My fiance owns a nice piece of property with an old run down home. The value of land and house right now is about 250,000. There is no mortgage on this property.
How do I borrow money to demolish the old house and build a new one?
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Shop around various local banks for a construction loan. It's essentially a line of credit that will be collateralized by the new house as it's built. Part of that loan will need to go towards the demolition of the old house (which won't be much compared to the cost of the new home) and the clearing of the lot.
One problem you might have is that your new mortgage will need to be 80% or less of the market value of the new home (to avoid PMI), so depending on how much it costs you to build the house, how much the land is worth, and how much demolition costs, you might still need to provide some cash at closing (or pay for the demo upfront) in order to stay under that 80% threshold.