Say I want to lease a car. Dealer says $350/month for 3-years.
The car is $23,000 if buying in cash.
Do I only need credit approval for $12,600 ($350 x 36-months), or for $23,000 ?
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Credit approval likely reflects the monthly payment more than the total amount borrowed. The bank will look at your ability to make that payment based on your salary, your other monthly obligations, (mortgage, credit card debt, other car payments, etc.) and they will price any risk if you are less likely to be able to make that payment.
You are doing an apples to oranges comparison when you look at the total amount paid over 3 years on a lease verses 5 years on a loan. If you borrow $23,000 on a 5 year note at 3%, with a monthly payment of $413 that uses a 5 year payoff (compared to your 3 year lease). A 5 year, 3% loan with a $350 payment would reflect a $19,500 loan. That's not that much lower than the $23,000 full purchase loan, except after 5 years you own the car outright, compared to the lease where you own nothing after three years.
So to try to answer your question, you would need to be approved for the equivalent of a $19,500 loan if you did the lease.