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If I get an Islamic mortgage for 200k, the bank might charge 210k, with the 10k being the profit they would take instead of interest. The monthly repayments for this mortgage will be split in two - one portion of it goes as rent, and the other portion is going towards buying more share of the house.

In a conventional mortgage, you would repay the bank for however much they have lent you towards the house plus interest.

So my question is, where is the rent portion of the repayment going? Is it going towards paying off the 200k that the bank used to buy the house? Why isn't rent counted towards me owning the share of the house? Or should I think of the rent as literally like rent which I would pay to my landlord if I rented a property?

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It depends as there are at least five varieties of Islamic home financing: Murabahah, Bay Bithaman Ajil, Musharakah Mutanaqisah, Ijarah Muntahiya Bit Tamlik, and Istisna (new constructions only). Most common (in US anyway) is probably Murabahah.

Assuming we're talking about Murabahah, what's happening is the bank acquires the property (directly or though a special purpose entity) on behalf of the customer and then resells the property to the customer at a mark up on an installment basis (the customer grants a security interest to the bank). That installment payment is not rent in the usual sense but is going towards paying down the obligation. The obligation ends once the final installment payment is made (prepayment is also permitted). There's no rent here, just installment payments.

Ijarah is different in that the bank acquires the property on behalf of the borrower and instead of selling it at a markup instead leases it to the customer. The lease ends with transfer of ownership of leased assets to the lessee (the customer). The lease payment will include a rental amount and an amount that goes toward the acquisition of the asset. Some of the recurring payment is rent.

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