  We investigate theoretically the $K^{0} p$ invariant mass spectrum of the
$K^{+} d \to K^{0} p p$ reaction and scrutinize how the signal of the "$\Theta
^{+}$" pentaquark, if it exists, emerges in the $K^{0} p$ spectrum. The most
prominent advantage of this reaction is that we can clearly judge whether the
"$\Theta ^{+}$" exists or not as a direct-formation production without
significant backgrounds, in contrast to other reactions such as photoproduction
and $\pi$-induced productions. We show that while the impulse or single-step
scattering process can cover the "$\Theta ^{+}$" energy region with an initial
kaon momentum $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ in the
laboratory frame, the contributions from double-step processes may have a
potential possibility to reach the "$\Theta ^{+}$" energy region with a higher
kaon momentum $k_{\mathrm{lab}} \sim 1 \, \mathrm{GeV} / c$. Assuming that the
full decay width of the "$\Theta ^{+}$" is around $0.5\, \mathrm{MeV}$, we
predict that the magnitude of the peak corresponding to the "$\Theta^+$" is
around a few hundred $\mu \mathrm{b}$ to $1\, \mathrm{mb}$ with the momentum of
the kaon beam $k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ while it is
around $\lesssim 1\, \mu \mathrm{b}$ with $k_{\mathrm{lab}} \approx 0.85\,
\mathrm{GeV} / c$. Thus, the "$\Theta^+$" peak is more likely to be seen at
$k_{\mathrm{lab}} \approx 0.40\, \mathrm{GeV} / c$ than at $k_{\mathrm{lab}}
\approx 0.85 \,\mathrm{GeV} / c$.
