Question

Hey. I am having trouble with a finance question:

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Consider two bonds. The first is a 6% coupon bond with six years to maturity, and a yield to maturity of 4.5% annual rate, compounded semi-annually. The second bond is a 2% coupon bond with six years to maturity and a yield to maturity of 5.0%, annual rate, compounded semi-annually.

Given the data for the first two bonds, now consider a third bond: a zero coupon bond with six years to maturity. Calculate the price per $100 of face value of the zero coupon bond. Calculate the yield to maturity for the zero coupon bond. (Express the yield as annual rate, compounded semi-annually).

HINT: Use the Value Additivity principle to answer. Create a synthetic zerocoupon bond, that is, a portfolio of the 6% coupon bond and the 2% coupon bond that has the same cash flows as a 6-year, zero coupon bond.

Answer 1

Let's call the two bonds A and B. As a first step we need to determine the price of each of the two bonds. We will make use of PV function of excel to determine the price.

Bond A:

Inputs of PV function are: Rate = YTM semi annual = 4.5% / 2 = 2.25%

Period = nos. of half years in time to maturity of 6 years = 2 x 6 = 12

PMT = Payment per period = semi annual coupon = 6% / 2 = 3% x Face value = 3% x 100 = 3

FV = future value = face value = par value = 100

Hence, Price of bond A = P_A = - PV (Rate, Period, PMT, FV) = - PV (2.25%, 12, 3, 100) = $ 107.81

Bond B:

Inputs of PV function are: Rate = YTM semi annual = 5% / 2 = 2.5%

Period = nos. of half years in time to maturity of 6 years = 2 x 6 = 12

PMT = Payment per period = semi annual coupon = 2% / 2 = 1% x Face value = 1% x 100 = 1

FV = future value = face value = par value = 100

Hence, Price of bond B = P_B = - PV (Rate, Period, PMT, FV) = - PV (2.5%, 12, 1, 100) = $ 84.61

Let's create a synthetic zero coupon bond, that is, a portfolio of N_A number of bond A and N_B number of Bond B that has the same cash flows as a 6-year, zero coupon bond.

Cash flow per period (semi annual) = N_A x Semi annual coupon of A + N_B x Semi annual coupon of Bond B = 3 x N_A + 1 x N_B = 3N_A + N_B

and cash flow on maturity = N_A x (Semi annual coupon of A + Face Value) + N_B x (Semi annual coupon of Bond B + Face Value) = (3 + 100) x N_A + (1 + 100) x N_B = 103N_A + 101N_B

Periodic cash flow from a zero coupon bond = 0 (it doesn't pay any coupon)

Hence, 3N_A + N_B = 0; Hence, N_B = - 3N_A -------Equation (1)

Cash flow on maturity from zero coupon bond = Face value = 100

Hence, 103N_A + 101N_B = 100 -------------Equation (2)

Substitute for N_B from Equation (1) in Equation (2) to get:

103N_A + 101 x (-3N_A) = 100 Or, -200N_A = 100, hence N_A = -100 / 200 =  - 0.5 and N_B = -3 x ( - 0.5) =  1.5

Thus price of the zero coupon bond = Price of the portfolio of 1.5 number of Bond B and - 0.5 number of bond A = 1.5 x P_B - 0.5x P_A = 1.5 x 84.61 - 0.5 x 107.81 = $  73.01