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?



1 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




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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.