  We study the weak-value amplification (WVA) in a phase measurement with an
optical interferometer in which shot noise limits the sensitivity. We compute
the signal and the shot noise including the full-order interaction terms of the
WVA, and show that the shot-noise contribution to a phase shift in a pointer
variable is always larger than the final variance of the pointer variable. This
yields difference in estimating noise level up to a factor of 1.5. To clarify
an advantage for practical uses of the WVA, we discuss signal-to-noise ratio
and its optimization in the presence of the shot noise.
