  The rare-earth intermetallic compound Ho$_5$Sn$_3$ demonstrates fascinating
magnetic properties which include temperature-driven multiple magnetic
transitions and field driven metamagnetism. We address the magnetic character
of this exciting compound through a combined experimental and theoretical
studies. Ho$_5$Sn$_3$ orders antiferromagnetically below $~28$ K, and shows
further spin reorientation transitions at 15 K and 12 K. We observe a sizable
amount of low-temperature magnetocaloric effect in Ho$_5$Sn$_3$ with a maximum
value of entropy change $\Delta S$ = -9.5 JKg$^{-1}$K$^{-1}$ for an applied
field of $H$ = 50 kOe at around 30 K. The field hysteresis is almost zero above
15 K where magneto-caloric effect is important. Interestingly, $\Delta S$ is
found to change its sign from positive to negative as the temperature is
increased above about 8 K, which can be linked to the multiple spin
reorientation transitions. The signature of the metamagnetism is visible in the
$\Delta S$ versus $H$ plot. The magnetic ground-state, obtained from the
density functional theory based calculation, is susceptible to the effective
Coulomb interaction ($U_{\rm eff}$) between electrons. Depending upon the value
of $U_{\rm eff}$, the ground-state can be ferromagnetic or antiferromagnetic.
The compound shows large relaxation (14\% change in magnetization in 60 min) in
the field cooled state with a logarithmic time variation, which may be
connected to the competing magnetic ground-states observed in our theoretical
calculations. The competing magnetic ground-states is also evident from the
small value of the paramagnetic Curie-Weiss temperature.
