Question

1. Suppose the gold spot price is $300/oz., the 1-year forward price is $310.686, and the continuously compounded risk-free rate is 5%. In class, we neglect the convenience yield for gold. In reality gold can may be lent and borrowed. Some entities operating in the wholesale gold market do lend gold and earn interest on such transactions. To sum up, there is a convenience yield for gold and it takes the name of “lease rate”.

   1. (a) What is the lease rate?

   2. (b) What is the return on a cash-and-carry if you cannot loan out the gold (i.e. you do not have access to the wholesale gold market)?

   3. (c) What is the return on a cash-and-carry if you do loan out the gold, earning the lease rate?

Answer 1

a). Solution :- Calculation of lease rate :-

Lease rate = Risk free rate - Natural logarithm (Forward price / Spot price)

= 0.05 - Natural logarithm (310.686 / 300)

= 0.05 - Natural logarithm (1.03562)

= 0.05 - 0.035 [ Natural logarithm (1.03562) = 0.035 (approx) ]

= 0.015

Conclusion:- Lease rate = 0.015

b). Solution :- Return on cash and carry = Forward price - Spot price (of buying gold)

= 310.686 - 300 * e^0.05(Continuous compounding, thus, exponential function i.e., "e" is applied)

= 310.686 - 300 * 1.051271 (e^0.05 = 1.051271)

= 310.686 - 315.3813

= (-) 4.6953 (Negative symbol represents loss).

Conclusion :- Return on cash and carry will be loss of $ 4.6953 (approx).

c). Answer :- If the gold is loan out then there will be break even position i.e., no profit no loss in the given question,  In other words, Return on cash and carry will be $ 0 only.