Coupon Payments: Assume that you wish to purchase a 20-year bond that...

Question

Assume that you wish to purchase a 20-year bond that has a maturity value of $1,000 and makes semi-annual interest payments of $40. If you require a 10% per annum nominal yield to maturity (5% effective semi-annual yield) on this investment, what is the maximum price you should be willing to pay for the bond?

• Answer

  828.410

• Step-by-step solutions

  To determine the maximum price you should be willing to pay for the bond, we calculate the present value of the bond’s cash flows, which include the semi-annual interest payments and the maturity value.

  Step 1: Define the Variables

  • Maturity Value (F): $1,000

  • Semi-annual Interest Payment (C): $40

  • Nominal Yield to Maturity (YTM): 10% per annum (i.e., 5% per semi-annual period)

  • Number of Years (N): 20 years

  • Total Number of Periods (n): 20 years × 2 = 40 periods

  • Periodic Rate (r): 5% or 0.05

  Step 2: Present Value Concept

  The value of the bond is the sum of:

  1. The present value of the interest payments (an annuity), and

  2. The present value of the lump sum maturity value.

  Step 3: Calculate Present Value of Interest Payments

  We use the present value of an annuity formula to calculate the value of the 40 interest payments of $40 each, discounted at 5% per period.

  Using financial calculator or spreadsheet functions, the present value of these payments is approximately $686.36.

  Step 4: Calculate Present Value of Maturity Value

  Next, we calculate the present value of the $1,000 received at maturity, 40 periods from now, discounted at 5%.

  The present value of the lump sum is approximately $142.05.

  Step 5: Add the Two Components

  Add the two present values:

  • Interest Payments: $686.36

  • Maturity Value: $142.05

  • Total Present Value (Price): $828.41