Comparison of Some Static Hedging Models of Agricultural Commodities Price Uncertainty

Abstract

In static framework, many hedging strategies can be settled following the various hedge ratios that have been developed in the literature. However, it is difficult to choose among them the best the appropriate strategy according the to preference or economic behavior of the decision-maker such as prudence and temperance. This is so even with the hedging effectiveness measure. After introducing a hedging ratio that take into account the prudence and temperance of the decision maker, we propose a ranking based approach to measure the effectiveness using L-moment to classify hedge portfolios, hence hedge ratios, with regard to their performance. Moreover, we deal with the hedging issue in presence of quantity and rollover risks and derive an optimal strategy that depends upon the basis and insurance contract. Such hedging issue includes the relevant risks encountered in practice and we relate how insurance contract, specially designed for production risk could affect the futures hedge. The application on some agricultural futures prices data at hands shows that taking into account quantity and rollover risks leads to better hedging strategy based on the L-performance effectiveness measure.

A Estimation of Hedge Ratios

In practice, the estimation of hedge ratio depends on the methods that is adopted to compute the hedge ratio. Herein, we describe some estimations methods for existing approach for hedge ratio with no quantity risk.

The minimum-variance hedge ratio is simply estimated by linear regression of spot returns on futures returns

$$\begin{aligned} r_{s,t} = a + \beta r_{f,t} + \varepsilon _t, \end{aligned}$$

(A.1)

where a the intercept, $\beta$ an estimate of $h_{\textsc {mv}}$, $\varepsilon$ the error term and t is the observation time. While the linear regression is easy to implement by ordinary least square technique, it relies on no exhaustive assumptions which makes the estimated hedge ratio critical on statistical basis. Error term in Eq. (A.1) is often heteroskedastic and ordinary least square approach is based on unconditional mean and variance instead.

In the expected utility approach, appropriate utility function and distribution are usually guested to achieve closed form solution. Otherwise, numerical approximation usually allows to derive the hedge ratio.

The estimation of the mean-extended-Gini hedge ratio, is usually based on empirical distribution function of $R_h$

$$\begin{aligned} {\widehat{\Gamma }}_{h}(\delta ) = \frac{\delta }{N}\left\{ \sum _{i=1}^Nr_{h,i}[1-{\widehat{G}}(r_{h,i})]^{\delta -1}- \frac{1}{N}\left( \sum _{i=1}^Nr_{h,i}\right) \left( \sum _{i=1}^N[1-{\widehat{G}}(r_{h,i})]^{\delta -1}\right) \right\} \end{aligned}$$

(A.2)

where N is the sample size and $r_{h,1},\ldots ,r_{h,N}$, the observations of hedge portfolio returns. Then mean-extended-Gini coefficient, ${\widehat{\Gamma }}_{h}(\delta )$ is minimized as risk measure function. Alternatively, Shalit (1995) had used another formula whose estimation is as follows

$$\begin{aligned} {\widehat{h}}_{\textsc {MEG}} = \frac{\sum _{i=1}^N(r_{s,i}-\bar{r_{s}})(d_i - {\bar{d}})}{\sum _{i=1}^N(r_{f,i}-{\bar{r}}_f)(d_i - {\bar{d}})} \end{aligned}$$

(A.3)

with $d_i = [1-{\widehat{G}}(r_{h,i})]^{\delta -1}$ and ${\bar{d}}=\sum _{i=1}^{N}d_i/N$.

The lower partial moment hedge is approximated either on basis of the empirical distribution or the kernel estimation, Lien and Tse (2000). The empirical distribution approach leads to

$$\begin{aligned} {\widetilde{\ell }}_{n}(c,r) = \frac{1}{N}\sum _{r_{h,i}<c}(c-r_{h,i})^n, \end{aligned}$$

(A.4)

and the kernel estimation consists in substituting the probability density function of the portfolio returns by a kernel density function,¹³

$$\begin{aligned} {\widehat{\ell }}(n,{\bar{r}},G) = \frac{1}{N\varpi }\sum ^{N}_{i=1}\int ^{{\bar{r}}}_{-\infty }({\bar{r}}-r)^n\,k\left( \frac{r-r_i}{\varpi }\right) \,\mathrm dr, \end{aligned}$$

(A.5)

with k is the kernel function and $\varpi$ is the bandwidth. By plugging $z=(r-r_i)/\varpi$ into the integral, we have

$$\begin{aligned} {\widehat{\ell }}_n({\bar{r}},G) = \frac{1}{N}\sum ^{N}_{i=1}l_{n}(c,r_{h,i}), \end{aligned}$$

(A.6)

with

$$\begin{aligned} l_{n}(c,r_{h,i}) = \int ^{(c-r_{h,i})/\varpi }_{-\infty }(c-z\varpi -r_{h,i})^n\,k(z)\mathrm dz. \end{aligned}$$

(A.7)

Setting $n=2$ and assuming that the portfolio returns and the futures returns are independent, then hedge ratio is the same as the of semi-variance will be the same as the minimum variance hedge ratio, Lien and Tse (2002).

Traditional way to estimate the hedge ratios from VaR and CVaR is numerical optimization, unless convenient distributions is use to get closed form solution (Sadefo Kamdem 2009; Sadefo 2005).