Introduction

This Chapter focuses on the concept of cross hedging and how to determine the hedge ratio when hedging with futures contracts. Cross hedging is a hedging strategy where an investor uses futures contracts that are not perfectly matched to the underlying asset being hedged. This technique is commonly employed when a suitable futures contract for the specific asset is not available or when the correlation between the asset and available futures is not perfect. We will explore the calculations involved in determining the hedge ratio and assess the effectiveness of cross hedging in managing risk.

Understanding Cross Hedging

Cross hedging involves using futures contracts that are related to, but not exactly the same as, the asset being hedged. While perfect hedges are ideal, cross hedging can be an effective alternative when perfect matches are unavailable. The success of cross hedging depends on the degree of correlation between the asset being hedged and the chosen futures contract.

Computing the Hedge Ratio

The hedge ratio (HR) represents the number of futures contracts required to hedge the exposure of one unit of the underlying asset. It is calculated using the correlation coefficient between the spot price of the underlying asset (S) and the price of the chosen gold futures contract (F). The formula for the hedge ratio is:

HR = Correlation(S, F) * (Standard Deviation(S0) / Standard Deviation(F))

where:

• Correlation(S, F) is the correlation between the underlying asset and the chosen futures contract.
• Standard Deviation(S) is the standard deviation of the spot price of underlying asset.
• Standard Deviation(F) is the standard deviation of the futures contract price.

Assessing Hedge Effectiveness

Hedge effectiveness measures how successful the cross hedge is in offsetting price movements of the underlying asset. Understanding the effectiveness of a cross hedge helps investors and businesses make informed decisions about their risk management strategies. To evaluate the effectiveness of the cross hedge, we use the coefficient of determination (R-squared), which indicates the proportion of the variance in the spot price of underlying asset (S) that is explained by the chosen futures contract (F) of the underlying asset. The formula for R-squared is:

R-squared = (Correlation(S, F))^2

R-squared will give us the percentage of price risk reduction achieved by the cross hedge. A higher R-squared value indicates a more effective hedge.

Now, let’s assume that the manufacturer’s analysis gives the following data:

• Correlation(G, F) = 0.85
• Standard Deviation(G) = 10 (measured in dollars per unit of gold used)
• Standard Deviation(F) = 15 (measured in dollars per futures contract)

Example: A jewelry manufacturer wants to hedge the price risk of gold used in their products. However, there is no available futures contract for the specific type of gold used. The manufacturer decides to use a gold futures contract that is correlated with the type of gold they use but not perfectly matched. The historical data shows that the correlation between the gold used and the chosen futures contract is 0.85. Calculate the hedge ratio and evaluate the effectiveness of the cross hedge. Assume that standard deviation of the gold spot is 10 and standard deviation of the gold futures is 15.

Solution: To calculate the hedge ratio and evaluate the effectiveness of the cross hedge, we can use the concept of the coefficient of determination (R-squared) to determine the percentage of price risk reduction achieved by the cross hedge.

Step 1: Calculate the Hedge Ratio
HR = 0.85 * (10 / 15) ≈ 0.5667

Step 2: Evaluate the Effectiveness of the Cross Hedge
R-squared = (0.85)^2 = 0.7225 ≈ 72.25%

• Correlation(G, F) = 0.85
• Standard Deviation(G) = 10 (measured in dollars per unit of gold used)
• Standard Deviation(F) = 15 (measured in dollars per futures contract)

Inerpreatation: The calculated hedge ratio is approximately 0.5667, and the R-squared value is approximately 72.25%. This means that the chosen gold futures contract, despite not being perfectly matched, can still explain about 72.25% of the variance in the spot price of gold used by the jewelry manufacturer. While it may not provide a perfect hedge, it can significantly reduce the price risk associated with the gold used in their products. The effectiveness of the cross hedge can be considered satisfactory, but the manufacturer should be aware that there may still be some residual risk due to the imperfect correlation.