Krzys’ Ostaszewski, http://www.math.ilstu.edu/krzysio/, Exercise 91, 2/10/7
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May 2005 Casualty Actuarial Society Course 8 Examination, Problem No. 23
(multiple choice answers added)
A bank offers a corporate client a choice between borrowing cash at 10% per annum and
borrowing platinum at 2.5% per annum. If platinum is borrowed, interest must be repaid
in platinum. Thus 100 ounces borrowed today would require 102.5 ounces to be repaid in
one year. The risk-free interest rate is 7% per annum, and storage costs for platinum are
0.4% per annum. Assume the following:
• The interest rates on the loans are expressed with annual compounding.
• The risk-free interest rate and storage costs are expressed with continuous
compounding.
• There is no income earned on platinum.
• There are no transaction costs for trading.
• The market participants can borrow money at the same risk-free rate of interest at which
they can lend money.
Determine whether the rate of interest on the platinum loan is too high or too low in
relation to the rate of interest on the cash loan, and then calculate the difference between
the current rate (2.5% continuously compounded) and the rate that would make the two
loans equivalent.
A. 0.35%
B. 0.15%
C. 0.00%
D. –0.15%
E. –0.35%
Solution.
Let us write r for the interest rate charged on platinum that would make the two loans
equivalent. The investor has two choices:
• Borrow money at 10%, buy platinum with the funds and pay 0.40% storage costs for
platinum. At the same time, enter into a forward contract to sell platinum at loan
maturity. At loan maturity, sell platinum at the forward price. Let us write S for the
current spot price of platinum. Then the forward price is F = Se0.07 + 0.004 = Se0.074 . Assume
100
the investor borrows $100. Then the investor will buy
ounces of platinum, and sell
S
100
them for
! Se0.074 = 100 ! e0.074 " 107.58. But the loan repayment is 110, so that the
S
investor is losing 110 ! 100 " e0.074 # 2.42 on the transaction, paid at loan maturity.
100 ! (1 + r )
100
• Borrow
ounces of platinum, and repay that loan with
ounces of
S
S
100r
platinum. The investor loses
ounces of platinum in this transaction, paid at loan
S
maturity. As the arbitrage-free forward price of platinum is Se0.074 , the dollar cost of that
100r
loss is
! Se0.074 = 100re0.074 .
S
For the two transactions to be equivalent, we must have
100re0.074 = 110 ! 100 " e0.074 ,
or
110 ! 100 " e0.074
r=
= 1.1e!0.074 ! 1 # 2.15%.
100e0.074
The difference between 2.5% and this rate is approximately 0.35%.
Answer A.
© Copyright 2007 by Krzysztof Ostaszewski.
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