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  Value of a Bond or Perpetuity

The Value of a Bond or Perpetuity:

One important use of discounting is to determine the value of bonds and perpetuities.

Bond - A debt security issued by a government or a firm where the purchaser lends the issuer a lump sum amount of money today in return for a promise from the issuer to pay a finite stream of future payments.

Coupon - The annual interest payment on a bond.

Perpetuity - A debt security where the purchaser lends the issuer a lump sum amount of money today in return for a promise from the issuer to pay a fixed amount of income to the purchaser forever.

To determine the value of a bond to its buyer, we calculate the present value of the payments stream. Suppose, for example, a bond is sold for $1,000 and pays a coupon of $100 per year for each of the next 10 years and a principal repayment in the tenth year of $1,000.

1159_principal repayment.jpg

At a discount rate of zero, the present value of the bond will be equal to PV =10($100) + $1,000 = $2,000 . As the discount rate increases above zero, the present value will continuously decline. Using Equations (3) and (5), we can simplify the calculation of the present value of the bond as follows:

419_present value of bond.jpg

$100/r : PV of an annual payment of $100 forever with the first payment being made in one year (from Eq.(3)) $100/r(1+ r)10 : PV of the payments that would be made from t =11 to t = ∞ on a bond that pays a $100 coupon forever.

Therefore the first two terms on the right hand side of the above calculation measure the present value of the 10 years of coupon payments on the bond. And the third term of the above calculation is the PV of the principal repayment that is received in 10 years.
Now, if the purchaser of the bond has a 5% discount rate the present value of the bond is:

592_discount rate of pV.jpg

• To find the present value of a perpetuity which beginning in year 1 pays $1000 per year forever, use Equation (3) to calculate:

881_PV.jpg

• Government and corporate bonds are sold in the bond market in the same way that shares of stock are sold in the stock market. Buyers and sellers agree to exchange a bond at an equilibrium price p. The yield on a bond is simply the rate of return on the market price (market value) of the bond. The face value of the bond equals the principal payment to the owner when the bond reaches maturity. To determine the yield or rate of return on a bond, we rewrite Equation (1) in the following form:

627_rate of return.jpg

Buyer knows bond price (p), the coupon payments (the values of xi for all i), and the principal payment; therefore, the only unknown is the yield or rate of return of the bond.

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