Question

The spot price of silver is $15.25 per ounce. The storage costs are $0.32 per ounce per year payable quarterly in advance. Assuming that interest rates are 8.5% per annum for all maturities with continuous compounding, calculate the futures price of silver for delivery in nine months

Answer 1

Future price for commodities with Storage Cost is given as follows:

F₀ = (S₀ + C) * exp(r*t) ---- (1)

Where:

F₀ = Future price today

S₀ = Spot price of silver

C = Present Value of Storage Cost

r = Interest Rate

t = Time until delivery (in years)

First we will calculate C i.e. present value of storage cost

Storage cost is $0.32 per ounce per year which is paid quarterly in advance which implies that for each quarter storage cost = 0.32/4 = $0.08 per ounce.

Storage cost is paid in advance every quarter which implies that storage cost is paid at the beginning of every quarter i.e.

T (0 month) = $0.08 per ounce (beginning of 1st quarter)  

T (3 month) = $0.08 per ounce (beginning of 2nd quarter)

T (6 month) = $0.08 per ounce (beginning of 3rd quarter)

We are considering storage cost for 3 quarters as the delivery of future is in 9 months

Present value of storage cost (C) is calculated by discounting storage cost at T (3 month) and storage cost at T (6 month) to current time i.e. T(0 month).

Discounting rate = 8.5%

C = 0.08 + 0.08 * exp (-8.5%*(3/12)) + 0.08 * exp (-8.5% * (6/12))  

= $ 0.2350 per ounce (rounded to 4 decimal places) ---(2)

Now replace the value calculated in (2) in eq (1)

F₀ = (S₀ + C) * exp(r*t)

= (15.25 + 0.2350) * exp (8.5% *(9/12))

= $ 16.5043 per ounce