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I think the consideration of renting is not purely in the hope of improving his credit score. They are now finding living with his parents more difficult, when you have no choice you tend not to think of any other options, when you do have options, it starts to make how you are living look unbearable. He had no income, so they were stuck. Now he does have an income.
I fully agree about being able to save if they stay where they are, and as far as advising them, I am torn between saying please stay where you are, and yes I understand you want to be independent. In fact, saving for a deposit is now no longer an issue, although of course they are going to need to furnish a house when they eventually get it. We are going to put the deposit amount into a savings bond, which allows them a 100% mortgage. The interest rates on both mortgage and savings are very reasonable, as far as the savings are concerned better than available anywhere else.
They went to the bank, expecting to be in a position to apply for this mortgage fairly soon, once the affordability issue was settled, which it was. But of course, met with the problem of his almost non existent credit record. I know people think that living with parents is the very best option anyone can have. If you've ever done it you know it isn't. We did it 50+ years ago and after 6 months were climbing the walls. And my parents were very reasonable and accommodating!!
So far as ideas of how to improve the credit score go, the credit card, used responsibly seems to be the best one, given that they are doing everything else. And currently, he doesn't yet have the information from the credit reference agency. I think when they get that they'll be in a better position to know how much improvement is needed, and how long this is likely to take.
They have been given a lot of advice about mortgage brokers also. I am in two minds myself about them. Either you have to pay them, or you have to rely on them to get the best deal for you, rather than the one that pays them the most commission. We have used one in the past, but we knew him from school, so had rather more trust in him than in someone we didn't know. Could have been misplaced but we had no option at the time as this was a buy to let mortgage which are very much more complicated to organise.