The Minimum Variance Hedge Ratio


One of the basic risks that investors face while dealing with international transactions is foreign exchange risk. In order to preserve ones position in a profitable place then a hedge can be taken. This is mostly in the form of futures contracts. They specify the price at which an asset or financial instrument can be bought or traded at a future date. The basic principle can be explained in this way. Consider a trader who sells groceries. They anticipate that the prices will change over time in an unfavorable manner. Hence they go to their supplier and agree that they will buy the groceries at a price they find to be favorable at some future date.

Futures contracts, especially when dealing with foreign exchange rates, assumes the some concepts to hold so as to avoid an arbitrage profit. These assumptions are there are no transaction costs, a constant tax rate is maintained and the same risk free rate is maintained for both lending and borrowing eventualities.

A short hedge are usually initiated by traders who are not sure of a product’s prices, and they speculate that the prices may decline, when the transaction date occurs. Long hedges are usually initiated by traders who intend to purchase an item but the positions of both traders is similar. Even with this assurance there are risks involved in a futures contract. This makes it paramount to also secure the futures contract. There are very many modern ways that have been established to do this. In order to come up with a close to ideal futures contract then the minimum variance hedge ratio is used.

It is also known as the optimal hedge ratio. It evaluates the correlation between an asset and liability and the hedging instrument that is meant to protect it against any unprecedented eventualities. It can also be said that the minimum variance hedge ratio is a formulae that compares the size of an entire protected position to the hedge used to secure it. It is mostly used when a futures contract is involved as a method of hedging. This is because there are no perfect hedges hence as much security and certainty should be acquired to prevent extreme loss due to the various calamities that may occur as time goes by.

This formulae is calculated as the multiplication product of the correlation coefficient between the changes in the current or spot and future prices. After calculating the ratio then the ideal number of future contracts can be established by dividing this ratio by the size of one futures.

The formulae for this method is given as shown below;

is the correlation and  is the standard deviation

So as to get the optimal number of futures contracts needed to properly secure an investors the following should be done after the ratio has been calculated.


In this paper an evaluation of the ratio on futures will be taken with regards to two currencies CAD and USD. So as to model the matrix two methods will be used. These are the bivariate error correction and the BEKK parametrization. So as to evaluate the performance of the hedging strategies four methods can be employed. These are native hedge portfolio, the unhedged portfolio and the dynamic bivariate GARCH.

The paramount assumptions of the minimum variance hedge ratio are that production is deterministic and that all the investor’s wealth is in the cash position. The optimal hedge ratio is decreased, greatly, by the occurrence of stochastic production. An alternative position other than cash, decreases the opportunity cost.

Because of using GARCH another assumption that can be made is that the current error term is a function of previous periods’ error terms. Also by applying the error corrections model an assumption that can be made is that if the spot and the future prices are integrated according to order one then an error is bound to occur that can be corrected by subjecting the result to more testing. The maximum period that a contract should be held depends on the cash position of the investor. There should be the existence of liquid futures contracts. The date that is being hedged against should also be close to the date of maturity of the futures contract so that the risk that a firm is exposed to due to time constraints can be avoided if not eliminated altogether.

Some of the pre-requisites of using this formulae is that it emphasizes the need for planning. So that one can take the proper number of futures contracts and put them in such a way that they can be beneficial to a company. The exchange rates between currencies should also be watched carefully to see whether there are any unexpected changes in the units of currency that are meant to be used. Risk managers should also have their ears on the ground to find any eventualities that are not necessarily financial in nature that may affect their position once a futures contract has been taken. Such eventualities may include political instability and changes in the regulations.


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