The hedging effectiveness of gold against US stocks in a post-financial crisis era

Figures & data

Figure 1. Price charts.

Figure 2. Return charts.

Table 1. Descriptive statistics—returns

	DJIA	S&P500	NASDAQ	GOLD
Mean	0.000239	0.000221	0.000366	0.000249
Median	0.000343	0.000323	0.000642	0.000397
Maximum	0.105083	0.109572	0.111594	0.102451
Minimum	−0.082005	−0.094695	−0.095877	−0.095121
Standard Deviation	0.011413	0.012426	0.013247	0.011570
Q1	−0.003628	−0.003746	−0.004582	−0.005247
Q3	0.004944	0.005282	0.006423	0.006348
Skewness	−0.109898	−0.352199	−0.287250	−0.259790
Kurtosis	14.30235	14.42968	10.94828	9.503091
Jarque-Bera	15,276.39	15,675.95	7591.511	5087.712
Probability	0.000000	0.000000	0.000000	0.000000
ARCH LM	112.8262	114.2004	92.63897	17.27686
Prob. F(10, 2848)	0.000000	0.000000	0.000000	0.000000
Sum of Deviations	0.684805	0.633971	1.050193	0.715815
Sum of Sq. Deviations	0.373597	0.442843	0.503269	0.383896
Q(36)	121.09	118.93	72.548	35.666
Correlation with Gold	0.015824	0.032400	0.011829	1
N	2869	2869	2869	2869

Table 2. Ng-Perron unit root test

		MZa	MZt	MSB	MPT
Level
DJIA		−1581.59	−28.1203	0.01778	0.01588
S&P500		−1580.67	−28.1121	0.01778	0.01588
NASDAQ		−1425.41	−26.6954	0.01873	0.01782
GOLD		−34.4447	−4.14997	0.12048	0.71133
Asymptotic critical values*:	1%	−13.8000	−2.58000	0.17400	1.78000
	5%	−8.10000	−1.98000	0.23300	3.17000
	10%	−5.70000	−1.62000	0.27500	4.45000
First Difference
ΔDJIA		−0.02825	−0.05252	1.85913	174.992
ΔS&P500		−0.06804	−0.14029	2.06190	211.270
ΔNASDAQ		0.06024	0.12879	2.13784	234.798
ΔGOLD		0.30649	0.99586	3.24921	569.592
Asymptotic critical values*:	1%	−13.8000	−2.58000	0.17400	1.78000
	5%	−8.10000	−1.98000	0.23300	3.17000
	10%	−5.70000	−1.62000	0.27500	4.45000

*Critical values are acquired from Table 1 of Ng and Perron (2001).

Table 3. Maximum likelihood estimates of the VAR-ADCC-BVGARCH model

	DJIA	Gold	S&P500	Gold	NASDAQ	Gold
${\mu }_{io}$	0.000	0.000	0.000	0.000	0.000	0.000
${\mu }_{i1}$	−0.104***	0.019	−0.106***	0.026	−0.078***	0.021
${\mu }_{\mathrm{i}2}$	−0.039*	0.004	−0.039*	0.004	0.000	0.004
${\omega }_{\mathrm{i}\mathrm{o}}$	−0.000***	0.000	0.000***	0.000***	−0.000***	0.000
${\alpha }_{\mathrm{i}1}$	0.031	0.063*	0.008	0.059	0.025*	0.059
${\alpha }_{\mathrm{i}2}$	0.097***	−0.020	0.061***	−0.013	0.072***	−0.014
${\beta }_{\mathrm{i}}$	0.829***	0.949***	0.904***	0.945***	0.856***	0.946***
${\delta }_{\mathrm{i}}$	0.000***	0.000	−0.000***	−0.000	0.000***	−0.000
${\theta }_i$	0.039***	0.945***	0.039***	0.944***	0.035***	0.948***
$\nu$	9.416***		9.986***		8.793***
LLF	−3878.103		−3822.704		−3854.125
Q(36)	65.892***	33.424	76.153***	33.470	52.558**	33.672
Qs(36)	33.175	31.002	33.635	30.551	30.897	31.053
ARCH(10)	1.395	1.734*	1.377	1.659*	0.869	1.746*
Sign Bias	0.000***	0.000*	0.000***	0.000*	0.000***	0.000*
Negative Size	−0.014***	−0.006***	−0.014***	−0.006***	−0.013***	−0.006***
Positive Size	0.002**	0.000	0.002**	0.000	0.002*	0.000
Joint Test	13.939	17.301*	13.762	16.556*	8.706	17.417*

*, **, and *** indicate the levels of significance at 10%, 5%, and 1%, respectively. The log likelihood function is denoted by LLF. The Ljung-Box test statistics based on 36 lags for standardized and squared standardized residuals are captured by Q(36) and Qs(36) respectively. The Lagrange multiplier test for conditional heteroskedasticity based on 10 lags is represented by ARCH(10).

Figure 3. Dynamic conditional correlations.

Figure 4. Optimal hedge ratios.

Table 4. Descriptive statistics—optimal hedge ratio

	Mean	Median	Std. Dev.	Minimum	Maximum
DJIA−Gold	0.005203	0.001923	0.129539	−0.48532	0.389958
NASDAQ−Gold	0.016103	0.012201	0.147318	−0.54268	0.362656
S&P500− Gold	0.026573	0.02036	0.138294	−0.51833	0.420838
Gold−DJIA	0.017314	0.002536	0.200722	−0.48446	0.67518
Gold−NASDAQ	0.027341	0.010097	0.160639	−0.42636	0.487357
Gold−S&P500	0.044871	0.026029	0.189958	−0.44851	0.637435

Figure 5. Optimal portfolio weights.

Table 5. Descriptive statistics—optimal portfolio weights

	Mean	Median	Std. Dev.	Minimum	Maximum
DJIA−Gold	0.606288	0.614807	0.126401	0.249724	0.969561
NASDAQ−Gold	0.511704	0.510087	0.121271	0.205219	0.920058
S&P500− Gold	0.578738	0.592309	0.135102	0.215212	0.951741

Table 6. Hedging effectiveness (HE)—optimal hedge ratio

	Optimal Hedge Ratio			Static OLS Hedge Ratio
	DJIA	NASDAQ	S&P500	DJIA	NASDAQ	S&P500
$Varianc{e}_{unhedged}$	4.6868	2.5871	4.1791	5.2688	2.1702	5.6784
$Varianc{e}_{hedged}$	1.6153	2.2440	1.9462	1.5977	1.4706	1.8252
$HE$	0.6553	0.1326	0.5338	0.6967	0.3223	0.6785

Ng, S., & Perron, P. (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica, 69(6), 1519–1554. doi:10.1111/ecta.2001.69.issue-6

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