How much higher of a intrest rate should I expect to pay with a 10% down and not a 20%
Good question! Unfortunately, the answer is a good deal more complicated than it used to be.
Until recently, if you applied for a conventional/conforming loan and the automated underwriting program gave a recommendation of "Approve/Eligible" for Fannie Mae Loans or "Accept" for Freddie Mac loans, there would be no difference in the rate for a loan with a 90% LTV versus one with 80% LTV. The only exception was for cash-out refinances where the LTV could affect the rate. In those days, the only difference between a loan at 90% LTV and one with 80% LTV was that the former required (and still requires) private mortgage insurance.
Nowadays, everything has changed. Just about every lender I have surveyed recently has come up with a matrix describing how much they will increase their base rate depending on the loan-to-value ratio (LTV) and the borrower's credit score. If your score is high enough, your rate might not get increased at all. If your score is below the benchmark the lender sets for eligibility to get the base rate, it all depends on how much lower the score is -- anywhere from .125% to 1% or more.
All of the above relates to conventional/conforming loans for which the automated underwriting system gives a recommendation of "Approve/Eligible" or "Accept." The automated underwriting system could give a less favorable underwriting recommendation. In that case, there might be no rate increase -- because the lender would not be willing to make the loan!
Let me explain, both Fannie Mae and Freddie Mac have automated underwriting systems. These automated underwriting systems -- called Desktop Underwriter and Loan Prospector, respectively -- are complicated computer programs that evaluate the likelihood of default for a given loan. The programs take into account credit score, length of employment, the loan-to-value ratio, the amount of money the borrower will have left in the bank after closing (for purchases) and their debt-to-income ratio, among other things. All of these things get compared to default rates for loans made in the past to people with similar profiles. If the borrower's profile meets a certain benchmark, the program will give an "Approve/Eligible" or "Accept" recommendation. Those are the loans for which you get the lender's best offered rate. For loans where the borrower's profile falls beneath that benchmark, the program assigns a risk grade. For Fannie Mae loans these risk grades are "Expanded Approval Level I," "Expanded Approval Level II," and "Expanded Approval Level III." The rates associated with those risk grades could be anywhere from 1% to 2% higher than those for associated with an "Approve/Eligible" recommendation. In addition, the private mortgage insurance rates are much higher for expanded approvals.
For a particular borrower, a change in loan-to-value ratio, i.e. from 80% to 90% could affect the recommendation given by the automated underwriting program. Thus, if you get an "Approve/Eligible" underwriting recommendation and later decide that you only want to put 10% down instead of 20%, you might get an expanded approval. In that case, your rate would go up as described above. Some lenders, however, will not make loans to borrowers whose application receives an automatic underwriting recomendation other than "Approve/Eligible" or "Accept" -- in which case the increase in loan-to-value isn't a matter of how much the rate will go up, it's a question of getting the loan in the first place.