Bond Valuation: Zero Coupon, Premium, and Discount Bond Pricing

Posted on Jun 8, 2025 in Finance

Problem 8: Bond Coupon Rate Calculation

To determine the annual coupon and coupon rate, we use the following financial calculator inputs and solve for the payment (PMT):

Financial Calculator Inputs (Problem 8)

  • N: 10.5 × 2 = 21 (semiannual periods)
  • I/Y: 6.2% / 2 = 3.1% (semiannual yield)
  • PV: -$945 (present value, entered as negative)
  • FV: $1,000 (face value)
  • PMT: Solve for $27.40 (semiannual coupon payment)

Based on the semiannual payment, we can calculate the annual coupon and coupon rate:

  • Annual coupon = $27.40 × 2 = $54.80
  • Coupon rate = $54.80 / $1,000 = 0.0548, or 5.48%

Problem 15: Zero Coupon Bond Pricing

You find a zero coupon bond with a par value of $10,000 and 17 years to maturity. If the yield to maturity (YTM) on this bond is 4.9 percent, what is the price of the bond? Assume semiannual compounding periods.

To find the price of a zero coupon bond, we need to find the present value of its future cash flows. With a zero coupon bond, the only cash flow is the par value at maturity. We find the present value assuming semiannual compounding to keep the YTM of a zero coupon bond equivalent to the YTM of a coupon bond:

P = $10,000(PVIF2.45%,34)

P = $4,391.30

Financial Calculator Inputs (Problem 15)

  • N: 17 years × 2 = 34 (semiannual periods)
  • I/Y: 4.9% / 2 = 2.45% (semiannual yield)
  • PMT: $0 (no coupon payments)
  • FV: -$10,000 (face value, entered as negative)
  • PV: Solve for $4,391.30 (bond price)

Problem 18: Bond Price Movements and Pull to Par

Bond X is a premium bond making semiannual payments. The bond has a coupon rate of 8.5 percent, a YTM of 7 percent, and has 13 years to maturity. Bond Y is a discount bond making semiannual payments. This bond has a coupon rate of 7 percent, a YTM of 8.5 percent, and also has 13 years to maturity. What are the prices of these bonds today assuming both bonds have a $1,000 par value? If interest rates remain unchanged, what do you expect the prices of these bonds to be in one year? In three years? In eight years? In 12 years? In 13 years? Illustrate your answers by graphing bond prices versus time to maturity.

Here, we are finding the price of semiannual coupon bonds for various maturity lengths. The bond price equation is:

P = C(PVIFAR%,t) + $1,000(PVIFR%,t)

Bond X (Premium Bond) Calculations

Coupon payment (C) = ($1,000 × 0.085) / 2 = $42.50

Semiannual YTM (R) = 7% / 2 = 3.5%

  • P0 (Today, 13 years to maturity) = $42.50(PVIFA3.5%,26) + $1,000(PVIF3.5%,26) = $1,126.68
  • P1 (In 1 year, 12 years to maturity) = $42.50(PVIFA3.5%,24) + $1,000(PVIF3.5%,24) = $1,120.44
  • P3 (In 3 years, 10 years to maturity) = $42.50(PVIFA3.5%,20) + $1,000(PVIF3.5%,20) = $1,106.59
  • P8 (In 8 years, 5 years to maturity) = $42.50(PVIFA3.5%,10) + $1,000(PVIF3.5%,10) = $1,062.37
  • P12 (In 12 years, 1 year to maturity) = $42.50(PVIFA3.5%,2) + $1,000(PVIF3.5%,2) = $1,014.25
  • P13 (In 13 years, 0 years to maturity) = $1,000

Bond Y (Discount Bond) Calculations

Coupon payment (C) = ($1,000 × 0.07) / 2 = $35

Semiannual YTM (R) = 8.5% / 2 = 4.25%

  • P0 (Today, 13 years to maturity) = $35(PVIFA4.25%,26) + $1,000(PVIF4.25%,26) = $883.33
  • P1 (In 1 year, 12 years to maturity) = $35(PVIFA4.25%,24) + $1,000(PVIF4.25%,24) = $888.52
  • P3 (In 3 years, 10 years to maturity) = $35(PVIFA4.25%,20) + $1,000(PVIF4.25%,20) = $900.29
  • P8 (In 8 years, 5 years to maturity) = $35(PVIFA4.25%,10) + $1,000(PVIF4.25%,10) = $939.92
  • P12 (In 12 years, 1 year to maturity) = $35(PVIFA4.25%,2) + $1,000(PVIF4.25%,2) = $985.90
  • P13 (In 13 years, 0 years to maturity) = $1,000

Financial Calculator Inputs (Bond X, Problem 18)

  • N: 13 years × 2 = 26 (semiannual periods)
  • I/Y: 7% / 2 = 3.5% (semiannual yield)
  • PMT: -$42.50 (semiannual coupon payment, entered as negative)
  • FV: -$1,000 (face value, entered as negative)
  • PV: Solve for $1,126.68 (bond price)

Financial Calculator Inputs (Bond Y, Problem 18)

  • N: 13 years × 2 = 26 (semiannual periods)
  • I/Y: 8.5% / 2 = 4.25% (semiannual yield)
  • PMT: -$35 (semiannual coupon payment, entered as negative)
  • FV: -$1,000 (face value, entered as negative)
  • PV: Solve for $883.33 (bond price)

All else held equal, the premium over par value for a premium bond declines as maturity approaches, and the discount from par value for a discount bond declines as maturity approaches. This phenomenon is known as “pull to par.” In both cases, the largest percentage price changes occur at the shortest maturity lengths.

Also, notice that the price of each bond when no time is left to maturity is the par value, even though the purchaser would receive the par value plus the coupon payment immediately. This is because we calculate the clean price of the bond.