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Minimum variance hedging: Levels versus first differences
Abstract
Nowadays it is widely accepted to estimate minimum variance hedge ratio regressions in first differences. There are both statistical and economic reasons for a first difference approach. However, no study has ever analyzed whether the first difference approach is also consistent with the theory of minimum variance hedging. In this paper we show, on the basis of a simulation study, that the first difference model with intercept does not provide hedge ratio estimates that are in line with the theory of minimum variance hedging. Only a linear regression model in levels provides theoretically consistent results.
Keywords: Minimum variance hedging, level regression, first differences, hedge ratio.
JEL classifications: Q11, Q13.
1. Introduction
Nowadays, minimum variance hedging (Johnson, 1960; Stein, 1961) is probably one of the best-known concepts in the literature on agricultural futures markets. Less well known, however, is probably that it took more than twelve years for the approach to become established in the literature. The approach only became a standard after Heifner (1972) showed that the minimum variance hedge ratio (i.e., the hedge ratio that minimizes the cash price risk for a hedger) can also be obtained from a linear regression model. One would only have to regress the cash price of a commodity on its futures price, and the slope parameter would then correspond to the minimum variance hedge ratio originally derived by Johnson (1960).
The exact formula of the minimum variance hedge ratio is:
๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐
where ๐๐๐๐๐๐ defines the correlation coefficient of the cash and futures price; and, ๐๐๐๐ and ๐๐๐๐ the
corresponding standard deviations of the cash and futures price.
Sรถren Prehn
Although, the minimum variance hedge ratio regression was initially also estimated in levels (e.g., Ederington, 1979; Berck, 1981; Dale, 1981), this changed quite quickly. Nowadays it is common practice to estimate the minimum variance hedge ratio regression in first differences (e.g., Park and Antonovitz, 1990; Brorsen et al., 1998; Brinker et al., 2009). There are both statistical and economic reasons for a first difference approach. For instance, first differences help to exclude the possibility of a spurious regression (Brown, 1985), and leading agricultural economists (Working, 1953; Peck, 1975) argued that hedgers (i.e., grain merchants or farmers) are more interested in the correlation of price changes than price levels.
However, no study has ever analyzed whether the first difference approach is also compatible with the theory of minimum variance hedging (Johnson, 1960; Stein, 1961). In this paper, however, we want to show on the basis of a simulation study that the last point is quite important, since the first difference model with intercept does not provide hedge ratio estimates that are consistent^1 with the theory of minimum variance hedging. As our results will show, only a linear regression model in levels provides hedge ratio estimates that are in line with the theory of minimum variance hedging.
In the following we will further analyze the last point. We will use a simple simulation study to show that only a linear regression model in levels provides hedge ratio estimates that are consistent with the theory of minimum variance hedging. Finally, we will conclude.
2. Levels versus first differences
For the sake of simplicity, we have limited our simulation study to a single futures contract. We decided to focus on an inverse market^2 , as Johnson (1960) did in his original article. However, our results also apply to a carry market.
(^1) When we talk about consistent in the following, we always mean consistent from a theoretical point of view and not from a statistical point of view. (^2) The term inverse market refers to a market situation in which the cash price is higher than the futures price. Inverse markets are known for discounting storage. In fact, short hedgers (i.e. those with a short position on the futures market) would suffer a hedging loss on an inverse market. The opposite of an inverse market is a carry market. Carry markets renumerate storage (Hieronymus, 1977).
More interesting, however, are the results for the first difference model. As Table 1 shows, the average minimum variance hedge ratio for the first difference model with intercept (column 3) is 0.9521. This value obviously not only differs significantly from the algebraic solution (the level regression) with 0.4982, but also comes (very) close to a full hedge.^5 In fact, if we had not assumed a random walk with drift for the basis but a deterministic trend, we would have obtained a value of one, i.e. a full hedge.
However, a full hedge only minimizes the cash price risk and not the basis risk (Hieronymus, 1977). The basis risk, however, was the main reason why Johnson (1960) originally developed the minimum variance hedge approach. In the 1950s, the New York coffee market, which was mainly investigated by Johnson (1957a,b), was chronically inverse^6 , and the majority of the New York coffee importers complained about the seasonally weakening basis, which gave them regular hedging losses as short hedgers; because of the weakening basis, the coffee importers could not hedge without having to buy coffee in Brazil for a stronger basis and resell it in New York for a weaker basis (Gray, 1960a). On the other hand, the coffee importers could not do without hedging either, since the U.S. banks always required a hedge as security for the interim financing of a forward transaction (Gray, 1960b). Johnson's main concern, therefore, was to develop a new hedging approach to minimize not only the cash price risk but also the basis risk causing the hedging losses.
As Table 1 shows, the first difference model with intercept is in clear contradiction to Johnsonโs efforts to also minimize the basis risk and thus the hedging losses. If anything, the model only minimizes parts of the basis risk, but not the entire basis risk. In fact, in the concrete example, the New York coffee importers would still make an average hedging loss of -4.5206 (column 3) due to the only partially hedged basis risk.^7 Consequently, we can conclude (at least on the basis of the simulation results) that the first difference model with intercept is not consistent with the theory of minimum variance hedging.
(^5) In fact, hedge ratios in the literature are usually close to one when first differences are used (Brown, 1985). The general rule is that the higher the correlation between the cash price and the futures price, the closer the hedge ratio is to one. (^6) Only once between 1950 and 1957 was the New York coffee market not inverse, compare Figure 2 in Gray (1960b). (^7) For comparison, the full hedge would have caused a hedging loss of -5.
The last point, however, can also be further substantiated econometrically. When two prices (such as the cash price and the futures price) converge^8 , the drift between the two prices is known to be captured by the intercept in a first difference model (Enders, 2014). In the concrete example, however, the drift corresponds to the basis, more precisely to the average change in the basis (a proof of this can be found in Appendix C). However, if the non- random, deterministic part of the basis risk (i.e., the average basis change) is already captured by the intercept, the slope parameter of the first difference model only provides a hedge ratio that minimizes the cash price risk and the random basis risk, but not the deterministic basis risk. Johnsonโs main concern, however, was never to minimize only the random basis risk but also the deterministic basis risk (i.e., the chance in the basis), especially since the deterministic basis risk was primarily responsible for the hedging losses the New York coffee importers experienced in the 1950s.
As it turns out, the only approach that is consistent with Johnsonโs theory of minimum variance hedging, is the minimum variance hedge ratio regression in levels. Only the level regression minimizes the total basis risk and thus the hedging losses (see Table 1, column 2), which were the main problem of the New York coffee importers in the 1950s (Gray, 1960a,b).^9
However, we have so far left out one important point, namely the measurement of hedging effectiveness. In the literature it is common practice to use the adjusted R-squared to measure hedging effectiveness (Ederington, 1979). According to Ederington, the R-squared measures the percentage of the cash price risk that is offset by a minimum variance hedge. For instance, in the concrete example, the level regression would offset 98.93 percent of the cash price risk, whereas the first difference model with intercept would only offset 94.70 percent, see Table
(^8) If the cash and futures price did not converge at contract maturity, arbitrageurs could make a risk-free profit. They could buy on the market with the lower price and sell on the market with the higher price (Hieronymus, 1977). (^9) In order to avoid a spurious regression, we recommend always testing for cointegration in advance. If the latter is given, there should be no statistical problems with a level regression. The slope parameter should measure the long-run minimum variance hedge ratio.
shows what the main problem of the first difference model is: The first difference model only partially minimizes the basis risk, which is in clear contradiction to Johnsonโs theory of minimum variance hedging.
The previous results are quite interesting, as they challenge the common practice of estimating minimum variance hedge ratio regressions in first differences (e.g., Brorsen et al., 1998; Brinker et al., 2009). As we have shown above, first differences are clearly in contradiction with the theory of minimum variance hedging. If minimum variance hedge ratios are really of interest, then there is no other way than to estimate the minimum variance hedge ratio regression in levels.^10 First differences would not provide hedge ratio estimates that are consistent with the theory of minimum variance hedging.
Our results, however, should not be misunderstood. We do not deny that there may be situations in which hedgers want to minimize only the random basis risk, but not the deterministic basis risk. In these situations, the first difference approach is clearly the right choice. Our point is, when the first difference approach is used for hedging, it should not be called a minimum variance hedge, which it clearly is not, but a random basis risk hedge. In particular, potential users of the hedge should be made aware that they are still exposed to a significant basis risk that may or may not work out in their favor.
3. Conclusions
In this paper we have investigated whether it is theory conform to estimate the minimum variance hedge ratio regression in first differences, as it is common in the literature. Interestingly, we were able to show that the first difference model with intercept does not provide hedge ratio estimates that are compatible with the theory of minimum variance hedging (Johnson, 1960). The first difference model minimizes only the random basis risk, but not the total basis risk. The total basis risk, however, was actually the original reason for Johnson to develop the minimum variance hedge approach. The only approach that provides hedge ratio estimates consistent with the theory of minimum variance hedging is a linear regression model in levels.
(^10) Note that to estimate the minimum variance hedge ratio consistently, it is necessary to use spread-adjusted data (i.e., spread-adjusted continuous futures and cash prices). Non-adjusted data would deliver inconsistent estimates (Prehn, 2020).
References
Berck, P. โPortfolio Theory and Demand for Futures: The Case of California Cottonโ, American Journal of Agricultural Economics , Vol. 63(3), (1981) pp. 466โ474.
Brinker, A.J., Parcell, J., Dhuyvetter, K. and Franken, J.R.V. โCross-Hedging Distillers Dried Grains Using Corn and Soybean Meal Futures Contractsโ, Journal of Agribusiness , Vol. 27(1-2), (2009) pp. 1โ15.
Brorsen, B.W., Buck, D.W. and Koontz, S.R. โHedging hard red winter wheat: Kansas City versus Chicagoโ, Journal of Futures Markets , Vol. 18(4), (1998) pp. 449โ466.
Brown, S.L. โA Reformulation of the Portfolio Model of Hedgingโ, A merican Journal of Agricultural Economics , Vol. 67(3), (1985) pp. 508โ512.
Dale, C. โThe Hedging Effectiveness of Currency Futures Marketsโ, Journal of Futures Markets , Vol. 1(1), (1981) pp. 77โ88.
Ederington, L.H. โThe Hedging Performance of the New Futures Marketsโ, Journal of Finance , Vol. 34(1), (1979) pp. 157โ170.
Enders, W. Applied Econometric Time Series (Hoboken, New Jersey: John Wiley & Sons, Inc., 2014).
Gray, R.W. โThe Characteristic Bias in Some Thin Futures Marketsโ, Food Research Institute Studies , Vol. 1(3), (1960a) pp. 296โ313.
Gray, R.W. The Importance of Hedging in Futures Trading; and the Effectiveness of Futures Trading for Hedging. In: H.H. Bakken, R.W. Gray, T.A. Hieronymus, and A.B. Paul (Eds.), Futures Trading Seminar, History and Development: Volume 1, 61โ120. (Madison, Wisconsin: Mimir Publishers Inc., 1960b).
Heifner, R.G. โOptimal Hedging Levels and Hedging Effectiveness in Cattle Feedingโ, Agricultural Economics Research , Vol. 24(2), (1972) pp. 25โ36.
Hieronymus, T.A. The Economics of Futures Trading (New York, New York: Commodity Research Bureau, 1977).
Johnson, L.L. โPrice Instability, Hedging, and Trade Volume in the Coffee Futures Marketโ, Journal of Political Economy , Vol. 65(4), (1957a) pp. 306โ321.
Appendix A
We used the software R for our simulation study.
Simulation study
set.seed(123)
Specify random walk function
RW <- function(N, x0, mu, variance) {
z <- cumsum(rnorm(n = N, mean = 0, sd = sqrt(variance))) t <- 1:N x <- x0 + t * mu + z return(x) }
Construction of vector
n <- 100000
vec_full_hedge_profit <- rep(NA, n)
vec_mvhr <- rep(NA, n)
vec_coef_levels <- rep(NA, n) vec_levels_adj_r2 <- rep(NA, n) vec_levels_profit <- rep(NA, n)
vec_coef_differences_with <- rep(NA, n) vec_differences_with_adj_r2 <- rep(NA, n) vec_differences_with_profit <- rep(NA, n)
For loop
for(i in 1:n){
construction futures price
drift <- 0. cash_price <- RW(21, 94.75, drift, 0.05)
construction basis
basis <- RW(21, -5.25, 0.25, 0.0025)
construction cash price
futures_price <- cash_price + basis
Calculating profit full hedge
vec_full_hedge_profit[i] <- -(cash_price[1] - futures_price[1]) + (cash_price[21] - futures_price[21])
Calculating MVHR algebraically
vec_mvhr[i] <- cor(cash_price, futures_price) * (sd(cash_price)/sd(futures_price))
level regression
fit_levels <- lm(cash_price ~ futures_price) vec_coef_levels[i] <- coef(fit_levels)[2] vec_levels_adj_r2[i] <- summary(fit_levels)$adj.r.squared
profit level regression
fit_aux <- lm(I(diff(cash_price) - (coef(fit_levels)[2] * diff(futures_price))) ~ 1) vec_levels_profit[i] <- coef(fit_aux)[1] * 20
first difference regression, with intercept
fit_differences_with <- lm(diff(cash_price) ~ diff(futures_price)) vec_coef_differences_with[i] <- coef(fit_differences_with)[2] vec_differences_with_adj_r2[i] <- summary(fit_differences_with)$adj.r.squared
profit first difference regression, with intercept
vec_differences_with_profit[i] <- coef(fit_differences_with)[1] * 20 }
Calculating average values
profits
mean(vec_levels_profit) mean(vec_differences_with_profit) mean(vec_full_hedge_profit)
MVHRs
mean(vec_coef_levels) mean(vec_coef_differences_with)
Adjusted R2
mean(vec_levels_adj_r2) mean(vec_differences_with_adj_r2)
Furthermore, if one considers the definition of the basis, ๐๐ = ๐๐๐๐ โ ๐๐๐๐, and rearranges the
former definition, then
๐ผ๐ผ๏ฟฝ = ๐๐ฬ
๐๐,๐ก๐ก โ ๐๐ฬ
๐๐,๐ก๐ก โ ๏ฟฝ ๐๐ฬ
๐๐,๐ก๐กโ1 โ ๐๐ฬ
๐๐,๐ก๐กโ1 ๏ฟฝ = โ๐๐๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐ก๐ก.
According to this definition, the intercept parameter ๐ผ๐ผ captures the average change of the basis over the time period of the hedge.
