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The down payment is for the lender's protection, not yours. You might have repair costs or emergencies. Keep the cash.
Eeeee-YUP.
"Keep your powder dry." Rates are good enough, unless you are swimming in cash, keeping a healthy reserve is smart.
You do need to calc that the monthly PITI is acceptable to your budget.
To the original question:Better to put a down payment or use money to buy down rate?
All the terms plus how long you have the loan will be needed to answer the question.
But, since this is your first loan and you will likely have many similar questions over your life, I encourage you to understand how these loans work so you can answer them yourself.
Use your favorite spreadsheet program and lay out each of the two proposed loans side by side and you will see when one loan becomes more favorable than the other. What you are measuring is principal balance. The pay down of the balance with the higher rate will start out better but the other loan will have more of its payment go to principal so at some point, it will overtake it. Of course, it is unlikely the payments will be identical so it is assumed for comparison purposes, the lower payment loan has additional principal to make the payments equal. Of course, the borrower can do whatever he wants with that spread.
So, here you go, which is better, both 15 year fixed: 3.5% $152,500 or 3.75% $150,000? In this case, the lower down (3.75% $150,000) is better all the way until the very end of the loan. If you were to keep the loan for the entire term, the lower interest loan would have saved you a few dollars. Probably not worth it because the likelihood of holding the loan to term is low.
Tax considerations omitted for simplicity.
The key is find where that crossover is for you and see if that makes sense.