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# Bond pricing based on market discount rate

Bond pricing is based on the discounted cash flow method whose mechanics depend on the features of the bond being valued. One method discounts the coupon payments and principal at a single discount rate while a more elaborate method discounts each cash flow at the relevant spot rate.

Under the single discount rate valuation method, bond price equals the present value of the bond cash flows i.e. coupon payments and principal determined at a single discount rate called the market discount rate (also called required yield and required rate of return) which represents the risk inherent in the bond cash flows.

## Bond price and market discount rate

Bond price is typically quoted per $100 of par value. If the price is lower (greater) than$100, the bond is said to be trading at a discount (premium). The price is lower (greater) than the par value if the coupon rate is lower (greater) than the market discount rate and the coupon payment is deficient (excessive).

## Pricing of a conventional bond

Since the coupon payments in case of a conventional bond are equal, equidistant and in arrears, the bond price can be determined using the formula for the present value of an ordinary annuity.

$P=\frac{c}{m}\times\frac{{1\ -\left(1+\frac{y}{m}\right)}^{-n\times m}}{\frac{y}{m}}+\frac{F}{\left(1+\frac{y}{m}\right)^n}$

Where P is the bond price, F is the face value, c is the annual coupon rate, y is the annual market discount rate, m is the number of coupon periods per year and n is the number of years till the maturity date.

A more general expression is as follows:

$P=\frac{PMT}{\left(1+r\right)^1}+\frac{PMT}{\left(1+r\right)^2}+…+\frac{PMT+FV}{\left(1+r\right)^N}$

Where PMT is the periodic coupon payment, FV is the face value, r is the periodic discount rate and n is the total number of coupon payments.

As evident from the equation above, in case of more than one coupon payments in a year, the periodic coupon rate and the periodic market discount rates are used to determine the bond price.