A bond is a debt instrument that pays periodic interest at a specified interest rate (called the coupon rate) and returns the principal on predetermined maturity date. Bond price also refers to the sum of the present value of the par value at maturity.
The cash flows of conventional bonds (bonds without embedded options) are fairly certain in amount and time and includes:
- Periodic interest payments, called coupon payments, each interest equals the face value of the bond multiplied by the periodic coupon rate, and
- Maturity period Payment due that is, the final bullet payment equal to the face value of the bond (maturity value) of the bond.
The value/price of the bond is equal to the present value of future coupon payments plus the present value of the maturity value, both of which are calculated at the current interest rate on the market. Since the coupon payment forms a cash flow that occurs after equal time intervals, the present value of the annuity present value formula is used to calculate the present value. Similarly, because the repayment of principal (maturity value) is a one-time payment at the end of the bond’s life, the present value of the maturity date is calculated using the present value formula of a sum that will occur in the future.
Therefore r is the interest rate prevailing in the market, c is the periodic coupon rate on the bond (i.e. annual coupon rate divided by number of coupon payments per year), and t is the total number of coupon payments outstanding till maturity and f is the face value of the bond (i.e. the principal balance),
Present value of coupon payments is calculated using the following formula:
PV of coupon payments = C*f *{1-(1+r)-t}/ r
The present value of the maturity value is calculated as follows:
PV of Maturity value = F / (1+r) t
Bond Price formula
Bond Price = c*f *{1-(1+r)-t}/ r + F / (1+r) t
Examples
Bond with Annual coupon payments
Company X has issued a bond having face value of $100,000 carrying coupon rate of 8% and maturing in 10 years. The market interest rate is 10%.
Price of Bond calculated as:
=8 % *$100,000 * {1-(1+10%)-10}/ 10% + $100,000/ (1+10%) 10
= $87,711
Example 2
Bond with Semiannual coupon payments
Company A has issued a bond having a bond which has face value of $100,000 carrying rate of 9% to be paid semiannually and maturing in 10 years. The market interest rate is 8%.
The company paid interest semi-annually the bond coupon rate per period is 4.5%(=9%/2), the market interest rate is 4 % (=8% /2) and the number of coupon payments ( time periods) is 20(=2*10). Hence, the price of the bond is calculated as the present value of all future cash flows as shown below:
Price of Bond
=4.5% *100,000 *1-(1+4%)-20/4% + $100,000/ (1+4%) 20
=$106,795
The relationship between coupon rate and market interest rate are shown below:
- If the market interest rate is higher than the coupon rate, the bond price is lower than the bond face value (i.e. trades at a discount );
- If the market interest rate and coupon rate are similar, bond price equals its face value (i.e trades at par); and
- If the market interest rate is lower than the coupon rate, the bond price is higher than the bond face value (i.e. trades at a premium)