The Finance Tutor

Return to Homepage

Semi-Annual Coupon Bond Valuation

The prior tutorial introduced several fundamentals of coupon bond valuation including the application of discount rates to derive the present value of future cash flows. The example and practice problems focused on coupon bonds that make annual coupon payments. However, the majority of coupon bonds issued in the United States pay coupons semi-annually. The proceeding tutorial will provide a comprehensive overview of semi-annual coupon bond valuation. In addition, a fundamental relationship between discount rates, coupon rates, and market value will be discussed.

Recall that the value of a bond is equal to the present value of its future cash flows. To calculate present value, an investor must forecast the bond’s future cash flows and discount them at a discount rate. The discounted cash flows can then be summed to derive the value of a bond. The basic formula of a coupon bond is detailed below. The formula assumes that an investor purchases the bond and holds it until maturity.

Market Coupon Coupon Coupon Coupon Principal
Value_Bond = (1+ r )¹ + (1 + r )² + (1 + r )³ … (1 + r )ⁿ + (1 + r )ⁿ

r = discount rate
n = periods until maturity

In contrast to annual coupon bonds, the annual coupon rate earned on semi-annual coupon bonds is divided into two, equal payments. For example, a semi-annual coupon bond with a $1,000 face value and a 9% coupon rate will pay a $45 coupon every six months. A semi-annual bond with a $10,000 face value and a 6% coupon rate will pay a $300 coupon every six months.

Consider the following example: Calculate the value of a semi-annual coupon bond with seven years to maturity, a $1,000 face value, an 8% coupon rate, and a 10% required rate of return.

Market $40 $40 $40 $40 $1,000
Value_Bond = (1.05)¹ + (1.05)² + (1.05)³ … (1.05)¹⁴ + (1.05)¹⁴

0.05 = discount rate/2
14 = 7 Years x 2 payments

The value of this bond can be derived using the Time Value of Money functions of a Hewlett Packard 12C:

HP 12C: N = 14 I = 5 PMT = $40 FV = $1,000

Solve = PV

PV = -$901.01

The present value of the bond equates to $901.01 or 90-3. Notice that the bond above is valued at a discount to its face value of $1,000. The market value of $901.01 is less than the bond's face value because its required rate of return (discount rate) is greater than the bond’s coupon rate. Calculate the market value of the bond if its required rate of return was decreased to 6%.

Market $40 $40 $40 $40 $1,000
Value_Bond = (1.03)¹ + (1.03)² + (1.03)³ … (1.03)¹⁴ + (1.03)¹⁴

0.03 = discount rate/2
14 = 7 Years x 2 payments

To Solve using a HP 12C:

N = 14 I = 3 PMT = $40 FV = $1,000 Solve = PV

PV = -$1,112.96

As you can see, the bond would now sell at a premium to par value. What if the required rate of return equaled the coupon rate of 8%? As you might guess the bond would sell for face value, or $1,000. The relationship between the required rate of return, the coupon rate, and market value is summarized below:

Required Rate of Return > Coupon Rate, the bond will sell at a discount to FV

Required Rate of Return < Coupon Rate, the bond will sell at a premium to FV

Required Rate of Return = Coupon Rate, the bond will sell at FV

The Following are additional examples that will aid in grasping the fundamentals of semi-annual coupon bond valuation:

A bond with a $100,000 face value offers a $5,000 coupon every six months until maturity. The bond’s length to maturity is 18 years. If the required rate of return is 9%, what is the current market value of the bond?

To solve using a HP 12C:

N = 36 I = 4.5 PMT = $5,000 FV = $100,000 Solve = PV

PV = -$108,833.02

The bond will sell for $108,833.02, or 108-27, in the market. What if the required rate of return increased to 12%? Immediately you should recognize that the bond will sell at a discount to face value because a 12% discount rate exceeds the coupon rate of 10%. Using a financial calculator, the value of the bond now equates to $85,379.01 or 85-12.

Consider a second example: An investor wishes to purchase a ten-year semi-annual coupon bond with a $1,000 face value and a 7% coupon rate. The market’s risk free rate is 3.0%. Inflation is expected to be 2.8%. The investor considers a 2.0% risk premium to be appropriate considering the bond’s various risks. Calculate the investor’s required rate of return and the price the maximum price he should pay for the bond.

Discount Rate (r) = 3.0% + 2.8% + 2.0% = 7.8%

To solve using an HP 12C:

HP 12C: N = 20 I = 3.9 PMT = $35 FV = $1,000

Solve = PV

PV = -$945.15

The investor’s required rate of return is 7.8%, or 3.9% per period. Thus, the maximum price he should pay for the bond is $945.15 or 94-16.

Questions:

1. Explain the relationship between a bond’s required rate of return, coupon rate, and market value.

2. Calculate the risk premium of a bond with a required rate of return of 9% and an inflation premium of 3%. Assume the market’s risk-free rate is 4%.

3. Calculate the market value of a semi-annual coupon bond with a $1,000 face value, a 4% required rate of return, a 7% coupon rate, and six years remaining until maturity.

4. Calculate the present value of a semi-annual coupon bond with a $10,000 face value, a 12.5% discount rate, a 10% coupon rate, and four years remaining until maturity.

5. Calculate the present value of a semi-annual coupon bond with a $10,000 face value, a 9% discount rate, a 9% coupon rate, and four years remaining until maturity.

Answers:

1. Reference

2. 2% r_Premium = 9% - 3% - 4%

3. $1,158.63 N = 12; I = 2; PMT = $35; FV = $1,000; PV = Solve

4. $9,231.40 N = 8; I = 6.25; PMT = $500; FV = $10,000; PV = Solve

5. $10,000.00 N = 8; I = 4.5; PMT = $450; FV = $10,000; PV = Solve