The economic value of volatility timing using a range-based volatility model

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The economic value of volatility timing using a range-based

volatility model

Ray Yeutien Chou

a,b,

, Nathan Liu

c a

Institute of Economics, Academia Sinica, #128, Yen-Jio-Yuan Road, Sec 2, Nankang, Taipei, Taiwan b

Institute of Business Management, National Chiao Tung University, Taiwan c

Department of Finance, Feng Chia University, Taiwan

a r t i c l e

i n f o

Available online 31 May 2010 JEL classiﬁcation: C5 C52 G11 Keywords: Asset allocation CARR DCC Economic value Range Volatility timing

a b s t r a c t

There is growing interest in utilizing the range data of asset prices to study the role of volatility in ﬁnancial markets. In this paper, a new range-based volatility model was used to examine the economic value of volatility timing in a mean–variance framework. We compared its performance with a return-based dynamic volatility model in both in-sample and out-of-sample volatility timing strategies. For a risk-averse investor, it was shown that the predictable ability captured by the dynamic volatility models is economically signiﬁcant, and that a range-based volatility model performs better than a return-based one.

1. Introduction

In recent years, there has been considerable interest in volatility. The extensive development of volatility modeling has been motivated by related applications in risk management, portfolio allocation, assets pricing and futures hedging. In discussions of econometric methodologies for estimating the volatility of individual assets, ARCH and GARCH have been

emphasized most. Various applications in ﬁnance and economics are provided as a review inBollerslev et al. (1992, 1994),

andEngle (2004).

Several studies have noted that range data based on the difference of high and low prices in a ﬁxed interval can offer a sharper estimate of volatility than the return data. Range data are available for most ﬁnancial assets and intuitively have more information than return data for estimating volatility. They utilize the two pieces of information (high and low)

comparing with the return data that use only the close to close price.Parkinson (1980)showed that it reduced the variance

of the volatility estimator by ﬁve times comparing with the traditional return-based volatility estimator. Furthermore,

range is an unbiased estimator of the standard deviation. There are quite a few extensions of Parkinson’s original results.1

More recently,Brandt and Jones (2006),Chou (2005, 2006), andMartens and van Dijk (2007). In particular,Chou (2005)

proposed a conditional autoregressive range (CARR) model which can easily capture the dynamic volatility structure, and

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Journal of Economic Dynamics & Control

Corresponding author at: Institute of Economics, Academia Sinica, #128, Yen-Jio-Yuan Road, Sec 2, Nankang, Taipei, Taiwan. Tel.: + 886 2 27822791x321; fax: + 886 2 27853946.

E-mail address: rchou@econ.sinica.edu.tw (R.Y. Chou). 1

See for example,Garman and Klass (1980),Wiggins (1991),Rogers and Satchell (1991),Kunitomo (1992),Yang and Zhang (2000), andAlizadeh et al. (2002).

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provides sharper volatility forecasts comparing with the return-based GARCH model. The CARR model is a conditional mean model and it is easily to incorporate other explanatory variables.

However, the literature above just focuses on the volatility forecast of a univariate asset. It should be noted that there

have been some attempts to establish a relationship between multiple assets, such as VECH (seeBollerslev et al., 1988),

BEKK (see Engle and Kroner, 1995) and a constant conditional correlation model (CCC) (seeBollerslev, 1990), among

others. VECH and BEKK allow time-varying covariance processes which are too ﬂexible to estimate, and CCC with a constant correlation is too restrictive to apply to general applications. Seminal work on solving the puzzle was carried out byEngle (2002a). A dynamic conditional correlation2_{(DCC) model proposed by}_{Engle (2002a)}_{provides another viewpoint}

to this problem. The estimation of DCC can be divided into two stages. The ﬁrst step is to estimate univariate GARCH, and

the second is to utilize the transformed standardized residuals to estimate time-varying correlations (see Engle and

Sheppard, 2001; Cappiello et al., 2006).

A new multivariate volatility, recently proposed byChou et al. (2009), combines the range data of asset prices with the

framework of DCC, namely range-based DCC.3_{The range-based DCC model is ﬂexible and easy to be estimated through the}

two-step estimation. It also has the relative efﬁciency of the range data over the return data in estimating volatility.

Through the statistical measures RMSE and MAE, based on four benchmarks of implied and realized covariance,4_they

concluded that the range-based DCC model performs better than other based models (MA100, EWMA, CCC, return-based DCC, and diagonal BEKK).

Asset allocation efﬁciency is closely linked to the predictions of asset returns and volatilities.West et al. (1993)was

the ﬁrst to focus on this insight and devise a way to use the utility function to derive the economic value of dynamic volatility models. The economic intuition is simple. A more accurate volatility prediction will render the investors a way to adaptively adjust their portfolio positions to achieve a higher utility level. Hence investors will be willing to pay a fee to switch from a fund manager with poor volatility prediction skill (or model) to another manager offering better volatility predictions. The maximum of such a switching fee is a measure of the difference of economic values of the two competing volatility models. The above described strategy of adjusting portfolio weights according to the prediction of volatility changes is called ‘‘volatility timing’’. This is different from the other type of ‘‘market timing’’ technique in which the portfolios are adjusted following the prediction of changes in expected returns. Market timing is usually not an effective tool given that an efﬁciency market implied the returns are unpredictable.

Following West et al. (1993), some studies have concentrated on whether some newly devised volatility models

have sufﬁciently high economic values (see Busse, 1999; Fleming et al., 2001, 2003; Marquering and Verbeek, 2004;

Thorp and Milunovich, 2007; Corte et al., 2009). The questions upon which we focused were two: ﬁrst, whether the range-based DCC model contains economic value comparing with a benchmark model using a static or buy-and-hold strategy; and second, whether economic value of range-based volatility model still exists comparing with a return-based DCC model.

In comparing the economic value of return-based and range-based models, it is helpful to use a suitable measure to capture the trade-off between risk and return. Most literature evaluates volatility models through error statistics and related applications but neglects the inﬂuence of asset expected returns. A more precise measurement should consider both of them, but only a few such studies have been made at this point. However, a utility function can easily connect them and build a comparable standard. Before entering into a detailed discussion for the economic value of volatility timing, it was necessary to clarify its deﬁnition in this paper. In short, the economic value of volatility timing is the gain compared with a static strategy. Our concern was to estimate the willingness of the investor with a mean variance utility to pay for a new volatility model rather than a static one.

In light of the success of the range-based volatility model, the purpose of this paper was to examine its economic

value in volatility timing by using the conditional mean–variance framework developed by Fleming et al. (2001).

We considered an investor with different risk-averse levels using conditional volatility analysis to allocate three assets:

stocks, bonds and cash. Fleming et al. (2001)extended the utility criterion derived from West et al. (1993) to test

the economic value of volatility timing for short-horizon investors with different risk tolerance levels.5_{In addition to the}

short-horizon forecast of selected models, we also examined the economic value of longer horizon forecasts and an asymmetric range-based volatility model in our empirical study. This study may lead to a better understanding of range volatility.

The reminder is laid out as follows. Section 2 introduces the asset allocation methodology, economic value measurement, and the return-based and the range-based DCC. Section 3 describes the properties of data used and evaluates the performance of the different strategies. Finally, the conclusion is showed in Section 4.

2_See_{Tsay (2002)}_and_{Tse and Tsui (2002)}_{for other related methods for estimating the time-varying correlations.}

3_{Fernandes et al. (2005)}_{propose another kind of multivariate CARR model using the formula Cov(X,Y) = [Var(x+ Y) Var(X) Var(Y)]/2. However, this} method can only apply to a bivariate case.

4

Daily data are used to build four proxies for weekly covariances, i.e. implied return-based DCC, implied range-based DCC, implied DBEKK, and realized covariances.

5

They found that volatility-timing strategy based on one-step ahead estimates of the conditional covariance matrix (seeFoster and Nelson, 1996) signiﬁcantly outperformed the unconditional efﬁcient static portfolios.

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2. Methodologies

To carry out this study we used the framework of a minimum variance strategy, which was conductive to determine the accuracy of the time-varying covariances. We wanted to ﬁnd the optimal dynamic weights of the selected assets and the implied economic value of a static strategy for a risk-adverse investor. Before applying the volatility timing strategies, we needed to build a time-varying covariance matrix. The details of the methodology are as follows.

2.1. Optimal portfolio weights in a minimum variance framework

Initially, we considered a minimization problem for the portfolio variance subjected to a target return constraint. To

derive our strategy, we let Rt be the k 1 vector of spot returns at time t.6 Its conditional expected return

l

t and

conditional covariance matrix

R

t were calculated by E½Rtj

O

t1 and E½ðRt

l

tÞðRt

l

tÞ 0

j

O

t1, respectively. Here,

O

t was

assumed as the information set at time t. To minimize the portfolio volatility subject to a required target return

m

target, it

1 t ð

l

tRf1Þ ð2Þ Under the cost of carry model, we regarded the excess returns as the futures returns by applying regular no-arbitrage

arguments.7_{It is clear that the covariance matrix}

_R

tof the spot returns is the same as that of the excess returns. Eq. (2) can

t ð3Þ

where the vector

l

tand the matrix

R

study. In addition, futures contracts are easy to be traded and have lower transaction cost compared to spot contracts. The above analysis pointed out that the optimal portfolio weights were time-varying. Here we assumed that the conditional

mean

l

twas constant.

8_{Therefore, the dynamics of weights only depend on the conditional covariance}

R

t. In this study, the

optimal strategy was obtained based on a minimum variance framework subject to a given return. The mean–variance framework above is used to derive the optimal portfolio weights under different target returns. In the following section, we

want to build criterion9to compare means and variances of the portfolios from the static and dynamic strategies. However,

it is not easy to decide the best strategy, especially for the investors with different risk aversions. In this study, we want to apply the quadratic utility function to calculate economic value under some settings.

2.2. Economic value of volatility timing

Fleming et al. (2001)uses a generalization of theWest et al. (1993)criterion which builds the relationship between a mean–variance framework and a quadratic utility to capture the trade-off between risk and return for ranking the

6_{Through out this paper, we have used blackened letters to denote vectors or matrices.} 7

There are no costs for futures investment. This means the futures return equals the spot return minus the risk-free rate. 8

The changes in expected returns are not easy to detect.Merton (1980)points out that the volatility process is more predictable than the return series.

9

The Sharpe ration is one of the candidates for comparison. However, it may underestimate the performance of dynamic strategies, seeMarquering and Verbeek (2004).

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performance of forecasting models. According to their work, the investor’s utility can be deﬁned as UðWtÞ ¼WtRp,t

a

W2 t 2 R 2 p,t ð5Þ

where Wtis the investor’s wealth at time t,

a

is his absolute risk aversion, and the portfolio return at period t is w0tRt.

the average realized utility U ðÞ can be used in estimating the expected utility with a given initial wealth W0.

U ðÞ ¼ W0 XT t ¼ 1 Rp,t

g

2ð1 þ

g

ÞR2 p,t " # ð6Þ

where W0is the initial wealth.

Therefore, the value of volatility timing calculated by equating the average utilities for two alternative portfolios is expressed as XT t ¼ 1 Rb,t

D

g

2ð1 þ

g

ÞðRb,t

D

Þ 2 ¼X T t ¼ 1 Ra,t

g

written as

Ht¼Dt

C

tDt ð8Þ

C

t¼diag Q1=2t Qtdiag Q 1=2

t ð9Þ

where Dtis the k k diagonal matrix of time-varying standard deviations from univariate GARCH models with

ffiffiffiffiffiffiffi hi,t

p

for the

i-th return series on the i-th diagonal.

G

t is a time-varying correlation matrix. The covariance matrix Qt= [qij,t] of the

standardized residual vector Zt¼ ðz1,t,z2,tÞ0is denoted as

Qt¼ ð1abÞQ þ a Zt1Z0t1þb Qt1 ð10Þ

where Q ¼ fqijgdenotes the unconditional covariance matrix of Zt. The coefﬁcients, a and b, are the estimated parameters

depicting the conditional correlation process. The dynamic correlation can be expressed as

r

12,t¼

ð1abÞq12þa z1,t1z2,t1þb q12,t1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1abÞq11þa z21,t1þb q11,t1½ð1abÞq22þa z22,t1þb q22,t1

q ð11Þ

We estimated the DCC model with a two-stage estimation through quasi-maximum likelihood estimation (QMLE) to get

consistent parameter estimates. The log-likelihood function can expressed as L=LVol+ LCorr, where LVol, the volatility

component, is 1

2

P

tðk logð2

p

Þ þlogjDtj2þr0tD2t rtÞ, and LCorr, the correlation component, is 12

P

tðk logjRtj þZ0tRt1ZtZ0tZtÞ.

The explanation is more fully developed inEngle and Sheppard (2001)andEngle (2002a).

In addition to using GARCH to construct standardized residuals, we can also build them by other univariate volatility models. In this paper, CARR was used as an alternative to verify whether the speciﬁcation selected adequately suits DCC or not.

The CARR model is a special case of the multiplicative error model (MEM) ofEngle (2002b). It can be expressed as

i,t1

zc i,t¼ ri,t

l

i,t where

l

i,t¼adji

l

i,t,adji¼

s

i

^

l

i

ð12Þ

10

West et al. (1993),Fleming et al. (2001), andCorte et al. (2009)also applied CRRA to their studies. 11

In our setting, we let the strategy pair (a,b) be (OLS, return-based DCC), (OLS, range-based DCC) and (return-based DCC, range-based DCC), respectively. Because the rolling sample method was adopted in the out-of-sample comparison, this type of OLS was named by rollover OLS.

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200 400 600 800 1,000 1,200 1,400 1,600 92 94 96 98 00 02 04 06 S&P 500 80 90 100 110 120 130 92 94 96 98 00 02 04 06 Tbond -15 -10 -5 0 5 10 92 94 96 98 00 02 04 06 S&P 500 -6 -4 -2 0 2 4 92 94 96 98 00 02 04 06 Tbond 0 4 8 12 16 92 94 96 98 00 02 04 06 S&P 500 0 1 2 3 4 5 92 94 96 98 00 02 04 06 Tbond

Fig. 1. S&P 500 index futures and T-bond futures weekly closing prices, returns and ranges, 1992–2006. These Panels A, B and C shows the weekly close prices, returns, and ranges of S&P 500 index futures and 10-year treasury bond (T-bond) futures over the sample period.

Table 1

Summary statistics for weekly S&P 500 and T-bond futures return and range data, 1992–2006.

S&P 500 futures T-bond futures

Return Range Return Range

Mean 0.158 3.134 0.016 1.306 Median 0.224 2.607 0.033 1.194 Maximum 8.124 13.556 2.462 4.552 Minimum 12.395 0.690 4.050 0.301 Std. dev. 2.112 1.809 0.855 0.560 Skewness 0.503 1.756 0.498 1.390 Kurtosis 6.455 7.232 4.217 6.462 Jarque–Bera 421.317 985.454 80.441 642.367 (0.000) (0.000) (0.000) (0.000)

The table provides summary statistics for the weekly return and range data on S&P 500 stock index futures and T-bond futures. The returns and ranges were computed by 100 logðpclose

t =popent Þand 100 logðphight =plowt Þ, respectively. The Jarque–Bera statistic is used to test the null of whether the return and range data are normally distributed. The values presented in parentheses are p-values. The annualized values of means (standard deviation) for S&P 500 and T-bond futures were 8.210 (15.232) and 0.853 (6.168), respectively. The simple correlation between stock and bond returns was 0.023. The sample period ranges from January 6, 1992 to December 29, 2006 (15 years, 782 observations) and all futures data were collected from datastream.

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where the range Ri,tis calculated by the difference between logarithm high and low prices of the i-th asset during a ﬁxed

time interval t, and it is also a proxy of standard deviation.

l

i,tand ^

l

iare the conditional and unconditional means of the

range, respectively. ui,tis the residual which is assumed to follow the exponential distribution.

s

i is the unconditional

standard deviation for the return series. In considering different scales in quantity, the ratio adjtwas used to adjust the

range to produce the standardized residuals.12

3. Empirical results

The empirical data employed in this paper consists of the stock index futures, bond futures and the risk-free rate. As to the above-mentioned method, we applied the futures data to examine the economic value of volatility timing for return-based and range-return-based DCC. Under the cost of the carry model, the results in this case can be extended to underlying spot

assets (seeFleming et al., 2001). In addition to avoiding the short sale constraints, this procedure reduces the complexity of

model setting. To address this issue, we used the S&P 500 futures (traded at CME) and the T-bond futures (traded at CBOT)

as the empirical samples. According toChou et al. (2009), the futures data were taken from datastream, sampling from

January 6, 1992 to December 29, 2006 (15 years, 782 weekly observations). Datastream provided the nearest contract and rolls over to the second nearby contract when the nearby contract approaches maturity. We also used the 3-month Treasury bill rate to substitute for the risk-free rate. The Treasury bill rate is available from the Federal Reserve Board.

Fig. 1shows the graphs for close prices (Panel A) returns (Panel B) and ranges (Panel C) of the S&P 500 and T-bond

futures over the sample period.Table 1presents summary statistics for the return and range data on the S&P 500 and

T-bond futures. The return was computed as the difference of logarithm close prices on two continuous weeks. The range was deﬁned by the difference of the high and low prices in a logarithm type. The annualized mean and standard deviation in percentage (8.210, 15.232) of the stock futures returns were both larger than those (0.853, 6.168) of the bond futures Table 2

Estimation results of return-based and range-based DCC model using weekly S&P500 and T-bond futures, 1992–2006.

S&P 500 futures T-bond futures

GARCH CARR GARCH CARR

Panel A: Volatilities estimation of GARCH and CARR models

c 0.188 0.008 (3.256) (0.242) ^ o 0.019 0.103 0.028 0.075 (1.149) (2.923) (1.533) (2.810) ^ a 0.051 0.248 0.060 0.157 (3.698) (9.090) (2.031) (5.208) ^ b 0.946 0.719 0.902 0.785 (71.236) (23.167) (18.645) (18.041) Q(12) 26.322 5.647 15.872 23.121 (0.010) (0.933) (0.197) (0.027)

S&P 500 and T-bond

Return-based DCC Range-based DCC

Panel B: Correlation estimation of return- and range-based DCC models ^ a 0.037 0.043 (4.444) (4.679) ^ b 0.955 0.951 (85.621) (80.411)

ri,t¼c þei,t,hk,t¼okþake2k,t1þbkhk,t1,ek,tjIt1Nð0,hk,tÞ, Ri,t¼ui,t,lk,t¼okþakRk,t1þbklk,t1,Rk,tjIt1expð1,Þ,k ¼ 1,2, Qt¼ ð1abÞQ þ a Z0

t1Zt1þb Qt1, and thenr12,t¼ ð1abÞq12þa z1,t1z2,t1þb q12,t1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1abÞq11þa z21,t1þb q11,t1½ð1abÞq22þa z22,t1þb q22,t1 q

where Rt is the range variable, Ztis the standard residual vector which is standardized by GARCH or CARR volatilities. Qt= {qij,t} and Qt¼ fqij,tgare the conditional and unconditional covariance matrix of Zt. The three formulas above are GARCH, CARR and the conditional correlation equations, respectively, of the standard DCC model with mean reversion. The table presents estimations of the three models using the MLE method. Panel A is the ﬁrst step of the DCC model estimation. The estimation results of GARCH and CARR models for two futures were presented here. Q(12) is the Ljung-Box statistic for the autocorrelation test with 12 lags. Panel B is the second step of the DCC model estimation. The values presented in parentheses are t-ratios for the model coefﬁcients and p-values for Q(12).

12

Parkinson (1980)derived the adjustment ratio as a constant, 0.361, but an asset price was required to follow a geometric Brownian motion with zero drift, which is not truly empirical.

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returns. This fact indicated that the more volatile market may have a higher risk premium. Both futures returns have negative skewness and excess kurtosis, indicating a violation of the normal distribution. The range mean (3.134) of the stock futures prices was larger than that (1.306) of the bond futures prices. This is reasonable because the range is a proxy of volatility. The Jarque–Bera statistic was used to test the null of whether the return and range data were normally distributed. Both return and range data rejected the null hypothesis. The simple correlation between stock and bond

returns was small13_{( 0.023), but this does not imply that their relationship was very weak. In our latter analysis, we}

showed that the dynamic relationship of stocks and bonds will be more realistically revealed by the conditional correlations analysis.

3.1. The in-sample comparison

To obtain an optimal portfolio, we used the dynamic volatility models to estimate the covariance matrices. The

parameters ﬁtted for return-based and range-based DCC, were both estimated and arranged inTable 2. We divided the

Volatility Estimates for the GARCH Model

0 4 8 12 16 20 1992 1994 1996 1998 2000 2002 2004 2006 S&P 500 Tbond

Volatility Estimates for the CARR Model

0 1 2 3 4 5 6 7 8 1992 1994 1996 1998 2000 2002 2004 2006 S&P 500 Tbond

Fig. 2. In-sample volatility estimates for the GARCH and CARR model. Panel A: volatility estimates for the GARCH model and Panel B: volatility estimates for the CARR model.

13

The results are different from the positive correlation value (sample period 1983–1997) inFleming et al. (2001). After 1997, the relationship between S&P 500 and T-bond presented a reverse condition.

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table into two parts corresponding to the two steps in the DCC estimation. In Panel A ofTable 2, one can use GARCH (ﬁtted

by return) or CARR (ﬁtted by range) with individual assets to obtain the standardized residuals. Fig. 2 provides the

volatility estimate of the S&P 500 futures and the T-bond futures based on GARCH and CARR. Then, these standardized

residuals series were brought into the second stage for dynamic conditional correlation estimating. Panel B of Table 2

presents the estimated parameters of DCC under the quasi-maximum likelihood estimation (QMLE).

The correlation and covariance estimates for return-based and range-based DCC are shown inFig. 3. It seems that the

correlation became more negative at the end of 1997. This means that it is more desirable to diversify in this period because the bond holding will help offset the volatility caused by the equity component in the portfolio. This conjecture is conﬁrmed in our latter analysis of the estimated portfolio weights. A deeper investigation is also given in

Connolly et al. (2005).

Following the model estimation, we constructed the static portfolio (built by OLS) using the unconditional mean and covariance matrices to get the economic values of dynamic models. Under the minimum variance framework, the weights of the portfolio were computed by the given expected return and the conditional covariance matrices estimated by return-based and range-return-based DCC. Then, we compared the performance of the volatility models on 11 different target annualized returns (5–15%, 1% in an interval).

Table 3presents how the performance comparisons varied with the target returns and the risk aversions. Panel A of

Table 3presents the annualized means ð

m

Þand volatilities ð

s

Þof the portfolios estimated from three methods, return-based

Correlation Estimates -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1992 1994 1996 1998 2000 2002 2004 2006 return DCC range DCC Covariance Estimates -3 -2 -1 0 1 2 1992 1994 1996 1998 2000 2002 2004 2006 return DCC range DCC

Fig. 3. In-sample correlation and covariance estimates for the return-based and range-based DCC model. Panel A: correlation estimates and Panel B: covariance estimates.

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DCC, range-based DCC and OLS. At a quick look, the annualized Sharpe ratios14_{calculated from return-based DCC (0.680)}

and range-based DCC (0.699) were higher than the static model (0.560). Panel B ofTable 3presents the average switching

fees ð

D

rÞfrom one strategy to another. The value settings of CRRA

g

were 1, 5, and 10. As for the performance fees with

different relative risk aversions, in general, an investor with a higher risk aversion should be willing to pay more to switch from the static portfolio to the dynamic ones. With higher target returns, the performance fees increased steadily. In

addition, Panel B ofTable 3also reports the performance fees for switching from return-based DCC to range-based DCC.

Positive values for all cases show that the range-based volatility model can give more signiﬁcant economic value in forecasting covariance matrices than return-based ones.

In the real practice, the transaction costs should be considered when the dynamic strategies are compared to the static one. For S&P 500 futures, the bid/ask spread and round-trip commission totally cost about $0.10 index unit. The annualized cost of a one-way transaction in our study can be calculated by 0.05/941.55 52= 0.28%, where 941.55 is an average index level from 1992 to 2006. It means the advantage of the dynamic strategies will not be offset by the transaction costs. For example, with a ﬁxed target return 10%, the economic advantage is about 6% for an investor with relative risk aversion of 5.

Fig. 4plots the weights of an in-sample minimum volatility portfolio derived from two dynamic models. OLS has constant weights for cash, stocks, and bonds, i.e. 0.1934, 0.7079, and 0.4855.

It is interesting to observe the dynamic patterns of the portfolio weights implied by the two dynamic models. In contrary to the OLS (buy-and-hold strategy), they have substantial ﬂuctuations across the sample periods. The two strategies (panel A for return-based DCC and panel B for range-based DCC) have roughly similar patterns in movements but

with noticeably quantitative differences. The stock portfolio weight is most stably ﬂuctuating around o0:8 before 1997

after which it drops to a lower level of about 0.65 with larger variations. It is interesting to observe that the bond weights have been negative or zero before 1997 and become positive after late 1997. The zero or negative weights are the result of the booming equity market in the mid 90s hence it is desirable to invest mostly in the equity market. The mid-crash in the Table 3

In-sample comparison of the volatility timing values in the minimum volatility strategy using different target returns, 1992–2006.

Target return (%) Return-based DCC Range-based DCC Rollover OLS

m s m s m s

Panel A: Means and volatilities of optimal portfolios

5 5.201 2.100 5.241 2.100 5.000 2.190 6 6.366 3.814 6.438 3.813 6.000 3.977 7 7.530 5.527 7.635 5.526 7.000 5.764 8 8.694 7.241 8.832 7.239 8.000 7.551 9 9.859 8.954 10.028 8.952 9.000 9.338 10 11.023 10.668 11.225 10.665 10.000 11.125 11 12.187 12.381 12.422 12.378 11.000 12.912 12 13.352 14.095 13.619 14.091 12.000 14.699 13 14.516 15.808 14.815 15.804 13.000 16.486 14 15.680 17.521 16.012 17.517 14.000 18.273 15 16.845 19.235 17.209 19.230 15.000 20.060

Target return (%) OLS to return DCC OLS to range DCC Return to range DCC

D1 D5 D10 D1 D5 D10 D1 D5 D10

Panel B: Switching fees with different relative risk aversions

5 0.303 0.376 0.393 0.343 0.417 0.434 0.040 0.041 0.041 6 0.703 0.950 1.008 0.777 1.025 1.084 0.074 0.076 0.076 7 1.244 1.771 1.897 1.353 1.883 2.009 0.109 0.112 0.112 8 1.929 2.845 3.063 2.073 2.994 3.213 0.144 0.149 0.151 9 2.761 4.173 4.507 2.940 4.360 4.696 0.180 0.189 0.191 10 3.739 5.753 6.224 3.956 5.979 6.453 0.217 0.230 0.233 11 4.866 7.578 8.206 5.121 7.846 8.477 0.255 0.273 0.277 12 6.142 9.641 10.441 6.434 9.951 10.754 0.294 0.318 0.324 13 7.565 11.932 12.914 7.897 12.283 13.270 0.334 0.365 0.373 14 9.135 14.436 15.609 9.507 14.831 16.009 0.375 0.414 0.424 15 10.851 17.142 18.509 11.262 17.580 18.952 0.418 0.466 0.479

The table reports the in-sample performance of the volatility timing strategies with different target returns. The target returns were from 5% to 15% (annualized). The weights for the volatility timing strategies were obtained from the weekly estimates of the conditional covariance matrix and the different target return setting. Panel A presents the annualized means ðmÞand volatilities ðsÞfor each strategy. The estimated Sharpe ratios for the return-based DCC model, the range-return-based DCC model, and the OLS strategy were 0.680, 0.699, and 0.560, respectively. Panel B presents the average switching annualized fees ðDrÞfrom one strategy to another. The values of the constant relative risk aversiongwere 1, 5, and 10.

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late 1997 has caused an increase of volatility which would cause a drop in investor’s utility and hence should be hedged

away. As is seen inFig. 3, this is a period when the correlation between stock and bond returns became negative. The

negative correlation in bond/equity return suggests an increase in the bond position would help to reduce the total portfolio volatilities. The lower level and higher variations of stock weights since then is also a reﬂection of the fact that the stock/bond correlations in the later periods are mostly negative but with wide swings. Finally, the cash position serves as a residual in the portfolio since the three asset weights add up to one. The movements will be related to the term spread or the term structure of interest rates and the bond volatility.

It is also useful to contrast the time-varying pattern of the bond position to the ﬁxed weight suggested by OLS. The latter suggest that roughly 48% should be invested in the bond market regardless of the movements in the volatility and correlation structures. This is obviously too naive given our discussion above that volatilities and correlations of stock and bond returns do vary over time. A buy-and-hold strategy will therefore yield a poor performance.

3.2. The out-of-sample comparisons

For robust inference, a similar approach was utilized to estimate the value of volatility timing in the out-of-sample analysis. Here the rolling sample approach was adopted for all out-of-sample estimations. This meant that the rollover OLS method replaced the conventional OLS method used in the in-sample analysis. Each forecasting value was estimated by 521 observations over about 10 years. Then, the rolling sample method provided 261 forecasting values for the one period ahead comparison. The ﬁrst forecasted value occurred the week of January 4, 2002.

In-sample Portfolio Weights Derived by the Return-based DCC Model

-4 -3 -2 -1 0 1 2 3 4 92 94 96 98 00 02 04 06

Cash S&P 500 Tbond

In-sample Portfolio Weights Derived by the Range-based DCC Model

-6 -4 -2 0 2 4 6 92 94 96 98 00 02 04 06

Cash S&P 500 Tbond

Fig. 4. In-sample minimum volatility portfolio weight derived by the dynamic volatility model. Panels A and B show the weights that minimize conditional volatility while setting the expected annualized return equal to 10%. The OLS model had constant weights for cash, stock, and bond, i.e. 0.1934, 0.7079, and 0.4855. Panel A: In-sample portfolio weights derived by the return-based DCC model and Panel B: in-sample portfolio weights derived by the range-based DCC model.

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Table 4reports how the performance comparisons varied with the target returns and the risk aversions for one period ahead

out-of-sample forecast. We obtained a consistent conclusion with Table 3. The estimated Sharpe ratios calculated from

return-based DCC, range-based DCC and rollover OLS were 0.540, 0.586 and 0.326, respectively. The performance fees switching

from rollover OLS to DCC were all positive. In total, the out-of-sample comparison supported the former inference.Fig. 5plots

the weights that minimize conditional volatility while setting the expected annualized return equal to 10%.

In addition to examining the performance of short-horizon investors, we further reported the results of the

long-horizon asset allocations.Table 5reports one to 13 periods ahead of out-of-sample performance for three methods. Here

the rolling sample approach provided 249 forecasting values for each out-of-sample comparison. The portfolio weights for all strategies were obtained from the weekly estimates of the out-of-sample conditional covariance matrices with a ﬁxed target return (10%). In general, the Sharpe ratios taken from range-based DCC were the largest, and return-based DCC were the next. For each strategy, however, we could not ﬁnd an obvious trend in the Sharpe ratios forecasting periods ahead. As for the result of the performance fees, it seems reasonable to conclude that an investor would still be willing to pay to switch from rollover OLS to DCC. Moreover, the economic value seems to indicate a decreasing trend for forecasting periods ahead. For a longer forecasting horizon (12–13 weeks), however, the results of estimated switching fees were mixed. Switching from return-based DCC to range-based DCC always remains positive.

Thorp and Milunovich (2007) show that a risk-averse investor holding selected international equity indices, with

g

¼2,5, and 10, would pay little for symmetric to asymmetric forecasts. In some cases, the switching fees would even be

negative. In order to further understand this argument, we examined it based on the range-based volatility model.Chou

(2005)provides an asymmetric range model namely CARRX:

l

t¼

o

þ

a

Rt1þ

b

l

t1þ

f

rett1. The lagged return in the

conditional range equation was used to capture the leverage effect. For building an asymmetric range-based volatility

model, CARR in the ﬁrst step of range-based DCC can be replaced by CARRX.Cappiello et al. (2006)introduced asymmetric

DCC: Qt¼ ð1abÞQ c N þ a Zt1Z0t1þb Qt1þc nt1n0t1. ntis the k 1 vector calculated by IðZto0Þ Zt to allow

Table 4

Out-of-sample comparison for the one period ahead volatility timing values in the minimum volatility strategy with different target returns, 1992–2006.

Target return (%) Return-based DCC Range-based DCC Rollover OLS

m s m s m s

5 4.691 1.698 4.747 1.661 4.344 1.749 6 5.438 3.083 5.540 3.016 4.808 3.176 7 6.186 4.468 6.333 4.370 5.273 4.603 8 6.933 5.853 7.127 5.725 5.737 6.030 9 7.681 7.239 7.920 7.080 6.202 7.456 10 8.428 8.624 8.714 8.435 6.667 8.883 11 9.176 10.009 9.507 9.790 7.131 10.310 12 9.923 11.394 10.300 11.145 7.596 11.737 13 10.671 12.779 11.094 12.500 8.060 13.164 14 11.418 14.165 11.887 13.854 8.525 14.591 15 12.166 15.550 12.680 15.209 8.990 16.018

Target return (%) OLS to return DCC OLS to range DCC Return to range DCC

D1 D5 D10 D1 D5 D10 D1 D5 D10

5 0.393 0.425 0.433 0.481 0.537 0.550 0.089 0.112 0.118 6 0.781 0.890 0.916 0.991 1.176 1.220 0.210 0.289 0.308 7 1.232 1.463 1.518 1.606 1.998 2.090 0.377 0.545 0.585 8 1.746 2.144 2.239 2.328 3.001 3.159 0.589 0.882 0.953 9 2.323 2.935 3.079 3.156 4.185 4.425 0.848 1.303 1.413 10 2.963 3.834 4.039 4.092 5.545 5.881 1.154 1.810 1.967 11 3.667 4.842 5.116 5.133 7.077 7.522 1.509 2.402 2.617 12 4.435 5.956 6.309 6.280 8.774 9.338 1.913 3.083 3.363 13 5.267 7.174 7.614 7.531 10.629 11.321 2.366 3.851 4.206 14 6.162 8.495 9.029 8.885 12.634 13.460 2.869 4.707 5.146 15 7.121 9.914 10.548 10.340 14.781 15.746 3.422 5.651 6.181

The table reports the one period ahead out-of-sample performance of the volatility timing strategies with different target returns. There were 521 observations in each of the estimated models and the rolling sample approach provided 261 forecasting values for each out-of-sample comparison. The ﬁrst forecasted value occurred the week of January 4, 2002. The target returns were from 5% to 15% (annualized). The weights for the volatility timing strategies were obtained from the weekly estimates of the one period ahead conditional covariance matrix and the different target return setting. Panel A presents the annualized means ðmÞand volatilities ðsÞfor each strategy. The estimated Sharpe ratios for the return-based DCC model, the range-based DCC model, and the rollover OLS strategy were 0.540, 0.586, and 0.326, respectively. Panel B presents the average switching annualized fees ðDrÞfrom one strategy to another. The values of the constant relative risk aversion were 1, 5, and 10.

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correlation to increase more in both falling returns than in both rising returns, and N ¼ Eðntn0tÞ, where denotes the

Hadamard matrix product operator, i.e. element-wise multiplication.Table 6presents the one period ahead performance of

the volatility timing values for asymmetric range-based DCC compared with rollover OLS. The switching fees from rollover

OLS to asymmetric range DCC seem to be smaller than the fees from rollover OLS to symmetric range DCC inTable 4. One of

the reasons for this may be the poor performance of the bond data. In this case, it is not valuable to switch the symmetric strategy to the asymmetric one.

Out-of-sample Portfolio Weight Derived by the Return-based DCC Model

-4 -3 -2 -1 0 1 2 3 4 2002 2003 2004 2005 2006

Cash S&P 500 Tbond

Out-of-sample Portfolio Weight Derived by the Range-based DCC Model

-6 -4 -2 0 2 4 6 2002 2003 2004 2005 2006

Cash S&P 500 Tbond

Out-of-sample Portfolio Weight Derived by the Rollover OLS Model

-3 -2 -1 0 1 2 3 2002 2003 2004 2005 2006

Cash S&P 500 Tbond

Fig. 5. Out-of-sample minimum volatility portfolio weight derived by the dynamic volatility model for one period ahead estimates. Panels A, B, and C show the one period ahead weights that minimize conditional volatility while the expected annualized return is set at 10%. Different from the in-sample case, the rolling sample method was used in estimating the portfolio weights. The portfolio weights in the rollover OLS model (Panel C) also vary with time. The ﬁrst forecasted weights occurred the week of January 4, 2002. Panel A: out-of-sample portfolio weight derived by the return-based DCC model, Panel B: out-of-sample portfolio weight derived by the range-based DCC model and Panel C: out-of-sample portfolio weight derived by the rollover OLS model.

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Table 5

Out-of-sample comparison for one to 13 periods ahead volatility timing values in the minimum volatility strategy, 1992–2006.

Periods ahead Return-based DCC Range-based DC Rollover OLS

m s SR m s SR m s SR

1 7.717 8.724 0.452 8.060 8.540 0.502 6.022 8.956 0.251 2 7.868 8.830 0.464 8.562 8.556 0.560 6.068 8.933 0.257 3 7.371 8.807 0.408 8.312 8.572 0.529 6.660 8.931 0.323 4 8.117 8.838 0.491 8.750 8.604 0.578 7.103 8.928 0.373 5 8.464 8.860 0.529 9.200 8.653 0.627 6.869 8.989 0.344 6 9.088 8.903 0.597 9.600 8.637 0.674 7.232 8.973 0.385 7 9.361 8.840 0.632 10.033 8.629 0.725 7.872 8.945 0.458 8 8.853 8.897 0.571 9.429 8.683 0.651 7.644 8.975 0.431 9 9.806 8.878 0.679 10.093 8.664 0.729 8.476 9.023 0.521 10 9.746 8.887 0.672 9.576 8.695 0.667 8.189 8.983 0.491 11 9.436 8.908 0.636 8.986 8.712 0.598 8.031 8.910 0.478 12 8.737 9.003 0.551 8.076 8.791 0.489 7.424 8.853 0.412 13 8.713 9.111 0.542 8.272 8.914 0.505 7.794 8.867 0.453

1 2.772 3.546 3.727 3.944 5.289 5.599 1.196 1.831 1.983 2 2.282 2.633 2.716 4.223 5.448 5.731 1.970 2.914 3.137 3 1.293 1.721 1.823 3.308 4.495 4.772 2.029 2.830 3.019 4 1.440 1.758 1.834 3.152 4.244 4.499 1.728 2.544 2.738 5 2.210 2.665 2.773 3.900 5.032 5.297 1.712 2.446 2.622 6 2.191 2.442 2.503 3.938 5.078 5.345 1.775 2.730 2.958 7 1.993 2.373 2.464 3.647 4.740 4.997 1.674 2.440 2.625 8 1.581 1.861 1.928 3.161 4.172 4.410 1.597 2.369 2.555 9 2.028 2.556 2.683 3.319 4.578 4.875 1.313 2.103 2.295 10 2.019 2.370 2.455 2.753 3.767 4.007 0.753 1.465 1.638 11 1.416 1.424 1.426 1.891 2.591 2.758 0.489 1.209 1.383 12 0.593 0.037 0.100 0.945 1.164 1.217 0.358 1.128 1.313 13 0.269 1.202 1.436 0.251 0.078 0.035 0.518 1.243 1.417

The table reports the one to 13 periods ahead out-of-sample performance of the volatility timing strategies with the ﬁxed 10% (annualized) target return. The weights for the volatility timing strategies were obtained from the weekly estimates of the one to 13 periods ahead conditional covariance matrix. There were 521 observations in each of the estimated models and the rolling sample approach provided 249 forecasting values for each out-of-sample comparison. The ﬁrst forecasted mean value occurred the week of January 4, 2002. Panel A presents the annualized means ðmÞ, volatilities ðsÞ, and Sharpe ratios (SR) for each strategy. Panel B presents the average switching annualized fees ðDrÞfrom one strategy to another. The values of the constant relative risk aversion were 1, 5, and 10.

Table 6

The one period ahead performance of the volatility timing values for the asymmetric range-based volatility model, 1992–2006. Target return (%) Means and volatilities of optimal portfolios for asymmetric

range-based DCC

Switching fees from rollover OLS to asymmetric range-based DCC m s D1 D5 D10 5 4.643 1.666 0.373 0.425 0.438 6 5.352 3.025 0.787 0.962 1.003 7 6.060 4.384 1.301 1.670 1.757 8 6.769 5.744 1.915 2.550 2.699 9 7.478 7.103 2.630 3.601 3.827 10 8.187 8.462 3.445 4.818 5.136 11 8.895 9.821 4.361 6.199 6.621 12 9.604 11.180 5.377 7.738 8.274 13 10.313 12.540 6.491 9.428 10.087 14 11.022 13.899 7.703 11.262 12.050 15 11.730 15.258 9.011 13.232 14.155

The table reports the one period ahead out-of-sample performance of the volatility timing strategies for the asymmetric range-based volatility model with different target returns. There were 521 observations in each of the estimated models and the rolling sample approach provided 261 forecasting values for each out-of-sample comparison. The ﬁrst forecasted value occurred the week of January 4, 2002. The target returns were from 5% to 15% (annualized). The weights for the volatility timing strategies were obtained from the weekly estimates of the one period ahead conditional covariance matrix and the different target return setting. The annualized means ðmÞand volatilities ðsÞof the optimal portfolio are presented here.Dris the average switching annualized fee from the rollover OLS model to the asymmetric range-based volatility model. The estimated Sharpe ratio for the asymmetric range-based DCC model was 0.521. The values of the constant relative risk aversion were set as 1, 5, and 10.

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4. Conclusion

In this paper, we examined the economic value of volatility timing for the range-based volatility model in utilizing range data which combines CARR with a DCC structure. Our analysis is carried out by utilizing S&P 500 and T-bond futures in a mean–variance framework with a no-arbitrage setting. By means of the utility of a portfolio, the economic value of dynamic models can be obtained by comparing it to OLS (a buy-and-hold strategy). Both the in-sample and out-of-sample results show that a risk-averse investor should be willing to switch from OLS to DCC with substantial high switching fees. Moreover, the switching fees from return-based DCC to range-based DCC were always positive. We concluded that the range-based volatility model has more signiﬁcant economic value than the return-based one. The results gave robust inferences for supporting the range-based volatility model in forecasting volatility. Future studies can consider more general type of utility functions and also include other asset classes such as commodity futures, REIT’s and VIX’s.

Acknowledgements

We would like to thank an anonymous referee and the editor Carl Chiarella for helpful comments and suggestions. Financial support for this paper is provided by the National Science Council of Taiwan (NSC 96-2415-H-001-019 and NSC 96-2420-H-009-006-DR). This paper has beneﬁted from the discussions with Chu-Sheng Tai, and the participants from the 15th Conference on the Theories and Practices of Securities and Financial Markets, Kaohsiung, Taiwan, the 15th Annual Global Finance Conference, Hangzhou, China, the Sixth China International Conference in Finance, Dalian, China, and the Second Risk Management Conference, Singapore. We are responsible for all remaining errors.

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