Question

Suppose the 1-year effective annual interest rate is 4.6% and the 2-year effective rate is 3.2%. Compute the fixed rate in a 2-year amortizing interest rate swap based on $440,000 of notional principal in the first year and $240,000 in the second year.

Please show steps

a. 4.11%

b. 3.91%

c. 4.69%

d. 3.22%

e. 3.63%

Answer 1

The interest rate swap comprises of two legs, the floating interest rate leg which has 4.6% as interest for Year 1 and 3.2 % as interest for Year 2 and the fixed lege whose interest rate needs to be determined. This interest rate of the fixed leg is known as the interest swap rate.

The initial notional is $ 440000 for the first year and amortizes to $ 240000 for the second year. The total swap tenure is 2 years

The fixed interest swap rate is that interest rate on the outstanding notional of the fixed leg that equalizes the present value of the fixed leg's annual interests plus amortization to the present value of the floating leg's annual interests plus amortization, all discounted at the given yearly annual interest rates.

Floating Leg:

Year 1: Interest Rate = 4.6 % and Notional = $ 440000

Year 1 Interest = 0.046 x 440000 = $ 20240 and Year 1 Amortization = $ 200000 (as notional goes down from $ 44000 at the beginning of the year to $ 240000 at the end of the year)

Total Year 1 Cash Flow = 20240 + 200000 = $ 220240

Year 2: Interest Rate = 3.2% and Notional = $ 240000

Year 2 Interest = 0.032 x 240000 = $ 7680 and Amortization = $ 240000 (as notional will go down to zero at the end of the swap)

Total Year 2 Cash Flow = 7680 + 240000 = $ 247680

PV of Year 2 Cash Flow at the end of Year 1 = 247680 / (1.032) = $ 240000

Total Cash Flow at the end of Year 1 = Year 1 Cash Flow + PV of Year 2 Cash Flow = 220240 + 240000 = $ 460240

PV of Totl Cash Flows = 460240 / (1.046) = $ 440000

Fixed Leg:

Let the fixed leg interest rate be R (in decimals)

Year 1 Interest = Fixed Interest Rate x Outstanding Year 1 Notional = R x 440000 and Amortization = $ 200000

Total Year 1 Cash Flow = (R x 440000) + 200000

Year 2 Interest = Fixed Interest Rate x Outstanding Year 2 Notional = R x 240000 and Amortization = $ 240000

Total Year 2 Cash Flow = (R x 240000) + 240000

Total Present Value of Interests of the Fixed Leg = PV (Fixed) = [(R x 440000 + 200000) / (1.046)] + [(R x 240000 + 240000) / (1.032)^(2)] = 191204.5889 + 420650.096R + 225347.034 + 225347.034R = 416551.6233 + 645997.13R

Now, PV(Fl) = PV(Fixed)

440000 = 645997.13R + 416551.6233

23448.3767 = R x 645997.13

R = 23448.3767 / 645997.13 = 0.036297 or 3.6297% ~ 3.63%

Hence, the correct option is (e)

NOTE: This problem assumes that the swap involves exchange of notional amortization, thereby them forming a cash flow. In real life the only cash flows in Interest Rate Swaps are the coupons which is what was assumed for the initial solution that gave 4.11% as the answer.