  We derive the asymptotic solutions for vacuum spacetimes with non-zero
cosmological constant $\Lambda$, using the Newman-Penrose formalism. Our
approach is based exclusively on the physical spacetime, i.e. we do not
explicitly deal with conformal rescaling nor the conformal spacetime. By
investigating the Schwarzschild-de Sitter spacetime in spherical coordinates,
we subsequently stipulate the fall-offs of the null tetrad and spin
coefficients for asymptotically de Sitter spacetimes such that the terms which
would give rise to the Bondi mass-loss due to energy carried by gravitational
radiation (i.e. involving $\sigma^o$) must be non-zero. After solving the
vacuum Newman-Penrose equations asymptotically, we propose a generalisation to
the Bondi mass involving $\Lambda$ and obtain a positive-definite mass-loss
formula by integrating the Bianchi identity involving $D'\Psi_2$ over a compact
2-surface on $\mathcal{I}$. Whilst our original intention was to study
asymptotically de Sitter spacetimes, the use of spherical coordinates implies
that this readily applies for $\Lambda<0$, and yields exactly the known
asymptotically flat spacetimes when $\Lambda=0$. In other words, our asymptotic
vacuum solutions with $\Lambda\neq0$ reduce smoothly to those where
$\Lambda=0$, in spite of the distinct characters of $\mathcal{I}$ being
spacelike, timelike and null for de Sitter, anti-de Sitter and Minkowski,
respectively. Unlike for $\Lambda=0$ where no incoming radiation corresponds to
setting $\Psi^o_0=0$ on some initial null hypersurface, for $\Lambda\neq0$, no
incoming radiation requires $\Psi^o_0=0$ everywhere.
