Futures Hedge Ratios: A Review

Abstract

This paper presents a review of different theoretical approaches to the optimal futures hedge ratios. These approaches are based on minimum variance, mean-variance, expected utility, mean extended-Gini coefficient, semivariance and Value-at-Risk. Various ways of estimating these hedge ratios are also discussed, ranging from simple ordinary least squares to complicated heteroscedastic cointegration methods. Under martingale and joint-normality conditions, different hedge ratios are the same as the minimum variance hedge ratio. Otherwise, the optimal hedge ratios based on the different approaches are in general different and there is no single optimal hedge ratio that is distinctly superior to the remaining ones.

This article is an updated version of the article entitled “Futures hedge ratios: a review” published in the Quarterly Review of Economics and Finance (Sheng-Syan Chen, Cheng-few Lee, and Keshab Shrestha, Vol. 43, 2003, pp. 433–465). We would like to thank the editor of the Quarterly Review of Economics and Finance, Professor H.S. Esfahani, and Elsevier for permission to reprint the paper.

Appendices

Appendix A: Theoretical Models

References	Return definition and objective function	Summary
Johnson (1960)	Ret1	The paper derives the minimum-variance hedge ratio. The hedging effectiveness is defined as E1, but no empirical analysis is done
	O1
Hsin et al. (1994)	Ret2	The paper derives the utility function-based hedge ratio. A new measure of hedging effectiveness E2 based on a certainty equivalent is proposed. The new measure of hedging effectiveness is used to compare the effectiveness of futures and options as hedging instruments
	O2
Howard and D’Antonio (1984)	Ret2	The paper derives the optimal hedge ratio based on maximizing the Sharpe ratio. The proposed hedging effectiveness E3 is based on the Sharpe ratio
	O3
Cecchetti et al. (1988)	Ret2	The paper derives the optimal hedge ratio that maximizes the expected utility function: $\int\limits_{{{{R}_s}}} {\int\limits_{{{{R}_f}}} {\log \left[ {1 + {{R}_s}(t) - h(t){{R}_f}(t)} \right]{{f}_t}\left( {{{R}_s},{{R}_f}} \right)d{{R}_s}d{{R}_f}} }$, where the density function is assumed to be bivariate normal. A third-order linear bivariate ARCH model is used to get the conditional variance and covariance matrix. A numerical procedure is used to maximize the objective function with respect to hedge ratio. Due to ARCH, the hedge ratio changes over time. The paper uses certainty equivalent (E2) to measure the hedging effectiveness
	O4
Cheung et al. (1990)	Ret2	The paper uses mean-Gini (v = 2, not mean extended-Gini coefficient) and mean-variance approaches to analyze the effectiveness of options and futures as hedging instruments
	O5
Kolb and Okunev (1992)	Ret2	The paper uses mean extended-Gini coefficient in the derivation of the optimal hedge ratio. Therefore, it can be considered as a generalization of the mean-Gini coefficient method used by Cheung et al. (1990)
	O5
Kolb and Okunev (1993)	Ret2	The paper defines the objective function as O6, but in terms of wealth (W) $U(W) = E\left[ W \right] - {{\Gamma }_v}(W)$ and compares with the quadratic utility function $U(W) = E\left[ W \right] - m{{\sigma }^2}$. The paper plots the EMG efficient frontier in W and ${{\Gamma }_v}(W)$ space for various values of risk aversion parameters (v)
	O6
Lien and Luo (1993b)	Ret1	The paper derives the multi-period hedge ratios where the hedge ratios are allowed to change over the hedging period. The method suggested in the paper still falls under the minimum-variance hedge ratio
	O9
Lence (1995)	O4	This paper derives the expected utility maximizing hedge ratio where the terminal wealth depends on the return on a diversified portfolio that consists of the production of a spot commodity, investment in a risk-free asset, investment in a risky asset, as well as borrowing. It also incorporates the transaction costs
De Jong et al. (1997)	Ret2	The paper derives the optimal hedge ratio that minimizes the generalized semivariance (GSV). The paper compares the GSV hedge ratio with the minimum-variance (MV) hedge ratio as well as the Sharpe hedge ratio. The paper uses E1 (for the MV hedge ratio), E3 (for the Sharpe hedge ratio) and E4 (for the GSV hedge ratio) as the measures of hedging effectiveness
	O7 (also uses O1 and O3)
Chen et al. (2001)	Ret1	The paper derives the optimal hedge ratio that maximizes the risk-return function given by $U\left( {{{R}_h}} \right) = E\left[ {{{R}_h}} \right] - {{V}_{{\delta, \alpha }}}\left( {{{R}_h}} \right)$. The method can be considered as an extension of the GSV method used by De Jong et al. (1997)
	O8
Hung et al. (2006)	Ret2	The paper derives the optimal hedge ratio that minimizes the Value-at-Risk for a hedging horizon of length $\tau$given by ${{Z}_{\alpha }}{{\sigma }_h}\sqrt {\tau } - E\left[ {{{R}_h}} \right]\tau$
	O10

1. Notes
2. A. Return model
3. (Ret₁) $\Delta {{V}_H} = {{C}_s}\Delta {{P}_s} + {{C}_f}\Delta {{P}_f} \Rightarrow {\text{hedge ratio = }}H = \frac{{{{C}_f}}}{{{{C}_s}}},{{C}_s} = {\text{units of spot\ \rm commodity\ and\ }}{{C}_f}{\text{ = units of futures contract }}$
4. (Ret₂) ${{R}_h} = {{R}_s} + h{{R}_f},$ ${{R}_s} = \frac{{{{S}_t} - {{S}_{{t - 1}}}}}{{{{S}_{{t - 1}}}}}$ (a) ${{R}_f} = \frac{{{{F}_t} - {{F}_{{t - 1}}}}}{{{{F}_{{t - 1}}}}} \Rightarrow {\text{hedge ratio: }}h = \frac{{{{C}_f}{{F}_{{t - 1}}}}}{{{{C}_s}{{S}_{{t - 1}}}}}$
5. (b) ${{R}_f} = \frac{{{{F}_t} - {{F}_{{t - 1}}}}}{{{{S}_{{t - 1}}}}} \Rightarrow {\text{ hedge ratio: }}h = \frac{{{{C}_f}}}{{{{C}_s}}}$.
6. B. Objective function
7. (O₁) Minimize $Var\left( {{{R}_h}} \right) = C_s^2\sigma_s^2 + C_f^2\sigma_f^2 + 2{{C}_s}{{C}_f}{{\sigma }_{{sf}}}{\text{ or }}Var\left( {{{R}_h}} \right) = \sigma_s^2 + {{h}^2}\sigma_f^2 + 2h{{\sigma }_{{sf}}}$
8. (O₂) Maximize $E\left( {{{R}_h}} \right) - \frac{A}{2}Var\left( {{{R}_h}} \right)$
9. (O₃) Maximize $\frac{{E\left( {{{R}_h}} \right) -{{R}_F}}}{{Var\left( {{{R}_h}} \right)}}({\text{Sharpe ratio),}}{{\text{R}}_{\rm{F}}} =\rm risk - free interest rate$
10. (O₄) Maximize $E\left[ {U(W)} \right],U\left(. \right){\text{ = utility function, W}} = {\text{ terminal wealth}}$
11. (O₅) Minimize ${{\Gamma }_v}\left( {{{R}_h}} \right),{{\Gamma }_v}\left( {{{R}_h}} \right) = - vCov\left( {{{R}_h},{{{\left( {1 - F\left( {{{R}_h}} \right)} \right)}}^{{v - 1}}}} \right){ }$
12. (O₆) Maximize $E\left[ {{{R}_h}} \right] - {{\Gamma }_v}\left( {{{R}_h}v} \right)$
13. (O₇) Minimize ${{V}_{{\delta, \alpha }}}\left( {{{R}_h}} \right) = \int_{{ - \infty }}^{\delta } {{{{\left( {\delta - {{R}_h}} \right)}}^{\alpha }}dG\left( {{{R}_h}} \right)}, \alpha > 0$
14. (O₈) Maximize $U\left( {{{R}_h}} \right) = E\left[ {{{R}_h}} \right] - {{V}_{{\delta, \alpha }}}\left( {{{R}_h}} \right)$
15. (O₉) Minimize $Var\left( {{{W}_t}} \right) = Var\left( {\sum\limits_{{t = 1}}^T {{{C}_{{st}}}\Delta {{S}_t} + {{C}_{{ft}}}\Delta {{F}_t}} } \right)$.
16. (O₁₀) Minimize ${{Z}_{\alpha }}{{\sigma }_h}\sqrt {\tau } - E\left[ {{{R}_h}} \right]\tau$
17. C. Hedging effectiveness
18. (E₁) $e = \left( {1 - \frac{{Var\left( {{{R}_h}} \right)}}{{Var\left( {{{R}_s}} \right)}}} \right)$
19. (E₂) $e = R_h^{{ce}} - R_{{ss}}^{{ce}},R_h^{{ce}}(R_s^{{ce}}) = {\text{ certainty equivalent return of hedged (unhedged) portfolio}}$
20. (E₃) $e = \frac{{\displaystyle\frac{{\left( {E\left[ {{{R}_h}} \right] - {{R}_F}} \right)}}{{Var\left( {{{R}_h}} \right)}}}}{\displaystyle{\frac{{\left( {E\left[ {{{R}_s}} \right] - {{R}_F}} \right)}}{{Var\left( {{{R}_s}} \right)}}}}$ or $e = \displaystyle\frac{{\left( {E\left[ {{{R}_h}} \right] - {{R}_F}} \right)}}{{Var\left( {{{R}_h}} \right)}} - \frac{{\left( {E\left[ {{{R}_s}} \right] - {{R}_F}} \right)}}{{Var\left( {{{R}_s}} \right)}}$
21. (E₄) $e = 1 - \displaystyle\frac{{{{V}_{{\delta, \alpha }}}\left( {{{R}_h}} \right)}}{{{{V}_{{\delta, \alpha }}}\left( {{{R}_s}} \right)}}$.

Appendix B: Empirical Models

References	Commodity	Summary
Ederington (1979)	GNMA futures (1/1976–12/1977), Wheat (1/1976–12/1977), Corn (1/1976–12/1977), T-bill futures (3/1976–12/1977) [weekly data]	The paper uses the Ret1 definition of return and estimates the minimum- variance hedge ratio (O1). E1 is used as a hedging effectiveness measure. The paper uses nearby contracts (3–6 months, 6–9 months and 9–12 months) and a hedging period of 2 weeks and 4 weeks. OLS (M1) is used to estimate the parameters. Some of the hedge ratios are found not to be different from zero and the hedging effectiveness increases with the length of hedging period. The hedge ratio also increases (closer to unity) with the length of hedging period
Grammatikos and Saunders (1983)	Swiss franc, Canadian dollar, British pound, DM, Yen (1/1974–6/1980) [weekly data]	The paper estimates the hedge ratio for the whole period and moving window (2-year data). It is found that the hedge ratio changes over time. Dummy variables for various sub-periods are used, and shifts are found. The paper uses a random coefficient (M3) model to estimate the hedge ratio. The hedge ratio for Swiss franc is found to follow a random coefficient model. However, there is no improvement in effectiveness when the hedge ratio is calculated by correcting for the randomness
Junkus and Lee (1985)	Three stock index futures for Kansas City Board of Trade, New York Futures Exchange, and Chicago Mercantile Exchange (5/82–3/83) [daily data]	The paper tests the applicability of four futures hedging models: a variance-minimizing model introduced by Johnson (1960), the traditional one to one hedge, a utility maximization model developed by Rutledge (1972), and a basis arbitrage model suggested by Working (1953). An optimal ratio or decision rule is estimated for each model, and measures for the effectiveness of each hedge are devised. Each hedge strategy performed best according to its own criterion. The Working decision rule appeared to be easy to use and satisfactory in most cases. Although the maturity of the futures contract used affected the size of the optimal hedge ratio, there was no consistent maturity effect on performance. Use of a particular ratio depends on how closely the assumptions underlying the model approach a hedger’s real situation
Lee et al. (1987)	S&P 500, NYSE, Value Line (1983) [daily data]	The paper tests for the temporal stability of the minimum-variance hedge ratio. It is found that the hedge ratio increases as maturity of the futures contract nears. The paper also performs a functional form test and finds support for the regression of rate of change for discrete as well as continuous rates of change in prices
Cecchetti et al. (1988)	Treasury bond, Treasury bond futures (1/1978–5/1986) [monthly data]	The paper derives the hedge ratio by maximizing the expected utility. A third-order linear bivariate ARCH model is used to get the conditional variance and covariance matrix. A numerical procedure is used to maximize the objective function with respect to the hedge ratio. Due to ARCH, the hedge ratio changes over time. It is found that the hedge ratio changes over time and is significantly less (in absolute value) than the minimum-variance (MV) hedge ratio (which also changes over time). E2 (certainty equivalent) is used to measure the performance effectiveness. The proposed utility-maximizing hedge ratio performs better than the MV hedge ratio
Cheung et al. (1990)	Swiss franc, Canadian dollar, British pound, German mark, Japanese yen (9/1983–12/1984) [daily data]	The paper uses mean-Gini coefficient (v = 2) and mean-variance approaches to analyze the effectiveness of options and futures as hedging instruments. It considers both mean-variance and expected-return mean-Gini coefficient frontiers. It also considers the minimum-variance (MV) and minimum mean-Gini coefficient hedge ratios. The MV and minimum mean-Gini approaches indicate that futures is a better hedging instrument. However, the mean-variance frontier indicates futures to be a better hedging instrument whereas the mean-Gini frontier indicates options to be a better hedging instrument
Baillie and Myers (1991)	Beef, Coffee, Corn, Cotton, Gold, Soybean (contracts maturing in 1982 and 1986) [daily data]	The paper uses a bivariate GARCH model (M2) in estimating the minimum-variance (MV) hedge ratios. Since the models used are conditional models, the time series of hedge ratios are estimated. The MV hedge ratios are found to follow a unit root process. The hedge ratio for beef is found to be centered around zero. E1 is used as a hedging effectiveness measure. Both in-sample and out-of-sample effectiveness of the GARCH-based hedge ratios is compared with a constant hedge ratio. The GARCH-based hedge ratios are found to be significantly better compared to the constant hedge ratio
Malliaris and Urrutia (1991)	British pound, German mark, Japanese yen, Swill franc, Canadian dollar (3/1980–12/1988) [weekly data]	The paper uses regression autocorrelated errors model to estimate the minimum-variance (MV) hedge ratio for the five currencies. Using overlapping moving windows, the time series of the MV hedge ratio and hedging effectiveness are estimated for both ex post (in-sample) and ex ante (out-of-sample) cases. E1 is used to measure the hedging effectiveness for the ex post case whereas average return is used to measure the hedging effectiveness. Specifically, the average return close to zero is used to indicate a better performing hedging strategy. In the ex post case, the 4-week hedging horizon is more effective compared to the 1-week hedging horizon. However, for the ex ante case the opposite is found to be true
Benet (1992)	Australian dollar, Brazilian cruzeiro, Mexican peso, South African rand, Chinese yuan, Finish markka, Irish pound, Japanese yen (8/1973–12/1985) [weekly data]	This paper considers direct and cross hedging, using multiple futures contracts. For minor currencies, the cross hedging exhibits a significant decrease in performance from ex post to ex ante. The minimum-variance hedge ratios are found to change from one period to the other except for the direct hedging of Japanese yen. On the ex ante case, the hedging effectiveness does not appear to be related to the estimation period length. However, the effectiveness decreases as the hedging period length increases
Kolb and Okunev (1992)	Corn, Copper, Gold, German mark, S&P 500 (1989) [daily data]	The paper estimates the mean extended-Gini (MEG) hedge ratio (M9) with v ranging from 2 to 200. The MEG hedge ratios are found to be close to the minimum-variance hedge ratios for a lower level of risk parameter v (for v from 2 to 5). For higher values of v, the two hedge ratios are found to be quite different. The hedge ratios are found to increase with the risk aversion parameter for S&P 500, Corn, and Gold. However, for Copper and German mark, the hedge ratios are found to decrease with the risk aversion parameter. The hedge ratio tends to be more stable for higher levels of risk
Kolb and Okunev (1993)	Cocoa (3/1952–1976) for four cocoa-producing countries (Ghana, Nigeria, Ivory Coast, and Brazil) [March and September data]	The paper estimates the Mean-MEG (M-MEG) hedge ratio (M12). The paper compares the M-MEG hedge ratio, minimum-variance hedge ratio, and optimum mean-variance hedge ratio for various values of risk aversion parameters. The paper finds that the M-MEG hedge ratio leads to reverse hedging (buy futures instead of selling) for v less than 1.24 (Ghana case). For high-risk aversion parameter values (high v) all hedge ratios are found to converge to the same value
Lien and Luo (1993a)	S&P 500 (1/1984–12/1988) [weekly data]	The paper points out that the mean extended-Gini (MEG) hedge ratio can be calculated either by numerically optimizing the MEG coefficient or by numerically solving the first-order condition. For v = 9 the hedge ratio of −0.8182 is close to the minimum-variance (MV) hedge ratio of −0.8171. Using the first-order condition, the paper shows that for a large v the MEG hedge ratio converges to a constant. The empirical result shows that the hedge ratio decreases with the risk aversion parameter v. The paper finds that the MV and MEG hedge ratio (for low v) series (obtained by using a moving window) are more stable compared to the MEG hedge ratio for a large v. The paper also uses a non-parametric Kernel estimator to estimate the cumulative density function. However, the kernel estimator does not change the result significantly
Lien and Luo (1993b)	British pound, Canadian dollar, German mark, Japanese yen, Swiss franc (3/1980–12/1988), MMI, NYSE, S&P (1/1984–12/1988) [weekly data]	This paper proposes a multi-period model to estimate the optimal hedge ratio. The hedge ratios are estimated using an error-correction model. The spot and futures prices are found to be cointegrated. The optimal multi-period hedge ratios are found to exhibit a cyclical pattern with a tendency for the amplitude of the cycles to decrease. Finally, the possibility of spreading among different market contracts is analyzed. It is shown that hedging in a single market may be much less effective than the optimal spreading strategy
Ghosh (1993)	S&P futures, S&P index, Dow Jones Industrial average, NYSE composite index (1/1990–12/1991) [daily data]	All the variables are found to have a unit root. For all three indices the same S&P 500 futures contracts are used (cross hedging). Using the Engle-Granger two-step test, the S&P 500 futures price is found to be cointegrated with each of the three spot prices: S&P 500, DJIA, and NYSE. The hedge ratio is estimated using the error-correction model (ECM) (M4). Out-of-sample performance is better for the hedge ratio from the ECM compared to the Ederington model
Sephton (1993a)	Feed wheat, Canola futures (1981–82 crop year) [daily data]	The paper finds unit roots on each of the cash and futures (log) prices, but no cointegration between futures and spot (log) prices. The hedge ratios are computed using a four-variable GARCH(1,1) model. The time series of hedge ratios are found to be stationary. Reduction in portfolio variance is used as a measure of hedging effectiveness. It is found that the GARCH-based hedge ratio performs better compared to the conventional minimum-variance hedge ratio
Sephton (1993b)	Feed wheat, Feed barley, Canola futures (1988/89) [daily data]	The paper finds unit roots on each of the cash and futures (log) prices, but no cointegration between futures and spot (log) prices. A univariate GARCH model shows that the mean returns on the futures are not significantly different from zero. However, from the bivariate GARCH canola is found to have a significant mean return. For canola the mean variance utility function is used to find the optimal hedge ratio for various values of the risk aversion parameter. The time series of the hedge ratio (based on bivariate GARCH model) is found to be stationary. The benefit in terms of utility gained from using a multivariate GARCH decreases as the degree of risk aversion increases
Kroner and Sultan (1993)	British pound, Canadian dollar, German mark, Japanese yen, Swiss franc (2/1985–2/1990) [weekly data]	The paper uses the error-correction model with a GARCH error (M5) to estimate the minimum-variance (MV) hedge ratio for the five currencies. Due to the use of conditional models, the time series of the MV hedge ratios are estimated. Both within-sample and out-of-sample evidence shows that the hedging strategy proposed in the paper is potentially superior to the conventional strategies
Hsin et al. (1994)	British pound, German mark, Yen, Swiss franc (1/1986–12/1989) [daily data]	The paper derives the optimum mean-variance hedge ratio by maximizing the objective function O2. The hedging horizons of 14, 30, 60, 90, and 120 calendar days are considered to compare the hedging effectiveness of options and futures contracts. It is found that the futures contracts perform better than the options contracts
Shalit (1995)	Gold, silver, copper, aluminum (1/1977–12/1990) [daily data]	The paper shows that if the prices are jointly normally distributed, the mean extended-Gini (MEG) hedge ratio will be same as the minimum-variance (MV) hedge ratio. The MEG hedge ratio is estimated using the instrumental variable method. The paper performs normality tests as well as the tests to see if the MEG hedge ratios are different from the MV hedge ratios. The paper finds that for a significant number of futures contracts the normality does not hold and the MEG hedge ratios are different from the MV hedge ratios
Geppert (1995)	German mark, Swiss franc, Japanese yen, S&P 500, Municipal Bond Index (1/1990–1/1993) [weekly data]	The paper estimates the minimum-variance hedge ratio using the OLS as well as the cointegration methods for various lengths of hedging horizon. The in-sample results indicate that for both methods the hedging effectiveness increases with the length of the hedging horizon. The out-of-sample results indicate that in general the effectiveness (based on the method suggested by Malliaris and Urrutia (1991)) decreases as the length of the hedging horizon decreases. This is true for both the regression method and the decomposition method proposed in the paper. However, the decomposition method seems to perform better than the regression method in terms of both mean and variance
De Jong et al. (1997)	British pound (12/1976–10/1993), German mark (12/1976–10/1993), Japanese yen (4/1977–10/1993) [daily data]	The paper compares the minimum-variance, generalized semivariance and Sharpe hedge ratios for the three currencies. The paper computes the out-of-sample hedging effectiveness using non-overlapping 90-day periods where the first 60 days are used to estimate the hedge ratio and the remaining 30 days are used to compute the out-of-sample hedging effectiveness. The paper finds that the naïve hedge ratio performs better than the model based hedge ratios
Lien and Tse (1998)	Nikkei stock average (1/1989–8/1996) [daily data]	The paper shows that if the rates of change in spot and futures prices are bivariate normal and if the futures price follows a martingale process, then the generalized semivariance (GSV) (referred to as lower partial moment) hedge ratio will be same as the minimum-variance (MV) hedge ratio. A version of the bivariate asymmetric power ARCH model is used to estimate the conditional joint distribution, which is then used to estimate the time varying GSV hedge ratios. The paper finds that the GSV hedge ratio significantly varies over time and is different from the MV hedge ratio
Lien and Shaffer (1999)	Nikkei (9/86–9/89), S&P (4/82–4/85), TOPIX (4/90–12/93), KOSPI (5/96–12/96), Hang Seng (1/87–12189), IBEX (4/93–3/95) [daily data]	This paper empirically tests the ranking assumption used by Shalit (1995). The ranking assumption assumes that the ranking of futures prices is the same as the ranking of the wealth. The paper estimates the mean extended-Gini (MEG) hedge ratio based on the instrumental variable (IV) method used by Shalit (1995) and the true MEG hedge ratio. The true MEG hedge ratio is computed using the cumulative probability distribution estimated employing the kernel method instead of the rank method. The paper finds that the MEG hedge ratio obtained from the IV method to be different from the true MEG hedge ratio. Furthermore, the true MEG hedge ratio leads to a significantly smaller MEG coefficient compared to the IV-based MEG hedge ratio
Lien and Tse (2000)	Nikkei stock average (1/1988–8/996) [daily data]	The paper estimates the generalized semivariance (GSV) hedge ratios for different values of parameters using a non-parametric kernel estimation method. The kernel method is compared with the empirical distribution method. It is found that the hedge ratio from one method is not different from the hedge ratio from another. The Jarque-Bera (1987) test indicates that the changes in spot and futures prices do not follow normal distribution
Chen et al. (2001)	S&P 500 (4/1982–12/1991) [weekly data]	The paper proposes the use of the M-GSV hedge ratio. The paper estimates the minimum-variance (MV), optimum mean-variance, Sharpe, mean extended-Gini (MEG), generalized semivariance (GSV), mean-MEG (M-MEG), and mean-GSV (M-GSV) hedge ratios. The Jarque-Bera (1987) Test and D’Agostino (1971) D statistic indicate that the price changes are not normally distributed. Furthermore, the expected value of the futures price change is found to be significantly different from zero. It is also found that for a high level of risk aversion, the M-MEG hedge ratio converges to the MV hedge ratio whereas the M-GSV hedge ratio converges to a lower value
Hung et al. (2006)	S&P 500 (01/1997–12/1999) [daily data]	The paper proposes minimization of Value-at-Risk in deriving the optimum hedge ratio. The paper finds cointegrating relationship between the spot and futures returns and uses bivariate constant correlation GARCH(1,1) model with error correction term. The paper compares the proposed hedge ratio with MV hedge ratio and hedge ratio (HKL hedge ratio) proposed by Hsin et al. (1994). The paper finds the performance of the proposed hedge ratio to be similar to the HKL hedge ratio. Finally, the proposed hedge ratio converges to the MV hedge ratio for high risk-averse levels
Lee and Yoder (2007)	Nikkei 225 and Hang Send index futures (01/1989–12/2003) [weekly data]	The paper proposes regime-switching time varying correlation GARCH model and compares the resulting hedge ratio with constant correlation GARCH and time-varying correlation GARCH. The proposed model is found to outperforms the other two hedge ratio in both in-sample and out-of-sample for both contracts
Lien and Shrestha (2007)	23 different futures contracts (sample period depends on contracts) [daily data]	This paper proposes wavelet base hedge ratio to compute the hedge ratios for different hedging horizons (1-day, 2-day, 4-day, 8-day, 16 day, 32-day, 64-day, 128-day; and 256-day and longer). It is found that the wavelet based hedge ratio and the error-correction based hedge ratio are larger than MV hedge ratio. The performance of wavelet based hedge ratio improves with the length of hedging horizon
Lien and Shrestha (2010)	22 different futures contracts (sample period depends on contracts) [daily data]	The paper proposes the hedge ratio based on skew-normal distribution (SKN hedge ratio). The paper also estimates the semi-variance (lower partial moment (LPM)) hedge ratio and MV hedge ratio among other hedge ratios. SKN hedge ratios are found to be different from the MV hedge ratio based on normal distribution. SKN hedge ratio performs better than LPM hedge ratio for long hedger especially for the out-of-sample cases

1. Notes
2. A. Minimum-variance hedge ratio
3. A.1. OLS
4. (M₁): $\Delta {{S}_t} = {{a}_0} + {{a}_1}\Delta {{F}_t} + {{e}_t}$:  Hedge ratio = ${{a}_1}$
5.     ${{R}_s} = {{a}_0} + {{a}_1}{{R}_f} + {{e}_t}$:  Hedge ratio = ${{a}_1}$
6. A.2.Multivariate skew-normal
7. (M₂): The return vector $Y = \left[ {\begin{array}{lll} {{{R}_s}} \\{{{R}_f}} \\\end{array} } \right]$ is assumed to have skew-normal distribution with covariance matrix V:
8.    Hedge ration = ${{H}_{{skn}}} = \frac{{V\left( {1,2} \right)}}{{V\left( {2,2} \right)}}$
9. A.3. ARCH/GARCH
10. (M₃): $\left[ {\begin{array}{lll} {\Delta {{S}_t}} \\{\Delta{{F}_t}} \\\end{array} } \right] = \left[ {\begin{array}{lll} {{{\mu}_1}} \\{{{\mu }_2}} \\\end{array} } \right] + \left[ {\begin{array}{lll}{{{e}_{{1t}}}} \\{{{e}_{{2t}}}} \\\end{array} } \right]$, ${{e}_t}|{{\Omega }_{{t - 1}}}\sim N\left( {0,{{H}_t}}\right),{{H}_t} = \left[ {\begin{array}{lll} {{{H}_{{11,t}}}} &{{{H}_{{12,t}}}} \\{{{H}_{{12,t}}}} & {{{H}_{{22,t}}}} \\\end{array}} \right]$, Hedge ratio $= {{H}_{{12,t}}}/{{H}_{{22,t}}}$
11. A.4. Regime-switching GARCH
12. (M₄): The transition probabilities are given by
13.   $\Pr \left( {{{s}_t} = 1|{{s}_{{t - 1}}} = 1} \right) = \frac{{{{e}^{{{{p}_0}}}}}}{{1 + {{e}^{{{{p}_0}}}}}}\& \Pr \left( {{{s}_t} = 2|{{s}_{{t - 1}}} = 2} \right) = \frac{{{{e}^{{{{q}_0}}}}}}{{1 + {{e}^{{{{q}_0}}}}}}$
14.   The GARCH model: Two-series GARCH model with first series as return on futures.
15.   ${{H}_{{t,{{s}_t}}}} = \left[ {\begin{array}{lll}{h_{{1,t,{{s}_t}}}} & 0 \\0 & {{{h}_{{2,t,{{s}_t}}}}} \\\end{array}} \right]\left[ {\begin{array}{lll} 1 & {{{\rho }_{{t,{{s}_t}}}}}\\{{{\rho }_{{t,{{s}_t}}}}} & 1 \\\end{array} } \right]\left[{\begin{array}{lll} {h_{{1,t,{{s}_t}}}} & 0 \\0 & {{{h}_{{2,t,{{s}_t}}}}}\\\end{array} } \right]$
16.   $h_{{1,t,{{s}_t}}}^2 = {{\gamma }_{{1,{{s}_t}}}} + {{\alpha }_{{1,{{s}_t}}}}e_{{1.t - 1}}^2 + {{\beta }_{{1,{{s}_t}}}}h_{{1,t - 1}}^2$, $h_{{2,t,{{s}_t}}}^2 = {{\gamma }_{{2,{{s}_t}}}} + {{\alpha }_{{2,{{s}_t}}}}e_{{2.t - 1}}^2 + {{\beta }_{{2,{{s}_t}}}}h_{{2,t - 1}}^2$
17.   ${{\rho }_{{t,{{s}_t}}}} = \left( {1 - {{\theta }_{{1,{{s}_t}}}} - {{\theta }_{{2,{{s}_t}}}}} \right)\rho + {{\theta }_{{1,{{s}_t}}}}{{\rho }_{{t - 1}}} + {{\theta }_{{2,{{s}_t}}}}{{\varphi }_{{t - 1}}}$
18.   ${{\phi }_{{t - 1}}} = \frac{{\sum\limits_{{j = 1}}^2 {{{\varepsilon }_{{1,t - j}}}{{\varepsilon }_{{2,t - j}}}} }}{{\sqrt {{\left( {\sum\limits_{{j = 1}}^2 {\varepsilon_{{1,t - j}}^2} } \right)\left( {\sum\limits_{{j = 1}}^2 {\varepsilon_{{2,t - j}}^2} } \right)}} }},{{\varepsilon }_{{i,t}}} = \frac{{{{e}_{{i,t}}}}}{{{{h}_{{it}}}}},{{\theta }_1},{{\theta }_2} \geq 0\& {{\theta }_1} + {{\theta }_2} \leq 1$,  Hedge ratio = $\frac{{{{H}_{{t,{{s}_t}}}}\left( {1,2} \right)}}{{{{H}_{{t,{{s}_t}}}}\left( {2,2} \right)}}$
19. A.5. Random coefficient
20. (M₅): $\Delta {{S}_t} = {{\beta }_0} + {{\beta }_t}\Delta {{F}_t} + {{e}_t}$
21.    ${{\beta }_t} = \bar{\beta } + {{v}_t}$ ,  Hedge ratio = $\bar{\beta }$
22. A.6. Cointegration and error-correction
23. (M₆): ${{S}_t} = a + b{{F}_t} + {{u}_t}$
24.    $\Delta {{S}_t} = \rho {{u}_{{t - 1}}} + \beta \Delta {{F}_t} + \sum\limits_{{i = 1}}^m {{{\delta }_i}\Delta {{F}_{{t - i}}}} + \sum\limits_{{j = 1}}^n {{{\theta }_i}\Delta {{S}_{{t - j}}}} + {{e}_j}$,  EC Hedge ratio = $\beta$
25. A.7. Error-correction with GARCH
26. (M₇): $\left[ {\begin{array}{lll} {\Delta {{{\log }}_e}\left( {{{S}_t}}\right)} \\{\Delta {{{\log }}_e}\left( {{{F}_t}} \right)}\\\end{array} } \right] = \left[ {\begin{array}{lll} {{{\mu }_1}}\\{{{\mu }_2}} \\\end{array} } \right] + \left[ {\begin{array}{lll}{{{\alpha }_s}\left( {{{{\log }}_e}\left( {{{S}_{{t - 1}}}} \right)- {{{\log }}_e}\left( {{{F}_{{t - 1}}}} \right)} \right)}\\{{{\alpha }_f}\left( {{{{\log }}_e}\left( {{{S}_{{t - 1}}}}\right) - {{{\log }}_e}\left( {{{F}_{{t - 1}}}} \right)} \right)}\\\end{array} } \right] + \left[ {\begin{array}{lll} {{{e}_{{1t}}}}\\{{{e}_{{2t}}}} \\\end{array} } \right]$, ${{e}_t}|{{\Omega }_{{t - 1}}}\sim N\left( {0,{{H}_t}} \right),{{H}_t} = \left[ {\begin{array}{lll} {{{H}_{{11,t}}}} & {{{H}_{{12,t}}}} \\{{{H}_{{12,t}}}} & {{{H}_{{22,t}}}} \\\end{array} } \right]$
27.    Hedge ratio = ${{h}_{{t - 1}}} = {{H}_{{12,t}}}/{{H}_{{22,t}}}$
28. A.8. Common stochastic trend
29. (M₈): ${{S}_t} = {{A}_1}{{P}_t} + {{A}_2}{{\tau }_t}$,  ${{F}_t} = {{B}_1}{{P}_t} + {{B}_2}{{\tau }_t}$,  ${{P}_t} = {{P}_{{t - 1}}} + {{w}_t}$,  ${{\tau }_t} = {{\alpha }_1}{{\tau }_{{t - 1}}} + {{v}_t},0 \leq \left| {{{\alpha }_1}} \right| < 1,$
30.    Hedge ratio for k-period investment horizon = $H_J^{*} = \frac{{{{A}_1}{{B}_1}k\sigma_w^2 + 2{{A}_2}{{B}_2}\left( {\frac{{\left( {1 - {{\alpha }^k}} \right)}}{{1 - {{\alpha }^2}}}} \right)\sigma_v^2}}{{B_1^2k\sigma_w^2 + 2B_2^2\left( {\frac{{\left( {1 - {{\alpha }^k}} \right)}}{{1 - {{\alpha }^2}}}} \right)\sigma_v^2}}.$
31. B. Optimum mean-variance hedge ratio
32. (M₉): Hedge ratio = ${{h}_2} = - \frac{{C_f^{*}F}}{{{{C}_s}S}} = - \left[ {\frac{{E\left( {{{R}_f}} \right)}}{{A\sigma_f^2}} - \rho \frac{{{{\sigma }_s}}}{{{{\sigma }_f}}}} \right]$, where the moments $E\left[ {{{R}_f}} \right],{{\sigma }_s}{\text{and}}{{\sigma }_f}$ are estimated by sample moments
33. C. Sharpe hedge ratio
34. (M₁₀): Hedge ratio = ${{h}_3} = - \frac{{\left( {\frac{{{{\sigma }_s}}}{{{{\sigma }_f}}}} \right)\left[ {\frac{{{{\sigma }_s}}}{{{{\sigma }_f}}}\left( {\frac{{E\left( {{{R}_f}} \right)}}{{E\left( {{{R}_s}} \right) - i}}} \right) - \rho } \right]}}{{\left[ {1 - \frac{{{{\sigma }_s}}}{{{{\sigma }_f}}}\left( {\frac{{E\left( {{{R}_f}} \right)\rho }}{{E\left( {{{R}_s}} \right) - i}}} \right)} \right]}}$, where the moments and correlation are estimated by their sample counterparts
35. D. Mean-Gini coefficient based hedge ratios
36. (M₁₁): The hedge ratio is estimated by numerically minimizing the following mean extended-Gini coefficient, where the cumulative probability distribution function is estimated using the rank function
37. $\Delta S_t = \rho u_{t-1} + \beta \Delta F_t + \sum\nolimits^m_{i=1}i\Delta F_{t-1}+ \sum\nolimits^n_{j=1}\phi_i\Delta S_{t-j}+ e_j,$ EC Hedge ratio = β
38. (M₁₂): The hedge ratio is estimated by numerically solving the first-order condition, where the cumulative probability distribution function is estimated using the rank function
39. (M₁₃): The hedge ratio is estimated by numerically solving the first-order condition, where the cumulative probability distribution function is estimated using the kernel-based estimates
40. (M₁₄): The hedge ratio is estimated by numerically maximizing the following function
41.    $U\left( {{{R}_h}} \right) = E\left( {{{R}_h}} \right) - {{\Gamma }_v}\left( {{{R}_h}} \right),$
42.    where the expected values and the mean extended-Gini coefficient are replaced by their sample counterparts and the cumulative probability distribution function is estimated using the rank function
43. E. Generalized semivariance based hedge ratios
44. (M₁₅): The hedge ratio is estimated by numerically minimizing the following sample generalized hedge ratio
45.    $V_{{\delta, \alpha }}^{{sample}}\left( {{{R}_h}} \right) = \frac{1}{N}\sum\limits_{{i = 1}}^N {{{{\left( {\delta - {{R}_{{h,i}}}} \right)}}^{\alpha }}} U\left( {\delta - {{R}_{{h,i}}}} \right)$, where $U\left( {\delta - {{R}_{{h,i}}}} \right) = \left\{{\begin{array}{lll} {1{\text{for}}\delta \geq {{R}_{{h,i}}}}\\{0{\text{for }}\delta < {{R}_{{h,i}}}} \\\end{array} } \right.$
46. (M₁₆): The hedge ratio is estimated by numerically maximizing the following function
47. $U\left( {{{R}_h}} \right) = {{R}_h} - V_{{\delta, \alpha }}^{{sample}}\left( {{{R}_h}} \right).$
48. E. Minimum Value-at-Risk hedge ratio
49. (M₁₇): The hedge ratio is estimated by minimizing the following Value-at-Risk
50.    $VaR\left( {{{R}_h}} \right) = {{Z}_{\alpha }}{{\sigma }_h}\sqrt {\tau } - E\left[ {{{R}_h}} \right]\tau$
51.    The resulting hedge ratio is given by
52.    ${{h}^{{VaR}}} = \rho \frac{{{{\sigma }_s}}}{{{{\sigma }_f}}} - E\left[ {{{R}_f}} \right]\frac{{{{\sigma }_s}}}{{{{\sigma }_f}}}\sqrt {{\frac{{1 - {{\rho }^2}}}{{Z_{\alpha }^2\sigma_f^2 - E{{{\left[ {{{R}_f}} \right]}}^2}}}}}$