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I'm currently unemployed and I'm coming off of a 3 year lease soon. My stepfather has agreed to cosign and pay for a car for me until I get employed again. Does it make any sense to have my name even on the lease? Will this help or hurt my fico score. It dropped significantly when I unfortunately was 30 days late with a credit card recently
Generally speaking any positive tradeline will help your credit report.
Not certain why you wouldn't want your name on the loan personally assuming you are able to make the payments. Trying to get fancy in this case can only hurt you and your father potentially, just do the standard cosign arrangement and go on with life.
I thought perhaps having a car loan would be seen as an additional expense and hurt my credit. I carry a significant balance on my credit card , yes I know that is bad but I can't afford to pay it off now that I'm unemployed. When I get a job, paying down my CC debt will be a prority (after 6-8 months of expenses are saved) As far as getting my stepfather in trouble, it would never happen, he would continue to pay the lease as he has stellar credit (I think his score is 770) and he wouldn't do anything to screw that up.
A car loan is an installment loan. The hard pull and the new tradeline will both hurt a bit, for example by lowering your AAoA.
In the long run, having such a tradeline on your reports is probably a net gain, for example in terms of "mix of credit types" for scoring purposes.
However, if you have trouble making the monthly payment, and your DTI is made worse, then that's another issue. These issues do not directly affect your score, unless they lead you to make late payments or default. A lender may take action if they are concerned about this.