Cost of Carry

The no-arbitrage forward price is constructed by asking what it costs to replicate delivery synthetically: borrow at the risk-free rate, buy the asset today, hold it to the delivery date. Under discrete compounding, take the spot price, add the present value of any storage and insurance outlays and subtract the present value of any income the asset throws off, then compound that adjusted spot at the risk-free rate over the contract horizon. Under continuous compounding the same logic collapses into a single exponent: the spot price grown at the financing rate plus the storage yield, less the income yield and any convenience yield, times the time to expiry.

Both routes give the same answer. The choice is a matter of whether carry items are quoted as cash amounts on fixed dates or as proportional yields.

Net carry is the algebraic sum of costs and benefits, so its sign is an empirical question, not a rule. Financing and storage push the forward above spot; dividends, coupons, lease rates and convenience yield push it below. When benefits dominate, the forward curve slopes downward and the market is in backwardation; when costs dominate, it slopes upward into contango.

Convenience yield is the residual that makes the relation hold for consumption commodities. It measures the non-monetary value of physical possession: the ability to keep a refinery or smelter running when spot supply tightens. Because it cannot be shorted or borrowed, convenience yield breaks the strict arbitrage argument for commodities, which is why commodity forward pricing is a bound rather than an equality in one direction.

Equity index futures carry is the financing rate less the dividend yield, so index futures typically trade slightly above spot but converge toward it as dividends accrue and expiry approaches. Currency forwards are the cleanest case: the benefit of holding foreign currency is the foreign interest rate, and the resulting relation is covered interest rate parity, with the forward premium approximately the interest rate differential. Bond futures carry compares repo financing against accrued coupon income, which is why the basis and the implied repo rate are standard analytics on the delivery-eligible bond. Gold and financial commodities have low storage costs relative to value and behave almost like financial assets, while agricultural and energy markets show carry dominated by physical constraints, seasonality and storage availability.

Expect to price a forward or futures contract on an equity, an index, a coupon bond, a currency and a storable commodity, given spot, a financing rate, and carry items expressed either as cash flows or as yields. Expect also to work the relation backwards: given an observed forward price, solve for the implied convenience yield, the implied repo rate, or the implied dividend yield.

The second skill is diagnosis. If the quoted forward exceeds the carry-implied value, the trade is to sell the forward, borrow, and buy and carry the asset; if it falls short, reverse the position where short-selling the underlying is feasible. Finally, be able to value an existing forward before expiry, where the mark to market is the difference between the current carry-implied forward price and the contracted price, discounted to today.