First National Bank recently made a 5-year,$100 million fixed-rate loan at 10%.Annual interest payments are $10 million,and all principal will be repaid in year 5.The bank wants to swap the fixed interest payment into floating-rate payments.If the bank could borrow at a fixed rate of 8% for 5 years,what is the notional principal of the swap?
A)$80 million
B)$100 million
C)$125 million
D)$180 million

Question

Quizplus · Accepted Answer

To create the swap, the bank would enter into a contract with a counterparty, agreeing to exchange the fixed interest payments of $10 million per year for floating rate payments. The notional principal of the swap is the amount on which these payments are based, and it is not exchanged between the parties.

To determine the notional principal of the swap, we need to find the present value of the fixed-rate payments, which will equal the present value of the floating-rate payments. We can then divide the present value of the fixed-rate payments by the fixed rate of 8% to find the notional principal.

The present value of the fixed-rate payments can be calculated using the formula:

PV = PMT x [1 - 1/(1 + r)^n] / r + FV/(1 + r)^n

where PV is the present value, PMT is the annual payment, r is the discount rate, n is the number of periods (in this case, 5), and FV is the future value (in this case, $100 million).

Plugging in the values, we get:

PV = $10 million x [1 - 1/(1 + 0.08)^5] / 0.08 + $100 million/(1 + 0.08)^5
   = $38,368,504.92

Dividing by the fixed rate of 8%, we get:

Notional principal = PV / 0.08
                  = $482,106,311.50

However, since the loan amount is only $100 million, it is unlikely that the bank would want to swap the entire amount. A more reasonable notional principal would be the amount of the loan, plus some additional amount that would ensure that the bank receives the fixed-rate payments it wants to swap.

For example, if the bank wanted to swap only $50 million of the loan, the notional principal of the swap would be:

Notional principal = $50 million + PV / 0.08
                  = $50 million + $482,106,311.50 / 0.08
                  = $6,076,826,393.75

This is still a very large notional principal, so another possible approach would be for the bank to enter into multiple swaps with different counterparties, each with a smaller notional principal.

First National Bank Recently Made a 5-Year,$100 Million Fixed-Rate Loan