Hedging via the futures markets is not always straightforward and one has to overcome problems such as mismatch of the maturity date, the underlying asset, or both. The former leads to a delta-hedge, the latter to a cross-hedge and when both are in place to a cross delta hedge. The seminal ideas go back almost half a century and the usual approach is due to Johnson (1960), Stein (1961) and Ederington (1979). A wider perspective on cross hedging in a risk return framework is described by Anderson and Danthine (1981). It is now a common subject in textbooks such as Stoll and Whaley (1993), Ritchken (1996), and Clark (2002), to name just a few, and yet there is still an ongoing debate about what techniques are the most useful.

In this paper we are concerned with commodity cross-hedging. Al- though it has been shown that it is possible to cross-hedge commodities with currencies as in Sadorsky (2000) and currencies with commodities as in Benet (1990), in this paper we consider only the case of two markets from the same class, one a futures market and the other a spot market with different but correlated commodities,

The most widely used methods for calculating the optimal hedge ratio are based on regression techniques. Three types of regression models can be used: price level, price change level and percentage change level. We denote by S(1) the spot price at time ¢ of the commodity targeted for hedging and by F,(1) the futures price at time f of the proxy hedging commodity with maturity time 7, The maturity of futures contracts usually comes after the hedging period. In other words if 7 is the exposure period for the hedge, the futures contracts used for cross hedging will have a maturity T>T,

In this paper we consider only the situation of minimizing the risk of holding a portfolio of the underlying commodity and futures contracts of one or several of its proxies. Shorting b, futures of one proxy for each long posi- tion in the spot leads to a cash flow at time 7, equal to

S(T) - BF (7) + BF O) () Minimizing the variance of this cash flow leads to cov,(S(T)),F.(7,)) var (FT)

The optimal coefficient 8, depends on the horizon of exposure to hedging T, and the futures maturity 7, In practice it is often assumed that it is constant with respect to time, although some studies suggest that this is not always the case. We shall treat this assumption with caution but for the sake of simplic- ity we drop the index and refer to the optimal hedge ratio as simply B.

The formula given in (2) can be recovered from a simple regression model such as

S(t) = + BF, (O +, (3)

where Foo (1) is the price of the futures contract at time ¢ with 7)(1) the nearest available maturity. The estimate of the optimal hedging ratio depends on the historical data.

In order to solve problems related to autocorrelation, many authors prefer a regression at the price change level. Thus, f is estimated from ()-F

S(t)- SU-1) = @+BCF -l) +e (4) T2 t a) TAL)

There may still be problems with the regression in (4) associated with heteroscedasticity. When this is the case, a percentage change level re- gression is often used in the form

S(t) Fray (O -l=a+b |—————- 1] te, (5) S(-1) F (1-1)

Tar!)

The hedge ratio is calculated from the estimated slope of this regression us- ing the formula where f* is the last day in the estimation sample.

When simple linear regression methods are used, the R? is a measure of the efficiency of the hedging. The theoretical support is provided by Ederington (1979) and Dale (1981) but some research by Chang and Fang (1990) indicated that this measure is not always appropriate and other mea- sures of efficiency may be more useful. However, the R? is still largely used in practice for its direct interpretability and ease of calculation, especially in conjunction with regression based methods.

The basic principles of hedging can be used for many commodities for which no futures contract exists, because often they are similar to commodities hay- ing futures that are traded. Wit! et al. (1987) compare the analytical approaches for estimating hedge ratios for agricultural commodities and discuss various issues related to regression based methods.

Our interest is jet fuel or kerosene, a commodity that could be hedged by using a number of other energy products. In energy markets, basis risk is an ever-present ingredient in hedge selection, On the one hand, the distribu- tion of prices in the energy markets is not as unbounded as in the foreign exchange market because operating costs and constraints tend to underpin downward price movements (Moonier and Potter 1998), However, on the other hand, other economic considerations such as transfer costs, storage costs and location may lead to very interesting cross hedging problems. One good example is Woo et al. (2001) where cross hedging in power utilities markets is discussed.

Another problem outlined by Pindyck and Rotemberg (1990) is that exchange rates seem to be a factor in commodity pricing, thereby adding a potential complication to the problem of cross hedging if sensitivities vary across commodities, They found that an equally weighted index of the dollar value of British pound, German mark, and Japanese yen negatively and sig- nificantly impacts the price of crude oil in both OLS regressions and latent variable models. Sadorsky (2000) also showed that futures prices for oil re- lated products are co-integrated with a trade-weighted index of exchange.

With these results in mind, we can use ordinary least squares (OLS) in the regression model (3) to fit the data, The estimate for the slope coefficient represents the hedge ratio. The results are reported in Table 5 where we see that the heating oil contract (NHO) is superior to the other proxies, both in terms of the significance of the hedge ratio measured by the t-statistic and by the overall goodness of fit measured by the R?. However, we can see from the value of the Durbin-Watson statistic that we have problems with positive autocorrelation.

This is a typical case where researchers will often change their re- gression model from a price level to a price change level. Although a large group of studies (Hill and Schneeweis (1981); Park et al. (1987); Braga et al. (1989) among many others) emphasized that using price level models is wrong due to obvious statistical problems we strongly agree with Witt et al. (1987) that the statistical first difference model is not congruent to the price change model. This is because the lag operator for price (percentage) change models takes differences as the change in prices over the time interval representing the hedge exposure and, as long as this exposure is not identical to the fre- quency of the data under the analysis. autocorrelation may still be a problem. Moreover, the exposure is not important for price level models because the same ratio can be used whereas when price change or percentage change models are employed the hedge ratio depends on the hedging period.

Furthermore, as we mentioned above, most airline companies, al- though they might have some storage capacity, cannot afford to stock vast amounts of kerosene for long periods of time. Therefore, their concern is the current futures price and the ending basis and this is where price level models are relevant. If there is any problem with autocorrelation, instead of resorting to a difference model, a more appropriate solution would be to correct the estimates of the regression coefficients using a Cochrane-Orcutt procedure (see Davidson and MacKinnon, 1993, Chapter 10). Then the adjusted esti- mates are BLUE.

Table 6 reports the results of the Cochrane-Orcutt adjusted estimates where f represents the new hedge ratio. One of the first things to be noticed is the improvement in the Durbin Watson statistic, We can also see that the - statistics for the hedge ratio and the R have degenerated considerably. In fact, the contracts on Brent crude (LCR) and liquid propane gas (NPG) are no longer relevant proxies. Based on Table 6, the heating oil contract (NHO) is clearly statistically superior to the others.*

Thus, we conclude that the heating oil contract is the best hedging instru- ment. One last problem remains, however. It is well-known that non-syn- chronization between spot data and futures data may affect the inference pro- cess because of a lead-lag relationship. To account for any synchronization problems in our data, we also calculate the Scholes and Williams (1977) in- strumental variable estimator, suggested by Sercu and Wu (2000), for the hedge ratio for the three contracts that remain relevant. This estimator is calcu- lated with the formula cov (AS(#),1V(1))

SW= (10)

cov (AF(t),1V(1))

where V(t) = AF(-1) + AF(t) + AF(tH+1) = F(t+1) - F(t-2) (11)

and AS(t) = S(1) - SQ-7) and AF(t) = F(t) - F(t-1).

Table 7 gives the results of the SW estimates for our data.

Jet fuel (kerosene) is a major cost for the airline companies. As a commodity linked to petroleum products its price is volatile and constitutes an important risk for these companies. Managing this risk is complicated by the fact the only futures contract traded on jet fuel is listed on the Tokyo market in Japa- nese yen, Thus, outside of Japan hedging on the organized exchanges re- quires the use of a proxy instrument to set up across-hedge. Numerous cross- hedging instruments have been proposed and this paper compares the effec- tiveness of the five major contracts suggested in the literature and used in practice - the Brent crude oil contract on the International Petroleum Ex- change (IPB), the light sweet crude oil contract on the New York Mercantile Exchange (NYMEX), the heating oil contract on NYMEX, the unleaded regu- lar gas contract on NYMEX, and the liquid propane gas contract on NYMEX -for hedging the spot jet fuel price on the Singapore spot market.

We find that the heating oil contract gives the best in-sample results. However, after correcting for autocorrelation, the fit of the new models to the data is not too good, explaining only about 24% - at best - of changes in the spot price. Since the problem may lie with the futures prices and poor synchronisation between spot data and futures data, we calculate the Scholes and Williams instrumental variable and then test out-of-sample. These tests cloud the picture. For the OLS based estimates, the Theil U and the Theil inequality coefficients suggest thal none of the contracts are very effective.

Furthermore, the heating oil contract is no longer the unambiguous front run- ner with the lowest R* and the worst Theil inequality coefficient, although its Theil-U is the best. When we perform the out of sample tests with the TV based estimates, the results are worse, thereby suggesting that there is noth- ing to be gained by following this route, although the heating oil contract seems to have the best performance.