Empirical Analysis - The Economic Value of Volatility Timing Using a

Chapter 3. The Economic Value of Volatility Timing Using a

4.3 Empirical Analysis

In this study, 887 weekly observations on the spot and futures for six classes (fifteen commodities), i.e., stock indices (FTSE 100, Nikkei 225 and S&P500 (SP)), currencies (British Pond (BP), Japanese Yen and Swiss Franc (SF)), metals (gold and silver), grains (corn, soybeans (Soy) and soybean oil (SO)), softs (coffee, cotton and sugar), and energy (crude oil (CL)), are obtained from Datastream. The detail of these data is described in Table 4.1. The time period of commodities is from January, 1, 1990 to December, 29, 2006. The futures data provided by Datastream are the nearest contract to deliver but rolled it over to the next nearest contract on the first day of the delivery month in order to avoid thin trading and expiration effects.

Table 4.2 gives summary statistics for returns and ranges of each spot and futures commodity. The returns are computed by 100×log(P_t/P_t₋₁), where P is the close _t price in each week. The ranges are computed by 100×log(P_t^High/P_t^Low), where P_t^High and P_t^Low are the maximum and minimum price respectively among the daily close prices in the t week and the last trading day close price in the ^th t 1− week^th ²⁷. The means of the returns are almost close to zero. As is noted by Fama (1965), this martingale behavior is often interpreted as being consistent with a weak form efficient market. Except soybeans, soybean oil and crude oil, the volatilities of all futures returns are somewhat higher than the volatilities of spot returns. The order of the magnitudes for the means of the range is roughly the same as that for the standard deviations of the

27 Unlike financial assets, the high-low price data of most commodities in a trading day are unavailable but close price data. In this study, however, the weekly data are used to examine the hedging performance.

Therefore, it is reasonable to use the measure as its proxy.

returns with the only two exceptions of corn and cotton. This reflects the fact that both range and standard deviations are measures of volatilities. Given that the range data are non-negative is present for all commodities.

< Table 4.2 is inserted about here >

In order to clarify the relative hedging performance, several models are used for comparison, including three buy-and-hold strategies (no hedge, naïve²⁸, and OLS) with fixed weights in the hedging period and five dynamic strategies (rollover OLS, return-based CCC, return-based DCC, range-based CCC, and range-based DCC) with time-varying weights in a framework of rolling sample.

The rolling sample approach here utilizes week-by-week updating to build the time-varying hedge ratios. There are 522 weekly observations (about ten years) in each of estimated period of the others. In addition, all cases provide 365 one period ahead out-of-sample forecasting values for comparison. The first forecasted value occurs on the week of January 3, 2000. Assume that there is one unit underlying asset in the beginning. No hedge means that the variance of its hedging portfolio is only decided by the underlying asset. Naïve here is the short hedge with selling one unit futures.

In order to formally compare the performances of each kind of hedging method, the hedging portfolios are applied by the estimated hedge ratios of each week. The variance of these portfolio returns can be written as Var(r_S_,_t −h_t^*r_F_,_t), where h are _t^* estimated optimal hedge ratios from different hedging methods. In this study, we focus on the out-of-sample forecasting results with one period ahead. Table 4.3 and Table 4.4 report the maximum likelihood estimations of the return-based and range-based DCC

28 The naïve hedging strategy is the simplest way to hedge the spot price risk. This strategy suggests that an investor who has a long position in the spot market should sell a unit of futures today and buy it back when he sells the spot. If the spot and futures prices both change by the same amount at all times, this will be a perfect hedge.

models respectively. Here the first out-of-sample parameter estimates²⁹ are provided.

In Table 4.3, Panel A and Panel B are the first step of the DCC model estimation, which are the GARCH model fittings of spot and futures returns respectively. In Table 4.4, they are for the CARR model fittings of ranges. Panel C is the second step of the DCC model estimation for both tables.

< Table 4.3 is inserted about here >

< Table 4.4 is inserted about here >

From the tables, the values of (α βˆ+ ˆ) are close to one except for soybean oil, indicating high persistence in volatility. As for correlation persistence, however, the values of (a bˆ+ exhibit inconsistent results. Some cases have high persistence in ˆ) correlation, but the others don’t. In addition, for the cases of gold and silver, the range-based DCC model shows stronger correlation persistence than the return-based one.

The comparisons of out-of-sample hedging performance are reported in Table 4.5.

Panel A of Table 4.5 shows the variances of the hedging portfolios. To further gauge the hedging efficiency among various methods, Panel B also reports the efficiency gain of each alternative method compared to no hedge. Furthermore, Panel C shows the percentage variance improvement compared with the range-based DCC model.

< Table 4.5 is inserted about here >

Several observations can be made from the reading of Table 4.5. First, the portfolio variances in stock indices, currencies, and metals are much smaller than those in grains, softs, and energy. It seems that the financial market³⁰ has active trade and visible information to reduce the price change between spot and futures. It is obvious that trading noises lead to worse hedging performance for the agriculture and energy

29 In total, we have 365 estimations. The first parameter estimates are provided in Table 3 and Table 4.

30 In general, gold and silver are viewed as financial assets.

markets. In respect of no hedge, all portfolio variances, with the single exception of currency, are very large, especially in silver and non-financial commodities.

As for the comparison of the seven hedging methods, the naïve method generally is the worst of all. This is not surprising as the assumption of perfect correlation between the spot and the futures returns underlying this method is clearly not supported empirically. In the cases of stock indices, however, naïve performs better than the other static and rollover OLS models³¹. Next, the fact that the dynamic strategies with time-varying hedge ratios outperform the buy-and-hold ones indicates that the traditional method assuming a constant hedge ratio through the hedging period has a lot of room for improvements.

Among the dynamic hedging methods, how do the range-based methods compare with the return-based methods? The results suggest that the range-based ones are better than their corresponding opponents with the return-based ones. Specifically, the variances of the hedging portfolio derived from the range-based volatility models are smaller than the return-based volatility ones in thirteen out of fifteen commodities. The finding has its exception only in the soybean and coffee cases.

Panel B of Table 4.5 shows the hedging effectiveness of all strategies. The simple naïve hedge for all commodities can reduce over about 75% variation of spot. In addition, there are over 90% high values of hedging effectiveness for all hedging strategies in silver and two classes, stock index and currency. Again, the difference of the hedging effectiveness between the static and dynamic models for these cases is small. However, the results in the other commodities still support the superiority of the dynamic hedging strategies over the static ones. It is noteworthy that the poor

31 Because the 10-year period might have some structural changes which would reduce the hedging efficiency of the OLS model, the 5-year OLS and rollover OLS models were considered as other comparison models. However, our empirical results indicated that the difference between two different estimated periods was very small.

effectiveness in crude oil seems to point out it is difficult to hedge by futures.

In order to more intuitively compare the performance of these hedging strategies, Panel C of Table 4.5 lists the hedging improvement ratio by the range-based DCC model. From the average percentage variance improvement reported in the last column, range-based DCC is the clear winner of all methods, with an improvement of about 30% over OLS, 27% over rollover OLS, 16% over return-based CCC, 10% over return-based DCC, and 5% over range-based CCC. It is valuable to take a more look at these values. In the same model setting, there are about 5% improvements by using the time-varying correlation strategies over the constant ones, and about 10%

improvements by using the range-based strategies over the return-based ones³². From the hedging point of view, the range is indeed a more efficient measure of volatility than the return. Furthermore, the additional effort in modeling the time-varying pattern of the conditional correlation is with rewards.

For illustration, Figure 4.1 plots the estimated hedge ratios using different methods for six cases, S&P 500, British Pond, gold, soybean oil, cotton, and crude oil, respectively. In addition to rollover OLS, the optimal return-based and range-based CCC or DCC models are put together for comparison. The rollover OLS model has the smoothest pattern in all cases, but still varies over time with a rolling-sample of ten years is used in the out-of-sample comparisons. To take cotton for example, there is an obvious jump in the middle of 2005. With the single exception of gold, the figures indicate that the hedge ratios from range are more volatile than those from return. To conclude, the dynamic methods provide wide variations of the hedge ratios around the

32 In fact, we need to redo the work of Panel C of Table 5 for our target model to get the accuracy value.

For simplicity, the related results are not listed in this study. Return-based (range-based) DCC has a 7.02% (4.96%) gain over return-based (range-based) CCC. Then, range-based DCC (CCC) has a 9.71%

(11.76%) gain over return-based DCC (CCC).

OLS estimates. A more flexible hedge ratio seems to be necessary in order to obtain a more effective hedging strategy.

4.4 Conclusion

Range is a more efficient estimator than return in forecasting volatility. However, few researches utilize its superiority in financial applications. This paper uses range-based hedging models for calculating optimal hedge ratios in six classes of commodity futures, totally fifteen commodities, and compares with hedging performance of other models.

For a one-period forecasting horizon, empirical findings indicate that hedging performances of range-based volatility models are significantly better than the other volatility models for most commodities. Based on minimum-variance hedge criterion, the hedging portfolio variances calculated from the range-based volatility models are smaller than the return-based ones in thirteen out of fifteen commodities.

In conclusion, the results mainly indicate the following three points: (1) static hedging strategies are not suitable for most futures hedging, especially for non-financial ones; (2) assuming constant correlation generally has an approximate 5 percent loss in hedging achievement; (3) in the same dynamic structure, hedging improvement for the range data compared with the return data is about 10 percent on average.

Table 4.1: The Source of Spot and Futures Data

The table reports the related information for the fifteen futures and spots in this paper. The exchanges for futures, Datastream names for spots and their codes are included in this table.

Futures Spot

Type Name Exchange Code Datastream Name Code

Stock Index FTSE 100 LIFFE LSXCS00 FTSE 100 - PRICE INDEX FTSE100 Nikkei 225 OSX ONACS00 NIKKEI 225 STOCK AVERAGE - PRICE INDEX JAPDOWA S&P 500 CME ISPCS00 S&P 500 COMPOSITE S&PCOMP

Currency British Pond CME IBPCS00 US $ TO UK (GTIS) BRITPUS

Japanese Yen CME IJYCS00 US $ TO JAPANESE YEN (GTIS) JAPYNUS Swiss Franc CME ISFCS00 US $ TO SWISS FRANC (GTIS) SWISFUS Metal Gold CMX NGCCS00 Gold, Handy & Harman Base $/Troy Oz GOLDHAR

Silver CMX NSLCS00 Silver, Handy & Harman (NY) cts/Troy OZ SILVERH

Grain Corn CBOT CC.CS00 Corn No.2 Yellow Cents/Bushel CORNUS2

Soybeans CBOT CS.CS00 Soyabeans, No.1 Yellow C/Bushel SOYBEAN Soybean Oil CBOT CBOCS00 Soya Oil, Crude Decatur Cents/lb SOYAOIL

Soft Coffee NYBOT NKCCS00 Coffee-ICO Composite Daily ICA c/lb COFDICA Cotton No. 2 NYBOT NCTCS00 Cotton,1 1/16Str Low -Midl, Memph C/Lb COTTONM Sugar No. 11 NYBOT NSBCS00 Raw Cane Sugar, World FOB Cents/lb SUGCNRW Energy Crude Oil NYMEX NCLCS00 Crude Oil-Brent Cur. Month FOB U$/BBL OILBREN

Table 4.2: Summary Statistics for Returns and Ranges of Spot and Futures

The table provides summary statistics for the weekly return and range data of the spot and futures samples in this study. The returns are computed by 100×log(P_t /P_t₋₁), where P is the close price in each week. The ranges are computed by _t

) / log(

100× P_t^High P_t^Low , where P_t^High and P_t^Low are the maximum and minimum price respectively among the daily close prices in the t week and the last trading day close ^th price in the t 1− week. The sample period ranges from Jan 1, 1990 to Dec 29, 2006 ^th (887 weekly observations).

Spot Futures

Return Range Return Range Mean Std Dev Mean Std Dev Mean Std Dev Mean Std Dev FTSE 0.105 2.084 2.304 1.418 0.103 2.186 2.469 1.483 Nikkei -0.090 2.955 3.297 1.870 -0.091 3.020 3.365 1.896 SP 0.157 2.073 2.239 1.410 0.157 2.117 2.309 1.474 BP 0.020 1.290 1.373 0.824 0.022 1.322 1.405 0.847 Yen 0.022 1.565 1.612 1.030 0.023 1.603 1.660 1.056 SF 0.026 1.579 1.734 0.904 0.027 1.605 1.760 0.920 Gold 0.050 1.879 1.863 1.377 0.050 1.967 1.968 1.388 Silver 0.101 3.295 3.257 2.326 0.098 3.412 3.422 2.447 Corn 0.053 3.367 3.485 2.272 0.056 3.373 3.345 2.283 Soy 0.018 3.150 3.173 2.161 0.019 3.078 3.135 2.024 SO 0.046 3.158 3.465 1.890 0.049 3.079 3.360 1.837 Coffee 0.059 4.479 3.909 3.325 0.050 5.527 5.704 3.905 Cotton -0.015 3.487 3.858 2.137 -0.019 3.750 3.775 2.515 Sugar -0.011 4.166 4.569 2.744 -0.021 4.462 4.762 3.015 CL 0.107 5.229 5.395 3.640 0.110 4.930 5.281 3.540

Note: BP (British Pond), SF (Swiss Franc), Soy (soybeans), SO (soybean oil), and CL (crude oil).

Table 4.3: Return-based DCC Model Estimations

This table shows the first estimation (totally 365 estimations) of the range-based DCC models using the MLE method for the out-of-sample forecast. The estimated period here is ranging from Jan 1, 1990 to Dec 31, 1999 (522 weekly observations). Panel A and Panel B are the first step of the DCC model estimation for spot and futures returns of commodities respectively. Panel C is the second step of the DCC model estimation.

LLF is the abbreviation for log likelihood function value. The values presented in parentheses are standard errors for the estimated coefficients.

The return-based DCC model is shown as follows:

, ,

k t k k t

r = +c ε , )ε_k_,_t |I_t₋₁ ~N(0,h_k_,_t ,k =1,2.

1 , 2

,_t = _k + _k _k_t−_i + _k _k_t−

k h

h ω α ε β ,

1 1

) 1

( − − + ₋ ′₋ + ₋

= _t _t _t

t a b Q aZ Z bQ

Q ,

where Z is the standard residual vector which is standardized by GARCH volatilities. _t Q and Q are the conditional and unconditional _t covariance matrix of Z . _t

Panel A: Estimates of GARCH(1,1) model for spot returns

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL c 1 0.238 -0.060 0.261 0.026 0.085 -0.014 -0.074 -0.085 0.191 0.098 -0.047 -0.131 -0.067 -0.057 -0.114

(0.086) (0.118) (0.068) (0.055) (0.0690 (0.071) (0.048) (0.129) (0.121) (0.108) (0.118) (0.167) (0.133) (0.170) (0.208) ˆ1

ω 0.069 1.046 0.052 0.096 0.066 0.283 0.083 0.434 0.413 0.712 5.467 2.505 1.791 0.583 1.202 (0.050) (0.473) (0.036) (0.074) (0.056) (0.218) (0.042) (0.264) (0.224) (0.232) (1.627) (1.728) (1.197) (0.344) (0.472) ˆ1

α 0.054 0.132 0.089 0.121 0.087 0.048 0.205 0.071 0.175 0.218 0.197 0.138 0.085 0.095 0.152 (0.020) (0.055) (0.025) (0.109) (0.032) (0.038) (0.098) (0.038) (0.039) (0.059) (0.074) (0.070) (0.035) (0.035) (0.053) ˆ1

β 0.930 0.757 0.897 0.831 0.896 0.848 0.787 0.889 0.799 0.709 0.137 0.755 0.723 0.876 0.811 (0.028) (0.086) (0.028) (0.121) (0.026) (0.100) (0.081) (0.050) (0.049) (0.061) (0.194) (0.095) (0.142) (0.047) (0.041)

Panel B: Estimates of GARCH(1,1) model for futures returns

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL c 2 0.236 -0.075 0.257 0.003 0.086 -0.014 -0.106 -0.096 0.184 0.111 -0.035 -0.173 -0.132 -0.162 -0.071

(0.093) (0.124) (0.069) (0.060) (0.070) (0.072) (0.055) (0.135) (0.124) (0.114) (0.119) (0.218) (0.141) (0.202) (0.196) ˆ2

ω 0.096 1.200 0.048 0.024 0.080 0.138 0.187 0.413 1.132 0.703 4.558 0.750 0.162 1.430 1.059 (0.064) (0.597) (0.034) (0.019) (0.058) (0.121) (0.081) (0.236) (0.500) (0.316) (1.420) (0.414) (0.177) (0.671) (0.547) ˆ2

α 0.054 0.120 0.084 0.055 0.092 0.044 0.274 0.066 0.204 0.186 0.200 0.151 0.105 0.119 0.111 (0.021) (0.054) (0.022) (0.034) (0.039) (0.028) (0.118) (0.031) (0.091) (0.074) (0.075) (0.050) (0.032) (0.044) (0.035) ˆ2

β 0.925 0.761 0.903 0.932 0.887 0.908 0.696 0.900 0.687 0.740 0.233 0.851 0.897 0.816 0.847 (0.030) (0.093) (0.024) (0.032) (0.028) (0.059) (0.098) (0.041) (0.096) (0.088) (0.184) (0.036) (0.022) (0.061) (0.035)

Panel C: Estimates of return-based DCC model

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL aˆ 0.021 0.054 0.024 0.043 0.019 0.096 0.119 0.039 0.215 0.247 0.328 0.046 0.218 0.388 0.151

(0.007) (0.008) (0.009) (0.005) (0.002) (0.014) (0.026) (0.016) (0.026) (0.024) (0.025) (0.010) (0.021) (0.026) (0.013) bˆ 0.976 0.942 -0.930 0.933 0.975 -0.142 -0.011 0.697 0.649 0.549 0.318 0.876 0.369 0.022 0.685

(0.010) (0.009) (0.043) (0.008) (0.004) (0.173) (0.232) (0.163) (0.045) (0.055) (0.037) (0.032) (0.125) (0.044) (0.038) LLF 809.399 914.249 850.669 818.568 898.616 901.638 395.667 522.703 353.391 577.613 535.826 194.930 186.348 375.383 376.728

Table 4.4: Range-based DCC Model Estimations

This table shows the first estimation (totally 365 estimations) of the range-based DCC models using the MLE method for the out-of-sample forecast. The estimated period here is ranging from Jan 1, 1990 to Dec 31, 1999 (522 weekly observations). Panel A and Panel B are the first step of the DCC model estimation for spot and futures ranges of commodities respectively. Panel C is the second step of the DCC model estimation.

LLF is the abbreviation for log likelihood function value. The values presented in parentheses are standard errors for the estimated coefficients.

The range-based DCC model is shown as follows:

t i t i_, =u_,

ℜ , ℜ_{k t}_, |I_t₋₁~ f(1, )⋅ , 2k =1, .

1 , 1

,_t = _k + _kℜ_k_t− + _k _k_t−

k ω α β λ

λ ,

1 1

) 1

( − − + ₋ ′₋ + ₋

= _t _t _t

t a b Q aZ Z bQ

Q ,

where ℜ is the range variable, _t Z is the standard residual vector which is standardized by CARR volatilities. _t Q and Q are the conditional _t and unconditional covariance matrix of Z . _t

Panel A: Estimates of CARR(1,1) model for spot ranges

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL ˆ1

ω 0.045 0.359 0.029 0.044 0.070 0.126 0.054 0.109 0.219 0.290 1.195 0.117 0.303 0.223 0.224 (0.024) (0.122) (0.018) (0.023) (0.038) (0.065) (0.028) (0.057) (0.073) (0.081) (0.362) (0.065) (0.125) (0.100) (0.098) ˆ1

α 0.092 0.212 0.109 0.110 0.105 0.076 0.177 0.093 0.208 0.233 0.213 0.107 0.140 0.151 0.185 (0.022) (0.041) (0.024) (0.032) (0.024) (0.028) (0.045) (0.028) (0.037) (0.045) (0.051) (0.036) (0.032) (0.032) (0.030) ˆ1

β 0.888 0.680 0.876 0.858 0.855 0.852 0.789 0.872 0.724 0.667 0.418 0.865 0.770 0.799 0.772 (0.027) (0.063) (0.027) (0.041) (0.037) (0.054) (0.054) (0.039) (0.050) (0.059) (0.137) (0.039) (0.059) (0.045) (0.037)

Panel B: Estimates of CARR(1,1) model for futures ranges

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL ˆ2

ω 0.063 0.335 0.039 0.028 0.087 0.083 0.064 0.109 0.278 0.318 0.835 0.234 0.115 0.262 0.174 (0.031) (0.115) (0.021) (0.017) (0.045) (0.048) (0.031) (0.063) (0.106) (0.099) (0.261) (0.107) (0.084) (0.114) (0.084) ˆ2

α 0.106 0.201 0.123 0.087 0.123 0.071 0.165 0.088 0.180 0.220 0.202 0.153 0.099 0.137 0.161 (0.025) (0.039) (0.025) (0.026) (0.027) (0.025) (0.043) (0.027) (0.040) (0.047) (0.052) (0.032) (0.023) (0.031) (0.027) ˆ2

β 0.869 0.701 0.859 0.893 0.829 0.882 0.797 0.880 0.728 0.669 0.533 0.810 0.866 0.805 0.804 (0.031) (0.057) (0.029) (0.032) (0.041) (0.044) (0.051) (0.039) (0.059) (0.067) (0.113) (0.042) (0.043) (0.046) (0.034)

Panel C: Estimates of range-based DCC model

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL aˆ 0.027 0.061 -0.026 0.051 0.005 0.092 0.046 0.032 0.243 0.289 0.337 0.110 0.241 0.439 0.142

(0.007) (0.009) (0.014) (0.005) (0.000) (0.013) (0.016) (0.011) (0.021) (0.023) (0.027) (0.009) (0.025) (0.029) (0.014) bˆ 0.978 0.933 0.818 0.930 0.996 -0.130 0.878 0.922 0.657 0.549 0.339 0.835 0.364 0.008 0.648

(0.008) (0.010) (0.153) (0.007) (0.000) (0.153) (0.040) (0.040) (0.032) (0.044) (0.041) (0.020) (0.133) (0.037) (0.051) LLF 804.909 914.491 849.783 859.567 876.946 898.055 412.295 516.787 336.278 577.734 532.714 180.451 188.785 377.413 369.952

Table 4.5: Comparisons of Out-of-Sample Hedging Performance

There are three parts in this table. Panel A shows the post-sample portfolio variances. Panel B shows the hedging effectiveness. Panel C shows hedging improvement ratio by range-based DCC for other methods. The models used for comparison include three buy-and-hold strategies (no hedging, naïve, and OLS) with fixed weights in the hedging period and five dynamic strategies (rollover OLS, return-based CCC, return-based DCC, range-based CCC, and range-based DCC) with time-varying weights in a framework of rolling sample. The rolling sample approach here utilizes week-by-week updating to build the time-varying hedge ratios. Assume that there is one unit underlying asset in the beginning. No hedging means that the variance of its hedging portfolio is only decided from the underlying asset. Naïve is the short hedge with selling one unit futures. There are 522 observations (ten years) in each of estimated period of the others. There are 365 one period ahead out-of-sample forecasting values provided for comparison. The first forecasted value occurs on the week of January 3, 2000.

Panel A: Portfolio variances for all strategies

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL no hedging 4.6695 7.8949 5.4291 1.4032 1.6720 2.1171 4.8753 12.1656 13.2761 12.4454 12.1466 15.7210 16.3349 17.7476 25.5334 naïve 0.0826 0.2366 0.1133 0.0643 0.1038 0.1293 1.1745 0.9274 2.6654 2.1024 1.2375 7.8405 5.8008 8.2354 6.5475 OLS 0.1110 0.2560 0.1262 0.0653 0.0979 0.1258 1.0777 1.0014 2.7249 2.1149 1.2713 5.8112 5.8424 6.8299 6.2694 rollover OLS 0.0901 0.2444 0.1179 0.0638 0.0971 0.1253 1.0616 0.9312 2.6168 2.0649 1.2475 5.7835 5.5008 6.8467 6.0101 return-based CCC 0.0705 0.2306 0.1252 0.0613 0.0829 0.1209 0.8342 0.7806 2.1103 1.2998 1.0443 4.3736 5.1678 6.3694 5.9494 return-based DCC 0.0704 0.2286 0.1258 0.0600 0.0824 0.1179 0.7644 0.7560 1.7623 1.1903 0.9536 4.1437 3.9894 5.0676 5.5862 range-based CCC 0.0575 0.2126 0.1022 0.0567 0.0709 0.1028 0.8654 0.5659 1.6215 1.6779 0.9557 4.7769 2.9810 5.2619 4.8775 range-based DCC 0.0570 0.2112 0.1052 0.0555 0.0711 0.0987 0.7538 0.5816 1.7809 1.7554 0.8273 4.4907 2.3862 3.7297 4.5918 Note: The number with an underline stands for the smallest hedging portfolio variance in each commodity column.

Panel B: Hedging effectiveness (1−Var_Hedgie/Var_no_hedging)

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL naive 0.9823 0.9700 0.9791 0.9542 0.9379 0.9389 0.7591 0.9238 0.7992 0.8311 0.8981 0.5013 0.6449 0.5360 0.7436

OLS 0.9762 0.9676 0.9767 0.9535 0.9414 0.9406 0.7789 0.9177 0.7948 0.8301 0.8953 0.6304 0.6423 0.6152 0.7545 rollover OLS 0.9807 0.9690 0.9783 0.9545 0.9419 0.9408 0.7822 0.9235 0.8029 0.8341 0.8973 0.6321 0.6632 0.6142 0.7646 return-based CCC 0.9849 0.9708 0.9769 0.9563 0.9504 0.9429 0.8289 0.9358 0.8410 0.8956 0.9140 0.7218 0.6836 0.6411 0.7670 return-based DCC 0.9849 0.9710 0.9768 0.9572 0.9507 0.9443 0.8432 0.9379 0.8673 0.9044 0.9215 0.7364 0.7558 0.7145 0.7812 range-based CCC 0.9877 0.9731 0.9812 0.9596 0.9576 0.9514 0.8225 0.9535 0.8779 0.8652 0.9213 0.6961 0.8175 0.7035 0.8090 range-based DCC 0.9878 0.9732 0.9806 0.9605 0.9575 0.9534 0.8454 0.9522 0.8659 0.8589 0.9319 0.7143 0.8539 0.7899 0.8202 Note: The number with an underline stands for the largest hedging effectiveness in each commodity column.

Panel C: Hedging improvement ratio by range-based DCC (1−Var_range_DCC/Var_other_model)

FTSE Nikkei SP BP Yen SF Gold Silver Corn Soy SO Coffee Cotton Sugar CL Averag naive 0.3098 0.1073 0.0715 0.1376 0.3148 0.2366 0.3582 0.3729 0.3318 0.1651 0.3315 0.4272 0.5886 0.5471 0.2987 0.3066

OLS 0.4868 0.1749 0.1664 0.1502 0.2736 0.2152 0.3006 0.4192 0.3464 0.1700 0.3493 0.2272 0.5916 0.4539 0.2676 0.3062 rollover OLS 0.3679 0.1357 0.1072 0.1307 0.2678 0.2122 0.2900 0.3754 0.3194 0.1499 0.3368 0.2235 0.5662 0.4553 0.2360 0.2783 return-based CCC 0.1916 0.0840 0.1593 0.0946 0.1417 0.1834 0.0964 0.2549 0.1561 -0.3505 0.2078 -0.0268 0.5383 0.4144 0.2282 0.1582 return-based DCC 0.1903 0.0760 0.1637 0.0754 0.1372 0.1625 0.0139 0.2307 -0.0106 -0.4748 0.1325 -0.0838 0.4019 0.2640 0.1780 0.0971 range-based CCC 0.0083 0.0067 -0.0300 0.0207 -0.0027 0.0399 0.1290 -0.0277 -0.0983 -0.0462 0.1343 0.0599 0.1995 0.2912 0.0586 0.0496

Panel A: S&P 500

0.7 0.8 0.9 1.0 1.1 1.2

2004 2005 2006

10-year rollover OLS return-based CCC range-based CCC

Panel B: British Pond

0.7 0.8 0.9 1.0 1.1 1.2 1.3

2004 2005 2006