I want to prove Determine the coupon rate $r$, such that the price of the bond, at $T_0$, equals its face value

Question

Consider a coupon bond, starting at $T_{0}$ , with face value $K$, coupon payments at $T_1, . . . , T_n$ and a fixed coupon rate $r$. Determine the coupon rate $r$, such that the price of the bond, at $T_0$, equals its face value.

Answer 1

For simplicity,we let \begin{align} &\delta=\frac{T_n-T_0}{n}\\ &T_i=T_0+i\delta, \end{align} for $i=1,2,...,n$ we have $$c_i=r\,\delta\,K.$$ The price, $p(t)$ at a time $t < T_1$, of the coupon bond is given by $$p(t)=KP(t,T_{n})+\sum_{i=1}^{n}c_i P(t,T_{i}),$$ we know the price of the bond, at $T_0$, equals its face value,thus we have $$p(T_0)=K=KP(T_0,T_{n})+r\,\delta\,K\sum_{i=1}^{n} P(T_0,T_{i}),$$ then $$r=\frac{1-P(T_0,T_{n})}{\delta\sum_{i=1}^{n} P(T_0,T_{i})}.$$ For more details, you can see this link

Answer 2

Are you familiar with the concept of yield-to-maturity (YTM)? Here you find all necessary steps. You first calculate using the current price and the cashflows. Then as you can see in the paper provided a bond with coupon rate equal to its YTM is priced at par (100) and thus the price equals its face value.

Answer 3

I always thought doing $y(t,T) = \frac{-ln(P(t,T))}{T-t}$ was a quick good approximation, it applies when the bond price is calculated in continuous time