bond valuation

Example 1.

The Xanth Co. were to issue a bond with 10 years to maturity. The Xanth bond has an annual coupon of $80, implying the bond will pay $80 per year for the next 10 years in coupon interest. In addition, Xanth will pay $1,000 to the bondholder in 10 years. Assuming similar bonds have a yield of 8%, what will this bond sell for?

First: 

At the going rate of 8%, the PV(Present Value) of the $1,000 paid in 10 years is:

PV = $1,000/1.08¹⁰

PV = $1,000/2.1589

PV = $463.19

Second:

Coupon interest pay in every year for 10 years is: $1,000 x 8% = $80, then the PV of this annuity:

Annuity PV = $80 x (1-1/1.08¹⁰)/.08

Annuity PV = $80 x (1-1/2.1589)/.08

Annuity PV = $80 x 6.7101

Annuity PV = $536.81

Total bond value = $463.19 + $536.81 = $1,000

Meaning: This bond sells for exactly its face value.

If the interest rate change, suppose that a year has gone by. The Xanth bond now has nine years to maturity. If the interest rate in the market has risen to 10%, what will this bond worth?

First:

PV = $1,000/ 1.10⁹

PV = $1,000/2.3759

PV = $424.10

Second:

Coupon interest pay in every year for  9 years is: $1,000 x 8% = $80, then the PV of this annuity:

Annuity PV = $80 x (1-1/1.10⁹)/.10

Annuity PV = $80 x (1 – 1/ 2.3579)/.10

Annuity PV = $80 x 5.7590

Annuity PV = $460.72

Total bond value = $424.10 + $460.2 = $884.82

Therefore, the bond should sell for about $885. We say this bond with its 8% coupon, is priced to yield 10% at $885. Because the bond sells for less than face value, it is said to be a discount bond.

The Xanth bond now has a coupon rate of 8% when market rate is only 6%. And there are nine years remaining. 

The PV of the $1,000 face amount is:

PV = $1,000/1.06⁹

PV= $1,000/1.6895

PV = $591.89

The PV of the coupon stream is:

Annuity PV = $80 x (1-1/1.06⁹)/.06

Annuity PV = $80 x (1-1 /1.6895)/ .06

Annuity PV = $80 x 6.8017

Annuity PV = $544.14

Total bond value : $591.89 +$544.14 = $1.136.03

Such a bond is said to sell at a premium and is called a premium bond.

Example 2.

Find value of bond if par value = $1,000 ; T=20 years ; c=8% semiannual payment, if:

R = 10%, P =( $40 x (1-1 /1.05⁴⁰)/0.05) + ($1,000/1.05⁴⁰) = $828.41

When R is greater than the coupon rate, the bond will sell at a discount (less than par).

R = 6 %, p= ( $40 x (1-1 /1.03⁴⁰)/0.03) + ($1,000/1.03⁴⁰) = $1,231.15

When YTM ® is less than the coupon rate, the bond will sell at a premium (greater than par)

Source: Corporate Finance Book, Stephen A.Ross, Randolph W.Westerfield and Jeffrey Jaffe, Ninth Edition.