Question

On January 21, gold is selling for $1387.25 per ounce in the spot market while the futures price for 3-month (90 day) April gold is $1402.50 per ounce on a 100 ounce contract. If the risk-free rate is 2.25%, determine what profit an investor will realize if he/she uses arbitrage (cash-and-carry or reverse cash-and-carry) to exploit the situation. (Assume the investor uses 2 gold futures contracts, and that no carrying costs are involved for storage or insurance.) Show all work. Highlight in bold your answer.

NOTE: Walk through the 4 steps employed in the process for carrying out a cash-and-carry or reverse cash-and-carry arbitrage by calculating or stating clearly the following items (in order):

1) What should the Futures Price be according to the cost-of-carry model?

2) Which arbitrage process (cash-and-carry or reverse cash-and-carry) should be use? WHY?

3) How would you set up the arbitrage process chosen? (Use a timeline to show the actions to be done at the start of the process (today) and at the processes maturity).

4) What will be the total profit from the arbitrage if you employ 2 contracts? Show all calculations, labeling each step used to find your answer. Indicate how you found the calculation values at each step. Again, show all work. Highlight in bold your answer.

Answer 1

Note: It is assumed that interest rate is continuosly compounded.

Theoretical Futures Price = Spot Price*e^[risk free rate*t]

Where e = constant (2.71828), t = years to expiry

Applying the above formula,

Spot Price = 1387.25, Risk Free Rate = 0.025/4 = 0.00625, t = 90/360 = 1/4

Therefore, Theoretical Futures Price(according to cost & carry) = 1387.25*e^[(0.00625)*1/4] = 1387.25*e^0.0015625 = 1387.25*1.001563(from table) = $1389.42

Actual Futures Price is $1402.50 i.e Greater than Theoretical Futures Price

Therefore, Future is Overvalued.

Therefore, to make an Arbitrage Gain, Buy Spot & Sell under Futures Contract i.e (Cash and Carry).

Steps to Arbitrage:

Now,

(1)   Borrow 1387.25*200 = $277450 for 3 months @ Risk Free Rate

(2)   Buy 200 ounces of gold @ current price i.e. $1387.25

(3)   Sell 2 Futures contacts, expiring in 3 months

After 3 months,

(4)   Sell 200 ounces of gold under Futures Contract and receive 1402.5*200 = $280500

(5)   Repay loan with interest 277450*1.001563(value from table, same as above) = $277883.65

Arbitrage Gain = Realized from sale-Repaid the loan = 280500-277883.65 = $2616.35