In Joshi's The Concepts and Practice of Mathematical Finance Page 323--324 I believe that there may be a mismatch of periods with forward rates:

Consider time partition $t_{0} < ... < t_{n}$ where $f_{j}(t)$ is the forward rate for period $[t_{j},t_{j+1}]$ for any $j = 0,..., n-1$ at time $t$. Therefore a ZCB $B_{n}$ that matures at $t_{n}$ can be valued for $j<n$ at time $t_{j}$ as $$B_{n}(t_{j})=\prod\limits_{k = j}^{n-1}\frac{1}{1+f_{k}(t_{j})(t_{k+1}-t_{k})}.$$

Let $B_{j}$ denote the ZCB maturing at $t_{j}$ for $j = 1,...,n$, then at time $t_{j-1}$ we purchase $B_{j}$ and then at time $t_{j}$ we reinvest some of the proceeds of the matured bond into $B_{j+1}$ and we proceed as such across the timeline until $t_{n}$.

This is correct and makes sense to me. But then it is further stated that:

At time $t_{1}$, the Numeraire $B_{2}$ will cost $\frac{1}{1+f_{1}(t_{1})(t_{1}-t_{0})}(*)$.

In my view this is incorrect and should rather cost $\frac{1}{1+f_{1}(t_{1})(t_{2}-t_{1})}$.

I do not understand why the period $[t_{0},t_{1}]$ is being used in $(*)$ rather than $[t_{1},t_{2}]$, since the forward rate $f_{1}$ runs over $[t_{1},t_{2}]$ by definition. Have I found a mistake or misunderstood it?


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