;{\text{and}}\;{ }\beta_{HML}$ for all stocks, we establish the following statements:

• If $\beta_{o} = { }\beta_{g} = \beta_{SMB} = \beta_{HML} = 0$, the addition of the four variables is not appropriate for this asset. In this case, the time–frequency CAPM⁴ is selected.

• If $\beta_{o} \ne 0{ },{ }\beta_{g} \ne 0,{ }\beta_{SMB} \ne 0{ }\;{\text{and }}\;\beta_{HML} \ne 0{ }$ the five additional variables are relevant and the Full Multi-Betas Model (FULL MB) is retained.

• If $\beta_{o} \ne 0{ },{ }\beta_{g} \ne 0,{ }\beta_{SMB} = 0{ }\;{\text{and }}\;{ }\beta_{HML} = 0,{ }\;{\text{only }}$ two additional variables are relevant and the Multi-Betas Model with Oil and Gold (MB OIL GOLD) is retained.

• If $\beta_{o} = 0{ },\,{ }\beta_{g} = 0,\,{ }\beta_{SMB} \ne 0{ }\,{\text{and }}\,{ }\beta_{HML} \ne 0,{ }$ only Fama–French factors are relevant and the Fama–French Model (FF) is retained.

• If at least one of the five Beta is significant, we always retain the Multi-Betas Model under a mixed version.

  We use this framework to summarize results in the Tables 3 and 13. We count for each frequency band the number of shares for which the CAPM or Multi-Betas is retained.

  By reading Tables 3 and 12 in Appendix, it is possible to establish the following comments:

• There is no stocks having the CAPM on all frequency bands, while lot of equities retained a version of Multi-Betas Model whatever the investment horizon.

• In the short-run (D1), the CAPM is valid for only five stocks (16.66% of the sample) and results are similar to the previous estimate of the Standard Model. This percentage decreases as the time horizon increase, so more stocks retain the Full Multi-Betas Model in the long-run. After D3 bands, there are no equities having non-significant additional variables. Stocks are therefore more impacted/affected by oil and/or gold in the long-term than in the short-term as the Fama–French Factors affected a large part of stocks.

Table 3 Analysis of Time–Frequency Multi-Betas Model results. Number of stocks for which the CAPM or the Multi-Betas is valid (in values and percentage)

The Time Frequency Multi-Betas Model is therefore of statistical interest for a majority of equities whatever the investment horizons. This model is complementary to the CAPM results as it introduces the decompositions of risk sources. It can be decision-making support for investors by synthesizing the results of the stock sensitivities.

Table 4 synthesizes, for each frequency band, the percentage of $\beta_{o} { },{ }\beta_{g} , \beta_{SMB} { },{ }\beta_{HML}$ significantly greater, lower, or equal to zero and also their means. To improve the analysis, we also indicate the mean of the corresponding $\beta_{m}$.

Table 4 Synthesis of $Betas$ signs and means

The number of Multi-Betas Model having a positive $\beta_{g}$ is relatively stable on all frequency bands (around 50%) whereas it increases for the model with positive $\beta_{o} .$ Furthermore, we note that their means increase from D1 (short-run) to D6 (long-run). Stocks with positive $\beta_{g}$ and/or positive $\beta_{0}$ have an average $\beta_{m}$ less or equal to one. The oil-sector stocks (Total, Technip) are strongly and positively sensitive to oil and gold for all frequency bands.

The number of stocks negatively sensitive to oil is higher in the short-term (D1–D2) than the long-term (D6), whereas it is stable for gold. The value of $\beta_{g}$ and $\beta_{0}$ increases as the investment horizon increases (in average). We note similar results for stocks with $\beta_{o } < 0$ as the mean of $\beta_{m}$ increases accross frequencies, it is less than one until the D3-Bands but is greater beyond. Financial Equities (SG, BNP, CA, and AXA) are not sensitive to oil short-run variations. However, these stocks have strongly negative $\beta_{o}$ when investment horizon increases (D5-D6). Financial stocks are also strongly negatively affected by gold prices both in the short- and long-run, however, the effect is greater. The equities are negatively sensitive to gold, which has an average $\beta_{m}$ greater than one except on D1. However, we notice that the intensity of negative $\beta_{g}$ is greater (on average) than the positive $\beta_{g}$ mean. Gold negatively affects the stocks with an important systematic risk in the financial sector and Alcatel. During an expansion period, a rise in the market leads to a stronger increase in the stock’s price and a decrease in the gold prices confirms the upward dynamic on stocks. Oppositely, during periods of crisis, the decreasing trend of the market pushes down the stock prices. In this context, investors close their positions and buy gold. Therefore, the demand for gold becomes increasingly important as its price naturally increases, confirming the investors’ choices, while consequently, stock prices decrease. The “safe haven” characteristic of gold is partially justified even if half the stocks have positive $\beta_{g} .$

We note similar conclusions for $\beta_{SMB} , \beta_{HML}$. The percentage of stocks with positive beta is quietly stable across frequencies (between 55–65%) even if a break appears for $\beta_{HML}$ while the value of the parameters increases as investment horizons increase with the average $\beta_{m}$ value. Same observations are made for negative $\beta_{SMB}$. Stocks with high $\beta_{m}$ tend to have a positive value of $\beta_{SMB} , \beta_{HML}$. The performances of small companies and high B/M ratio firms then have a stronger impact on equities, especially in the long-run. This result is coherent with the Fama–French Model.

To highlight the usefulness of time–frequency parameters for portfolios managers, we present in the third part, a portfolios approach.

Portfolios application

As the betas are differentiated across frequency bands and investment horizons, a portfolio, initially constructed with proper specificity, could get lost in the long-run as exposure to risk factors are different. Consequently, we build three portfolios respectively that are negatively sensitive to oil price variations (Pf1), insensitive to oil fluctuation (Pf2) and positively sensitive to oil (Pf3). In a similar way, we also create three portfolios that indicate negatively sensitive (Pf4), insensitive (Pf5), and positively sensitive (Pf6) to gold price variations. An insensitive to oil and gold variation portfolios (Pf7) is also created. To finish, we elaborate the last portfolio (Pf8) as a tracker of the Market ($\beta_{m}$ = 1). All stocks included in a portfolio are equally weighted.

Table 5 presents the results of the Standard Multi-Beta estimation for the eight portfolios.

We note that all portfolios respect their specificity as Pf1 has a negative significant $\beta_{o}$ while it is non-significant for Pf2, and significantly positive for Pf3. Note also that the gold sensitivities have some similar signs to oil sensitivities for these three portfolios. Concerning the Portfolio 4, 5, and 6, all gold sensitivities respect the expected signs. For the last portfolio, Pf8, it $\beta_{m}$ is equal to 1 and it is insensitive to oil and gold prices fluctuations.

Then, we re-estimate the Model in the Time–Frequency space to check if these characteristics are still valid according to investment horizons (see Table 14). Table 6 provides the parameters estimated on each band and indicates if they are significantly different from the parameters of the Standard Model estimations in Table 5. One star indicates a significant difference at a 10% risk level while two stars is for a 5% risk level.

Table 5 Standard Multi Beta Model for Portfolios

Table 6 Time–Frequency sensitivities of Portfolios compared to Standard Multi Beta Model

For the Pf8, we note that the $\beta_{m}$ keeps its property in the short-run (D1-D2) while it is greater than one for D3 to D5 frequency bands and lesser than one on D6. These results indicate that the initial tracker profile of Pf8 is only valid for short-run investments and so an adjustment of portfolio allocation is required if a long-run tracker profile is expected. In addition, this portfolio has sensitives to oil and gold price variations in the long-run as this exposure is non-observed in the Standard Model.

For Pf1 (initially negatively sensitive to oil), it conserves this characteristic for all frequencies, but this initial exposure is significantly greater in the long-run. A similar result is noted for Pf3 (initially positively sensitive to gold) as the sensitivities are significantly greater in the long-run than in the short-run. The Pf2 keeps its insensitive property until D4 bands but not in the long-run as it becomes negatively sensitive to oil (and positively to gold). We also observe similar conclusions concerning the sensitivities to Gold for these three portfolios.

For Pf4, the initial profile with negative $\beta_{g}$ is conserved no matter the investment horizons, but in the long-run, its exposure to gold is significantly greater than estimated by the Standard Model. Similar observations are noted for the Pf6, however for some horizons, the sensitivity is greater than estimated in Table 5. Results for a gold insensitive portfolio (Pf5) highlight significant difference between parameters on D2, D4 and D6 frequency bands as $\beta_{g}$ is positive.

Concerning the Pf7, initially non sensitive to oil and gold, the insensitive characteristic is lost as the $\beta_{o}$ is significantly negative starting D3 bands even if $\beta_{g}$ is still equal to 0. To conserve this property for long-run investments, a new allocation is required.

For all portfolios, we note a differentiation of parameters across frequency bands confirming previous observations.

The results obtained by wavelet estimators of risks sensibilities are useful for investors to support their decision-making on portfolio allocation and could be completed by multiple criteria decisions making (MCDM) methods and clusters algorithms based on different risks measured in the assessment of financial risks or in predictions of variables (Kou et al. 2014, 2021).

Conclusion

The Time–Frequency Multi-Betas Model effectively complements the different instruments used by stock investors to build their portfolios. In the first hand, it can substitute the CAPM by considering the residual anomalies by using ARMA-EGARCH processes to model the errors of the regression. On the other hand, it improves the CAPM by adding exogeneous variables and it considers the heterogeneity of agents’ behaviors by the wavelet decompositions. Despite some statistical shortcomings, particularly those concerning the characteristics of its frequency residuals, this model brings a significant gain of information to model the risk premiums.

In the short-run, the $\beta_{m}$ parameter of the Time–Frequency Multi-Betas Model, measuring the sensitivity to market fluctuation, is not significantly different to the Standard Multi-Betas Model and the CAPM. For a short-run investor, the use of the CAPM can be sufficient to make investment choices based on the $\beta_{m}$. However, he can consider the Multi-Betas Model and the sensitivities to gold and oil in order to modulate its choices.

The Standard Multi-Betas Model (without wavelets) is retained for a majority of the stocks in its full or one mixed version. The stock sensitivities to oil and gold are lower than the sensitivity to the market, but we can appreciate potential positive effects on some sectors such as the Petroleum/Gaz-stocks, and Financial sectors. For example, oil negatively affects the majority of stocks, however, its impact is stronger for high $\beta_{m}$ equities than low $\beta_{m}$ equities. Gold negatively and more strongly affects equities with a high systematic risk (such as the Financial sector) but the effect is reversed for equities with a $\beta_{m}$ lower than one. The Time–frequency Multi-Betas Model multiplies the possibilities of analysis by crossing the betas and the sectors with the investment horizons. We confirm the differentiations of risk according to investment horizons observed by Gençay et al. (2005) and Mestre and Terraza (2018). We also find similar results with the Mestre and Terraza analysis, as the Standard Multi-Beta and Time–Frequency Model provide slightly similar Beta coefficients in the short-run with CAPM estimates, but the more the investment horizon increases the more the differences between Models coefficients are significant.

The Time–Frequency Multi-Betas Model is more useful for fundamentalist investors (in the long-run) as there are significant differences with Standard Model estimations. At low-frequencies (D6), the CAPM is not retained whereas for the others, oil and gold variables and Fama–French factors have significant effects on equities. Their effects increase as the time horizons increase. The application to portfolios highlights the potential effect of variation of risk exposure across frequencies on the property and characteristic of portfolios in the long-run, and some initial features do not hold.

Wavelets represent a powerful tool to differentiate the stock sensitivities to various factors according to the agents investment horizons. The combination of the time–frequency estimates of the Multi-Betas Model improves the investment choice possibilities and risk analysis.

Appendix

See Tables 7, 8, 9, 10, 11, 12, 13 and 14.

Table 7 Equities characteristics

Table 8 Multicolineartity analysis

Table 9 CAPM-EGARCH Estimates

Table 10 Comparison of $\beta_{m}$ between CAPM-EGARCH and Multi-Betas-EGARCH

Table 11 Frequency Bands corresponding days

Table 12 Time–frequency multi-beta-AR-EGARCH estimates

Table 13 Synthesis of time–frequency model for each stocks

Table 14 Time–frequency multi-beta-AR-EGARCH estimates