Hedge Ratio Theorem in Excel

The Hedge Ratio Theorem is a concept in risk management that helps investors to hedge their positions using futures or options contracts. It states that the optimal hedge ratio, which is the proportion of the hedged position to the total position, is equal to the correlation between the spot and futures prices, multiplied by the ratio of the standard deviations of the spot and futures prices.

In other words, the Hedge Ratio Theorem tells us how many futures or options contracts we need to buy or sell to minimize the variance of the hedged position’s value. The theorem is especially useful for cross-hedging, which is when the underlying asset and the hedging instrument are not identical, but have some degree of correlation.

For example, suppose an airline company wants to hedge its exposure to jet fuel prices using crude oil futures contracts. The company can use the Hedge Ratio Theorem to calculate the optimal number of crude oil futures contracts to buy or sell, based on the correlation and standard deviations of the jet fuel and crude oil prices. By doing so, the company can reduce the risk of losing money due to fluctuations in jet fuel prices.

Basic Theory:

The Hedge Ratio Theorem is primarily used in the context of hedging, which involves taking a position in one
asset to offset the risk associated with another asset. The basic theory revolves around finding the optimal
ratio of the two assets to create a hedge that minimizes risk exposure.

Let’s consider two assets, Asset A and Asset B. The Hedge Ratio (H) is the ratio of the change in the value of
Asset A to the change in the value of Asset B. Mathematically, it can be expressed as:

Procedures:

1. Collect Data: Gather historical data for the two assets you want to analyze. This data
   should include the prices or returns of both assets over a specified period.
2. Calculate Returns: Calculate the returns for both assets by taking the percentage change in
   prices over the chosen period.
3. Compute Hedge Ratio: Use the formula mentioned earlier to compute the Hedge Ratio.
4. Optimize Hedge Ratio: Analyze the historical data to determine the optimal Hedge Ratio
   that minimizes risk. This can be done through statistical methods or optimization techniques.
5. Apply Hedge Ratio: Implement the calculated Hedge Ratio to create a hedge by taking
   positions in both assets.

Scenario:

Let’s consider a scenario with fictional stock prices for Asset A and Asset B over a period of five days:

Calculation in Excel:

1. Calculate the daily returns for Asset A and Asset B.

                   =(B2-B1)/B1
                   =(C2-C1)/C1
   

2. Compute the Hedge Ratio.

                   =CORREL(D2:D6, E2:E6)
   

   Here, D2:D6 represents the returns for Asset A, and E2:E6 represents the returns for Asset B.

3. Optimize the Hedge Ratio based on historical data analysis.
   • For simplicity, let’s assume the correlation itself is the optimized Hedge Ratio in this scenario.
4. Apply the Hedge Ratio to create a hedged position.

                   H = -0.76 (as an example)
   

   This means that for every 1 unit change in Asset B, you would short 0.76 units of Asset A to create an
   effective hedge.

Result:

In this scenario, the calculated Hedge Ratio of -0.76 suggests a negative correlation between Asset A and Asset
B. Implementing this hedge ratio in a portfolio would help mitigate the overall risk exposure.

Other Approaches:

1. Regression Analysis: Use linear regression to analyze historical data and determine the
   slope coefficient as the Hedge Ratio.
2. Variance-Covariance Method: Calculate the variance and covariance between the two assets
   to derive the optimal Hedge Ratio.
3. Advanced Statistical Methods: Employ more sophisticated statistical methods or machine
   learning algorithms for a robust analysis of historical data and optimal Hedge Ratio determination.