684 Part Eight Risk Management
bre44380_ch26_673-706.indd 684 09/30/15 12:09 PM
Spot and Futures Prices—Commodities
The difference between buying commodities today and buying commodity futures is more
complicated. First, because payment is again delayed, the buyer of the future earns inter-
est on her money. Second, she does not need to store the commodities and, therefore, saves
warehouse costs, wastage, and so on. On the other hand, the futures contract gives no con-
venience yield, which is the value of being able to get your hands on the real thing. The
manager of a supermarket can’t burn heating oil futures if there’s a sudden cold snap, and
he can’t stock the shelves with orange juice futures if he runs out of inventory at 1 p.m. on
a Saturday.
Let’s express storage costs and convenience yield as fractions of the spot price. For com-
modities, the futures price for t periods ahead is^17Ft = S 0 (1 × rf + storage costs − convenience yield)tIt’s interesting to compare this formula with the formula for a financial future. Conve-
nience yield plays the same role as dividends or interest foregone (y) on securities. But
financial assets cost nothing to store, and storage costs do not appear in the formula for finan-
cial futures.
Usually you can’t observe storage cost or convenience yield, but you can infer the difference
between them by comparing spot and futures prices. This difference—that is, convenience
yield less storage cost—is called net convenience yield (net convenience yield = convenience
yield − storage costs).(^17) This formula could overstate the futures price if no one is willing to hold the commodity, that is, if inventories fall to zero or some
absolute minimum.
Suppose the six-month CAC futures contract trades at 5,000 when the current (spot) CAC index
is 5,045.41. The interest rate is 1% per year (about .5% over six months) and the dividend yield on
the index is 2.8% (about 1.4% over six months). These numbers fit the formula perfectly because
Ft = 5,045.41 × (.005 − .014) = 5,000
But why are the numbers consistent?
Suppose you just buy the CAC index for 5,045.41 today. Then in six months you will own
the index and also have dividends of .014 × 5,045.41 = 70.64. But you decide to buy a futures
contract for 5,000 instead, and you put €5,045.41 in the bank. After six months, the bank
account has earned interest at .5%, so you have 5,045.41 × 1.005 = 5,070.64, enough to buy
the index for 5,000 with €70.64 left over—just enough to cover the dividend you missed by
buying futures rather than spot. You get what you pay for.^16
EXAMPLE 26.1^ ●^ Valuing Index Futures
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(^16) We can derive our formula as follows. Let S 6 be the value of the index after six months. Today S 6 is unknown. You can invest S 0 in
the index today and get S 6 + yS 0 after six months. You can also buy the futures contract, put S 0 in the bank, and use your bank balance
to pay the futures price F 6 in six months. In the latter strategy you get S 6 − F 6 + S 0 (1 + rf) after six months. Since the investment is
the same, and you get S 6 with either strategy, the payoffs must be the same:
S 6 + yS 0 = S 6 − F 6 + S 0 (1 + rf)
F 6 = S 0 (1 + rf − y)
Here we assume that rf and y are six-month rates. If they are monthly rates, the general formula is Ft = S 0 (1 + rf − y)t, where t is the
number of months. If they are annual rates, the formula is Ft = S 0 (1 + rf − y)t/12.