The theories of Hedging and the Cost of Carry
Basic concept in futures contracts pricing is the hedging, which aims to reduce the
price risk for the investors, but in fact is not necessarily leading to positive financial
income ( Hull, 1997)1. A short hedge is called situation when a company knows that it is
due to sell an asset in the future and is taking respective short future position. If a
company knows that it has to buy the underlying commodity in the future, it could
take long future position , also called long hedge. The future spot price could go up or
down and the hedger could lose or gain from his position. The main function of the
hedging is to reduce the risk by making the outcome more certain. Possible reasons
for negative outcome from hedging might be misunderstandings related with the
specification of the asset, the maturity date of the future contract, and eventual closing
of the contract before expiration.
According to Hull, the basis risk in hedging is determined by the difference in spot and
future price of the asset which is hedged :
Formula 4 : Basis = Spot price of asset to be hedged – Future price of contract used
If there is no difference between the spot and the future price, than the basis risk is zero
at the expiration of the future contract. Strengthening of the basis is called in situation
when the spot price increases more than the future contract price. In the opposite case,
we say that the basis is declining or weakening. The basis risk for an investment
asset increases from the uncertainty of the level of the free- risk rate, the asset’s yield in
the future, and big imbalances in the supply and demand and storage of commodities.
Holders and processors of the underlying commodity of future contracts are usually
using this financial instrument to hedge against the price fluctuation risk. For one
commodity in future contract , we could have several maturity dates with different prices.
The spot price of the underlying commodity could be lower, on average the same, or
higher than the determined in the future contract price on the maturity date. The
difference depends a lot on the storage costs, transportation costs and grade
differences. In case the future price at the maturity date is lower than the spot price at
the maturity date, the investor would purchase future contracts, take delivery and sell
the delivered commodity at the spot price. In case the future price at the maturity date is
bigger than the spot price at the same time, the investor would be likely to sell futures,
buy spot commodity and make delivery against the future contract ( Stoll and Whaley,
1993). Such strategies are possible also prior to the maturity date, in case the
underlying commodity is available, and if the future price exceeds the spot price with
more than the cost of the carrying the commodity to maturity:
Hull C, John, “ Options, Futures and Other derivatives”, Third Edition, Prentice Hall International Inc.,
1997, p.31-33
1
Formula 52 : Ft > St + B , where :
Ft is the future price
St is the spot price
B is the cost of carrying the commodity until maturity of the future contract.
If Formula 1 holds, the investor would sell the future contracts at price Ft, buy the
underlying commodity at price St, and carry the commodity to maturity against the
futures contract.
In case the cost of carrying is too big, it might result in situation , where the amount by
which the future price can exceed the spot price is limited. Than it is better to buy at
spot price and holding the commodity than purchasing a future contract. The situation is
expressed by Formula 2 :
Formula 6 : Ft ≤ St + B
In futures contracts we have situation called “reverse arbitrage”, which is a situation
where the investor is selling the spot commodity and purchasing future contracts. It is
possible only if there is enough supply of the underlying commodity. The underlying
commodity in this case should be sold by the its owner who can replace it with future
contracts, or might be loaned to someone like him, who sells the commodity and buys
futures. The sale of the loaned commodity is called short sale, and the short seller is
obliged to return the commodity on demand. The reverse arbitrage equilibrium, called
also : “the cost of carry relation”, requires the future price to be equal to the spot price
and the cost of carrying:
Formula 7 : Ft = St + B
When commodities are not held as investments, but for consumption needs, then
formula 6 holds, and the “ individuals and companies are reluctant to sell commodities
and buy future contracts, since future contracts can not be consumed ”( Hull, 1997 )3.
As already mentioned, Formula 7 is related with the cost of carry of the underlying
commodities. According to Hull, the cost of carry as a term represents the relation
between future and spot prices. It is the summarized cost for storage and interest
rate, deducted by the income earned by the asset. The Future price can not exceed
the spot price by more than the cost of carrying the commodity to the maturity of the
futures contract. As long as there is enough supply, the future prices can not be lower
than the cost of carry equilibrium price. But in case it is, than the investors will be shortselling the spot- commodity and buying futures. When the storage and interest costs
are continuous, the relation between spot and future prices is represented by:
Formulas 1, 2 and 3 : Stoll H., Whaley R., “ Futures and options – Theory and Applications”, SouthWestern Publishing Co., 1993, pp.32-33
3
Hull C, John, “ Options, Futures and Other derivatives”, Third Edition, Prentice Hall International Inc.,
1997, p.66
2
Formula 8 : Ft = St eᵇ(T-t), where
b is the cost of carrying the commodity, including interest
T-t is the time to maturity of the future contract
Ft – the future price at time t
St – the spot price at time t
The Cost of Carry is different when the storage costs are paid at maturity:
Formula 9 : Ft = St ( 1+ b*), where
b* is the cost of carrying the underlying commodity as proportion of the commodity
price, b* = B/ St. This rate corresponds to the life of the future contract.
When the future contract is entered as an investment, than when the future price is
greater than the spot price, it reflects the cost of carry. In case the asset is with
consumption purpose and the future price is again higher than the spot price, the
difference is representing the cost of carry net of the convenience yield.