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## Market volatility timing

It has been proved in Chapter 4 that the closing price (Ct) of an asset and its technical range (TRt) are cointegrated and that where PR, is the Parkinson range. Let the unconditional mean of ln(PRt) be C, E[ln(PRt)] = C. Adding C to both sides of Eq. (7.1) yields From Eq. (7.2) we can see that ln(Ct) = ln(TRt) - Cis the equilibrium relation between ln(C,) and ln(TR,), and that the absolute value of -ln(PRt) + 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 GJR-GARCH (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, ARCH-like models are return-based. 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 range-based 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 intra-day price range in place of the absolute return. They find much better predicting power with the range-based volatility model over the return-based model for out-of-sampling forecasts. For a comprehensive review on range-based volatility, see Chou et al. (2009). A critical issue with both the CARR model and the range-based 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 range-based EGARCH model does not explain the variation in the price range, so both the CARR model and the range-based EGARCH model are partial (incomplete) models.

Recently, a new range-based 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 mean-variance 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 GJR-GARCH (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 in-sample and out-of-sample forecast comparisons between the GARCH@CARR model and the EGARCH and GJR-GARCH 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 et is assumed to be distributed with a log-normal 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( PRt.XP t ,) is scaled by a factor p to make sure that pX, is an unbiased conditional volatility' estimator of rt. 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 log-likelihood 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 log-likelihood function can be rewritten as where The partial log-likelihood, Hr) can be used to compare with that of a return- based GARGH-like 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 mean-variance utility' function and composing his portfolio from a risky asset and a risk-free asset. For a given level of (initial) wealth, the investor’s optimization problem is given by where U( •) is the quadratic utility function, rt, r^t and rp c are respectively the return on the risky asset, the return on the risk-free 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,_(rr-r^t) is the expected risk premium over t- to t, and Vt.i(rt- Vf t) is the expected variance of risk premium over t-1 to t. Following Marquering and Verbeek (2004), we assume that short selling and borrowing at the risk-free rate is not allowed, the portfolio weights must lie between 0 and 1, and the optimal portfolio weight becomes Over the out-of-sample period, the investor realizes an average utility level of where fiQ and a are the sample mean and variance, respectively, over the out-of- sample period for the return on the benchmark portfolio. In this chapter, the popular EGARCH (Nelson, 1991) and GJR-GARCH (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 ig and are the sample mean and variance, respectively, over the out-of- 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 in-sample and the out-of- 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 risk-free 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 high-low 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 risk-free interest rate where rp, and rj-t are respectively the excess return and the risk-free 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 Ljung-Box <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

 Range Excess return Absolute excess return Squared excess return Mean 6.724 0.000 3.190 18.647 Median 5.692 0.417 2.530 6.402 Std 4.190 0.012 2.915 44.606 Min 2.110 -25.417 0.011 0.000 Max 41.847 11.562 25.417 646.045 Skewness 3.371 -1.024 2.343 8.751 Kurtosis 21.526 6.708 13.570 108.287 Jarquc-Bera 6607.443*** 305.026*** 2272.566*** 193658*** P(l) 0.476 0.052 0.185 0.158 P( 2) 0.381 -0.042 0.096 0.042 P( 3) 0.279 0.025 0.175 0.076 p(6) 0.210 -0.060 0.094 0.028 P( 12) 0.123 0.029 0.033 -0.010 Q( 12) 325.53*** 7.528 61.436*** 23.338**

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 In-sample volatility timing

Maximum likelihood estimation of the GARCH@CARR model is performed over the whole sample for in-sample volatility timing.4 For comparison we use the

Table 7.2 Estimates of GARCH@CARR, GJR-GARCH and EGARCH

 Sample GARCH&CARR GJR-GARCH EGARCH Panel A: Point Estimates and Log-Likelihood (0 -0.335 0.611E - 04 -2.281 (0.053) (0.351) (0.447) «1 0.610 0.843 0.643 (0.002) (0.047) (0.067) A 0.263 0.112 0.296 (0.019) (0.065) (0.098) К -0.097 0.046 -0.344 (0.014) (0.054) (0.064) 0.619 p (0.025) 0.369 G (0.013) Panel B: Auxiliary Statistics V 0.873 0.978 0.643 К r) 742.574 724.375 730.285

Notes: The return-based volatility model is specified as In this chapter, we assume constant expected risk premium, which is removed by demeaning the excess return.

For the GJR-GARCH 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 GJR-GARCH 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).

GJR-GARCH (Glosten, et al. 1993) and EGARCH (Nelson, 1991) as benchmark models. We favor the GJR-GARCH 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 GJR-GARCH model has the highest persistence (y/) in volatility dynamics, then comes the GARCH@CARR model, while the EGARCH model reports the lowest persistence. The log-likelihood 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 GJR-GARCH 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 GJR-GARCH model reports no significant leverage effect.

Figure 7.2 presents the in-sample 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 in-sample volatility timing, the optimal weight allocated on the risky asset is presented as where wint is the in-sample optimal weight, an t is the in-sample 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 in-sample 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

• 7.3 presents the cumulative portfolio returns formulated using different volatility models. Obviously, portfolio returns formulated using volatility models are less volatile.
• 7.4.3 Out-of-sample volatility timing

In practice, the ultimate way to evaluate a model is through its performance in out-of-sample 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 in-sample portion composed of the first m observations and an out-of-sample portion composed of the last q observations. The initial out-of-sample forecast a'mll is based on the first m observations. The next out-of-sample forecast (fm+2 is based on the first m+1 observations. Proceeding in this manner through the end of the out-of-sample period, we generate a series of # out-of-sample forecasts. The recursive predicting Table 7.3 Summary statistics of the portfolio returns formulated using different volatility models

 GARCH@CARR Market Portfolio EGARCH GJR- GARCH Panel A: In-sample volatility timing Mean 5.87E-03 6.78E-03 5.87E-03 5.59E-03 Median 9.79E-03 0.011 8.6E-03 8.34E-03 Std 0.029 0.043 0.030 0.029 Min -0.154 -0.245 -0.165 -0.168 Max 0.085 0.124 0.075 0.082 Skewness -0.755 -1.027 -0.892 -1.041 Kurtosis 2.770 3.733 3.252 4.672 Utility (%) 0.462 0.397 0.455 0.437 Annual utility gain (%) - 0.774 0.072 0.298 Sharp ratio 0.094 0.084 0.092 0.086 Panel B: Out-of-sample volatility timing Mean 3.54E-03 4.61E-03 2.83E-03 2.91E-03 Median 6.52E-03 9.625E-03 5.621E-03 5.395E-03 Std 0.029 0.045 0.032 0.032 Min -0.158 -0.186 -0.158 -0.158 Max 0.082 0.102 0.085 0.092 Skewness -0.859 -0.801 -0.951 -0.808 Kurtosis 3.323 1.518 2.866 2.636 Utility (%) 0.226 0.163 0.130 0.135 Annual utility gain (%) - 0.763 1.152 1.101 Sharp ratio 0.061 0.064 0.034 0.036

Notes: (1) For in-sample volatility timing, the market portfolio is formulated by allocating all the wealth to the risky asset = 1), which is consistent with the buy-and-hold trading strategy. (2) For out-of-sample 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 csout m+k >s the out-of-sample volatility forecasts, and  which is the historical mean of the excess return. Goyal and Welch (2003, 2008) show that Jiout mk k is a stringent benchmark: predictive regression forecasts based on macroeconomic variables frequently fail to outperform the historical average forecast in out-of-sample tests.

Figure 7.4 presents the time series plots of the optimal allocation weights on risky asset. Compared with the GJR-GARCH 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 out-of-sample 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 out-of-sample 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 non-distinguishable out-of-sample 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- of-sample 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 (GJR-GARCH) volatility forecast. These two empirical facts reveal that the GARCH@CARR model has sharper volatility timing ability than the EGARCH and GJR-GARCH 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 range-based volatility model, GARCH@CARR model is more informative than the return-based EGARCH and GJR-GARCH models for both in-sample and out-of-sample volatility timing.  Market volatility timing 61

Notes

• 1 Parkinson (1980) forcefully argued and demonstrated the superiority of using range as a volatility estimator as compared with the standard methods. Beckers (1983), among others, further extended the range estimator to incorporate information about the opening and closing prices and the treatment of a time-varying drift, as well as other considerations. Other references concerning price range volatility include Carman and Klass (1980), Wiggins (1991), Rogers and Satehcll (1991), Kunitomo (1992) and Yang and Zhang (2000). Alizadeh, ct al. (2002) presented theoretically and empirically that a range-based volatility estimator is not only highly efficient but also approximately Gaussian and robust to microstructure noise. Degiannakis and Livada (2013) even found the range-based volatility estimator is more accurate than the realized volatility estimator.
• 2 Other research papers concerning the economic value of volatility timing include Busse (1999), Fleming et al. (2001), Thorp and Milunovich (2007) and Chou and Liu (2010).
• 3 Only the data from 1983 to 2016 is used for the reason that a change in the data compilation occurred around the end of April, 1982 (Chou, 2005).
• 4 In this paper, vvc assume the expected excess return (risk premium) to be constant, thus we demean the excess return when performing maximum likelihood estimation.
• 5 Since £(|x,|) = Jbт if xt ~ N(0, a), so we use | rtJnj‘l as the volatility proxy.
• 6 For in-sample volatility timing, the market portfolio is formulated by allocating all the wealth to the risky asset, which is consistent with the buy-and-hold trading strategy.

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