What Should be an Equilibrium Forward Price?

Example 2.3. (Creating a Synthetic Long Gold Forward):

The following steps create a situation with the delivery of gold and cash flow identical to a long gold forward contract – creates synthetically a gold forward long contract. The spot price of gold today, S₀, is $1,000/oz. The risk-free interest rate, R₀, is 6% and the gold lease rate, I₀, is 2%/year, payable at the end of the lease period.

1. Borrow money S₀ = $1,000 for one year
2. Buy gold in the spot market
3. Lend the gold for one year
4. At the end of one year, pay the loan, collect the gold and the fee from lending the gold, and you have the gold

This is equivalent to a long gold forward contract with zero cash flow at the beginning and a payment for the underlying at the settlement:

→    t = 0: CF₀ = +$1,000 (loan) - $1,000 (pay for Gold) = 0
        No gold in possession – b/c the gold bought was lent

→    t = 1: Pay back the loan and collect the gold rental fee
       CF₁ = - 1,000 · (1 + 6%) + 1,000 · 2% = -$1,040
       Get back the gold: + Gold(1oz)

→   The net amount to pay is:
       F_SYNTHETIC = (spot price) + (interest cost) – (gold lease fee)
       = S₀ · (1 + R₀ - I₀) = 1,000 · (1 + 6% - 2%) = $1,040