Buying a home vs. Stocks for appreciation.

So we currently own and live in a townhomr with roughly $100k equity. We loved apartment style living and would love to go back to it. We have roughly 100k also saved up for another down-payment.

One option is to look for an apt and rent out the townhome with property management company. We likely make a few hundred per month taking into account PM fees, repairs, unoccupied months etc. I will mention the home is in a high demand area. We invest the 100k or so amount that would have been used in a down-payment in some type of index fund.

Other option is to keep the townhome and purchase another home that is more to our liking. Make the 20% down-payment and live in the new home while continuing to rent out old townhome.

Purely from a financial perspective, does it boil down to the stocks outperforming the 2nd home appreciation over 10 or 15 years? Assuming, home costs with taxes etc month to month same as apt. I am missing some major variables here, what are they?