Question

A bank offers a client a choice between borrowing cash at 3.75% per annum or borrowing gold at 2% per annum.

If gold is borrowed, interest must be repaid in gold. Thus, 100 ounces borrowed today would require 102 ounces to be repaid in one year. The interest rates on the two loans are expressed with annual compounding.

The risk-free interest rate is 2.25% per annum and storage costs are 0.5% per annum. The risk-free interest rate and storage costs are expressed with continuous compounding. We assume that the price of gold is USD 2671 per ounce and the client wishes to borrow USD 2,671,000.

a. Discuss whether the rate of interest on the gold loan is too high or too low in relation to the rate of interest on the cash loan.

b. What should the interest rate of gold be so that no arbitrage profit can be made? (What should be the correct rate of interest for the gold loan?)

Answer 1

If the client takes a cash loan the amount that has to be repaid is 2,671,000 * 3.75% + 2,671,000 = 2771163.

If 100 ounces of gold is borrowed, 102 ounces of gold must be repaid. The price of gold per ounce is USD 2671.

Number of ounces the client can borrow = 2671000/2671 = 1000.

Given Risk free rate = 2.5 % and storage costs are 0.5 %. (Both are continously compounded).

Forward price of 1 ounce of gold = 2671e^(0.025+0.005)*1 = 2671*2.71828^(0.03)*1 = 2752.344

We need to find the forward cost of 1020 ounces (1000*2%). So by buying 1,020 ounces of gold in the forward market the client can repay the gold loan.

The amount to be repaid for gold loan = 1020 * 2752.344 = 2807391

The cost of gold loan is more as compared to the cash loan by (2807391 - 2771163) = 36,228 $.

Including all the costs the interest on gold loan is ((2807391 - 2671000)/2671000) = 5.10 %

The interest on gold loan is higher by 5.1 - 3.75 = 1.35%. This might be due to the adminstrative costs associated with a gold loan. The correct interest rate on the gold loan to avoid this kind of arbitrage should give us the amount of 2771163 i.e. what we would ideally receive on the simple interest loan.

The number of ounces that can be bought with 2771163 is this amount divided by the forward price of one ounce of gold.

2771163/2752.344 = 1006.837.

So ideal interest on gold to avoid arbitrage opportunities = ((1006.837 - 1000)/1000)*100 = 0.6837 %.