  We systematically study several versions of the disjunction and the existence
properties in modal arithmetic. First, we newly introduce three classes
$\mathrm{B}$, $\Delta(\mathrm{B})$, and $\Sigma(\mathrm{B})$ of formulas of
modal arithmetic, and study basic properties of them. Then, we prove several
implications between the properties. In particular, among other things, we
prove that for any consistent recursively enumerable extension $T$ of
$\mathbf{PA}(\mathbf{K})$ with $T \nvdash \Box \bot$, the
$\Sigma(\mathrm{B})$-disjunction property, the $\Sigma(\mathrm{B})$-existence
property, and the $\mathrm{B}$-existence property are pairwise equivalent.
Moreover, we introduce the notion of the $\Sigma(\mathrm{B})$-soundness of
theories, and prove that for any consistent recursively enumerable extension of
$\mathbf{PA}(\mathbf{K4})$, the modal disjunction property is equivalent to the
$\Sigma(\mathrm{B})$-soundness.
