7.7 Commodity futures preview: storage and convenience yield

Motivation

Gold, silver, crude and natural gas trade on MCX through futures, and their prices are governed by the same carry logic with two extra terms. M10 develops this in depth; here we set up the formulas.

Commodities differ from equity indices in two ways. Holding them costs money (storage, insurance, financing), and for some of them, holding the physical good has a value that holding a futures contract does not (a refinery that runs out of crude stops). Distinguish:

Investment assets, held mainly for investment by many people (gold, to a degree silver). Enough holders are willing to sell spot and buy futures that both arbitrage directions work.
Consumption assets, held mainly for use (crude oil, natural gas, base metals, agricultural goods). Holders of inventory will not lend or sell it for a small arbitrage profit, so the reverse cash-and-carry fails.

7.7.1Investment assets with storage#

Proposition 7.10investment asset with storage

If storage costs are a proportion $u$ of value per year (paid continuously, like a negative dividend yield), or have present value $U$ when they are fixed amounts, then

F_0 = S_0e^{(r+u)T} \qquad \text{or} \qquad F_0 = (S_0 + U)e^{rT}. \tag{7.20}

Proof

Identical to Props 7.1 and 7.2 with $q$ replaced by $-u$ (storage is a cost of carry, not an income), or with $-I$ replaced by $+U$ : the cash-and-carry arbitrageur must finance the asset and its storage; the reverse arbitrageur sells the asset short and saves the storage.∎

7.7.2Consumption assets and convenience yield#

Proposition 7.11consumption asset

For a consumption asset, no arbitrage implies only

F_0 \le S_0e^{(r+u)T}.

Define the convenience yield $y \ge 0$ by

F_0 = S_0e^{(r+u-y)T}. \tag{7.21}

Proof

The cash-and-carry argument needs only the ability to buy, store and deliver, so $F_0 > S_0e^{(r+u)T}$ remains an arbitrage. The reverse argument needs someone to sell the physical good short or out of inventory and buy it back later via futures. A user of the commodity values the inventory for production, so they will not do this even if $F_0 < S_0e^{(r+u)T}$ . Hence no lower bound follows from arbitrage. The gap, written as a yield, is $y$ .∎

7.7.3Backwardation, contango and roll yield#

$y$ measures the market's valuation of having the physical good now. It is high when inventories are low and supply is tight: the curve is in backwardation, futures below spot, and a long futures holder who rolls earns a positive roll yield (sells the expiring contract high, buys the next one lower). It is near zero when inventories are ample: the curve is in contango at roughly full carry, and rolling longs lose money every month. Section 7.8.1 is a story about what happens when a hedger assumes $y$ stays high, and 7.8.3 about what happens when storage itself runs out.

MCX notes (dated; verify on mcxindia.com, which blocked automated access during research) MCX gold futures (GOLD) have a 1 kg lot quoted in ₹ per 10 g; bullion contracts expire around the 5th of the contract month, with a tender (delivery-intention) period of a few days before expiry during which open positions may lead to delivery (secondary summaries, Oct 2026; verify with the exchange). MCX crude oil futures have a 100-barrel lot quoted in ₹ per barrel, expire around the 19th–20th of the month, and are cash settled at a "due date rate" equal to the settlement price of the NYMEX WTI front-month contract on the last trading day, converted to rupees (as described in MCX's April 2020 circulars, reported by Business Standard). MCX also lists options on these futures, which are priced with Black's model (M08 · 8.8).