# Lending Commodities

All rates given are in continuous time.

If the current spot price of a commodity is \$ 23.26 , the commodity discount rate is 7.3 % , the risk free rate is 3.1% and the commodity is expected to appreciate by 1.2 %, what is the NPV of lending the commodity for 1 year, in dollars?

If the lender wanted the borrower to compensate fora negative NPV , so the lender’s loss was zero ( in case he had a loss ) , how much payment would he need at the end of the loan period?

NPV = C0 + PV(C1) = (\$23.26) + \$23.36 × 1.012 ÷ 1.073 = (\$1.32)

\$1.32 * 1.073 = \$1.42.

The borrower would have to pay \$1.32 now or \$1.42 in one year.

NPV = 23.26*e^(0.012-0.073) - 23.26 = -1.376

( using the formula NPV = S0*e^(g-alpha)-S0 )

to break even the lender should demand a payment of :

S0*e^(alpha-g) = 23.26*e^(0.061) = 24.723 ( which is not the same as 23,26+1.376=24.636)

My mistake; I missed the continuous compounding.

Also, apparently by “lender” and “borrower” you mean, respectively, “short” and “long” in a forward contract. If you really have a lender and a borrower, the borrower doesn’t pay for the purchase price of the commodity; he returns it at the end of the loan period.

lender and borrower are standard terms in a commodity business . The producer( hedger ) is a lender , and the speculator is a borrower. ( see CFA text on Risk Mgmt of commodities )

And the consumer( hedger) is a borrower while the speculator is a lender in that deal.

Long / Short is a position they’re in.

The economics of a trade i.e. the NPV of a deal, is usually considered from long postion point of view , but again , that is merely a communication convenience .

Cool. I’ve never been in the commodity business, so I was unfamiliar with the vernacular. Always like learning new things. Thanks!