SEARCH

SEARCH BY CITATION

Keywords:

  • Hedging performance;
  • Market conditions;
  • Stock market liquidity
  • G14;
  • G15;
  • G18

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

This paper examines the impact of stock market liquidity on the hedging performance of stock index futures, and extends the conditional OLS model described by Miffre [Journal of Futures Markets 24 (2004) 945] by including stock market liquidity in the regression model. The empirical results indicate that information regarding stock market liquidity is useful in predicting the optimal hedge ratio under different market conditions. In a bear market, the conditional OLS model with stock market liquidity provides the best hedging performance for the out-of-sample period. Although the OLS model outperforms the generalized autoregressive conditional heteroskedasticity and conditional OLS models for the out-of-sample period in a bull market, the conditional OLS model with stock market liquidity outperforms the conditional OLS model without stock market liquidity in terms of downside risks (lower partial moment).


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

This paper examines the impact of stock market liquidity on the hedging performance of stock index futures. The debate regarding the best econometric models for estimating the optimal hedge ratio has run for many years. Previous studies have found that complex econometric models such as the error correction and generalized autoregressive conditional heteroskedasticity (GARCH) approaches outperform the OLS model (e.g. Yang & Allen, 2004; Hsu et al., 2008). Furthermore, Miffre (2004) reports that the conditional OLS model outperforms the OLS and GARCH models. However, Alexander & Barbosa (2007) find no evidence that complex econometric models, including time-varying conditional covariance and error correction methods, can improve upon the OLS hedge ratio.

Liquidity is attracting increased attention from traders, policy-makers, and academics. Chordia et al. (2000) and Korajczyk & Sadka (2008) find the existence of commonality in liquidity. Moreover, Chakrabarty & Chung (2008) and Chung & Kim (2009) indicate that different market structures affect market liquidity. In addition, the impact of market liquidity on the relationship between futures and spot prices has received much attention in recent years. Roll et al. (2007) find that aggregate stock market liquidity affects the futures–cash basis and, in turn, the futures–cash basis affects liquidity.1 Meanwhile, Fung (2007) and Fung & Yu (2007) also indicate that stock market liquidity affects the dynamic relationship between stock and futures prices. Although previous studies examine the relationship between stock market liquidity and the futures–cash basis, no work has investigated the impact of stock market liquidity on hedging performance. Therefore, the present paper attempts to fill the gap.

It is likely that stock market liquidity will affect the hedging performance of stock index futures contracts. Figlewski (1984) indicates that the return and risk for an index futures hedge depend upon the basis behavior. Figlewski (1984) also suggests that the hedge ratio determines the overall risk and return characteristics of the hedged positions. Roll et al. (2007) report that the futures–cash basis is affected by spot market liquidity. Fung & Yu (2007) also suggest that stock market liquidity affects the error correction dynamics of index and futures prices. Moreover, Fung & Yu (2007) report that this relationship was stronger during the 1997 financial market crisis in Hong Kong. Indeed, as indicated by Roll et al. (2007), the financial crisis suggested that market conditions can be severe and liquidity can decline or even disappear, implying that liquidity is more important during a financial crisis. According to the above discussion, if stock market liquidity affects basis risks and hedge ratios determine the overall risk and return characteristics of the hedged positions, it is expected in the present study that stock market liquidity affects the hedging performance of futures contracts through hedge ratios, especially for a bear market (Fung & Yu, 2007; Roll et al., 2007).

This paper contributes to the published literature in two ways. First, to the best of our knowledge, this is the first paper to examine the impact of stock market liquidity on hedging performance. Specifically, the present paper extends the conditional OLS model described by Miffre (2004) by including aggregate stock market liquidity in the regression model. Second, this paper examines the hedging models under different market conditions. As prior work suggests that stock market liquidity has a greater impact on the dynamic relationship between stock and futures prices during a financial crisis (see Fung & Yu, 2007), this paper further divides the out-of-sample period into bull- and bear-market conditions to examine the impact of stock market liquidity on hedging performance.

The empirical results indicate that stock market liquidity contains valuable information for the determination of the optimal hedge ratio under different market conditions. For the out-of-sample period with a bear market, the conditional OLS model with stock market liquidity provides the best hedging performance. For the out-of-sample period with a bull market, the conditional OLS model with stock market liquidity outperforms the conditional OLS model without stock market liquidity with regard to the LPM, whereas the OLS model provides the best hedging performance.

The remainder of this research is organized as follows. Section 2 presents a literature review and Section 3 describes the data. Sections 4 and 5 discuss the methodology used in this paper, including hedging models and performance-evaluation metrics respectively. Section 6 reports the empirical results. Section 7 presents conclusions.

2. Literature Review

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

Hedging of futures contracts has received a great deal of attention from academics and practitioners, and the futures literature on the determination of the optimal hedge ratio can be traced back to the late 1970s. Ederington (1979) proposes that the optimal hedge ratio can be estimated using the OLS-based method. However, Ederington’s (1979) approach has received increasing criticism, particularly with respect to the constant hedge ratio over time. Hence, recent studies have used GARCH-type models to characterize the behavior of hedge ratios over time. Myers (1991), Baillie & Myers (1991), and Yang & Allen (2004) find that GARCH-type time-varying optimal hedge ratios outperform constant hedge ratios, indicating that a GARCH approach appears ideally suited to estimating time-varying optimal hedge ratios. Hsu et al. (2008) suggest that copula-based GARCH models perform more effectively than OLS, constant conditional correlation (CCC) GARCH, and dynamic conditional correlation (DCC) GARCH models.2 Moreover, prior research uses the asymmetric GARCH model to estimate time-varying hedge ratios, and finds that optimal hedge ratios are not insensitive to the asymmetric volatility model (e.g. Brooks et al., 2002; Meneu & Torro, 2003).

The rationale behind the use of GARCH models lies in the fact that these models readjust the hedging position as new information becomes available. However, the estimation procedure of GARCH models requires tedious maximization of the log-likelihood function (see Miffre, 2004). To overcome this problem, Miffre (2004) proposes using the conditional OLS model to estimate optimal hedge ratios. As with GARCH models, optimal hedge ratios based on the conditional OLS model can estimate time-dependent hedge ratios when new information is revealed. Miffre (2004) finds that the conditional OLS model reduces the hedged portfolio volatility better than the OLS and GARCH models. Similarly, Alexander & Barbosa (2007) find that sophisticated econometric models (such as GARCH models and vector error correction models [VECM]) do not outperform the OLS model.

The impact of stock market liquidity on the relationship between index futures and the corresponding underlying stock index has attracted considerable attention in the field of market microstructure. Roll et al. (2007) argue that deviations from no-arbitrage relations are related to market liquidity because liquidity facilitates arbitrage. Roll et al. (2007) test this idea by studying the dynamic relationship between stock market liquidity and the index futures basis. Roll et al. (2007) report that the dynamics of the absolute futures–cash basis and liquidity are jointly determined and that liquidity shocks can indicate futures shifts on a long-term basis. Fung (2007) examines whether the aggregate order imbalance for index stock can explain the arbitrage spread between index futures and the underlying cash index, and finds that the arbitrage spread is positively related to the aggregate order imbalance in the spot market. Additionally, Fung (2007) indicates that negative order imbalance has a stronger impact than positive order imbalance. Fung & Yu (2007) investigate the impact of stock market order imbalance on the dynamic behavior of index futures and cash index prices, and find that order imbalance in the stock market affects the error correction dynamics of index and futures prices. Specifically, they report that this relationship was stronger during the 1997 financial market crisis in Hong Kong.

In summary, various econometric models have been developed in previous studies to estimate the optimal hedge ratio. Furthermore, such studies indicate that stock market liquidity affects the relationship between index futures and the underlying spot market. Tong (1996) suggests that the determination of an optimal hedge ratio is affected by arbitrage forces. Hence, if stock market liquidity can explain the arbitrage spread between index futures and the underlying cash index, it is likely that stock market liquidity will affect the optimal hedge ratio. However, no research has investigated the impact of stock market liquidity on hedging performance. Accordingly, the present paper attempts to fill this gap.

3. Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

The data used in the present study includes the Taiwan Weighted Stock Index Prices, the number of transactions, and trading volume in shares and dollars for the Taiwan Stock Exchange (TWSE), and futures contracts traded on the Taiwan Futures Exchange (TAIFEX). The number of 1-min transactions, trading volume in shares and dollars, and daily index prices for the TWSE are obtained from the TWSE, and the daily data with regard to futures contracts are taken from the TAIFEX. To avoid thin markets and expiration effects, the nearby futures contract is rolled over to the next nearest contract when it emerges as the most active contract. The sample encompasses 3 years, or a total of 736 trading days, from 2 January 2006 to 21 December 2008.

To compute the daily order imbalance, the tick test is adopted to classify the trade (see Lee & Ready, 1991; Chan, 2000; Fung, 2007).3 Under the tick test, each trade is classified into one of the following categories: an uptick, a downtick, a zero uptick, or a zero downtick. If the current traded price is above (below) the previous price, the trade is classified as an uptick (a downtick), and a zero tick occurs when the current price is the same as the immediately previous trade. In this case, the trade is classified according to the trade before the previous price. As indicated by Chordia et al. (2002), the number of transactions and trading volume in shares and dollars are used as proxies of stock market liquidity. The natural logarithm of the number of transactions and trading volume in shares and dollars are used to calculate the order imbalance. Therefore, OIBNUM, OIBSH, and OIBDOL measure the order imbalance in number of transactions, shares, and dollars respectively.

4. The Hedging Models

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

The hedging strategies used in the present paper are the naïve hedge and the optimal hedge ratios estimated using the OLS model, the CCC GARCH model, the conditional OLS model without stock market liquidity, and the conditional OLS model with stock market liquidity.

4.1. Naïve Hedge

This approach involves taking a futures position that exactly offsets the spot position; hence, the hedge ratio is one at all times. The advantage of this method is its simplicity, in that it assumes that both the spot and futures positions are perfectly correlated.

4.2. OLS Model

The optimal hedge ratio estimated by the OLS model can be written as the following regression:

  • image(1)

where SRt and FRt are the stock and futures returns at time t, respectively, and et is the residual at time t. The optimal hedge ratio is equivalent to the slope coefficient, β0.

4.3. Constant Conditional Correlation Generalized Autoregressive Conditional Heteroskedasticity Model

The current study uses the CCC GARCH model suggested by Kroner & Sultan (1993) to estimate the optimal time-varying hedge ratio. Previous researches have extensively applied the CCC GARCH model in a hedging context (e.g. Miffre, 2004; Cotter & Hanly, 2006; Hsu et al., 2008). The CCC GARCH model is:

  • image(2)
  • image(3)
  • image(4)
  • image(5)
  • image(6)

where FRt and SRt are the futures and stock returns at time t, respectively, Ft and St are the logarithms of the futures and spot prices at time t, respectively, (St−1 − c − ηFt−1 is the error correction term, and ρ is the correlation coefficient between the returns on the stock and futures contracts. The coefficients βs and βf are the speeds of adjustment parameters: the larger is βf, the greater is the response of Ft to the previous period’s deviation from the long-run equilibrium, implying that the spot plays a more important role in price discovery. The dynamic optimal hedge ratio at time t in the CCC GARCH model is calculated as inline image

4.4. Conditional OLS Model without Stock Market Liquidity

The conditional OLS time-varying hedge ratio suggested by Miffre (2004) is estimated as follows:

  • image(7)

where Zt−1 is the information variable at time − 1. As indicated by Miffre (2004), the optimal time-varying hedge ratio can be computed as β0 + β1Zt−1. As noted in Miffre (2004), the present paper uses the basis and futures returns as the information variables at time t − 1.4 Hence, the conditional OLS model without stock market liquidity can be rewritten as:

  • image(8)

where FRt and SRt are the futures and stock returns at time t, respectively, and BASISt−1 is the basis at time − 1.

4.5. Conditional OLS Model with Stock Market Liquidity

This paper extends the conditional OLS model by including stock market liquidity as an information variable to estimate the dynamic hedge ratio. The conditional OLS model with stock market liquidity is specified as:

  • image(9)

where FRt and SRt are the futures and stock returns at time t, respectively, BASISt−1 is the basis at time − 1, and LIQt−1 is the stock market liquidity at time − 1. OIBNUM, OIBSH, and OIBDOL are used as proxies for stock market liquidity, and measure the order imbalance in the number of transactions, shares, and dollars respectively.

The in-sample period covers the 180 trading days from 2 January 2006 to 25 September 2006; the sample is then rolled over to the next trading day, and the models, including the OLS model, the CCC GARCH model, the conditional OLS model without stock market liquidity, and the conditional OLS model with stock market liquidity, are re-estimated over the new sample period to produce one-period-ahead hedge ratios. This generates a time series of 556 rolling hedge ratios for each model over the out-of-sample period from 26 September 2006 to 31 December 2008.

5. Performance Metrics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

The in-sample and out-of-sample hedging strategies can be evaluated and compared using three different performance metrics: variance, semi-variance, and lower partial moment (LPM). Semi-variance and LPM focus particularly on the downside risk (see Cotter & Hanly, 2006). Hedging performance with regard to the downside risk is very important for practitioners, especially in a bear market (or a financial crisis). Consequently, semi-variance and LPM are used in the present paper.

5.1. Performance Metric 1: Variance

The variance of the hedged portfolio is computed each day as σ2(SRt − HRt × FRt), where HRt is the hedge ratio estimated using the OLS model, the CCC GARCH model, the conditional OLS model without stock market liquidity, and the conditional OLS model with liquidity. The lower the variance is, the better the hedging performance.

5.2. Performance Metric 2: Semi-variance

The semi-variance used in the present paper is calculated as follows (see Cotter & Hanly, 2006):

  • image(10)

where τ is the target return and R is the return on the hedged portfolio. F(R) is the distribution function of R. The return on the hedged portfolio is computed as SRt − HRt × FRt. However, as indicated by Cotter & Hanly (2006), the semi-variance does not distinguish between investors who may have different risk preferences.

5.3. Performance Metric 3: Lower Partial Moment

Development of the LPM is an important approach for measuring the downside risk. The LPM is calculated as:

  • image(11)

where n reflects the investor’s risk tolerance with regard to downside risk. An investor who is more concerned with extreme shortfalls would require a higher n. Cotter & Hanly (2006) suggest that the LPM is a good indicator of downside risk, and, as indicated in their study, = 3 is used in the present study, and corresponds to a risk-averse investor. More advantages of using the LPM to examine hedging performance are discussed by Cotter & Hanly (2006).

6. Empirical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

6.1. Summary Statistics for the Data

Table 1 presents a summary of the statistics for the data, including the mean, the median, the maximum, the minimum, and the standard deviation for the SR, the FR, the OIBNUM, the OIBSH, and the OIBDOL. Panel A of Table 1 displays the summary statistics for the in-sample period, from which it can be seen that the daily mean values for the SR and FR are 0.0001 and 0.0001 respectively. The daily order imbalance measures for OIBNUM, OIBSH, and OIBDOL are 1.0186, 1.0278, and 1.2646 respectively.

Table 1.   Summary of statistics for the data The mean, median, maximum, minimum, and standard deviation for the in-sample and out-of-sample periods are shown. The out-of-sample period is divided into two subperiods: the first is a bull market and runs from 26 September 2006 to 20 May 2008; the second is a bear market (a so-called subprime crisis) and runs from 21 May 2008 to 31 December 2008. SR is the stock return and FR the futures return. OIBNUM, OIBSH, and OIBDOL measure the order imbalance in number of transactions, shares, and dollars respectively.
 ReturnsOrder imbalance
SRFROIBNUMOIBSHOIBDOL
Panel A: In-sample (2 January 2006 to 25 September 2006)
 Mean0.00010.00011.01861.02781.2646
 Median0.00030.00033.74213.65624.3944
 Maximum0.01290.01546.84606.52577.8412
 Minimum−0.0189−0.0245−6.0460−6.6538−7.2229
 Standard deviation0.00490.00574.31624.41945.2314
Panel B: Out-of-sample, subperiod 1 (26 September 2006 to 20 May 2008)
 Mean0.00030.00031.62041.75431.7673
 Median0.00070.00103.42443.84003.6355
 Maximum0.02230.02895.73866.64806.7113
 Minimum−0.0293−0.0317−5.9902−6.7526−7.1890
 Standard deviation0.00590.00693.60094.15473.9452
Panel C: Out-of-sample, subperiod 2 (21 May 2008 to 31 December 2008)
 Mean−0.0019−0.0019−0.5090−0.5750−0.4884
 Median−0.0013−0.0016−2.8826−3.6137−2.9762
 Maximum0.02650.02995.96226.89287.4151
 Minimum−0.0258−0.0381−5.5685−6.2707−6.2754
 Standard deviation0.01000.01273.98624.85334.3622

In addition, the present paper divides the out-of-sample trading days into two subperiods: the first is a bull market, and runs from 26 September 2006 to 20 May 2008 (a total of 398 trading days); the second is a bear market (a so-called subprime crisis), and runs from 21 May 2008 to 31 December 2008 (a total of 158 trading days). According to Alizadeh & Nomikos (2004), the optimal hedge ratio might depend on the state of the market. Hodgson et al. (2003) indicate that the nature of the price-discovery process for stock and futures prices varies with market conditions. Fung & Yu (2007) report that stock market liquidity had a greater impact on the dynamic behavior of index futures and cash index prices during the 1997 financial market crisis in Hong Kong. Therefore, the present study divides the out-of-sample period into bull- and bear-market conditions to examine the impact of stock market liquidity on hedging performance.

Panels B and C of Table 1 present a summary of the statistics for the first and second subperiods respectively. From Panel B, it is evident that the mean values of SR, FR, OIBNUM, OIBSH, and OIBDOL are all positive during the bull market. Specifically, the SR and FR are 0.0003 and 0.0003, respectively, and the OIBNUM, OIBSH, and OIBDOL are 1.6204, 1.7543, and 1.7673 respectively. By contrast, Panel C of Table 1 reports that the mean values of SR, FR, OIBNUM, OIBSH, and OIBDOL are all negative during the bear market. Furthermore, Panels B and C of Table 1 show that the standard deviations for all variables increase in a bear market.

6.2. Estimation of the Conditional OLS Models

Table 2 presents the estimation results of the regression models for the OLS, the conditional OLS without stock market liquidity, and the conditional OLS with stock market liquidity. The OLS regression model results show that the hedge ratio with a value of 0.8230 is significant at the 1% level, and the adjusted R2 and Durbin–Watson values are 0.9129 and 2.2501 respectively. The results obtained using the conditional OLS model without stock market liquidity indicate that the regression coefficients for BASISt−1 and FRt × BASISt−1 are all significant at the 5% level, and the regression coefficients for the FRt−1 and FRt × FRt−1 are not significant. These findings are similar to the results reported by Miffre (2004). The adjusted R2 for the conditional OLS model without stock market liquidity is 1.1220% greater than that obtained using the OLS model.

Table 2.   Conditional OLS model with stock market liquidity The estimates of the OLS and conditional OLS models are shown. The OLS model is estimated as: inline image where SRt and FRt are the stock and futures returns at time t, respectively, and et is the residual at time t. The conditional OLS model without stock market liquidity (LIQ) is estimated as: inline image where BASISt−1 is the basis at time − 1. The conditional OLS model with stock market liquidity is estimated as follows: inline imageinline image where LIQt−1 is the stock market liquidity at time − 1. OIBNUM, OIBSH, and OIBDOL are used as proxies of stock market liquidity and measure the order imbalance in number of transactions, shares, and dollars respectively. NA, not applicable. Significance at ***1, **5, and *10% levels.
CoefficientsOLS modelConditional OLS model
LIQ = NoLIQ = OIBNUMLIQ = OIBSHLIQ = OIBDOL
α00.00010.0008***0.0010***0.0010***0.0010***
α1NA−0.2020***−0.2244***−0.2275***−0.2258***
α2NA0.03400.0563**0.0577**0.0569**
α3NA NA−0.0002**−0.0002**−0.0001**
β00.8230***0.8573***0.8469***0.8479***0.8472***
β1NA−5.4401**−5.2512*−5.1872*−5.4192**
β2NA−2.7833−5.1567**−5.0684**−5.2316**
β3NA NA0.0155***0.0148***0.0130***
inline image0.91290.92510.93200.93260.9328
Durbin–Watson2.25012.05121.96781.94891.9600

Table 2 also shows the estimation results of the conditional OLS model with stock market liquidity, from which it can be seen that the regression coefficients LIQt−1 for the OIBNUM, OIBSH, and OIBDOL are −0.0002, −0.0002, and −0.0001, respectively, and these values are all significant at the 5% level. Moreover, the regression coefficients FRt × LIQt−1 for OIBNUM, OIBSH, and OIBDOL are 0.0155, 0.0148, and 0.0130, respectively, and are all significant at the 1% level. In addition, the adjusted R2 for the conditional OLS model with stock market liquidity is higher at 0.70% (compared with 0.69% for OIBNUM, 0.75% for OIBSH, and 0.77% for OIBDOL) than that of the conditional OLS model without stock market liquidity. All the Durbin–Watson values for the conditional OLS models (without and with liquidity) are close to 2.

Overall, these findings imply that the hedge ratio is related to stock market liquidity. This finding is consistent with those of Roll et al. (2007), Fung (2007), and Fung & Yu (2007), in that stock market liquidity would affect the dynamic relationship between stock and futures prices.

6.3. Estimation of the Constant Conditional Correlation Generalized Autoregressive Conditional Heteroskedasticity Model

Table 3 presents the estimation results obtained using the CCC GARCH model. The regression coefficients for βs and βf are −0.0643 and 0.3737 respectively. Furthermore, βf is significant at the 5% level and βs is not significant at the 10% level, implying that the stock market plays an important role in price discovery.5 The correlation between stock and futures return is 0.9655 and is significant at the 1% level. Additionally, ai and bi are positive and ai + bi is close to 1, indicating that shocks in stock and futures markets have high persistence in volatility. The persistence of autoregressive conditional heteroskedasticity effects implies that current information remains important for forecasting conditional variances.

Table 3.   Estimation of the GARCH model The estimates of the constant conditional correlation generalized autoregressive conditional heteroskedasticity (GARCH) model are presented, which is estimated as follows: inline imageinline imageinline imageinline imageinline image where FRt and SRt are the futures and stock returns at time t, respectively, Ft and St are the logarithms of the futures and spot prices at time t, respectively, (St−1 − c − ηFt−1) is the error correction term, and ρ is the correlation coefficient between the returns on the stock and the futures contracts. Significance at ***1, **5 and *10% levels. Thumbnail image of

Finally, the results of Ljung–Box tests show that the Q-statistics computed from the autocorrelation coefficients of residuals and squared residuals are not significant at the 10% level for the CCC GARCH regression model, indicating that this model successfully accounts for the serial correlation in residuals and squared residuals.

6.4. In-sample Hedging Performance

The conditional OLS model with stock market liquidity is compared with the naïve hedge, the OLS model, the CCC GARCH model, and the conditional OLS model without stock market liquidity. The hedging performance metrics used in this paper are variance, semi-variance, and LPM. The in-sample hedged portfolio variance, the semi-variance, and the LPM and hedging effectiveness for all regression models are summarized in Table 4.6

Table 4.   In-sample hedging effectiveness The in-sample hedged portfolio variance and the hedging performance for the OLS, conditional OLS, and generalized autoregressive conditional heteroskedasticity (GARCH) model are shown. The hedging performance metrics used in this research are variance, semi-variance, and lower partial moment. HE1, HE2, and HE3 measure hedging effectiveness with regard to the conditional OLS with the liquidity proxy. Hedging effectiveness (HE1, HE2, or HE3) is measured as the percentage increase (if positive, decrease if negative) in risk of each hedge ratio relative to the conditional OLS hedge ratios with stock market liquidity. OIBNUM, OIBSH, and OIBDOL are used as proxies for stock market liquidity and measure the order imbalance in number of transactions, shares, and dollars respectively. NA, not applicable.
2 January 2006 to 25 September 2006Variance × 104Semi-variance × 104Lower partial moment (LPM × 104)
VarianceHE1HE2HE3SemiHE1HE2HE3LPMHE1HE2HE3
Naïve hedge0.16400.54440.54810.54920.07530.49090.49540.49140.00050.88710.89440.8891
Static OLS hedge0.10980.03390.03650.03710.05260.04090.04400.04120.00030.07660.08070.0777
GARCH hedge0.11150.05030.05290.05360.05250.03980.04290.04020.00030.11780.12210.1190
Conditional OLS hedge without a liquidity proxy0.1044−0.0166−0.0142−0.01360.05120.01440.01740.01470.00030.02880.03280.0299
Conditional OLS hedge with the liquidity proxy: OIBNUM0.1062NA0.00250.00310.0505NA0.00300.00040.0003NA0.00390.0010
Conditional OLS hedge with the liquidity proxy: OIBSH0.1059−0.0024NA0.00070.0503−0.0030NA−0.00260.0003−0.0039NA−0.0028
Conditional OLS hedge with the liquidity proxy: OIBDOL0.1058−0.0031−0.0007NA0.0505−0.00040.0027NA0.0003−0.00100.0028NA

Regarding the hedged portfolio variance, Table 4 indicates that the smallest variance of the hedged portfolio is obtained using the conditional OLS model without stock market liquidity of 0.1044. The second best of the hedging strategies examined is the conditional OLS model with stock market liquidity, the hedged portfolio variances for the OIBNUM, OIBSH, and OIBDOL being 0.1062, 0.1059, and 0.1058 respectively.

In practice, the downside risk is more important for hedgers and practitioners. Table 4 also presents the hedging performance with regard to the downside risk for all regression models. The semi-variance and LPM results all indicate that the conditional OLS model with stock market liquidity outperforms the naïve hedge, the OLS model, the GARCH model, and the conditional OLS model without stock market liquidity. For instance, when utilizing the semi-variance to measure the hedging effectiveness, the naïve hedge increases downside risk by 49.09% more than the conditional OLS model using OIBNUM as a proxy for stock market liquidity. In addition, for the semi-variance and the LPM, the results of the conditional OLS model with stock market liquidity indicate that using the OIBSH as a proxy for the stock market liquidity outperforms the OIBNUM and OIBDOL.

6.5. Out-of-sample Hedging Performance

Table 5 presents the out-of-sample hedged portfolio variance, the semi-variance, the LPM and the hedging effectiveness for the two subperiods. Panel A of Table 5 shows the hedging performance in a bull market, from which it can be seen that the best hedging strategy is the OLS model with regard to the variance, the semi-variance, and the LPM. The second best hedging strategy for downside risk with respect to LPM is the CCC GARCH model. Although the OLS and CCC GARCH models outperform the conditional OLS models, the conditional OLS model with stock market liquidity outperforms the conditional OLS model without stock market liquidity in terms of the LPM. For instance, the LPM for the conditional OLS model without stock market liquidity increases the basis risk by 5.45% more than the conditional OLS model when using the OIBSH as a proxy for stock market liquidity. This finding indicates that stock market liquidity is useful for improving the hedging performance with regard to downside risk.

Table 5.   Out-of-sample hedging effectiveness for the two subperiods The out-of-sample hedged portfolio variance and the hedging performance of the OLS, conditional OLS, and generalized autoregressive conditional heteroskedasticity (GARCH) models for the two subperiods are shown. The first subperiod is a bull market and runs from 26 September 2006 to 20 May 2008; the second subperiod is a bear market (a so-called subprime crisis) and runs from 21 May 2008 to 31 December 2008. The hedging performance metrics used in this research are variance, semi-variance, and lower partial moment. HE1, HE2, and HE3 measure hedging effectiveness with regard to the conditional OLS with the liquidity proxy. Hedging effectiveness (HE1, HE2, or HE3) is measured as the percentage increase (if positive, decrease if negative) in risk of each hedge ratio relative to the conditional OLS hedge ratios with stock market liquidity. OIBNUM, OIBSH, and OIBDOL are used as proxies for stock market liquidity and measure the order imbalance in number of transactions, shares, and dollars respectively. NA, not applicable.
 Variance × 104Semi-variance × 104Lower partial moment (LPM × 104)
VarianceHE1HE2HE3SemiHE1HE2HE3LPMHE1HE2HE3
Panel A: Out-of-sample subperiod 1 (26 September 2006 to 20 May 2008)
Naïve hedge0.20970.49110.49210.49070.10220.37090.37610.37040.00090.24200.26360.2365
Roll-over OLS hedge0.1288−0.0836−0.0831−0.08390.0682−0.0856−0.0821−0.08590.0006−0.2162−0.2025−0.2196
GARCH hedge0.14560.03540.03600.03500.0708−0.0506−0.0470−0.05100.0007−0.0384−0.0217−0.0426
Conditional OLS hedge without a liquidity proxy0.1369−0.0263−0.0257−0.02660.0727−0.0245−0.0208−0.02490.00080.03650.05450.0319
Conditional OLS hedge with the liquidity proxy: OIBNUM0.1406NA0.0006−0.00030.0746NA0.0038−0.00040.0007NA0.0174−0.0044
Conditional OLS hedge with the liquidity proxy: OIBSH0.1405−0.0006NA−0.00090.0743−0.0038NA−0.00420.0007−0.0171NA−0.0215
Conditional OLS hedge with the Liquidity proxy: OIBDOL0.14060.00030.0009NA0.07460.00040.0042NA0.00080.00440.0219NA
Panel B: Out-of-sample subperiod 2 (21 May 2008 to 31 December 2008)
Naïve Hedge1.08401.39531.40251.39500.58041.87011.88511.87970.01998.53488.60548.6050
Roll-over OLS hedge0.56370.24560.24930.24550.34710.71610.72510.72190.00973.64093.67533.6750
GARCH hedge0.56380.24570.24940.24550.26520.31140.31830.31580.00461.21661.23301.2329
Conditional OLS hedge without a liquidity proxy0.45870.01360.01660.01350.20480.01280.01810.01620.00220.04120.04900.0489
Conditional OLS hedge with the liquidity proxy: OIBNUM0.4526NA0.0030−0.00010.2022NA0.00520.00340.0021NA0.00740.0074
Conditional OLS hedge with the liquidity proxy: OIBSH0.4512−0.0030NA−0.00310.2012−0.0052NA−0.00190.0021−0.0074NA0.0000
Conditional OLS hedge with the liquidity proxy: OIBDOL0.45260.00010.0031NA0.2016−0.00330.0019NA0.0021−0.00730.0000NA

Panel B of Table 5 presents the hedging performance for all hedging strategies during economic recession. For the variance, the semi-variance, and the LPM, the conditional OLS model using OIBSH as a proxy for stock market liquidity provides the best hedging strategy. Specifically, the conditional OLS model with stock market liquidity outperforms the other hedging methods in terms of the variance, the semi-variance, and the LPM. For example, the conditional OLS model using the OIBNUM as the stock market liquidity reduces the downside risk by 364.09% more than the OLS model. Overall, the results indicate that stock market liquidity can improve hedging performance during a bear market.

Figure 1 shows the out-of-sample hedge ratios for the OLS model, the GARCH model, and the conditional OLS model using the OIBSH, from which it can be seen that the OLS hedge ratios are less volatile than the GARCH model and the conditional OLS model with the OIBSH. Furthermore, Figure 1 indicates that the hedge ratio standard deviation for the GARCH model is greater than the conditional OLS model with the OIBSH. Panel B of Table 6 shows that the hedge ratio standard deviations for OHR, GHR, and COIBSHHR are 0.0200, 0.0578, and 0.0539, respectively, for the bull market and 0.0444, 0.0973, and 0.0681 for the bear market. These results indicate that the volatility of the hedge ratio increases in a bear market. Finally, Panel A of Table 6 reports the correlations between the conditional OLS model with the OIBSH hedge ratio and its counterparts, and indicates that the correlations range from 0.4832 to 0.3506 for the two subperiods. This finding implies that the conditional OLS model with the OIBSH does not capture the same variation in the hedge ratio as other hedge models (see Miffre, 2004).

image

Figure 1.  Out-of-sample hedge ratios.

Download figure to PowerPoint

Table 6.   Behavior of hedge ratios for the out-of-sample period The behavior of hedge ratios for the out-of-sample period is shown. OHR is the OLS hedge ratio. GHR is the generalized autoregressive conditional heteroskedasticity hedge ratio. COIBSHHR is the conditional OLS hedge ratio with the OIBSH. OIBSH is the order imbalance in number of shares. The first subperiod is a bull market that runs from 26 September 2006 to 20 May 2008; the second subperiod is a bear market (a so-called subprime crisis) and runs from 21 May 2008 to 31 December 2008.
 Out-of-sample, full (26 September 2006 to 31 December 2008)Out-of-sample subperiod 1 (26 September 2006 to 20 May 2008)Out-of-sample subperiod 2 (21 May 2008 to 31 December 2008)
OHRGHRCOIBSHHROHRGHRCOIBSHHROHRGHRCOIBSHHR
Panel A: Correlation matrix
 OHR1.00001.00001.0000
 GHR0.47051.00000.34471.00000.57971.0000
 COIBSHHR0.42940.41271.00000.48320.35061.00000.39140.48001.0000
Panel B: Hedge ratio
 Average0.81250.82430.82710.80330.82420.82270.83520.83920.8397
 Median0.80910.83350.83920.80760.83420.82830.85220.86210.8608
 Maximum0.87971.00941.16710.84281.16711.00940.87971.14880.9294
 Minimum0.74670.40600.51080.76200.57320.58610.74670.51080.4060
 Standard deviation0.03420.05950.07470.02000.05780.05390.04440.09730.0681

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References

This paper investigates the impact of stock market liquidity on hedging effectiveness. Previous studies find that the use of advanced econometric models to estimate the optimal hedge ratio can improve the hedging performance (e.g. Alizadeh & Nomikos, 2004; Yang & Allen, 2004; Hsu et al., 2008). In addition, Miffre (2004) finds that the hedging performance for the conditional OLS model outperforms the OLS and the GARCH models. By contrast, Alexander & Barbosa (2007) suggest that the hedging effectiveness of the sophisticated econometric models (such as the GARCH and VECM models) do not outperform the OLS model. Roll et al. (2007), Fung (2007), and Fung & Yu (2007) find that stock market liquidity affects the dynamic relationship between stock and futures prices. However, no work has investigated the influence of stock market liquidity on hedging performance. Hence, this paper is an attempt to fill that gap.

This paper extends the model described by Miffre (2004) by including stock market liquidity in the regression model. The empirical results indicate that stock market liquidity contains information useful for predicting the optimal hedge ratio and enhances the hedging performance during a bear market. The conditional OLS model with stock market liquidity provides the best hedging performance for the out-of-sample period with a bear market, with regard to the variance, the semi-variance, and the LPM. The conditional OLS model with stock market liquidity outperforms the conditional OLS model without stock market liquidity for the out-of-sample period with a bull market with regard to the LPM, whereas the OLS model provides the best hedging performance in terms of the variance, the semi-variance, and the LPM. Therefore, our findings are consistent with those of Fung & Yu (2007), who report that stock market liquidity has a greater impact on the dynamic relationship between stock and futures prices during a financial crisis.

Footnotes
  • 1

    Chung et al. (2009) provide evidence that past day-trading activity negatively affects liquidity and past liquidity negatively affects future day-trading activity.

  • 2

    In addition, other more advanced models such as Markov switching models (e.g. Alizadeh & Nomikos, 2004) also appear useful for hedging stock prices.

  • 3

    As the TWSE data set does not contain information on bid-ask quotes, the quote-based approach cannot be applied in this paper.

  • 4

    The information variables at time − 1 used by Miffre (2004) are the futures returns, basis, dividend yield, spread of corporate bond yield, and the term structure of interest rates. Miffre (2004) finds that the basis (and the futures returns) is (are) particularly crucial in predicting the hedge ratio one period ahead when using the conditional OLS model with a time-varying (constant) basis. Accordingly, the futures returns and the basis are used as the information variables.

  • 5

    Hodgson et al. (2003) suggest that stock prices contain additional information regarding price discovery during the bull market. The stock and futures returns are 0.0001 and 0.0001, implying that this trading phase is a bull market. Hence, the finding that the stock market plays an important role in price discovery is consistent with Hodgson et al. (2003).

  • 6

    The in-sample and out-of-sample hedging strategies all indicate that the hedged portfolio variance is smaller than the unhedged variance. These results are not presented here but are available upon request.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Literature Review
  5. 3. Data
  6. 4. The Hedging Models
  7. 5. Performance Metrics
  8. 6. Empirical Results
  9. 7. Conclusions
  10. References