We present an analytical solution of the Bloch-McConnell equations for the case of chemical exchange saturation transfer between hyperpolarized nuclei in cavities and in solvent (HyperCEST experiment). This allows quantitative investigation of host-guest interactions by means of nuclear magnetic resonance spectroscopy and, due to the strong HyperCEST signal enhancement, even NMR imaging. Hosts of interest can be hydrophobic cavities in macromolecules or artificial cages like cryptophane-A which was proposed as a targeted biosensor. Relevant system parameters as exchange rate and host concentration can be obtained from the monoexponential depolarization process which is shown to be governed by the smallest eigenvalue in modulus. For this dominant eigenvalue we present a useful approximation leading to the depolarization rate for the case of on- and off-resonant irradiation. It is shown that this rate is a generalization of the longitudinal relaxation rate in the rotating frame. We demonstrate for the free and cryptophane-A-encapsulated xenon system, by comparison with numerical simulations, that HyperCEST experiments are precisely described in the valid range of this widely applicable analytical approximation. Altogether, the proposed analytical solution allows optimization and quantitative analysis of HyperCEST experiments but also characterization and optimal design of possible biosensors.