Financial Math FM/Bonds

Bond

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

Basic formulas

 $\ {P}=F(r)a_{{\overline {n|}}i}+Cv^{n}$

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.

Examples

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.

Solution

 $\ {P}={v^{20}}_{.02}F$ 
 $\ {P}={1.02}^{-20}1000$ 
 $\ {P}=672.97$

Using a BA II, press:
2ND FV; 1000 +

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.

Solution

For this problem we are told that it has an 8% coupon convertible semiannually. To find the value of the coupon, it is simply 4% of the par value, $5,000 which is $200. Thus this bond will pay $200 every 6 months for 20 years and $5,000 at the redemption date, 20 years in the future. The present value of the bond is the sum of the value of the annuity paying every 6 months plus the discounted value of the $5,000.

 $\ {P}=Fra_{{\overline {n|}}i}+Cv^{n}$ 
 $\ {P}=5,000(.04)a_{{\overline {40|}}.03}+5,000v^{40}$ 
 $\ {P}=(200){\frac {1-{1.03}^{-40}}{0.03}}+5,000({1.03}^{-40})$ 
 $\ {P}=6,155.74$

Or, on your calculator press: 2ND FV; 5000 FV; 200 PMT; 3 I/Y; 40 N; CPT PV
The first command clears the financial calculator. The second command enters 5000 as the future value (FV). The third command enters 200 as the coupon payment (PMT). The fourth command enters 3 as the interest rate (I/Y). The final command computes (CMPT) the present value (PV).

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.

Solution

 $\ {P}=Cv^{2n}$ 
 $\ {1487.11}=2000{v_{.025}}^{2n}$ 
 $\ 2n=-{\frac {ln({\frac {1487.11}{2000}})}{ln(1.025)}}$ 
 $\ n=6$

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, $\ {\frac {r}{i}}$ , is 1.03125.
The present value of the redemption value is 381.50.

Given $\ v^{n}=0.5889$ , what is the price of bond X?

Solution

The price for any bond is the present value at the yield rate of the coupons plus the present value of the yield rate of the redemption value. Given $r=$ semi-annual coupon rate and $i=$ the semi-annual yield rate. Let $C=$ redemption value. Then the price P for bond X is $P=1000ra+Cv^{2n}$ (using a semi-annual yield rate throughout.)

 $=1000{\frac {r}{i}}(1-v^{2n})+381.50$

Callable

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