 • Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers ## 4.2.2 Basic Bond Valuation

Basic bond valuation formula

A bond's value is the present value of the payments the issuer is contractually obligated to make -- from the present until maturity. The discount rate depends on the prevailing interest rate for debt obligations with similar risks and maturities.

Using the basic DCF method, a bond's value is --

 B0 = Sum (1 to n) I / (1+i)n + M/ (1+i)n OR B0 = I *[(1+i)n - 1] / [(1+i)n * i] + M/ (1+i)n B0 = bond's value at time zero I = annual interest payments i = discount rate n = number of years to maturity M = par value (payment at maturity)

Look at the example to the right to see the formula in action.

Computing bond price. If you know the bond's par value, coupon rate, time to maturity and current yield, you can compute its price. See attached spreadsheet for computing prices and yields for bonds paying semi-annual interest.

Example

Suppose Play Now, Inc. issues ten-year bonds (par \$1,000) with an annual coupon of 8.6%. Similar ten-year bonds are paying 8.0% interest. What is the value of one of Play Now's new bonds -- that is, what should be its price?

Answer (using tables). You can use present value and present value annuity tables:

 B0 = C/(1+i)n + M/(1+i)n = \$86 x (PV of ten year \$1 annuity at 8.0%) + \$1,000 x (PV of \$1 to be paid in 10 years at 8.0%) = \$86 x 6.710) + (\$1,000 x .4632) = 577.06 + \$463.20 = \$1,040.26

 End of year Interest Principal Present value 1 \$86 \$79.63 2 \$86 \$73.73 3 \$86 \$68.27 4 \$86 \$63.21 5 \$86 \$58.53 6 \$86 \$54.19 7 \$86 \$50.18 8 \$86 \$46.46 9 \$86 \$43.02 10 \$86 \$1000 \$503.03 Discount Rate 8.0% Bond Value \$1,040.26

Semiannual interest

Most bonds, although the coupon rate is stated as an annual interest rate, actually pay interest semiannually. Valuing bonds that pay interest semiannually involves three steps:

• Convert bond's annual interest (I) to semiannual interest -- divide I by 2
• Convert the years to maturity (n) to semiannual periods -- multiply n by 2
• Convert annual required return (i) to semiannual discount rate -- divide i by 2

The bond valuation formula for a bond paying interest semiannually is:

 B0 = Sum (1 to n) I/2 / (1+i/2)2n + M/ (1+i/2)2n OR B0 = I/2 *[(1+i/2)2n - 1] / [(1+i/2)2n * i/2] + M/ (1+i/2)2n B0 = bond's value at time zero I = annual interest payments i = discount rate n = number of years to maturity M = par value (payment at maturity)

Computing bond price. If you know the bond's par value, coupon rate, time to maturity and current yield, you can compute its price. See attached spreadsheet for computing prices and yields for bonds paying semi-annual interest.

Continuous compounding. [Example of continuous compounding]

Example

Returning to our example, suppose Play Now issues ten-year bonds (par \$1,000) with an annual coupon rate of 8.6% that pay interest semiannually. Similar ten-year bonds are paying 8.0% interest. What is the value of one of Play Now's new bonds -- that is, what should be its price?

Answer. You can use present value and present value annuity tables:

 B0 = C/2/(1+ i/2)2n + M/(1+i/2)2n = \$43 x (PV of 20-period \$1 annuity at 4.0% per period) + \$1,000 x (PV of \$1 after 20 periods at 4.0% per period = (\$43 x 13.590) + (\$1,000 x .4564) = \$584.37 + \$456.40 = \$1,040.77

Notice that this bond is identical to the bond in the previous example with the exception that it pays interest semiannually. The effect of the semiannual payments is to increase the price of the bond -- from \$1,040.26 to \$1,040.77.

Answer. You can also use a spreadsheet.
 End of year Interest Principal Present value 0.5 \$43 \$41.35 1.0 \$43 \$39.76 1.5 \$43 \$38.23 2.0 \$43 \$36.76 2.5 \$43 \$35.34 3.0 \$43 \$33.98 3.5 \$43 \$32.68 4.0 \$43 \$31.42 4.5 \$43 \$30.21 5.0 \$43 \$29.05 5.5 \$43 \$27.93 6.0 \$43 \$26.86 6.5 \$43 \$25.82 7.0 \$43 \$24.83 7.5 \$43 \$23.88 8.0 \$43 \$22.96 8.5 \$43 \$22.08 9.0 \$43 \$21.23 9.5 \$43 \$20.41 10.0 \$43 \$19.62 10.0 \$1000 \$456.39 TOTAL \$1,040.77

 This page was last updated on: April 2, 2004