  Vaidman pointed out the importance of modular values, and related the modular
value of a Pauli spin operator to its weak value for specific coupling
strengths [Phys. Rev. Lett. 105, 230401 (2010)]. It would be useful if this
relationship is generalized since a modular value, which assumes a finite
strength of the measurement interaction, is sometimes more practical than a
weak value, which assumes an infinitesimally small interaction. In this paper,
we give a general expression that relates the weak value and the modular value
of an arbitrary observable in the 2-dimensional Hilbert space for an arbitrary
coupling strength. Using this expression, we show the "failure of sum rule" for
modular values, which has a resemblance to the "failure of product rule" for
weak values. We give examples of "failure of sum rule" for some interesting
cases, i.e., paradoxes based on nonlocality, which include EPR paradox, Hardy's
paradox, and Cheshire cat experiment.
