# How is the integral relationship between current yield curve and forward yield curve derived?

$$y(\tau) = \frac{1}{\tau} \int_{0}^{\tau} du \Big(f(u)\Big)$$ As far as I understand the forward rate is the future rate based on the expectation hypothesis. But it is unclear how many years into the future is the forward yield curve. Is it one year ahead? Is the future horizon unimportant at all and the results will always be the same?

It would therefore be good to see a derivation of this relationship.