forward vs spot simply-compounded spot interest rate

Question

Question about forward vs spot simply-compounded spot interest rate.Some definitions

• $P(a,b)$ a zero coupond price at time $a$ and maturity $b$
• $L(a,b)$ simply compounded spot interest rate set at time $a$ and payement at time $b$
• $F(t,a,b)$ simply compounded forward interest rate at time $t$, set at time $a$ and payement at time $b$

For a timeline $t<T<S$, we have formula for spot rate \begin{equation} L(t,T) = \frac{1}{T-t}[\frac{1}{P(t,T)}-1] \hspace{1cm} L(T,S) = \frac{1}{S-T}[\frac{1}{P(T,S)}-1] \end{equation} By arbitrage free hypothesis, we have the zero coupond relation $P(t,S) = P(t,T)P(T,S)$, hence \begin{equation} P(t,S)(S-T)L(T,S) = P(t,T) - P(t,S) \end{equation} On the other hand, we have the formula for forward interest rate \begin{equation} F(t,T,S)=\frac{1}{S-T}[\frac{P(t,T)}{P(t,S)}-1] \leftrightarrow P(t,S)(S-T)F(t,T,S) = P(t,T) - P(t,S) \end{equation} The idea is $F(t,T,S)$ has to be a forward rate at time $t$ of the spot rate $L(T,S)$, but what i've proved, they are equal. I see that is not logic, but i can not see what was wrong in my formula.

Can someone help me please?

Thanks

Answer 1

The flaw is $L(T,S)$ is a future spot rate that is determined at time $T>t$ and unknown at present.

It is correct that

$$F(t,T,S)=\frac{1}{S-T}\left[\frac{P(t,T)}{P(t,S)}-1\right] \iff P(t,S)(S-T)F(t,T,S) = P(t,T) - P(t,S), $$

as this is just the definition of the forward rate.

However, you are saying that

$$\frac1{P(t,T)}\frac1{P(T,S)}=\frac1{P(t,S)},$$

where

$$P(T,S)=\frac1{1+L(T,S)(S-T)}$$

The zero-coupon bonds maturing at $T$ and $S$ have known prices at present. You don't know that buying the $T$-maturity bond today and rolling into a new $S-T$-maturity zero-coupon bond at future time $T$ will produce the same return as buying an $S$-maturity bond today.