Statistical properties of post-sample hedging effectiveness

International Review of Financial Analysis 16 (2007) 293 – 300

Statistical properties of post-sample hedging effectiveness ☆ Donald Lien ⁎ University of Texas at San Antonio, USA Received 11 January 2007; accepted 19 January 2007 Available online 27 January 2007

Abstract This paper examines the mean and the variance of post-sample hedging effectiveness. It is shown that, the hedging effectiveness measure adopted in the current literature is a biased estimator of the true hedging effectiveness. Moreover, it underestimates the true hedging effectiveness. Empirical results base upon twentyfour futures markets for the error correction hedge ratio, however, suggest the bias is negligible. On the other hand, in some markets, the variance of the hedging effectiveness is too large for the estimate to be reliable. © 2007 Elsevier Inc. All rights reserved. Keywords: Futures markets; Hedging effectiveness; Post-sample analysis

1. Introduction A futures market serves as a financial instrument for risk reduction. To evaluate the usefulness of a futures market, one frequently turns to the hedging effectiveness measure proposed by Ederington (1979). Specifically, one constructs a hedge strategy such that the resulting hedged portfolio obtains the minimum variance of the return among all possible portfolios. Hedging effectiveness is measured by the percentage reduction of the minimum variance from the variance of the spot return. Different assumptions on the statistical behaviors of spot and futures prices lead to different minimum variance hedge strategies. Regardless, the true hedging effectiveness is derived from a true minimum variance hedge ratio. ☆ Part of this paper was written when the author was visiting Shanghai Finance University. The author wishes to acknowledge Keshab Shrestha for research assistance and James Groff for suggestions and comments. The usual disclaimer applies. ⁎ Contact address: International Business Program, College of Business, University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, Texas 78249 USA. E-mail address: [email protected].

1057-5219/$ - see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2007.01.002

294

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

In empirical studies, hedging effectiveness is usually computed with the estimated hedge ratio replacing the true unknown hedge ratio. Lien (2006) demonstrates analytically that the method produces a downward biased estimator of the true hedging effectiveness. It therefore underestimates the usefulness of a futures contract. An unbiased estimator that corrects the bias is then proposed. This paper investigates empirically the size of the bias using weekly data from twentyfour financial and commodity futures markets. Specifically, error-correction models are applied to generate estimated minimum variance hedge ratios and calculate the hedging effectiveness. It is found that the downward bias is very small. Consequently, bias correction seems unnecessary at least for direct hedging scenarios examined in the current paper. When examining the variance of hedging effectiveness, we find that, in hogs and cotton markets, the variance of the hedging effectiveness is too large for the estimate to be reliable. The remainder of this paper is organized as follows. The next section discusses the true minimum variance hedge ratio and the true hedging effectiveness. We then describe the estimation methods of the hedge ratio and the hedging effectiveness. The downward bias of the estimated hedging effectiveness is characterized and the correction method is proposed. The subsequent section provides a detailed analysis for the error correction hedge ratio. The variance of the estimated hedging effectiveness is derived in the following section. An empirical study using weekly data from twenty-four futures markets is performed and analyzed. The final section concludes the paper. 2. Analytical hedging effectiveness When we know the true data generation process (DGP), we can calculate the true minimum variance (MV) hedge ratio as the ratio of the conditional covariance between spot and futures prices over the conditional variance of the futures price: h⁎ ¼ CovðP ; F jI Þ=VarðF jI Þ; ð1Þ t

tþ1

tþ1 t

tþ1 t

where Pt+1 and Ft+1 are, respectively, spot and futures prices at time t + 1; It is the information available at time t. Frequently, spot and futures prices are found to contain a unit root. As a result, we have h⁎ ¼ Covðp ; f jI Þ=Varðf jI Þ; ð2Þ t

tþ1

tþ1 t

tþ1 t

where pt+1 = Pt+1 − Pt and ft+1 = Ft+1 − Ft. When conditional variances and the conditional covariance are stationary, the true MV hedge ratio is constant over time and we can replace h⁎t with h⁎. The usefulness of a futures contract is calculated by the hedging effectiveness measure (Ederington, 1979), which is the percentage reduction in variance when the MV hedge ratio is applied. Let Ik be a (k × k) dimensional identity matrix and let ek be a k-dimensional vector such that all elements equal to one. Define M = Ik − (ekek′ / k). The hedging effectiveness of a futures contract is H ⁎ ¼ 1−

w VMw ; p VMp

ð3Þ

where p is a k-dimensional vector consisting of k spot returns and w is a k-dimensional vector consisting of k hedged portfolio returns. A hedged portfolio return at time s is calculated as ps − h⁎fs. As a result, p′Mp and w′Mw are sample variances of spot returns and hedged portfolio returns, respectively. In the following, we consider the statistical properties for the estimator of the Ederington measure. Nonetheless, it should be pointed out this measure has its own limitations (Lien, 2005a,b).

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

295

3. Estimation of hedging effectiveness In reality, the true DGP is unknown and at best one can only derive estimators for the true MV hedge ratio and the true hedging effectiveness. Lien (2006) provides statistical properties for the estimator of hedging effectiveness. This section follows the analysis of the paper. Let hˆ denote the MV hedge ratio estimator derived from a given sample and let Ĥ denote the corresponding hedging effectiveness estimator. That is, wˆ VM wˆ Hˆ ¼ 1− ; p VMp

ð4Þ

where ŵ = p − hˆf. Note that ŵ = (ŵ − w) + w = (h⁎ − hˆ )f + w, after algebraic manipulations, we obtain the estimated hedging effectiveness as follows:

 f VMw 2 f VMf ⁎ ⁎ ⁎ ˆ ˆ ˆ H ¼ H −2ðh − hÞ −ðh − hÞ : ð5Þ pVMp p VMp In this paper, we consider post sample evaluations. That is, the hedged portfolio is constructed from a completely different sample from that applied to derive the estimated hedge ratio. Given a fixed post sample and assume that ĥ is an unbiased estimator of h⁎, then
 ˆ f VMf pH ⁎ : ˆ ¼ H ⁎ −Varð hÞ ð6Þ EðHÞ p VMp Thus, Ĥ is a biased estimator of H⁎. The estimation bias is
 f VMf ˆ B ¼ −VarðhÞ b0: p VMp To correct the bias, the true hedging effectiveness should be measured by
 ˆ f VMf : H˜ ¼ Hˆ þ Est:VarðhÞ p VMp

ð7Þ

ð8Þ

where Est.Var(hˆ )is an estimator of Var(hˆ ) and the second term is the estimated bias. Note the above result is derived under the assumption that hˆ is an unbiased estimator of h⁎. This is the minimal assumption for a hedging strategy to be useful. If the estimated hedge ratio is biased, any further analysis makes little sense. In all empirical work, this assumption is implicitly adopted and sometimes justified by some model specification tests. 4. Error correction hedge ratio Consider the following error correction model for spot and futures returns: pt ¼ ap0 þ

m X

bpi pt−i þ

i¼1

ft ¼ af 0 þ

mV X i¼1

bfi pt−i þ

n X

gpj ft−j þ hp bt−1 þ ept ;

ð9Þ

j¼1 nV X j¼1

gfj ft−j þ hf bt−1 þ eft ;

ð10Þ

296

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

where bt−1 = Pt−1 − cFt−1 describes the cointegration relationship. This model encompasses the conventional hedge ratio estimation methods. The OLS hedge ratio is derived when we choose βpi = βfi = γpj = γfj = θp = θf = 0. We obtain the generalized hedge ratio of Myers and Thompson (1989) if θp = θf = 0. In empirical work, it is frequently found that c is close to one. As a consequence, we set bt−1 = Pt−1 − Ft−1, which is the basis at time t − 1. The MV hedge ratio can be estimated from the following regression equation: pt ¼ a þ ZtVb þ dft þ et ;

ð11Þ

where Zt is a q-dimensional column vector of exogenous variables including bt−1 and lagged spot and futures returns, β is the corresponding q-dimensional column vector for coefficients and εt is a white noise with variance σε2. More specifically, δ corresponds to the MV hedge ratio, h⁎. Suppose that we have T random samples for estimation purpose. Let p⁎ = (p1 … pT)′ and f⁎ = ( f1 … fT)′ both be T-dimensional column vectors and let Z⁎ = (Z1 … ZT)′ be a (T × q) matrix. The OLS estimator for (β′δ)′ is



bˆ ¼ Z⁎VM⁎ Z⁎ f⁎VM⁎ Z⁎ dˆ

Z⁎VM⁎ f⁎ f⁎VM⁎ f⁎

−1


 Z⁎VM⁎ p⁎ ; f⁎VM⁎ p⁎

ð12Þ

where M⁎ = IT − (eT eT′ / T) where IT is a (T × T) dimensional identity matrix and eT is a Tdimensional vector such that all elements equal to one. Upon applying the inverse of a partitioned matrix, we have f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎ p⁎ dˆ ¼ : f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎ f⁎

ð13Þ

Let ε⁎ = (ε1 … εT)′. Denote δ by h⁎ (the true EC hedge ratio) and denote δˆ by hEC (the estimated EC hedge ratio), respectively. Then Eq. (13) can be rewritten as follows: hEC ¼ h⁎ þ

f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎ e⁎ f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎ f⁎

:

ð14Þ

:

ð15Þ

From which we obtain E(hEC) = h⁎ and VarðhEC Þ ¼

r2e

f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎  f⁎

The estimated variance, Est.Var(hEC) replaces σε2 by its estimator σˆ ε2. Upon substituting the above relationship into Eq. (7), the unbiased hedging effectiveness for the error correction hedge strategy, H˜ EC, is derived:
 f VMf ˜ ˆ H EC ¼ H EC þ Est:VarðhEC Þ ; ð16Þ pVMp where ĤEC = 1 − {[( p − hEC f )′M( p − hECf )] / p′Mp} is the hedging effectiveness currently adopted in the literature.

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

297

5. Variance of hedging effectiveness From Eq. (5), we can derive the variance for the estimated hedging effectiveness as follows:       f VMw 2 f VMf 2 f VMf 2 4 ⁎ ˆ ˆ ˆ ˆ 2 VarðHÞ ¼ 4 ½VarðhÞ þ E½ðh −hÞ − ½VarðhÞ p VMp p VMp pVMp ð f VMwÞð f VMf Þ ˆ 3 : þ4 E½ðh⁎ − hÞ ð17Þ ð pVMpÞ2 To find the higher moments of h⁎ − ĥ, let f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎  ¼ ðQ1 N QT Þ: ð18Þ Q¼ f⁎V½M⁎ −M⁎ Z⁎ ðZ⁎VM⁎ Z⁎ Þ−1 Z⁎VM⁎ f⁎ Eq. (15) leads to T X E½ðhEC −h⁎ Þ3  ¼ Q3t Eðe3t Þ; ð19Þ t¼1

E½ðhEC −h⁎ Þ4  ¼

T X

Q4t Eðe4t Þ þ 6

t¼1

T X

Q2t Q2s ½Eðe2t Þ2 :

ð20Þ

s;t¼1 spt

In empirical studies, we assume stationary higher order moments and, therefore, E(εtj) is T P eˆtj =T where εˆt is the residual from the estimated result of Eq. (11). estimated by t¼1

Table 1 Summary of 24 futures contracts

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 29 20 21 22 23 24

Commodity

Sample period

Spot Commodity

S&P 500 TSE 35 Nikkei 225 TOPIX FTSE 100 CAC40 All Ordinary Soybean Oil Soybeans Soybean Meal Corn Wheat Cotton Cocoa Coffee Hogs Crude Oil Silver Gold Japanese Yen Deutsche Mark Swiss Franc British Pound Canadian Dollar

6/1/82–12/31/97 3/1/91–12/31/97 9/5/88–12/31/97 9/5/88–12/31/97 3/3/84–12/31/97 3/1/89–12/31/97 1/3/84–12/31/97 1/2/79–12/31/97 1/2/79–12/31/97 1/2/79–12/31/97 1/2/79–12/31/97 3/30/82–12/31/97 1/3/80–12/31/97 11/1/83–12/31/97 2/2/79–12/31/97 3/30/82–12/31/97 4/4/83–12/31/97 1/2/79–12/31/97 1/2/79–12/31/97 1/2/86–12/31/97 1/2/86–12/31/97 1/2/86–12/31/97 1/2/86–12/31/97 11/30/87–12/31/97

S&P 500 TSE 35 Nikkei 225 TOPIX FTSE 100 CAC40 All Ordinary Soybean Oil, Crude Decatur Soybean, No. 1 Yellow Soybean Meal, 44% protein Corn, No. 2 Yellow Wheat, No. 2 Soft Red Cotton 1 1/16 Low–Middling Cocoa Coffee, Brazilian Hogs, Omaha West Texas Int. Silver, Handy & Harmon Gold Handy & Harmon Yen Deutsche Mark Swiss Franc British Pound Canadian Dollar

Note: This table lists the commodities, sample periods, Spot Commodities, and sample sizes for the 24 different futures contracts used for empirical analyses in this study.

298

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

6. An empirical study We use the weekly data on twenty-four futures markets to examine hedging effectiveness measures of the EC hedge ratio. The list of the commodities/assets and sample periods are given in Table 1. The data set, obtained from Datastream includes, among others, seven stock index futures, two precious metals and five currencies. To avoid the rollover risk when the futures contract switches, we delete the crossover returns from the dataset. Table 1 also specifies the spot asset/commodity that is being hedged by a corresponding futures contract. The post sample performances are based on the last two years worth of data (i.e., the last 104 weeks of data for the weekly samples). To estimate the EC hedge ratio, we consider the following regression equation: pt ¼ a þ dft þ bbt−1 þ

N X i¼1

gi pt−i þ

N X

di ft−i þ et ;

ð21Þ

i¼1

where all variables are defined in the previous sections except that we measure spot and futures prices in logarithms. Terry (2005) compared hedging effectiveness when price differences, percentage returns and logarithmic returns are adopted to derive minimum variance hedge ratios. Akaike Information Criterion (AIC) is applied to determine the lag order, N (Frino, deB Harris, McInish, & Thomas, 2004; Ghosh & Gilmore, 1997). Lien and Shrestha (2005) compared implications of AIC and FIC (focus information criterion) for futures hedging.

Table 2 Hedge ratios and hedging effectiveness Commodity

Hedge ratio

ĤEC

˜ EC H

Standard deviation

S&P 500 TSE 35 Nikkei 225 TOPIX FTSE 100 CAC40 All Ordinary Soy Bean Oil Soya Bean Soya Meal Corn Wheat Cotton Cocoa Coffee Hogs Crude Oil Silver Gold Japanese Yen Deutsche Mark Swiss Franc British Pound Canadian Dollar

0.86302 0.97045 0.98424 0.93676 0.88618 0.94502 0.71502 0.94230 0.97353 1.01278 1.03302 0.96583 0.87179 0.83673 0.56779 0.63194 1.02233 0.80614 0.88866 0.98096 0.99779 0.99409 0.97897 0.93028

0.97332 0.98892 0.98065 0.96263 0.96842 0.98752 0.92219 0.95849 0.84582 0.77673 0.75145 0.81835 0.60609 0.86585 0.68789 0.28523 0.83109 0.74332 0.86314 0.98210 0.98619 0.97826 0.98871 0.97784

0.97336 0.98906 0.98073 0.96276 0.96848 0.98760 0.92230 0.95864 0.84594 0.77688 0.75172 0.81901 0.60768 0.86607 0.68831 0.28596 0.83124 0.74384 0.86329 0.98213 0.98622 0.97829 0.98876 0.97793

0.00265 0.00469 0.00332 0.00575 0.00316 0.00316 0.00714 0.01000 0.00755 0.00985 0.01833 0.04066 0.10510 0.01285 0.03089 0.04416 0.00837 0.03647 0.01039 0.00141 0.00100 0.00141 0.00265 0.00412

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

299

Table 3 The bias of estimated hedging effectiveness Commodity

S&P 500 TSE 35 Nikkei 225 TOPIX FTSE 100 CAC40 All Ordinary Soy Bean Oil Soya Bean Soya Meal Corn Wheat Cotton Cocoa Coffee Hogs Crude Oil Silver Gold Japanese Yen Deutsche Mark Swiss Franc British Pound Canadian Dollar

EC hedge ratio ˜ EC ĤEC − H

˜ EC) / H˜ EC (ĤEC − H

−0.00004 −0.00014 −0.00008 −0.00013 −0.00006 −0.00008 −0.00011 −0.00015 −0.00012 −0.00015 −0.00027 −0.00066 −0.00159 −0.00022 −0.00042 −0.00073 −0.00015 −0.00052 −0.00015 −0.00003 −0.00003 −0.00003 −0.00005 −0.00009

−0.00411 −0.01415 −0.00816 −0.01350 −0.00619 −0.00810 −0.01193 −0.01565 −0.01419 −0.01931 −0.03592 −0.08059 −0.26165 −0.02540 −0.06102 −0.25528 −0.01805 −0.06991 −0.01738 −0.00305 −0.00304 −0.00307 −0.00506 −0.00920

The numbers in column 3 are expressed in percentages.

Table 2 presents the estimation results. The EC hedge ratio ranges from 0.568 (coffee) to 1.033 (corn). For all currency markets, the ratio is close to one except Canadian dollar (0.930). Among the stock index markets, Australian All Ordinary Index has the smallest hedge ratio (0.715). Nikkei 225 has the largest hedge ratio (0.984). Commodities have the largest and the smallest hedge ratios among all twenty-four markets. On average, hedge ratios for agricultural commodities are the largest, comparable to the currency markets. Soft commodities (i.e., cotton, cocoa, and coffee), on the other hand, have the smallest hedge ratios. Currency futures markets have the greatest hedging effectiveness followed by stock index futures markets. On average, the commodity market has the smallest hedging effectiveness. It also has the largest fluctuation, ranging from 0.285 (hogs) to 0.958 (soy bean oil) for the EC strategy. Turning to the relationship between Ĥ and H˜, we note that f ′Mf / p′Mp is close to one and the variances of both hedge ratios are very small. Consequently, there is little difference between Ĥ and H˜ for each futures contract. That is, the estimated bias is very small. The results are provided ˜ occurs in the cotton market (0.0016) in Table 3. The greatest difference between Ĥ and H followed by the hogs market (0.0007). Overall, our empirical results indicate that, although the hedging effectiveness measure currently adopted in the literature is downward biased, the bias is negligible and there is no need to correct the bias at least for direct hedging purposes. Estimated standard deviation for hedging effectiveness of the EC ratio ranges from 0.001 (Deutsche Mark) to 0.105 (cotton); see the last column of Table 2. Currency futures markets have

300

D. Lien / International Review of Financial Analysis 16 (2007) 293–300

the smallest variances followed by the stock index futures markets. In these markets, three standard deviations account only around 1% of the estimated hedging effectiveness. The story is different for some commodity markets. For example, the hedging effectiveness for the cotton futures market is 0.606. A confidence interval based upon three standard deviations lies between 0.291 and 0.921. Similarly, the hedging effectiveness for the hog futures market is 0.285. With three standard deviations on each side, the resulting confidence interval ranges from 0.153 to 0.418. 7. Conclusions This paper provides a statistical analysis of the post-sample hedging effectiveness. It is shown that the hedging effectiveness currently adopted in the finance literature is a downward biased estimator of the true hedging effectiveness. Using weekly data from twenty-four markets, we construct the minimum variance hedge ratio from the error correction model. The empirical results, however, indicate that the bias is negligible. As a consequence, bias correction seems to be theoretically sound but empirically redundant. The variance of the hedging effectiveness is then discussed. It is found that, for some commodities, the estimated hedging effectiveness is too imprecise to be reliable for conclusion drawing. References Ederington, L. (1979). The hedging effectiveness of the new futures markets. Journal of Finance, 34, 157−170. Frino, A., deB Harris, F. H., McInish, T. H., & Tomas, M. J., III (2004). Price discovery in the pits: the role of market makers on the CBOT and the Sydney Futures Exchange. Journal of Futures Markets, 24, 785−804. Ghosh, A., & Gilmore, C. G. (1997). The rolling spot futures contract: An error correction model analysis. Journal of Futures Markets, 17, 117−128. Lien, D. (2005). The use and abuse of the hedging effectiveness measure. International Review of Financial Analysis, 14, 277−282. Lien, D. (2005). A note on the superiority of the OLS hedge ratio. Journal of Futures Markets, 25, 1121−1126. Lien, D. (2006). Estimation bias of futures hedging performance: A note. Journal of Futures Markets, 26, 835−841. Lien, D., & Shrestha, K. (2005). Estimating optimal hedge ratio with focus information criterion. Journal of Futures Markets, 25, 1011−1024. Myers, R. J., & Thompson, S. R. (1989). Generalized optimal hedge ratio estimation. American Journal of Agricultural Economics, 71, 858−868. Terry, E. (2005). Minimum-variance futures hedging under alternative return specifications. Journal of Futures Markets, 25, 537−552.