  We examine the neutrino and cosmic ray spectra resulting from two models of
fitting the spectral energy distribution (SED) of the blazar Mrk~421 using a
self-consistent leptohadronic code. The $\gamma$-ray emission is attributed to
either synchrotron radiation of ultra-high energy protons (LHs model) or to
synchrotron radiation from electrons that result from photopion interactions of
lower energy protons (LH$\pi$ model). Although both models succeed in fitting
satisfactorily the SED, the parameter values that they use result in
significantly different neutrino and cosmic-ray spectra. For the LH$\pi$ model,
which requires high proton energy density, we find that the neutrino spectrum
peaks at an energy $E_{\rm \nu,peak}= 3.3$ PeV which falls well within the
energy range of recent neutrino observations. While at the same time its peak
flux is just under the sensitivity limit of IC-40 observations, it cannot
produce ultra-high energy cosmic rays. In the LHs model, on the other hand,
neutrinos are far from being detectable because of their low flux and peak
energy at $E_{{\rm \nu,peak}} \simeq 100$ PeV. However, the propagation of
protons produced by the decay of escaping neutrons results in an ultra-high
energy cosmic ray flux close to that observed by Pierre Augere, HiRes and
Telescope Array at energies $E_p \simeq 30$ EeV.
