Investments, tenth edition

Action

Initial Cash Flow

CF at Time T

Buy asset; pay carrying costs at T

2

P

0

P

T

 2 C

Borrow P

0

; repay with interest at time T

P

0

2

P

0

(1 1 r

f

)

Short futures position

0

F

0

 2 P

T

  TOTAL

0

F

0

 2 P

0

(1 1 r

f

) 2 C

Because market prices should not allow for arbitrage opportunities, the terminal cash flow 

of this zero-net-investment, risk-free strategy should be zero. 

 If the cash flow were positive, this strategy would yield guaranteed profits for no invest-

ment. If the cash flow were negative, the reverse of this strategy also would yield profits. In 

practice, the reverse strategy would involve a short sale of the commodity. This is unusual 

but may be done as long as the short sale contract appropriately accounts for storage costs. 

Thus,  

9

   we conclude that

 

   

F

0

 5 P

0 

(1 1 r

f 

) 1 C 

Finally, if we define  c   5   C / P  

0

 , and interpret  c  as the percentage “rate” of carrying costs, 

we may write   

 

F

0

 5 P

0 

(1 1 r

f

 1 c

 

) 

 (23.3)  

which is a (1-year) parity relationship for futures involving storage costs. Compare 

Equation 23.3 to the parity relation for stocks, Equation 22.1 from the previous chapter, and 

you will see that they are extremely similar. In fact, if we think of carrying costs as a “nega-

tive dividend,” the equations are identical. This result makes intuitive sense because, instead 

of receiving a dividend yield of  d,  the storer of the commodity must pay a storage cost of  c.  

Obviously, this parity relationship is simply an extension of those we have seen already. 

    23.5 

Commodity Futures Pricing  

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  C H A P T E R  

2 3

  Futures, Swaps, and Risk Management 

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 Although we have called  c  the carrying cost of 

the commodity, we may interpret it more gener-

ally as the  net  carrying cost, that is, the carrying 

cost net of the benefits derived from holding the 

commodity in inventory. For example, part of the 

“convenience yield” of goods held in inventory 

is the protection against stocking out, which may 

result in lost production or sales. 

 It is vital to note that we derive Equation 23.3 

assuming that the asset will be bought and stored; 

it therefore applies only to goods that currently 

 are 

 being stored. Two kinds of commodities 

cannot be expected to be stored. The first kind 

is commodities for which storage is technologi-

cally not feasible, such as electricity. The second 

includes goods that are not stored for economic 

reasons. For example, it would be foolish to buy 

an agricultural commodity now, planning to store 

it for ultimate use in 3 years. Instead, it is clearly 

preferable to delay the purchase until after the 

harvest of the third year, and avoid paying stor-

age costs. Moreover, if the crop in the third year 

is comparable to this year’s, you could obtain it 

at roughly the same price as you would pay this 

year. By waiting to purchase, you avoid both interest and storage costs. 

 Because storage across harvests is costly, Equation 23.3 should not be expected to apply 

for holding periods that span harvest times, nor should it apply to perishable goods that are 

available only “in season.” Whereas the futures price for gold, which is a stored commodity, 

increases steadily with the maturity of the contract, the futures price for wheat is seasonal; 

its futures price typically falls across harvests between March and July as new supplies 

become available. 

  Figure 23.9  is a stylized version of the seasonal price pattern for an agricultural product. 

Clearly this pattern differs from financial assets such as stocks or gold for which there is 

no seasonal price movement. Financial assets are priced so that holding them in portfo-

lio produces a fair expected return. Agricultural prices, in contrast, are subject to steep 

periodic drops as each crop is harvested, which makes storage across harvests generally 

unprofitable.  

 Futures pricing across seasons therefore requires a different approach that is not based 

on storage across harvest periods. In place of general no-arbitrage restrictions we rely 

instead on risk premium theory and discounted cash flow (DCF) analysis.