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IV ApplicationsMarket volatility timingIt has been proved in Chapter 4 that the closing price (C_{t}) of an asset and its technical range (TR_{t}) are cointegrated and that
where PR, is the Parkinson range. Let the unconditional mean of ln(PR_{t}) be C, E[ln(PR_{t})] = C. Adding C to both sides of Eq. (7.1) yields
From Eq. (7.2) we can see that ln(C_{t}) = ln(TR_{t})  Cis the equilibrium relation between ln(C,) and ln(TR,), and that the absolute value of ln(PR_{t}) + C measures the deviation from the equilibrium. Larger deviation from equilibrium implies higher risk in this system. Since PR, is a volatility estimator, we will show in this chapter that PR, can be used in market volatility timing strategy'. 7.1 Introduction Volatility has always been an interesting and important topic in financial econometrics. Ever since the seminal paper of Engle (1982), a large number of ARCH like models have been proposed. Among them, the GARCH (Bollcrslev, 1986), the EGARCH (Nelson, 1991) and the GJRGARCH (Glosten, et al. 1993) have been emphasized most in volatility estimating and forecasting. For a critical review with a thorough survey of the ARCH literature, see Bollerslev, et al. (1992). Traditionally, ARCHlike models are returnbased. Recent literature shows a rising interest in using price range to estimate volatility. The price range is far more informative about the current level of volatility than is the squared return.^{1} Chou (2005) proposed the conditional autoregressive range model (henceforth CARR) to describe the dynamics of rangebased volatility. They find the CARR model does provide a sharper volatility estimate compared with the standard GARCH model. Brandt and Jones (2006) formulated a model that is analogous to Nelson’s (1991) EGARCH model, but used the square root of the intraday price range in place of the absolute return. They find much better predicting power with the rangebased volatility model over the returnbased model for outofsampling forecasts. For a comprehensive review on rangebased volatility, see Chou et al. (2009). A critical issue with both the CARR model and the rangebased EGARCH model of Brandt and Jones (2006) is that they are “incomplete”. The CARR model only explains the variation in the price range that is indirectly related to the return volatility, while the rangebased EGARCH model does not explain the variation in the price range, so both the CARR model and the rangebased EGARCH model are partial (incomplete) models. Recently, a new rangebased volatility model, GARCH@CARR is proposed by Xie, et al. (2019) to study the dynamics of asset volatility. The GARCH@CARR model is a joint modeling of the asset return and the price range. In this model, the dynamics of asset volatility are specified to be subordinated to the CARR model. This chapter examines the economic value of GARCH@CARR model in volatility timing. Following Marquering and Verbeek (2004), we consider an investor with meanvariance utility function who uses the conditional volatility forecast to allocate his wealth between two assets: the risky asset and the riskfree asset.^{2} Empirical studies are performed on the monthly S&P500 stock index, and the results show that the GARCH@CARR model produces a valuable volatility forecast. Using the EGARCH (Nelson, 1991) and GJRGARCH (Glosten et al. 1993) as benchmark models, we find portfolio returns formulated using the GARCH@CARR model report larger mean and smaller standard deviation, and thus the higher Sharp ratio. The utility statistics show that an investor would be willing to pay 1.41% (1.09%) portfolio management fee to have access to the GARCH@CARR volatility forecast relative to the EGARCH (GJR GARCH) volatility forecast. The chapter is organized as follows. Section 2 presents the econometric methodologies with some discussions. Section 3 presents the volatility timing strategy. In Section 4, an empirical example is performed on the S&P500 index to investigate the volatility timing ability of the GARCH@CARR model. Both insample and outofsample forecast comparisons between the GARCH@CARR model and the EGARCH and GJRGARCH models are presented. Section 5 concludes. 7.2 GARCH@CARR model In this chapter the GARCH@CARH model of Xie, et al. (2019) is used to estimate the assets’ volatility. The GARCH@CARR model of order (/>, q) is presented as follows where r, and PR, are respectively the return and the Parkinson range. The distribution of the disturbance term z, is assumed to be distributed with a standard normal. The distribution of the disturbance term e_{t} is assumed to be distributed with a lognormal density' with unit mean. Further it is assumed that z, and c, are mutually independent. Parameter к is used to capture the leverage effect which has been widely documented in empirical literature. The first two equations are referred to as the return equation and the GARCH equation, and the last equation is referred to as the measurement equation as it relates the observed realized measure (price range) to the latent volatility'. The measurement equation is an important component because it “completes” the model. In the return equation, X, = E( PR_{t}.XP _{t} ,) is scaled by a factor p to make sure that pX, is an unbiased conditional volatility' estimator of r_{t}. The persistence parameter of the GARCH@CARR model is calculated as ^T^_{=1}«, + Pj The GARCH@CARR can be estimated by maximum likelihood estimation (MLE) method. The loglikelihood function of the GARGH@GARR model (conditionally on JF,_,) is given by
where 0 is the parameter set to be estimated. With the assumption that z, is independent of £„ we can factorize the joint conditional density' for (r„ PR,) by
Thus the loglikelihood function can be rewritten as
where
The partial loglikelihood, Hr) can be used to compare with that of a return based GARGHlike model. More discussions about the GARCH@GARR model are available in Xie, et al. (2019). 7.3 Economic value of volatility timing Consider an investor maximizing a meanvariance utility' function and composing his portfolio from a risky asset and a riskfree asset. For a given level of (initial) wealth, the investor’s optimization problem is given by where U( •) is the quadratic utility function, r_{t}, r^_{t} and r_{p c} are respectively the return on the risky asset, the return on the riskfree asset and the portfolio return, со, is the wealth proportion allocated on the risky asset, and у is the risk aversion parameter. In this paper, we take у = 3, which is also commonly used in empirical studies. E,_i(•) and are respectively the conditional expected portfolio return and the conditional expected portfolio variance. It is straightforward that the optimal proportion allocated on the risky asset is given by where E,_(r_{r}r^_{t}) is the expected risk premium over t to t, and V_{t}.i(r_{t} Vf _{t}) is the expected variance of risk premium over t1 to t. Following Marquering and Verbeek (2004), we assume that short selling and borrowing at the riskfree rate is not allowed, the portfolio weights must lie between 0 and 1, and the optimal portfolio weight becomes
Over the outofsample period, the investor realizes an average utility level of
where fi_{Q} and a are the sample mean and variance, respectively, over the outof sample period for the return on the benchmark portfolio. In this chapter, the popular EGARCH (Nelson, 1991) and GJRGARCH (Glosten, et al. 1993) models are used as our benchmark models, and the portfolios formed using these two models are used as the benchmark portfolios. For the same investor when he or she forecasts the asset volatility using the GARCH@CARR model, the realized average utility level is given by
where i_{g} and are the sample mean and variance, respectively, over the outof sample period for the return on the portfolio formed using the forecasts of the asset volatility based on the GARCH@CARR model. The economic value of volatility timing is measured by utility gain which is defined as the difference between Equation (7.13) and Equation (7.12). We multiply the utility gain by 1200 to get the average annualized percentage return. The utility gain can be interpreted as the portfolio management fee that an investor would like to pay to have access to the additional information available in a predictive model relative to the benchmark model. 7.4 Empirical results This section describes the data and presents both the insample and the outof sample results for volatility timing performance of the GARCH@CARR model relative to the benchmark models. 7.4.1 The data We collected the monthly data of the S&P500 stock index for the sample period from January 1983 to December 2016 with 408 observations.^{3} For each month four pieces of price information, the high, low, opening and closing prices were reported. The S&P500 index data sets were downloaded from the website www. finance.yahoo.com. The riskfree interest rate was downloaded from the home page of Amit Goyal at http://www.hec.unil.ch/agoyal/. In this analysis, the S&P500 stock index is used as the risky asset. The highlow Parkinson range, PR, is calculated by where ln( •) is natural logarithm, H, and L, are respectively the high and low prices. The risky asset return is calculated as where r, and C, are respectively the risky asset return and the closing price. Excess return is obtained by the risky asset return less the riskfree interest rate where rp, and rj_{t} are respectively the excess return and the riskfree interest rate. Figure 7.1 presents the time series plots of the Parkinson price range and the excess return. It is clear from Figure 7.1 that price range is clustering and is more volatile in negative return than in positive return (the leverage effect). We multiply both the range and the excess return by 100. Table 7.1 presents the summary statistics for the range, the excess return, the absolute excess return and the squared excess return. The range, the absolute excess return and the squared excess return can be used as volatility estimator candidates. Consistent with the widely documented facts, we find high kurtosis and large negative skewness of the excess return, which indicates a strong deviation from the normal distribution. The LjungBox <2statistics indicate strong persistence in both range, absolute excess return and the squared excess return, which implies the predictability of the volatility. It is more interesting to compare the Q statistics for the range with those for the absolute excess return and for the squared excess return. The statistics show that there is a significant difference between the range and the absolute excess return (squared excess return), which means the volatility information in the range is different from those in the absolute excess return and in the squared excess return. Figure 7.1 Time series plots of Parkinson price range and excess return Table 7.1 Summary statistics of the range, the excess return, the absolute excess return and the squared excess return
Notes: We use p(i) as the sample autocorrelation function coefficient at lag i, Q as the Ljung Box Q statistics. The symbols *, **, *** mean respectively significance at 10%, 5%, anti 1%. 7.4.2 Insample volatility timing Maximum likelihood estimation of the GARCH@CARR model is performed over the whole sample for insample volatility timing.^{4} For comparison we use the Table 7.2 Estimates of GARCH@CARR, GJRGARCH and EGARCH
Notes: The returnbased volatility model is specified as
In this chapter, we assume constant expected risk premium, which is removed by demeaning the excess return. For the GJRGARCH model, the dynamics of the volatility is presented as
where I,__{t} is an indicator function ofe,_*: /_{r}_j =0 iff_{;}_i<=0 and i = 1 if£,_[>(). The persistence parameter lor the GJRGARCH model is y/=a+/3+ic/2. For the EGARCH model, the volatility dynamics is given by
The persistence parameter for the EGARCH model is y/=a. The numbers in the brackets are the standard errors. When performing estimation, we only consider the models of order (1,1) because there is evidence that the order (1,1) is sufficient for capturing the persistence in the range and in the volatility (Chou, 2005). GJRGARCH (Glosten, et al. 1993) and EGARCH (Nelson, 1991) as benchmark models. We favor the GJRGARCH and EGARCH models for two reasons: (»') the flexibility in capturing the leverage effect and (it) the ability in fitting the volatility dynamics. Table 7.2 reports the estimation results. In this section, we only consider models of order (1,1) because it has been well noted that order (1, 1) is sufficient for capturing the persistence in volatility. The estimation results show that the GJRGARCH model has the highest persistence (y/) in volatility dynamics, then comes the GARCH@CARR model, while the EGARCH model reports the lowest persistence. The loglikelihood statistic /(r) shows that the GARCH@CARR model, although it does not maximize the log likelihood function of return, still produces a better empirical fit than both the GJRGARCH and the EGARCH models. The estimates also report significant leverage effect in both the GARCH@CARR model and the EGARCH model since ks are reported to be negative and statistically significant. Interestingly, the GJRGARCH model reports no significant leverage effect. Figure 7.2 presents the insample volatility forecasts reported by the the GJR GARCH, EGARCH and the GARCH@CARR models. It is clear from Figure 7.2 that the GARCH@CARR model is more adaptive to the price changes.^{5} For insample volatility timing, the optimal weight allocated on the risky asset is presented as where w_{int} is the insample optimal weight, a_{n t} is the insample volatility forecast, and ft is the sample mean of the excess return which is presented as Table 7.3 presents in Panel A the summary statistics of the insample portfolio returns formulated using different volatility models. It is clear that portfolio returns formulated using volatility models have lower standard deviations than the market portfolio.^{6} Both the utility and the Sharpe ratio statistics show the outperformance of the GARCH@CARR model over the other three benchmark models, which are consistent with the estimation results. Figure
In practice, the ultimate way to evaluate a model is through its performance in outofsample forecasting. In this section a recursive (expanding) window forecasting procedure is carried out on the S&P500 stock index. To be specific, the whole T data observations are divided into an insample portion composed of the first m observations and an outofsample portion composed of the last q observations. The initial outofsample forecast a'_{mll} is based on the first m observations. The next outofsample forecast (f_{m+2} is based on the first m+1 observations. Proceeding in this manner through the end of the outofsample period, we generate a series of # outofsample forecasts. The recursive predicting Table 7.3 Summary statistics of the portfolio returns formulated using different volatility models
Notes: (1) For insample volatility timing, the market portfolio is formulated by allocating all the wealth to the risky asset = 1), which is consistent with the buyandhold trading strategy. (2) For outofsample volatility timing, the wealth proportion allocated to the risky asset is determined by Equation (7.11). (3) The utility gain is obtained by the GARCH@CARR utility less the benchmark model utility. For example, if the benchmark model is EGARCH, the utility gain is calculated bv (0.226%0.130%), and the annual utility gain is given bv (0.226%0.130%) x 1200 = 1.152. procedure simulates the situation of a forecaster in real time. In this section, we choose q = 240 which expands twenty years of data observations, from January 1997 to December 2016. The wealth proportion allocated to the risky asset is determined by where cs_{out m}+k >^{s} the outofsample volatility forecasts, and which is the historical mean of the excess return. Goyal and Welch (2003, 2008) show that Ji_{out mk k} is a stringent benchmark: predictive regression forecasts based on macroeconomic variables frequently fail to outperform the historical average forecast in outofsample tests. Figure 7.4 presents the time series plots of the optimal allocation weights on risky asset. Compared with the GJRGARCH model, the optimal weights reported by the EGARCH and GARCH@CARR models are more volatile, which indicates that the GARCH@CARR model and EGARCH model are more adaptive to the volatility changes. Figure 7.5 presents the time series plots of the outofsample cumulative portfolio returns. We find the GARCH@ CARR model not only reports a larger cumulative return but also lower volatility compared with other competing models. The summary statistics for the outofsample portfolio returns formulated using different volatility models are reported in Panel В in Table 7.3. From the results two empirical facts emerge: (1) the EGARCH and the GJR GARCH models have nondistinguishable outofsample volatility forecasting performance since the summary statistics of the portfolio returns produced by these two models are very close. This finding is consistent with Figure 7.5 as the time series plots of the cumulative portfolio returns reported by these two models are quite similar. (2) The GARCH@CARR model delivers better out ofsample volatility forecasts. The summary statistics show that the portfolio returns formulated using the GARCH@CARR model report larger sample mean, lower standard deviation, higher utility and sharp ratio compared with the other two competing models. For example, the annual utility gain statistic shows that an investor would be willing to pay 1.15% (1.10%) portfolio management fee to have access to the GARCH@CARll volatility forecast relative to the EGARCH (GJRGARCH) volatility forecast. These two empirical facts reveal that the GARCH@CARR model has sharper volatility timing ability than the EGARCH and GJRGARCH models have. 7.5 Summary From Proposition (2) (see Chapter 4, p.20) we obtain that the Parkinson range measures the deviation from the equilibrium between closing price and technical range. Therefore, the we can use the Parkinson range as a risk measure of the system of closing price and technical range. Taking the Parkinson range as a measure of volatility risk, this chapter investigates its economic value in volatility timing strategy. An empirical study is performed on the monthly S&P500 stock index, and the results show that the rangebased volatility model, GARCH@CARR model is more informative than the returnbased EGARCH and GJRGARCH models for both insample and outofsample volatility timing. Market volatility timing 61 Notes

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