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They don't look at the value, they look at the purchase price, most of the time.
The times when they don't are when you are buying from family as a family gifted deposit purchase or when buying from landlord or council right to buy.
Commercial properties or renovations and self builds have slightly different rules, for mainstream residential, most lenders will go by the above rules.
If you are buying under value and not getting a gifted deposit, then Getting the property cheaper doesn't matter, though some lenders will refuse if they decide that it is a 'distressed' sale.
Assuming we wanted to buy a property from someone who agreed to sell it to us at less than market value would banks be shitty about that from a mortgage perspective?
Say value is 300k and prospective sale price is 240k. Say we had 40k savings plus stamp duty/legal costs and wanted a mortgage of 200k. Would we be OK getting that full 200 or would they question it because of the market value of the property?