How would you convert PMI to an APR?

[Frugalicious]

How would you go about converting the dollar amount of PMI into an APR?  Let's say you have a house and the balance on the mortgage is $200k.  The PMI on the house is $95 a month.  What is the math to convert?

Is it as simple as (95X12)/200,000?  Or am I missing something?

Thanks.  For some reason this feels like it shouldn't be that hard, but I'm doubting my approach.

[Hotstreak]

That would be accurate for your first payment.  Since PMI doesn't decrease as your loan balance decreases, you will find a higher % after each payment.


I'm curious what the practical application of this would be.  Do you have the choice between a higher APR loan, and one with PMI?

[Frugalicious]

The application is to figure out which mortgage is costing the borrower more money.

Assuming, loan 1's balance is $80k, with an interest rate of 6.75% (yikes!), but no PMI.

Loan 2's balance is 200k, with an interest rate of 4.375%, PMI of $95 per month.

The borrower was attempting to pay down loan 2 to get rid of PMI, but the effective rate is lower on loan 2, even with PMI.

That's what I was trying to get at.

[Hotstreak]

Okay, that makes sense.  Can you take the calculated APR's, including PMI, for each month until 80% LTV and average those?  That would be a great comparison to the other fixed rate loan with no PMI.

[Frugalicious]

The only issue is that the PMI amount seems to reset yearly.  But, there's enough of a difference in rates between the two to make a convincing case for focusing on loan 1 (if excess cash MUST be sent to a mortgage, that is). 

[retired?]

This is similar to the Q how much does an HOA fee cost?  What is the buying power that it costs.

My answer, in Excel format, would be:

mortgage rate is r
PMI is i
Principle is P
Number of months is N

The mortgage payment for the loan only is pmt(r/12,N*12,-P) = K

You are actually paying K + i

So, to determine the effective rate, you want to solve for R such that

K + i = pmt(R/12,N*12, -P)

It's a useful exercise b/c it answers the question "by not putting down 20% to avoid PMI, I am effectively paying R rather than r".

The twist that makes it a little harder is that PMI will disappear as soon as you reach 20%+ equity (not automatically, you need to contact the bank and then cancel).  If you are interested, I can provide some formula for that.  No PMI would be reached within 5-7 years.  In this case you have:

Principle = sum over time with PMI (discount K+i at rate R) + sum over time after PMI is not required(discount K at rate r)