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First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$. If the coupon rate $r_i$ is set to the spot LIBOR rate $L(T_{i−1}, T_i)$ ,then \begin{align} c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K \end{align} We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate \begin{align} c_i=\frac{1}{P(T_{i−1}, T_i)}-1 \end{align} The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
2. At time $T_{i−1}$ you will receive the amount $1$.
3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly $\frac{1}{P(T_{i−1}, T_i)}$ bonds.
4. At $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon $c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then, by summing up all values of $c_i$ and the value of the notional amount (i.e. 1) at $T_n$, we have $$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})=p(t,T_0)$$

Edit for Gordon. First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$.At time $T_n$ the owner receives the face value K.We now go on to compute the price of this bond, and it is obvious that the coupon bond can be replicated by holding a portfolio of zero coupon bonds with maturities $T_i$, $i = 1,...,n$.So the price,$P(t)$, at a time $t < T_1$, of the coupon bond is given by $$P(t)=KP(t,T_{n})+\sum_{i=1}^{n}c_i P(t,T_{i})$$

If the coupon rate $r_i$ is set to the spot LIBOR rate $L(T_{i−1}, T_i)$ ,then \begin{align} c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K \end{align} We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate \begin{align} c_i=\frac{1}{P(T_{i−1}, T_i)}-1 \end{align} The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
2. At time $T_{i−1}$ you will receive the amount $1$.
3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly $\frac{1}{P(T_{i−1}, T_i)}$ bonds.
4. At $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon $c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then, by summing up all values of $c_i$ and the value of the notional amount (i.e. 1) at $T_n$, we have $$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})=p(t,T_0)$$

First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$. If the coupon rate $r_i$ is set to the spot LIBOR rate $L(T_{i−1}, T_i)$ ,then \begin{align} c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K \end{align} We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate \begin{align} c_i=\frac{1}{P(T_{i−1}, T_i)}-1 \end{align} The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
2. At time $T_{i−1}$ you will receive the amount $1$.
3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly $\frac{1}{P(T_{i−1}, T_i)}$ bonds.
4. At $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon $c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then $$P(t)=KP(t,T_{n})+\sum_{i=1}^{n}c_i P(t,T_i)$$ $K=1$ and the value $P(t,T_i)$ at $t$ equals $1$ and we prove $c_i=P(t, T_{i−1})−P(t, T_i)$, thus we have $$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})=p(t,T_0)$$

First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$. If the coupon rate $r_i$ is set to the spot LIBOR rate $L(T_{i−1}, T_i)$ ,then \begin{align} c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K \end{align} We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate \begin{align} c_i=\frac{1}{P(T_{i−1}, T_i)}-1 \end{align} The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
2. At time $T_{i−1}$ you will receive the amount $1$.
3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly $\frac{1}{P(T_{i−1}, T_i)}$ bonds.
4. At $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon $c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then, by summing up all values of $c_i$ and the value of the notional amount (i.e. 1) at $T_n$, we have $$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})=p(t,T_0)$$

First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$. If the coupon rate $r_i$ is set to the spot LIBOR rate $L(T_{i−1}, T_i)$ ,then \begin{align} c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K \end{align} We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate \begin{align} c_i=\frac{1}{P(T_{i−1}, T_i)}-1 \end{align} The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
2. At time $T_{i−1}$ you will receive the amount $1$.
3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly $\frac{1}{P(T_{i−1}, T_i)}$ bonds.
4. At $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon $c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then $$P(t)=KP(t,T_{n})+\sum_{i=1}^{n}c_i P(t,T_i)$$ $K=1$ and the value $P(t,T_i)$ at $t$ equals $1$ and we prove $c_i=P(t, T_{i−1})−P(t, T_i)$, thus we have $$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})$$

First, fix point in time $T_0,...,T_n$ whereas $T_1,...,T_n$ are the coupon dates and $T_0$ is interpreted as the emission date of the bond. At time $T_i$, $i = 1,...,n$ the owner of the bond receives $c_i$. If the coupon rate $r_i$ is set to the spot LIBOR rate $L(T_{i−1}, T_i)$ ,then \begin{align} c_i=(T_i-T_{i−1})L(T_{i−1}, T_i)K \end{align} We now go on to compute the value of this bond at some time $t < T_0$, in the case when the coupon dates are equally spaced, with $T_i−T_{i−1}=\delta$, and to this end we study the individual coupon $c_i$. Without loss of generality we may assume that $K = 1$, and inserting the definition of the LIBOR rate \begin{align} c_i=\frac{1}{P(T_{i−1}, T_i)}-1 \end{align} The value at $t$, of the term $−1$ , is of course equal to $-P(t, T_i)$ and it remains to compute the value of the term$\frac{1}{P(T_{i−1}, T_i)}$ which is paid out at $T_i$.This is, however, easily done through the following argument.

1. Buy, at time $t$, one $T_{i−1}$-bond. This will cost $P(t, T_{i−1})$.
2. At time $T_{i−1}$ you will receive the amount $1$.
3. Invest this unit amount in $T_{i−1}$-bond. This will give you exactly $\frac{1}{P(T_{i−1}, T_i)}$ bonds.
4. At $\,T_i$ the bonds will mature, each at the face value $1$. Thus, at time $T_i$, you will obtain the amount $\frac{1}{P(T_{i−1}, T_i)}$

Thus the value at $t$, of obtaining $\frac{1}{P(T_{i−1}, T_i)}$ at $T_i$, is given by $P(t, T_{i-1})$, and the value at t of the coupon $c_i$ is $P(t, T_{i−1}) − P(t, T_i)$. then $$P(t)=KP(t,T_{n})+\sum_{i=1}^{n}c_i P(t,T_i)$$ $K=1$ and the value $P(t,T_i)$ at $t$ equals $1$ and we prove $c_i=P(t, T_{i−1})−P(t, T_i)$, thus we have $$P(t)=P(t,T_{n})+\sum_{i=1}^{n}P(t,T_{i-1})-P(t,T_{i})=p(t,T_0)$$