Question

	Year 1	Year 2	Year 3
Treasury Zero-Coupon Bond Price	0.976	0.952	0.928
Interest Rate Swap	a	b	c
Oil Forward Price	57	58	59.5
Oil Swap Price	d	e	f

Use the information in the table above and construct the set of fixed rates of the interest rate swaps and the oil swap prices for 1 through 3 years. (Find a-f)

Answer 1

As a first step we will have to calculate the spot rates and using the spot rates, we will have to calculate the forward rates.

Please see the table below, where calculations have been made. These are not the final answers yet. Please note the second column titled "Linkage" where formula corresponding to each row has been explained. "i" is the counter for time. N is the year no.

Parameter	Linkage
Year	N	1	2	3
Treasury Zero-Coupon Bond Price, P (0, ti)	P	0.976	0.952	0.928
Spot rate, S (0, ti)	S = (1/P)1/N-1	0.024590	0.024900	0.025221
1 year forward rate, F (ti-1, ti)	F = (1+SN)^N / (1 + SN-1)N-1 - 1	0.024590	0.025210	0.025862

Fixed rates of the interest rate swaps will be given by,

Let's now calculate a, b, c one by 1

Set N = 1, there will be just 1 term

R₁ = a = (0.976 x 0.024590) / (0.976) = 0.024590 = 2.46%

R₂ = b = (0.976 x 0.024590 + 0.952 x 0.025210) / (0.976 + 0.952) =  0.024896 = 2.49%

R₃ = c = (0.976 x 0.024590 + 0.952 x 0.025210 + 0.928 x 0.025862) / (0.976 + 0.952 + 0.928) = 0.025210 = 2.52%

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Now we have to calculate the fixed rate of the oil swap price. The methodology remains same except that we are now going to use the Oil Forward Price in stead of interest rate futures.

R₁ = d = (0.976 x 57) / (0.976) = 57.00

R₂ = e = (0.976 x 57 + 0.952 x 58) / (0.976 + 0.952) =  57.49

R₃ = f = (0.976 x 57 + 0.952 x 58 + 0.928 x 59.5) / (0.976 + 0.952 + 0.928) = 58.15