Question

Interest Rate Swap

a) Based on the following 3-month forward (futures) rates, calculate the 1.5year swap rate on a $100,000,000 notional swap

Period Number	Days in Period	Forward Rate	Quarterly Forward Rate
1	90	0.50%	0.1250%
2	90	1.00%	0.2500%
3	90	1.25%	0.3125%
4	90	1.50%	0.3750%
5	90	1.90%	0.4750%
6	90	2.15%	0.5375%

b) Initially, you agree to receive the fixed swap rate, just determined in a. (above), every 90 days for a quarterly payment (assumes a 360 day-count year) and pay a floating payment based on 3-month Libor. Assume the notional amount is $100,000,000. What is your termination value after 6 months (180 days) assuming the following swap rate curve at that time? (Rates are expressed on an annual basis).

6m = 1.60% 9m = 1.80% 1y = 2.00% 1.5y = 2.25% 2.0y = 2.55% 3.0y = 3.30%

Answer 1

a) Based on the following 3-month forward (futures) rates, calculate the 1.5 year swap rate on a $100,000,000 notional swap

Let's assume the swap rate to be R.

PV of all the future payments at fixed rate = PV of all the future floating rate payment

A typical fixed rate payment at the end of period i = R x Notional Principal, P x 90 / 360 = Px R / 4 = 0.25 x P x R

It's discounted value =  0.25 x P x R x D_i where D_i = discount factor for the payment at the end of period i

Final payment = 0.25 x P x R + P at the end of period n

Hence, it's discounted value = (0.25 x P x R + P) x D_n

PV of all the future payments at fixed rate

= 0.25 x P x R x sum of discount factor + P x last period's discount factor

PV of all future floating rate payment = Value of the loan today = notional principal, P = $ 100,000,000

Hence, 0.25 x P x R x sum of discount factor + P x last period's discount factor = P

Hence, 0.25 x R x sum of discount factor + last period's discount factor = 1

Please see the table below:

Period Number	Days in Period	Forward Rate	Quarterly Forward Rate	Discount factor
N	A	B	C	D = 1 / (1 + N x C)
1	90	0.50%	0.125%	0.998751561
2	90	1.00%	0.250%	0.995024876
3	90	1.25%	0.313%	0.990712074
4	90	1.50%	0.375%	0.985221675
5	90	1.90%	0.475%	0.976800977
6	90	2.15%	0.538%	0.968757568
Total				5.915268731

Hence, 0.25 x R x sum of discount factor + last period's discount factor = 1

Or, 0.25 x R x 5.915268731 + 0.968757568 = 1

Hence, R = (1 - 0.968757568) / (0.25 x 5.915268731) = 0.021126635 = 2.11%

Hence, 1.5 year swap rate, R = 2.11%

Part (b)

Initially, you agree to receive the fixed swap rate, just determined in a. (above), every 90 days for a quarterly payment (assumes a 360 day-count year) and pay a floating payment based on 3-month Libor. Assume the notional amount is $100,000,000. What is your termination value after 6 months (180 days) assuming the following swap rate curve at that time? (Rates are expressed on an annual basis).

6m = 1.60% 9m = 1.80% 1y = 2.00% 1.5y = 2.25% 2.0y = 2.55% 3.0y = 3.30%

Please note that the value of the floating rate bond will be restored to the Par amount = Notional principal = P upon the next reset date. Hence, the value the floating rate bond = PV of floating rate interest + Notional Principal over 1 period.

As per applicable swap rate curve, 6 months LIBOR = 1.60% and applicable LIBOR for quarter 3 = 1.25%

Hence, valuation of the floating leg = 100,000,000 x (1 + 1.25% x 90 / 360) / (1 + 1.60% x 90 / 360) = $ 99,912,849

Valuation of fixed leg = PV of pending four quarterly fixed coupon + PV of principal = 0.25 x P x R x sum of the discount factor + P x last period discount factor

Swap rate will now be as shown below:

Period Number	Days in Period	Forward Rate	Quarterly Forward Rate	Discount factor
N	A	B	C	D = 1 / (1 + N x C)
1	90	1.25%	0.313%	0.996884735
2	90	1.60%	0.400%	0.992063492
3	90	1.80%	0.450%	0.986679822
4	90	2.00%	0.500%	0.980392157
Total				3.956020207

Hence,  Valuation of fixed leg = 0.25 x P x R x sum of the discount factor + P x last period discount factor = 100,000,000 x [0.25 x 2.11% x 3.956020207 + 0.980392157] = $100,126,016

Hence, termination value after 6 months (180 days) = Valuation of the balance fixed rate payments - valuation of the floating rate payments = $100,126,016 - $ 99,912,849 = $ 213,168

Hence, the termination value = $ 213,168