You can and should shop around.
The credit system is actually set up to allow you to have your credit run several times in a short period for the exact purpose of shopping around, and all of those inquiries show up as a single one on your credit.
What to know about "rate shopping."
Looking for a mortgage or an auto loan may cause multiple lenders to request your credit report, even though you’re only looking for one loan. To compensate for this, the score ignores all mortgage and auto inquiries made in the 30 days prior to scoring. So if you find a loan within 30 days, the inquiries won't affect your score while you're rate shopping. In addition, the score looks on your credit report for auto or mortgage inquiries older than 30 days. If it finds some, it counts all those inquiries that fall in a typical shopping period as just one inquiry when determining your score. For FICO scores calculated from older versions of the scoring formula, this shopping period is any 14 day span. For FICO scores calculated from the newest versions of the scoring formula, this shopping period is any 45 day span. Each lender chooses which version of the FICO scoring formula it wants the credit reporting agency to use to calculate your FICO score.
· 1 decade ago