Economic gains of realized volatility in the Brazilian stock market/Ganhos economicos da volatilidade realizada no mercado Brasileiro de acoes.
| Autor | Garcia, Marcio Gomes Pinto |
| Cargo | Texto en ingl |
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Introduction
Given the growth of financial markets and the increasing complexity of its securities, volatility models play an essential role to help the task of risk management and investment decisions. Since realization of volatility returns based on daily data is not observable, the traditional approach is to invoke parametric assumptions regarding the evolution of the first and second moments of the returns, which is the idea behind ARCH and stochastic volatility models. Nevertheless, these models fail to capture some stylized facts such as autocorrelation persistence and fat tail of returns. The availability of intraday data opens up the possibility of approximating volatility directly from asset returns. The use of an observable variable, in turn, facilitates the task of dealing with problems that involves a significant number of assets. Indeed, traditional methods suffer from the curse of dimensionality, which is to say, the difficulty of these methods to handle with a wider range of assets.
The advantages of realized measures have been extensively analyzed in the recent period, when technological barriers have been gradually surmounted so as to provide the kind of data necessary to its calculation. Andersen et al. (2003) compared it to traditional methods and confirmed its superiority in terms of forecasting performance. Identical conclusion has been reached by many concurrent studies, like Engle et al. (2008). Previous findings relating microstructure parameters and volatility were revisited considering realized measures. Chan & Fong (2006), for instance, found that trading volume is the main factor driving the relationship between volume and realized volatility, as opposed to studies that pointed out order imbalance as the most important one. In Brazil, Carvalho et al. (2006) found that returns displayed a normal distributional when standardized by realized measures, a useful property concerning risk management purposes, namely Value-at-Risk statistics. The authors based their conclusions on a sample of the five most liquid stocks traded at the domestic stock exchange, sampling at a 15-min frequency. In spite of the apparent consensus over the subject, there are many relevant issues that deserve attention, in particular the bias originated by microstructure noise and measurement errors. McAleer & Medeiros (2008) documented a review of the literature, stressing the future improvements that must be made in order to deal with such biases.
The objective of this paper is to evaluate the economic gains associated with the use of realized measures in the context of an investment decision where investors take conditional volatility forecasts as the main parameter in the portfolio optimization problem: the so-called volatility timing strategy proposed by Fleming et al. (2001). We examine this issue by comparing forecasts of the covariance matrix obtained by means of a multivariate version of Corsi's Heterogeneous Auto-Regressive (HAR) model with the ones provided by traditional volatility models. Our database consists of high frequency transactions prices from the twenty most liquid stocks from the Brazilian Stock Exchange, BMF&Bovespa, and covers the period from February 2006 to January 2011.
This paper contributes to the attempt of applying models of realized volatility to a multivariate framework. Alone, this is not an innovation to the literature. However, we employ a greater than usual number of assets and it is not straightforward to infer that the results will continue to hold. Moreover, the fact that Brazil is an emerging market raises the question whether adaptations to models originally designed to fit consolidated markets are required.
We find that economic gains associated with realized volatility increase proportionally to the target return. We also show that, when target returns are close to the risk-free, portfolios weights are heavily dependent on the risk-free asset. This finding allows us concluding that realized volatility performs better for increasing levels of risk. Using the unconditional mean as a reference for expected returns, an investor would be willing to pay substantial positive fees to switch from a portfolio based on forecasts taken from traditional volatility methods to one based on realized volatility forecasts, when target returns are superior to 15% per year. When expected returns are bootstrapped, although fees are still positive on average, high standard deviation values lead us to conclude that utility gains are equal at a statistical viewpoint. Actually, when estimation risk is significant, average portfolio returns are far apart from target returns irrespective of the volatility measure used, which is an indication that economic gains of realized volatility are offset by estimation risk. We also perform robustness checks that confirm that, when estimation risk is negligible, economic gains are robust to changes in the parameters of the economic utility and of the optimization problem. Finally, we conclude that the inclusion of an external risk factor, aimed at adapting the model to an emerging economy, do not add in terms of utility gain.
Our results represent an important input to the applicability of realized volatility as a reference for risk estimates for the stock market in Brazil. It provides additional evidence to the literature that links positive economic values associated to realized volatility, as Christoffersen et al. (2012), in their study of the benefits of realized volatility measures for option pricing, and Fleming et al. (2003), which used it as an input for a volatility timing strategy based on four assets traded at the US futures market.
The text is organized as follows. In Section 2, we provide key realized volatility concepts. Next, we document the database sources and how we compute multivariate volatility. In Section 4, we present our version of the HAR model and describe its application to a multivariate setting. Then, we present the methodology behind the evaluation of the economic gains and, in Section 6, discuss results and robustness checks. Finally, we offer our concluding remarks in Section 7.
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Theoretical background: Realized Volatility
We will provide a brief review of the theoretical framework underlying the realized volatility (RV) measure. The theory of Quadratic variation is the baseline to understand how we obtain this direct measure of volatility.
We begin by assuming that logarithm prices ([p.sub.t]) follow a continuous-time diffusion process given by:
[P.sub.t] = [p.sub.0] + [[integral].sup.t.sub.0] [mu](s)ds + [[integral].sup.t.sub.0] [sigma](s)dW (s)
where W(t) is a standard Brownian motion, [mu](t) is the mean process with finite variation and [sigma](t) is the instantaneous volatility which, by definition, is a positive process.
Over the time interval [t-k, t], the continuous compound return ([r.sub.t,k] = [p.sub.t] - [p.sub.t]-k) is given by the following process:
[r.sub.t] = [[integral].sup.t.sub.t-k] [mu](s)ds + [[integral].sup.t.sub.t-k] [sigma](s)dW(s) (1)
When we sum up the contribution of the mean component, [mu](t), to the variation of returns we will find out it can be ignored. This is because [mu](t)dt is of lower order of magnitude when compared to the second term [sigma](t)dW (t) in terms of second order properties (Andersen & Benzoni, 2009). Then, Quadratic Variation (QV) is defined as follows:
QV(t, k) = [[integral].sup.t.sub.t-k] [sigma][(s).sup.2]ds (2)
Suppose that one has all available information on intraday returns of an asset making it possible to calculate the sum of the squared returns sampled at a given frequency, over a trading day:
[RV.sub.t] = [T.summation over (i=1)][r.sup.2.sub.t] (3)
where RV is the Realized Volatility measure.
With no microstructure noise, Andersen et al. (2003) showed that QV converges in probability to RV. So, RV, defined as the sum of squared intraday returns, is the discrete version of the quadratic variation process. However, it does not come without a cost as it raises a set of issues related to microstructure of transaction that will be discussed in section 3, as we describe the construction of the database.
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Database Construction
We use a database that contains all intraday trading prices of the stocks traded at BM&FBovespa. The time series ranges from February/2006 to January/2011 and we select the twenty stocks listed in Table 1. There are two main reasons behind the outcome of this selection. First of all, we want to work at the highest possible frequency and minimize microstructure biases that arise when working with stocks with low liquidity. As we will see, all of the selected stocks meet this liquidity criterion. Besides, since we are doing out-of-sample forecasts that require a large number of days to work properly, we rule out stocks that belong to the database for less than 300 trading days. In fact, we also benefit from the longer time period of the database (1) by performing robustness checks with different estimation windows.
Stocks in Brazil are divided into preferred (PN) and common (ON) shares. The main difference is that the first type has the priority over dividend distributions, but does not give voting rights. As you can see in the following table, both types of shares are well represented in our database which consists of actively traded assets whose gaps between consecutive trades do not exceed 26.9 seconds. The range of sectors imposed by our stocks' selection just reflects the diversification of Brazilian industry. Thus, the concentration on the basic materials' industry is not a surprise, but other important industries such as financial and utilities are represented as well.
Before constructing the realized volatility estimates, the first decision concerns the sampling frequency. The choice of the optimal frequency involves a trade-off between microstructure issues and loss of information as we will discuss below. If we...
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