Question

please solve for (c), the answer is 0.3020

structure for (annual effective) interest rates is as The term for corresponding maturities: 8-1 follows 5%, 2 year: 10%, 3 year: 15%, 4 year: 20% 1 ycar he swap rate for a 4-year interest rate swap of floating : 5%, Find the swap ra erest for fixed rate interest if the notional amount is level for the four year swap tenor. Suppose that the notional amount is $1,000,000 for the first ears and $2,000,000 for the third and fourth years. Find the swap rate. A two-year deferred swap with a tenor of two years is (c) arranged at the beginning of the first year. If the notional amount is level for the swap, determine the swap rate.

Answer 1

Part (c)

Spot rates, S₁ = 5%; S₂ = 10%, S₃ = 15%, S₄ = 20%

Using the spot rates, we will have to calculate the forward rates F_{n-1, n}

F_{n-1, n} = (1 + S_n)ⁿ / (1 + S_n-1)^n-1 - 1

Hence, F_0,1 = S₁ = 5%

F_{1, 2} = (1 + S₂)² / (1 + S₁)¹ - 1 = (1 + 10%)² / (1 + 5%) - 1 = 15.24%

F_{2, 3} = (1 + S₃)³ / (1 + S₂)² - 1 = (1 + 15%)³ / (1 + 10%)² - 1 = 25.69%

F_{3, 4} = (1 + S₄)⁴ / (1 + S₃)³ - 1 = (1 + 20%)⁴ / (1 + 15%)³ - 1 = 36.34%

Please note that this is a deferred swap with level notional amount.

The fixed swap rate is price of a zero coupon bond, weighted average of implied forward rates. If the text is confusing, please see the formula below:

$R = \frac{\sum_{i=k}^{n}P_{i}\times F_{i-1,i}}{\sum_{i=k}^{n}P_{i}}$

where

R = Fixed rate of the swap

k = period when swap starts, in our case it's 2 year deferred swap. So, the swap starts from k = 3

n = 2 + 2 = 4 (2 year deferred swap with tenor of 2 years) \

P_i = Price of a zero coupon bond today with maturity at i for a maturity value of $ 1 = 1 / (1 + S_i)ⁱ where S_i is the spot rate corresponding to year i

F_i-1, _i = Forward rate between period i - 1 & i

Please see the table below now:

• Please see the second row. That will guide you how each of the columns have been calculated.
• You can now locate the variables P & F in the columnar form (please see second row)

Term	Spot Rate	1 year forward rate	Price of a zero coupon bond with maturity value of $ 1
i	Si	Fi-1,i	Pi = 1 / (1 + Si)i
1	5.00%	5.00%	0.952381
2	10.00%	15.24%	0.826446
3	15.00%	25.69%	0.657516
4	20.00%	36.34%	0.482253

For this question, k = 3 and n = 4

Hence we need to apply the sum over i = 3 to i = 4.

Hence, Fixed swap rate, R = (0.657516 x 25.69% + 0.482253 x 36.34%) / (0.657516 + 0.482253) = 0.344193195 /  1.139769 =  0.3020

I sincerely hope this helps you understand the question and its solution.