Question – Montgomery/Harris Country Prop Tax Asssement on Home Purchase

Hey y'all

Asking out here as I can't find a straight answer elsewhere… My wife and I are casually looking at houses in Harris and Montgomery counties around $1.2M range. Looking at the public tax record for these homes, many of these (esp. in Montgomery) have assessments around $600-800k. This is significantly lower than the last sale price of these properties and ridiculously lower than the market value and doesn't make sense.

I am aware of the homestead exemption and the 10% annual cap, still this doesn't compute. How does a home sold in 2018 ($850k) have an assessed value of $675k and is on the market for $1.15M?

If we purchase one of these properties, will the tax assessment automatically increase to the sale price ( >$35k/yr in Prop tax in some areas?!?) after the first assessment year?