# Mismatch of periods with numeraire compared to the forward rates

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 $$j at time $$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?