# Pricing of Bonds Payable

Bonds are priced at their net present value, taking into account the value of the bond itself at maturity as well as any payments associated with periodic interest payments based on the stated rate.

Bonds often sell at a price that differs from the face value. To determine the price of a bond, the net present value of both the face value and the future interest payments are calculated and added together. Interest payments can be calculated using the present value of an ordinary annuity. The face value of the bond itself can be calculated using present value of a single sum. For example, the town of Blue issues $10,000,000, 30-year bonds, which pay out annual interest (once per year) at a stated rate of 8%. The market rate of interest is 10%. r is the market interest rate, c is the coupon rate on the bond, t is the bond term, and F is the face value of the bond. In this case, r is 10%, c is 8%, t is 30, and F is$10,000,000.
${\rm{PV}}=\frac{\rm{PMT}}i\ \times\left(1-\frac1{\left(1+i\right)^n}\right)$
The first step is to solve for the PV of the coupon payments.
\begin{aligned}\rm{PV}&=\frac{\800\rm{,}000}{0.1}\ \times\left(1-\frac1{\left(1+{0.1}\right)^{30}}\right)\\\rm{PV}&=\frac{\800\rm{,}000}{0.1}\times\left(1-\frac1{\left(17.4494\right)}\right)\\\rm{PV}&=\8\rm{,}000\rm{,}000\times\rm 0.94269\\\rm{PV}&=\7\rm{,}541\rm{,}532\end{aligned}
The next step is to solve for the PV of the face value of the bond. To find the present value of the face value of the bond, use the lump sum time value of money formula. F is \$10,000,000, r is 10%, and n is 30.
${\rm{PV}}\;{\rm{of}}\;{\rm{face}}\;{\rm{value}}\;{\rm{of}}\;{\rm{the}}\;{\rm{bond}}=\frac{F}{(1+r)^{30}}$
Solve for the PV of the face value of the bond in the following steps.
\begin{aligned}\rm{PV}&=\10\rm{,}000\rm{,}000/(1+0.10)^{30}\\\rm{PV}&=\10\rm{,}000\rm{,}000/(1.10)^{30}\\\rm{PV}&=\10\rm{,}000\rm{,}000/17.4494022\\\rm{PV}&=\573\rm{,}086\end{aligned}
Finally, the PV of the face value and PV of the coupon payments are added together to arrive at the bond price: $\7\rm{,}541\rm{,}532 + \573\rm{,}086 = \8\rm{,}114\rm{,}618$. This bond is being sold at a discount, as its coupon rate is less than the market rate.