# Bond Price

A bond is a debt instrument that pays periodic interest payments based at a stated interest rate called coupon rate and returns the principal at a pre-determined maturity date.

Cash flows of a conventional bond (a bond with no embedded options) are fairly definite in amount and timing and comprise of:

• Periodic interest payments called coupon payments each of which equals the face value of the bond multiplied by the periodic coupon rate, and
• Maturity payment i.e. final bullet payment which equal to the face value (maturity value) of the bond.

The value/price of a bond equals the present value of future coupon payments plus the present value of the maturity value both calculated at the interest rate prevailing in the market. Since coupon payments form a stream of cash flows that occur after equal interval of time, their present value is calculated using the formula for present value of an annuity. Similarly, since the repayment of principal (maturity value) is a one-off payment at the end of the bond life, the present value of 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 periodic coupon rate on the bond (i.e. annual coupon rate divided by number of coupon payments per year), 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), the 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

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

 Bond Price = 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 = \$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 coupon rate per period is 4.5% (= 9% ÷ 2), the market interest rate is 4% (= 8% ÷ 2) and number of coupon payments (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 = \$106,795

This uncovers an important relationship between coupon rate and market interest rate:

• If the market interest rate is higher than the coupon rate, the bond price is lower than the bond face value (i.e. it trades at a discount);
• If the market interest rate and coupon rate are exactly same, bond price equals its face value (i.e. it 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. it trades at premium)