@daniel_ma
We do have a lot in the forum on COC and convenience yield, but it's always fun to attempt the briefest possible explanation given all that we've learned. The cost of carry model says the theoretical future price should be a fully-loaded function of owning ("carrying") the commodity: in a no-arbitrage world with good information (including somewhat predictive information), if all costs/benefits are accounted for, informed participants should be nearly indifferent to the choice between owning the commodity and promising to buy/sell it in the future (the futures contract) because the futures contract price will adjust to preclude arbitrage.
If for example the spot price (today's price) of corn is $3.36 and its storage cost is 9% per annum, when the riskfree rate is 1.0% per annum, the cost of carry model predicts a theoretical one year futures price given by F(1.0) = S(0) * exp[(1% Rf 9% storage)*1.0 year] = $3.71, because COC says
F(T) = S(0)*exp[(r+u)*T] where (r) is the cost to finance the purchase of the commodity and (u) is the cost to store it. So, according to this, if you bought a bushel today, funding the purchase by borrowing at 1.0%, and stored the corn for one year, your fully loaded cost in one year will be $3.71. So that's the theoretical futures price. More specifically, if the futures price is too much greater or less than this value, arbitrageurs can profit; e.g., if the actual traded
futures price is too much greater than $3.71, arbitrageurs can "cash and carry" for a guaranteed profit (borrow to buy the relatively under-priced commodity, store it, then deliver against the short futures position taken because the futures price is trading too rich). BTW, as it's not certain (we are predicting the future!), if you are risk-averse you might only go long on this contract, for speculation purposes, if you expect a profit, such that F(1.0) = $3.71 < E[S(T)], is the theory of normal backwardation. The key point here is merely that F(T) is distinct from E[S(T)], F(1.0) is an observable price today, whereas E[S(1.0)] is the unobservable expected (future) spot price in one year.
Convenience yield, denoted in Hull by (y), distinguishes a consumption commodity from an investment commodity. Convenience yield confers an intangible benefit that is analogous to the tangible benefit of a divided or income produced by the commodity. It can be viewed as ownership's
optionality benefit: is there an quantifialbe advantage to owning the commodity rather than buying the futures contract. If yes, that's the yield of
convenience (e.g., I own oil, I've got some in case I need it!). Continuing the above example, say the 1-year futures price trades at $3.64 rather than $3.71 but we assume COC holds. The explanation is the convenience yield:
F(T) = S(0)*exp[(r+u - y)*T; in this case, (1.0) = S(0) * exp[(1% + 9% - 2%)*1.0] = $3.64
McDonald calls convenience yield "the extra nonmonetary return" conferred on the OWNER of the asset. It subtracts because the long future position doesn't gain that additional benefit so reduced his/her future price. The cost of carry model, in words, increases the futures price for all costs of ownership (including financing cost: a futures contract is unfunded) and subtracts for all (forgone) benefits of ownership. While not contradicting McDonald's view, Hull treats it more as a plug variable: i.e, the intangible benefit that is explained by the traded futures price. Hull on convenience yield: "In general, ownership of the physical asset enables a manufacturer to keep a production process running and perhaps profit from temporary local shortages. A futures contract does not do the same. The benefits from holding the physical asset are sometimes referred to as the convenience yield provided by the commodity. ... The convenience yield simply measures the extent to which the left-hand side [i.e,. F(0)] is less than the right-hand side [ie., S*exp^(r+u)] in [the cost of carry] equation 5.15 or 5.16" -- Hull, Page 123. Prentice Hall. Kindle Edition. For reasons beyond my scope, the introduction of a convenience yield further implies that the theoretical futures price will trade within a range (an interval) rather than at the point estimate given by COC for an investment (non consumption) commodity. I hope that's a useful summary!
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