Financial Math FM/Bonds
A bond is a debt security, in which the issuer, usually a corporation or public institution, owes the holders a debt and is obliged to pay interest (the coupon) and to repay the principal at a later date. A bond is a formal contract to repay borrowed money with interest at fixed intervals. There are two main kinds of bonds: accumulation bonds (zero coupon bonds) and bonds with coupons. An accumulation bond is where the issuer of the bond agrees to pay the face value at a later redemption date, but they are sold at a discount.
Example: a 20 year $1000 face value bond with a 3.5% nominal annual yield would have a price of $502.56.
Bonds with coupons are more common and it's where the issuer of the bond makes period payments (coupons) and a final payment.
Example: a 10 year $1000 par value bond with a 8% coupon convertible semiannually would pay $40 coupons every 6 months and then $1000 at the end of the 10 years.
Terminology and variable naming convention
- P, Price: The price of a bond P is the amount that the lender, the person buying the bond, pays to the government or corporation issuing the bond.
- F, Face value, par value: This is the amount by which the coupons are calculated and usually the value of the bond upon maturity.
- C, Redemption value: Value of a bond upon maturity and equal to it's par value if it is redeemable at par. Unless stated otherwise a bond is redeemable at par.
- r, coupon rate of a bond: It is important to note how often the coupon is convertible. A $100 8% bond convertible quarterly would pay $20 every three months while a $100 8% bond convertible semiannually would pay $40 every 6 months.
- Fr, the amount of the coupon. A $1000 6% bond convertible semiannually bond would have a coupon of $30 every 6 months.
- P - C = the premium if P > C
- C - P = the discount if C > P
In this formula, the present value of the bond is simply the sum of the present value of the coupons and the present value of the redemption value. The present value of the coupons is represented as an annuity over n periods while the final redemption value is discounted over n periods. It is important to note that the yield rate and the coupon rate are not always the same.
- Example 1
- Find the price of a 10-year zero coupon bond with a par value $1000 and a yield rate of 4% convertible semiannually.
- Example 2
- Find the price of a 20-year bond, with a par value of $5,000 and a coupon rate of 8%, convertible semiannually, at a yield of 6%, convertible semiannually.
- Example 3
- The price of a zero coupon bond $2000 face value bond is $1487.11. The yield rate convertible semiannually is 5%. Find the maturity date.
- Example 4
- You have decided to invest in Bond X, an n-year bond with semi-annual coupons and the following characteristics:
- Par value is 1000.
- The ration of the semi-annual coupon rate to the desired semi-annual yield rate, , is 1.03125.
- The present value of the redemption value is 381.50.
Given , what is the price of bond X?