Back in Pricing of futures, we discussed about the theoretical pricing of futures. But the futures price (or more technically correct, the forward price) of gold is calculated differently. This is because there is a lease rate for gold. As we mentioned before in Get paid to borrow gold and silver?,
But for a certain class of gold owners, they DO earn interests on gold. Right now, instead of receiving interest for lending out gold, they are paying people to borrow gold.
The best way to explain gold forward pricing is to use an example. To understand this, we assume that you have already read and understood Pricing of futures beforehand. Let’s suppose the spot price of gold is $1000 per ounce. The lease rate for 180 days is 2 percent per annum while the carry cost (which includes storage and interests) is 5% per annum.
So, we borrow $1000 for 180 days. At the carry cost of 5%, we have to repay $1000 * (1+.05(180/365)) = $1024.66 in 180 days time. With the borrowed $1000, we buy 1 ounce of gold and lease it out. At the end of the 180 days lease period, we expect to get back 1 * (1+.02(180/365) = 1 (1.01) = 1.01 ounce of gold.
Therefore, 1 ounce of gold has grown to 1.01 ounce in 180 days time at a value of $1024.66. Therefore, the forward price of gold will have to be $1024.66/1.01 = $1014.51. If the 180-day forward price of gold is not at $1014.51, then an arbitrage opportunity exists (see How futures price affect market price for more details).
To sum it all up with an equation, if the spot price is S, the forward price is F(T) for a time-horizon of T days, the carry cost is r, and the gold lease rate is r*, we have:
F(T) = S [1 + r (T/365)] / [1 + r* (T/365)]