I was asked this in an interview.
The correct answer, I was told, follow from this argument
Let $L_0[0,t]$ denote the time 0 price of Libor for period $0$ to $t$.
Let $L_0[t,t+\delta_t]$ denote the time 0 price of Libor for period $t$ to $t+\delta t$
$(1+L_0[0,t])^{-1}$ dollar today is worth $1$ dollar at time $t$ hence $1+L_0[t,t+\delta_t]$ dollar at time $t+\delta t$
So the correct answer should be $(1+L_0[0,t])^{-1}-(1+L_0[0,t+\delta t])^{-1}$ by discounting the extra 1 dollar back to time 0.
I understand this, but I thought
$1$ dollar today is worth $1+L_0[0,t]$ dollar at time $t$, hence $(1+L_0[0,t])(1+L_0[t,t+\delta_t])$ dollar at time $t+\delta t$
That should be the same as investing 1 dollar for period 0 to $t+\delta t$. This should be $1+L_0[0,t+\delta t]$. Equating these give a different answer.
$(1+L_0[0,t])(1+L_0[t,t+\delta_t])= 1+L_0[0,t+\delta t]$
This gives $L_0[t,t+\delta_t] = (1+L_0[0,t+\delta t])(1+L_0[0,t])^{-1} - 1$
This is out by a factor of $1+L_0[0,t+\delta t]$. why is that the case? what did I miss?