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# Forward and Par rates

#### Taunk

##### Member
Hi,
Can some one please explain the solution to P1.T4.313.3

313.3. We are given the 1.5 year discount function (i.e., set of discount factors) below. Also, below is the mathematical definition of the semi-annual par rate: the par rate, C(T), is the coupon rate such that the present value of the bond equals par.

Given these discount factors, which is nearest to the 1.5 year par rate, C(1.5)?
a) 1.33% b) 1.50% c) 1.80% d) 2.40%

Thanks

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @taunk,

The par rate is that coupon rate (C) that causes the bond price to sell at par. That is, it is that coupon rate such that Bond Price = Par. The Swap rates are all par rates.

Using the formula given:

C/2*[d(0.5) + d(1.0) + d(1.5)] + d(1.5) = 1
C/2*[0.99700 + 0.98510 + 0.96460] + 0.96460 = 1
C/2*[2.94670] + 0.96460 = 1
C/2*[2.94670] = 1 - 0.96460 = 0.03540
C = 2*[0.03540/2.94670] = 2.402%

In order to check whether the answer is right, we can compute the spot rate r(1.5). This is because the par rate is only slightly lower than the corresponding spot rate.

d(1.5) = 1/(1 + r(1.5)/2)^3
(1 + r(1.5)/2)^3 = 1/d(1.5)
1 + r(1.5)/2 = 1/[d(1.5)]^1/3
r(1.5) = 2*[(1/(0.96460)^1/3) - 1] = 2.417%

Hope that explains your query
Thanks!

#### Taunk

##### Member
Thank you for the detailed explanation.