Question

Suppose the storage cost for gold is $70 per ounce per year and the interest rate for borrowing or lending is 3% per annum, compounded continuously. Storage costs are assessed when you take delivery of the gold, but you can pay them at a later date with accumulated interest.

1. Show how you could make an arbitrage profit if the June and December futures contracts for a particular year trade at $1,350 (spot price) and $1,400 per ounce (spot price), respectively, and show how the arbitrage works assuming a contract size of 100 ounces. Ignore daily settlement (marking to market) in answering this question.

2. What storage cost would eliminate this arbitrage opportunity?

This is the only information that I received.

Answer 1

Answer: PART 1

1) Lets assume we buy Gold at Current Spot Price i.e price of June Month which is $1350.

2) We will short the gold future contract, expiring 180days from now i.e. December spot price which is $1400

3) Pay a storage fee of (180days/360)($70) = $35 per ounce. [note here that we reduce the annual cost to 6 Months the period over which the contract runs] {Assume 360days in yeear, assume contract is for 6month difference between June & Dec.}

4) Rate of Return will be {$1400-$1350}/$1350= 3.70% for half yearly hence for annual rate of return will be 3.70*2= 7.4% is Rate of Return.

5) There is an arbitrage opportunity because you can borrow at 3% and lend at 7.4%. Thus, we should borrow money, buy gold in the spot market, go short futures, and complete a cash-and-carry strategy, with an implied repo rate of: 7.4% - 3% or 4.4% on an annual basis.

Answer: PART 2

Now let us calculate at what price storage cost would eliminate this arbitrage opportunity by using the following formula

= Pt (1 + rt,T ) + FV (Storage)

here:

Pt = Current Spot Price i.e June prices $1350

rt = Time of Contract as assumed Halfyearly i.e. 180 days

T = Borrowing or Lending Interest Rate i.e. 3%

FV = Storage Cost i.e. $35

= $1350(1 + 180/360 (.03)) + $35 = $1405.25

Notice that at this price the implied repo rate would be:

Implied Repo Rate = ($1405.25 - $35 - $1350)/1350 = 1.5% for 180 days, which implies: 1.5% x 2 = 3% for 1 year.