# Bond Price

A bond is a debt instrument: it pays periodic interest payments based on the stated (coupon) rate and return the principal at the maturity.

Cash flows on a bond with no embedded options are fairly certain and the price of bond equals the present value of future interest payments plus the present value of the face value (which is returned at maturity) based on the interest rate prevailing in the market.

The present value of interest payments is calculated using the formula for present value of an annuity and the present value of the face value (also called the maturity value) is calculated using the formula for present value of a single sum occurring in future.

If **r** is the interest rate prevailing in the market, **c** is the coupon rate on the bond, **t** is the time periods occurring over the term of the bond and **F** is the face value of the bond, the present value of interest payments is calculated using the following formula:

Present Value of Interest Payments = c × F × | 1 − (1 + r)^{-t} |

r |

The present value of the face value (i.e. the maturity value) is calculated as follows:

Present Value of Face Value of a Bond = | F |

(1+r)^{t} |

Therefore, the price of a bond is given by the following formula:

Present Value of Interest Payments = c × F × | 1 − (1 + r)^{-t} | + | F |

r | (1 + r)^{t} |

## Examples

**Example 1:** Bond with annual coupon payments

Company A has issued a bond having face value of $100,000 carrying annual coupon rate of 8% and maturing in 10 years. The market interest rate is 10%.

The price of the bond is calculated as the present value of all future cash flows:

Price of Bond = 8% × $100,000 × | 1 − (1 + 10%)^{-10} | + | $100,000 |

10% | (1 + 10%)^{10} |

Price of Bond = $87,711 |

**Example 2:** Bond with semiannual coupon payments

Company S has issued a bond having face value of $100,000 carrying coupon rate of 9% to be paid semiannually and maturing in 10 years. The market interest rate is 8%.

Since the interest is paid semiannually the bond interest rate per period is 4.5% (= 9% ÷ 2), the market interest rate is 4% (= 8% ÷ 2) and number of time periods are 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} | + | $100,000 |

4% | (1 + 4%)^{20} |

Price of Bond = $106,795 |

Written by Obaidullah Jan, ACA, CFA