Question

A $100 bond with 4% coupon rate matures in 25 years. It bears semiannual coupons and is purchased for $117.50 to yield i⁽²⁾.

A $100 bond with 5% coupon rate also matures in 25 years. It also bears semiannual coupons, but is purchased for $135.00 to yield i⁽²⁾.

What is i⁽²⁾?

1. Less than 1.75%

2. At least 1.75%, but less than 2.25%

3. At least 2.25%, but less than 2.75%

4. At least 2.75%, but less than 3.25%

5. 3.25% or more

Answer 1

Bond 1: Face Value = $ 100, Coupon Rate = 4 % per annum payable semi-annually, Purchase Price = $ 117.5 and Tenure = 25 years or (25 x 2) = 50 half-years

Semi-Annual Coupons = 0.04 x 100 x 0.5 = $ 2

Let discount rate be denoted by r1

Therefore, 117.5 = 2 x (1/r1) x [1-{1/(1+r1)^(50)}] + 1000 / (1+r1)^(50)

Using EXCEL's Goal Seek Function/ a financial calculator/hit and trial method to solve the above equation, we get:

r1 = 0.015 or 1.5 %

Bond 2:

Face Value = $ 100, Coupon Rate = 5 % per annum payable semi-annually, Purchase Price = $ 135 and Tenure = 25 years or (25 x 2) = 50 half-years

Semi-Annual Coupons = 0.05 x 100 x 0.5 = $ 2.5

Let discount rate be denoted by r2

Therefore, 135 = 2.5 x (1/r2) x [1-{1/(1+r2)^(50)}] + 1000 / (1+r2)^(50)

Using EXCEL's Goal Seek Function/ a financial calculator/hit and trial method to solve the above equation, we get:

r2 = 0.015 or 1.5 %

Therefore, i(2) = (1+r1/r2)^(2) - 1 = (1.015)^(2) - 1 = 0.03022 or 3.022 %

Hence, the correct option is (d)