ارزیابی مبتنی بر موجک ریسک بازار: در مورد بازارهای نوظهور

The measurement of market risk poses major challenges to researchers and different economic agents. On one hand, it is by now widely recognized that risk varies over time. On the other hand, the risk profile of an investor, in terms of investment horizon, makes it crucial to also assess risk at the frequency level. We propose a novel approach to measuring market risk based on the continuous wavelet transform. Risk is allowed to vary both through time and at the frequency level within a unified framework. In particular, we derive the wavelet counterparts of well-known measures of risk. One is thereby able to assess total risk, systematic risk and the importance of systematic risk to total risk in the time-frequency space. To illustrate the method we consider the emerging markets case over the last twenty years, finding noteworthy heterogeneity across frequencies and over time, which highlights the usefulness of the wavelet approach.

The assessment of market risk has long posed a challenge to many types of economic agents and researchers (see, for instance, Granger, 2002 for an overview). Market risk arises from the random unanticipated changes in the prices of financial assets and measuring it is crucial for investors. Besides its interest to portfolio managers, the assessment of market risk is relevant for the overall risk management in banks and bank supervisors. Although bank failures are traditionally related with an excess of non-performing loans (the so-called credit risk), the failure of the Barings Bank in 1995 showed how market risk can lead to bankruptcy. Furthermore, market risk has received increasing attention in recent years as banks’ financial trading activities have grown. Although the measurement of market risk has a long tradition in finance, there is still no universally agreed upon definition of risk. The modern theory of portfolio analysis dates back to the pioneering work of Harry Markowitz in the 1950s. The starting point of portfolio theory rests on the assumption that investors choose between portfolios on the basis of their expected return, on the one hand, and the variance of their return, on the other. The investor should choose a portfolio that maximizes expected return for any given variance, or alternatively, minimizes variance for any given expected return. The portfolio choice is determined by the investor's preferred trade-off between expected return and risk. Hence, in his seminal paper, Markowitz (1952) implicitly provided a mathematical definition of risk, that is, the variance of returns. In this way, risk is thought in terms of how spread-out the distribution of returns is. Later on, the Capital Asset Pricing Model (CAPM) emerged through the contributions of Sharpe (1964) and Lintner, 1965a and Lintner, 1965b. According to the CAPM, the relevant risk measure in holding a given asset is the systematic risk, since all other risks can be diversified away through portfolio diversification. The systematic risk, measured by the beta coefficient, is a widely used measure of risk. In statistical terms, it is assumed that the variability in each stock's return is a linear function of the return on some larger market with the beta reflecting the responsiveness of an asset to movements in the market portfolio. For instance, in the context of international portfolio diversification, the country risk is defined as the sensitivity of the country return to a world stock return. Traditionally, it is assumed that beta is constant through time. However, empirical research has found evidence that betas are time varying (see, for example, the pioneer work of Blume, 1971 and Blume, 1975). Such a finding led to a surge in contributions to the literature (see, for example, Alexander and Benson, 1982, Collins et al., 1987, Fabozzi and Francis, 1977, Fabozzi and Francis, 1978, Ferson and Harvey, 1991, Ferson and Harvey, 1993, Ghysels, 1998, Harvey, 1989, Harvey, 1991 and Sunder, 1980, among others). One natural implication of such a result is that risk measurement must be able to account for this time-varying feature. Besides the time-variation, risk management should also take into account the distinction between the short and long-term investor (see, for example, Candelon, Piplack, & Straetmans, 2008). In fact, the first kind of investor is naturally more interested in risk assessment at higher frequencies, that is, short-term fluctuations, whereas the latter focuses on risk at lower frequencies, that is, long-term fluctuations. Analysis at the frequency level provides a valuable source of information, considering that different financial decisions occur at different frequencies. Hence, one has to resort to the frequency domain analysis to obtain insights into risk at the frequency level. In this paper, we re-examine risk measurement through a novel approach, wavelet analysis. Wavelet analysis constitutes a very promising tool as it represents a refinement in terms of analysis in the sense that both time and frequency domains are taken into account. In particular, one can resort to wavelet analysis to provide a unified framework to measure risk in the time-frequency space. As both time and frequency domains are encompassed, one is able to capture the time-varying feature of risk while disentangling its behavior at the frequency level. In this way, one can simultaneously measure the evolving risk exposure and distinguish the risk faced by short and long-term investors. Although wavelets have been more popular in fields such as signal and image processing, meteorology, and physics, among others, such analysis can also shed fruitful light on several economic phenomena (see, for example, the pioneering work of Ramsey and Lampart, 1998a, Ramsey and Lampart, 1998b, Ramsey and Zhang, 1996 and Ramsey and Zhang, 1997). Recent work using wavelets includes that of, for example, Kim and In, 2003 and Kim and In, 2005, who investigate the relationship between financial variables and industrial production and between stock returns and inflation, Gençay et al., 2005 and Gençay et al., 2003 and Fernandez, 2005 and Fernandez, 2006, who study the CAPM at different frequency scales, Connor and Rossiter (2005) focus on commodity prices, In and Kim (2006) examine the relationship between the stock and futures markets, Gallegati and Gallegati (2007) provide a wavelet variance analysis of output in G-7 countries, Gallegati, Palestrini, and Petrini (2008) and Yogo (2008) resort to wavelets for business cycle analysis, Rua (2011) focuses on forecasting GDP growth in the major euro area countries, and others (see Crowley, 2007, for a survey). However, up to now, most of the work drawing on wavelets has been based on the discrete wavelet transform. In this paper we focus on the continuous wavelet transform to assess market risk (see also, for example, Aguiar-Conraria and Soares, 2011a, Aguiar-Conraria and Soares, 2011b, Aguiar-Conraria and Soares, 2011c, Crowley and Mayes, 2008, Raihan et al., 2005, Rua, 2010, Rua, 2012, Rua and Nunes, 2009 and Tonn et al., 2010). We provide an illustration by considering the emerging markets case. The new equity markets that have emerged around the world have received considerable attention in the last two decades, leading to extensive recent literature on this topic (see, for example, Bekaert and Harvey, 1995, Bekaert and Harvey, 1997, Bekaert and Harvey, 2000, Bekaert and Harvey, 2002, Bekaert and Harvey, 2003, Chambet and Gibson, 2008, De Jong and De Roon, 2005, Dimitrakopoulos et al., 2010, Estrada, 2000, Garcia and Ghysels, 1998 and Harvey, 1995, among others). The fact that the volatility of stock prices changes over time has long been known (see, for example, Fama, 1965), and such features have also been documented for the emerging markets. The time variation of risk comes even more naturally in these countries due to the changing economic environment resulting from capital market liberalizations or the increasing integration with world markets and the evolution of political risks. In fact, several papers have acknowledged time varying volatility and betas for the emerging markets (see, for example, Bekaert and Harvey, 1997, Bekaert and Harvey, 2000, Bekaert and Harvey, 2002, Bekaert and Harvey, 2003, Estrada, 2000 and Santis and Imrohoroglu, 1997). Moreover, the process of market integration is a gradual one, as emphasized by Bekaert and Harvey (2002). Therefore, methods that allow for gradual transitions at changing speeds, such as wavelets, are preferable to segmenting the analysis into various subperiods. Hence, the emerging markets case makes an interesting example for measuring risk with the continuous wavelet transform. This paper is organized as follows. In Section 2, the main building blocks of wavelet analysis are presented. In Section 3, we provide the wavelet counterpart of well-known risk measures. In Section 4, an application to the emerging markets case is provided. Section 5 concludes.

Although most textbook models assume volatilities and covariances to be constant, it has long been acknowledged among both finance academics and practitioners that market risk varies over time. Besides taking into account such time-varying feature, the risk profile of an investor, in terms of investment horizon, makes it also crucial to assess risk at the frequency level. Naturally, a short-term investor is more interested in the risk associated with high frequencies whereas a long-term investor focuses on lower frequencies. This paper provides a new look into market risk measurement by resorting to wavelet analysis, as it allows one to evaluate the time and frequency-varying features within a unified framework. In particular, we derive the wavelet counterpart of well-known measures of market risk. We consider total risk, as measured by the variance of returns, the systematic risk, captured by the beta coefficient, and we provide the tools to assess the importance of systematic risk on total risk in the time-frequency space. To illustrate the method, we consider the emerging markets case, which has received a great deal of attention in the literature over the last twenty years. As those countries have experienced a changing economic environment, it is particularly interesting to see how market risk has changed across frequencies and over time. We find that the variance of monthly returns is determined essentially by short-run fluctuations and that the volatility has changed over time. In particular, the periods of higher volatility are associated with several economic crises that hit the emerging markets. Regarding the systematic risk, we find that the beta coefficient is relatively stable at low frequencies, presenting a value of around 1. In contrast, at higher frequencies, the beta coefficient varies considerably, attaining values as high as 3 in some economic episodes. Additionally, we assessed the importance of the systematic risk in explaining total risk in emerging markets. Again, we find noteworthy variation in the results across frequencies and over time. We conclude that the importance of systematic risk in emerging markets is relatively high and stable over time at low frequencies. At higher frequencies, the influence of the systematic part was relatively low before the mid-1990s, but increased gradually thereafter, attaining values also relatively high at the end of the sample. All of these results highlight the importance of considering time and frequency-varying features in risk assessment. Hence, wavelet analysis can be a valuable tool for obtaining additional insights that may influence risk-taking decisions.