Alternative Methods for Estimating the Optimal Hedge Ratio

Transcription

1 ISSN: International Journal of Advances in Management and Economics Available online at RESEARCH ARTICLE Alternative Methods for Estimating the Optimal Hedge Ratio Mihai-Cristian Dinica* Bucharest Academyof Economic Studies, Romania. *Corresponding Author: Abstract This paper presents a comparison between different methods used to estimate the static and time-varying optimal hedge ratio, based on their hedging effectiveness, for the U.S. wheat market.the results show that the OLS method using expanding estimation windows outperformsin terms of variance reduction the methods considered for the analysis: the static OLS and error-correction model, the rolling window OLS and the bivariate GARCH errorcorrection model. Keywords: Hedging, Hedging effectiveness, Optimal hedge ratio, Risk management. Introduction The naive recommendation for hedging is to use a hedge ratio equal with the unit value. If the spot and futures prices would be perfectly correlated, the changes in the spot price would be perfectly netted by the changes of the futures price. However, the basis risk exists and the spot and futures prices do not exhibit perfect correlation, converging only at the maturity of the contract. In this case, the naive one-to-one hedge ratio does not minimize the risk of the hedged portfolio and the optimal hedge ratio (OHR) has to be computed. This paper presents a comparison of different methods used to estimate the static and timevarying OHR, based on their hedging effectiveness for the U.S. wheat market. Initially, it is presented the spot and futures price evolution of wheat during the selected timeframe, together with the price main characteristics. Then, the evolution of the basis is discussed. The methods used to estimate the OHR are: the static OLS, the static error-correction model (ECM), the rolling-window OLS with different window lengths (RW OLS), the expanding estimation window OLS (EEW OLS), and the bivariate GARCH error-correction model (B-GARCH).The results show that the OHR estimated through EEW OLS performs better than the static and the rest of time-varying OHRs. The remainder of the paper is organized as follows: the next section briefly presents the main findings in the existing literature; in the third section the methodology is described, followed by the results and discussion section. Finally, the conclusions are given in the last section. Literature Review Numerous papers have the objective of finding the best method to estimate the OHR. The determination of the OHR can be done from a utility maximizing perspective [1-5] or by minimizing a certain risk measure [6-12]. Lence [13] derives the necessary and sufficient conditions for the OHR to be independent of riskaverse agents utility functions and Rao [14] points out that the conditions are satisfied by a wide class of models of spot and futures returns. The OHR that is estimated in the literature is either static (usually estimated through OLS regressions and error-correction models) or timevarying. The OHR that vary through time is estimated using complex methods such as GARCH and regime-switching models or by simple OLS regressions with different rolling windows lengths (RW OLS). Lien et al. [15]reestimated on a day by day rollover the OLS hedging ratio in the post sample period by simultaneously augmenting the next observation and dropping the first observation. Compared with a constant correlation vector GARCH, this RW OLS method provided better results. In a study focused on the electricity market, Bystrom [16] found that the static OLS hedge ratio performed better than the time varying one, Mihai-Cristian Dinica Sep.-Oct Vol.2 Issue

2 estimated through GARCH or RW OLS with 50 periods length method. Alexander and Barboza [17] found no evidence that complex econometric models can improve on a simple OLS regression for estimating the OHR. Moon et al. [18] applied the RW OLS method on the Korean securities market and showed that it is superior to other multivariate GARCH models. Bhattacharya et al. [19] used rolling windows of 30, 60 and 90 days in their estimation for the Indian stock market and found that the moving least squares approach outperforms the static OLS in terms of hedging efficiency. Miffre [20] proposed the conditional OLS model to estimate the time-varying OHR. Dinica and Armeanu [21] showed that the OHR is constant over different estimation periods for copper and aluminium, while Kostika and Markellos [22] estimated the OHR using the ARCD model. Methodology The initial step of the methodology consists in the analysis of the spot and futures price evolution during the selected period. Next, the descriptive statistics of the price levels, logarithmic returns and basis are given and discussed. The basis is computed as the difference between the logarithms of the futures price and spot price. The descriptive statistics that are computed are: mean, median, maximum, minimum, skewness, kurtosis, standard deviation and annualized volatility. The use of non-stationary data in regressions can lead to spurious results [23]. For testing the stationarity of the data series, the augmented Dickey-Fuller test with trend and intercept was applied. Also, for testing the cointegration between the spot and futures prices I used the Johansen cointegration test. The static OHR is estimated using OLS method and error-correction model (ECM). The OHR that vary through time is estimated using the following models: RW OLS with different rolling window lengths, the EEW OLS and the bivariate GARCH (1,1)error-correction model (B-GARCH). The OLS method sets the following: (1) Where and are the spot and futures logarithmic returns, is the estimation of the OHR and represents the error term. The second model used to estimate static OHR is the ECM. The long-run relationship between the logarithm of the spot and futures prices is represented by the following equation: (2) The ECM regression that estimates the OHR is the following: (3) Where the lagged error is term from the long-run relationship and is the error term. The coefficient is the estimation of the OHR using the ECM. The first type of time varying OHR is estimated using the OLS method with different rolling window lengths. The RW OLS equation has the following form: (4) Where is the OHR estimated at time t, based on the information from moment t n+1 to moment t and n represents the number of periods from the rolling window. In this paper, n is set to 50, 150, 300 and 364 periods. The last rolling window length (364 periods) represents the number of weekly returns from the sample period, a methodology that is consistent with the one followed in [15], [17] and [18]. The methodology of EEW OLS consists in reestimation of the OLS hedge ratio after each new passing period, without dropping any observation. The EEW OLS has the following form: (5) Where is the optimal hedge ratio estimated at time t, based on the information from the entire sample. The OHR is also estimated using the bivariate GARCH (1,1) error-correction model. The conditional mean and conditional variancecovariance equations are given by: (6) (7) (8) (9) (10) Mihai-Cristian Dinica Sep.-Oct Vol.2 Issue

3 The OHR is given by the ratio between the conditional covariance and the conditional variance of the futures returns. (11) In order to quantify the efficiency of the hedging it is computed the Ederington (1979) hedging effectiveness indicator (HE). The HE shows the proportion of the variance of the unhedged portfolio that is reduced through hedging. Where portfolio and unhedged spot position. (12) is the variance of the hedged is the variance of the The database consists in weekly spot and futures prices of U.S. wheat for the period from to (522 weekly price observations and 521 weekly returns). In order to Results and Discussion The evolution of spot and futures prices, emphasized in Fig. 1, is characterized by an important volatility during the analyzed period. After a relative calm period between the years 2002 and 2007, the prices faced a violent upsurge caused by the drought from the summer of Thus, in less than a year, the prices increased from a level of 4 USD/bushel to values higher than 12 USD/bushel at the beginning of Given the better weather conditions, the prices retreated in 2008 to previous levels around 4 USD/bushel. After the financial crisis, both the spot and futures prices of wheat continued an upward trend characterized by increased volatility. construct the futures price series, we consider the nearby wheat contract price traded on CBOT, with rollover at the beginning of the expiration month. Also, the day of the week that was chosen is Wednesday and in the case that Wednesday was not a business day, the next business day was considered. The database was split into twosub-samples. The sample period consists in the first 7 years (from to ) and contains 365 weekly price observations and 364 weekly returns (spot or futures). This period is used in order to estimate the static OHR (OLS and ECM), the parameters of the B-GARCH model and the first value of the RW OLS and EEW OLS optimal hedge ratio. The second period is used to estimate the time-varying OHRs (RW OLS, EEW OLS and B-GARCH) and to compute the variances of the hedged and unhedged portfolios. These values are next used to compute the HE indicator for each model SPOT FUTURES Fig. 1: Spot and futures price evolution Table 1: Descriptive statistics Spot Futures Basis Price Return Price Return Mean % % 8.35% Median % % 7.10% Maximum % % 45.72% Minimum % % -7.52% Skewness Kurtosis St dev % % 8.08% Volatility 39.22% 35.21% 58.25% Table 1 presents the main descriptive statistics of the price levels, returns and basis. The data come to confirm the initial findings regarding the increased volatility in the wheat market. Thus, Mihai-Cristian Dinica Sep.-Oct Vol.2 Issue

4 the maximum and minimum weekly returns show that prices can have extreme changes during short periods of time. For example, the spot price exhibits the maximum increase in a week of 23.69%, while the maximum decrease is 22.37% for the same period. It can also be observed that the variations of the spot prices are higher than those of the futures prices. This shows that the spot is more affected and responds faster to the changes in the different factors such as the weather conditions or the supply-demand imbalances. The values of the basis show that the futures price is, on average, higher than the spot price. Also, the volatility of the basis is higher than that of the spot and futures prices. This shows that the hedgers are faced to important basis risk in the wheat market that can make their hedges ineffective. Another finding is that the return and basis distributions are not normal, being asymmetric and leptokurtic. Table 2 presents the results of the ADF test. While the price series are unit root processes, the logarithmic returns represent stationary series. In this case, in order to avoid spurious results, the regressions are estimated based on the return series. Table 2: ADF test results Spot Futures t-stat p-value t-stat p-value Level Return Critical values: 1%: ; 5%: ; 10%: Table 3: Johansen cointegration test results Hypothesis No cointegrating vector At most one Value Critical values: None: 1%: 20.04; 5%: 15.41; At most one: 1%: 6.65; 5%: 3.76 weekly returns vary around an average near 0% Spot returns Fig. 2: Spot logarithmic returns evolution Futures returns Fig. 3: Futures logarithmic returns evolution In Fig. 5 is evidenced the evolution of the basis. During the analyzed period, the basis is positive most of the time, the futures contract trading at a premium compared to spot. The basis reached extreme values, especially during the peak of the financial crisis in the last part of year 2008 when the futures price of wheat was more than 40% higher than the spot price. This may be explained by the increased volatility in the markets during the crisis, the spot price adjusting faster than the futures price to the new conditions. After the end of the financial crisis, the basis returned to normal levels around 5-10%. Another remark to be made is the fact that the basis is characterized by a stochastic and unpredictable evolution. This characteristic induces important risks for the hedger, making the estimation of the OHR even more valuable. The results of the Johansen cointegration test, evidenced in Table 3, show that the spot and futures prices are not cointegrated at the 5% significance level. Thus, it is expected for the ECM model to perform worse than the simple OLS. Next, in Fig. 2 and 3 are emphasized the evolutions of the weekly spot and futures returns. In both cases it can be noticed that periods of high variations alternate with periods of low volatility. Also, the Fig. 4: Basis evolution Mihai-Cristian Dinica Sep.-Oct Vol.2 Issue BASIS

5 As can be easily seen in Fig. 5, the OHRs estimated through the B-GARCH and the RW OLS with the smallest window length (50 weeks) exhibit the greatest volatility during the post sample period. The evolution of the OHR becomes smoother with the increase in the rolling window length, producing in this way smaller rebalancing costs. Also, the static OHRs are greater than the unit value while the time-varying hedge ratios are higher than one in most of the cases IV I II III IV I II III IV I II III IV Naïve OLS ECM RW 50 RW 150 RW 300 RW 364 EW OLS B-GARCH Fig. 5: Optimal hedging ratios Table 4 presents the hedging effectiveness indicators for several models. On the analyzed period, these indicators range between 92.39% and 93.05%. The greatest hedging effectiveness is achieved by the EEW OLS method, while the RW OLS with the window length of 50 weeks and the B-GARCH model ranks the second, respectively the third. Also the RW OLS method with window lengths of 150, 300, respectively 364 weeks does not provide better results that the static OHRs. The results show that the EEW OLS outperforms the RW OLS and the B-GARCH in terms of hedging effectiveness for the case of wheat and it is a method that can be considered in future comparisons regarding the most appropriate model to estimate the OHR. Conclusions In the case that spot and futures prices would be perfectly correlated, the naive one-to-one hedge ratio would reduce entirely the variation of a hedged position, the changes in the spot price being perfectly netted by the changes of the futures price. Table 4: Hedging effectiveness Hedge ratio OLS ECM RW OLS 50 RW OLS 150 RW OLS 300 RW OLS 364 EEW OLS B-GARCH HE The basis risk consists in the fact that the difference between futures and spot prices is not constant over time, the spot and futures prices not being perfectly correlated. This paper showed that the US wheat prices are characterized by a high volatility, the spot and futures returns are not normally distributed and extreme variations are likely. Thus, both wheat producers and buyers face important price risks, and the need for hedging appears in this case. Also, the evolution of the basis is stochastic and volatile. This characteristic induces important risks for the hedgers, making the estimation of the OHR even more valuable. This paper presented a comparison between different methods used to estimate the static and time-varying OHR, based on their hedging effectiveness, for the U.S. wheat market. The analyzed methods are: the static OLS, the static error-correction model (ECM), the rolling-window OLS with different window lengths (RW OLS), the expanding estimation window OLS (EEW OLS), and the bivariate GARCH error-correction model (B-GARCH). The OHRs estimated through the B-GARCH and the RW OLS with the smallest window length (50 weeks) exhibit the greatest volatility during the post sample period. Also, the evolution of the OHR becomes smoother with the increase in the rolling window length and the estimated OHRs are generally higher than the naive hedge ratio. Regarding the hedging effectiveness, the best results are achieved by the EEW OLS method, while the RW OLS with the window length of 50 weeks and the B-GARCH model ranks the second, respectively the third. Acknowledgment This work was cofinanced from the European Social Fund through Sectorial Operational Programme Human Resources Development , project number POSDRU/107/1.5/S/77213, Ph.D. for a career in interdisciplinary economic research at the European standards. Mihai-Cristian Dinica Sep.-Oct Vol.2 Issue

6 References 1. Cecchetti SG, Cumby RE, Figlewski S (1988) Estimation of the optimal futures hedge. Review of Economics and Statistics. 70: Kolb RW, Okunev J (1993) Utility maximizing hedge ratios in the extended mean Gini Framework. J. Futures Markets. 13: Lence SH (1996) Relaxing the assumptions of minimum variance hedging. J. Agricultural and Resource Economics. 21: Bessembinder H, Lemmon ML (2002) Equilibrium pricing and optimal hedging in electricity forward markets. J. Finance. 57: Cotter J, Hanly J (2012) A utility based approach to energy hedging. Energy Economics. 34: Johnson L (1960) The Theory of Hedging and Speculation in Commodity Futures. Review of Economic Studies. 27: Ederington LH (1979) The hedging performance of the new futures markets. Journal of Finance. 34: Benet BA (1992) Hedge period length and ex-ante futures hedging effectiveness: The case of foreign exchange risk cross hedges. J. Futures Markets. 12: Chou WL, Fan KK, Lee CF (1996) Hedging with the Nikkei index futures: The conventional model versus the error correction model. The Quarterly Review of Economics and Finance. 36: Ripple RD, Moosa IA (2007)Hedging effectiveness and futures contract maturity: the case of NYMEX crude oil futures. Applied Financial Economics.17: Chang C-L, McAleer M, Tansuchat R (2011) Crude oil hedging strategies using dynamic multivariate GARCH. Energy Economics Juhl T, Kawaller IG, Koch PD (2012) The Effect of the Hedge Horizon on Optimal Hedge Size and Effectiveness When Prices are Cointegrated. J. Futures Markets.32: Lence SH (1995) On the optimal hedge under unbiased futures prices. Economics Letters. 47: Rao VK (2000) Preference-free optimal hedging using futures. Economics Letters. 66: Lien D, Tse YK, Tsui AKC (2002) Evaluating the Hedging Performance of the Constant Correlation GARCH Model. Applied Financial Economics. 12: Bystrom HNE (2003) The hedging performance of electricity futures on the Nordic power exchange. Applied Economics. 35: Alexander C, Barboza A (2007) Effectiveness of Minimum-Variance Hedging: The Impact of Electronic Trading and Exchange-traded Funds. Journal of Portfolio Management. 33: Moon GH, Yu WC, Hong CH (2009) Dynamic hedging performance with the evaluation of multivariate GARCH models: evidence from KOSTAR index futures. Applied Economics Letters. 16: Bhattacharya S, Singh H, Alas RM (2011) Optimal hedge ratio with moving least squares an empirical study using Indian single stock futures data. International Research Journal of Finance and Economics. 79: Miffre J (2004) Conditional OLS Minimum Variance Hedge Ratio. Journal of Futures Markets. 24: Dinica MC, Armeanu, D (2013) Optimal risk management at metals market. Actual Problems of Economics. 145: Kostika, E. and Markellos, R.N. (2013) Optimal Hedge Ratio Estimation and Effectiveness Using ARCD. Journal of Forecasting. 32: Cotter J, Hanly J (2006) Revaluating Hedging Performance. The J. Futures Markets. 26: Mihai-Cristian Dinica Sep.-Oct Vol.2 Issue