We derive the coefficient equation for the first-order orbital angular momentum-polarization mode dispersion (OAM-PMD) of OAM mode in a ring fiber by using the OAM-PMD dynamic equation and the fixed birefringence cascaded model. We also numerically calculate the influences of geometrical birefringence on the effective refractive index difference between the odd and even modes that constitute the OAM mode and calculate the variation of first-order OAM-PMD coefficient with the angular frequency under different ellipticities of ring fibers. The results show that the first-order OAM-PMD coefficient is determined not just by the effective refractive index difference between the odd and even modes that constitute the OAM mode, but also by its variation with the angular frequency. We find that the ellipticity of the ring fiber increases the first-order OAM-PMD coefficient, particularly for relative lower-order OAM modes, and thus severely limit the property and propagation distance of the OAM modes.