Mortgage prepay math with PMI

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Topic Author
allanj37
Posts: 7
Joined: Mon Feb 23, 2009 10:29 am

Mortgage prepay math with PMI

Post by allanj37 »

I know that prepaying a mortgage can be a controversial topic, but I'd like some help with checking my math so I can make the decision a little easier.

We just moved to a new house and put 5% down (so our loan includes a PMI payment). We're about to put our old house on the market, and we have a good chunk of equity built up there. My thought is that after the house sells, we will put enough money into the new house to bring our equity up to 20% — thereby getting rid of our PMI payment.

Loan details:
$261,100 principal
4.125% interest
$80.51 monthly PMI payment
$41,180 additional payment required to bring the loan up to 20% equity

So the standard math is that a prepayment of 41k would be "earning" our 4% loan interest rate. The rate should actually be a lot better than that though since it would also get rid of our PMI payment.

So my quick estimation was that (80.51 * 12) / 41,180 = 2.346% yearly interest. Added on to our loan's interest, the 41k prepayment would be "earning" 6.471% (for the first 10 years 94 months we're scheduled to pay PMI; then back to 4.125% for the remainder of the life of our loan).

Does that sound correct? Or at least ballpark range?
Last edited by allanj37 on Wed May 24, 2017 12:28 pm, edited 1 time in total.
alex_686
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Joined: Mon Feb 09, 2015 1:39 pm

Re: Mortgage prepay math with PMI

Post by alex_686 »

Math is your friend, and a HP BA II Plus calculator is cool. So here are the numbers.

Step 1
PV = 261,000
FV = 0
I = 4.125% / 12 = .34375%
N = 360

CPT PMT = -1264.93

Step 2
One can remove PMI at 80%, so when the loan hits $219,789

PV = -261,000
FV = -219,789
PMT = -1264.93
I = 4.125% / 12 = .34375%

CPT N = 94 Months

Step 3, calculate the effective interest.

PV = -261,000
FV = -219,789
N = 94 Months
PMT = -1264.93 - 80.51 PMI = -1,345.44

CPT I = .37689 monthly interest, or 4.5264% annually.

So, no, not 6.471%. After this you want to consider taxes which will knock down those numbers somewhat. I hope this helps.
Former brokerage operations & mutual fund accountant. I hate risk, which is why I study and embrace it.
Topic Author
allanj37
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Joined: Mon Feb 23, 2009 10:29 am

Re: Mortgage prepay math with PMI

Post by allanj37 »

I think I mostly follow what you did, but I'm not sure that's the same calculation I'm wanting to do. I think the difference is that you're calculating the PMI payment as a percentage of the overall loan amount (and I'm doing it based on the extra one-time payment).

The way I was thinking of it is:
I have $41k to invest. If I put it into my mortgage, it eliminates PMI. So I'm investing $41,180 to save $966.12 a year on PMI, which is a 2.346% yearly return. Plus the normal 4.125% I would have been paying on that amount.
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#Cruncher
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Re: Mortgage prepay math with PMI

Post by #Cruncher »

allanj37 in original post wrote:So my quick estimation was that (80.51 * 12) / 41,180 = 2.346% yearly interest. Added on to our loan's [4.125%] interest, the 41k prepayment would be "earning" 6.471% (for the first 10 years we're scheduled to pay PMI ...). Does that sound correct? Or at least ballpark range?
Yes, it's in the ballpark. I get a somewhat lower figure of 6.14%, calculated as follows:
  • With normal payments it would take about 94 months [ * ] to work the loan down to 80% of value so that Private Mortgage Insurance (PMI) would no longer be required.
  • At 4.125%, $41,180 would grow to $56,856 over 94 months.

    Code: Select all

    56,856 = 41180 * (1 + 4.125% / 12) ^ 94
  • This along with the $80.51 monthly PMI savings represents a 6.14% annual return as calculated with the Excel RATE function:

    Code: Select all

    6.14% = 12 * RATE(94, 80.51, -41180, 56856, 0)
alex_686 wrote:Step 3, calculate the effective interest.
PV = -261,000
FV = -219,789
N = 94 Months
PMT = -1264.93 - 80.51 PMI = -1,345.44
CPT I = .37689 monthly interest, or 4.5264% annually.
I agree with the original poster, alex, that you don't seem to be calculating the return on his contemplated $41,000 prepayment. Rather you seem to be calculating the overall loan rate including PMI.

* I don't know why you say you're scheduled to pay PMI for "10 years", Allan. alex_686's estimate of 94 months seems much closer:
Step 2
One can remove PMI at 80%, so when the loan hits $219,789
PV = -261,000
FV = -219,789 [ = 261000 / 95% * 80% ]
PMT = -1264.93
I = 4.125% / 12 = .34375%
CPT N = 94 Months
alex_686
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Re: Mortgage prepay math with PMI

Post by alex_686 »

I strongly believe that my calculations are correct. You are mixing up assets and liabilities.

When doing calculations like this you want to focus on 2 things.
First you want to compare apples to apples.
Second you want to look at your portfolio as a single unit. This allows you to evaluate the opportunity costs.

One more set of calculations. What would you monthly mortgage payment be? In order to compare apples to apples we must assume that the difference from the lower mortgage payment will be saved and invested.

PV = 261,000 - 41,180 = $219,820
N = 360 Months
FV = $0
I = 4.125 / 12 = .34375% monthly

CPT PMT = 1063.35

So $1,264.93 - $1,063.35 = $199.57.

Now we are read to compare:

Universe #1
Home = $275,000, returning ?%
Negative mortgage = -$261,000 at 4.5264%
Liquid assets, $41,180 at ?%

Universe #2
Home = $275,000, returning ?%
Negative mortgage = -$$219,820 at 4.125%
Liquid assets, $0
Additional Cash For Investing: $199.57, @?%

Now the big question - which portfolio will generate the biggest risk adjusted return?

We can cancel out the Home asset. They will return the same under either choice. FYI, I would say about the rate of inflation but your millage may vary.

This leaves the big question of what the $41,180 will be invested in. Not bonds, it would be hard to find something that would have a guaranteed rate above 4%. Stocks carry high risks but also higher returns. Stocks also carry a contingent benefit that they can be liquidated to meet any emergency you might have.

When I was in a similar situation I chose to higher down payment. It is hard to beat 4% on a risk adjusted basis. Plus I had adequate emergency funds so the contingent benefit of holding liquid assets was low.
Former brokerage operations & mutual fund accountant. I hate risk, which is why I study and embrace it.
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Abe
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Re: Mortgage prepay math with PMI

Post by Abe »

OP: Your calculation looks okay to me.
Slow and steady wins the race.
Braje
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Re: Mortgage prepay math with PMI

Post by Braje »

Could depend on the lender, but I thought you were require to keep the PMI for a minimum amount of time (I think 2 years) regardless of how much you pay down.
Topic Author
allanj37
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Re: Mortgage prepay math with PMI

Post by allanj37 »

Braje wrote:Could depend on the lender, but I thought you were require to keep the PMI for a minimum amount of time (I think 2 years) regardless of how much you pay down.
Probably does depend on the lender. We went with a local credit union, and this was something we specifically asked about before getting the loan. If we pay up to 20% equity after selling our old house, they'll cancel the PMI, and they'll even re-amortize the loan for free.
Topic Author
allanj37
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Re: Mortgage prepay math with PMI

Post by allanj37 »

Alex, forget the overall loan for a moment. Just consider that I have the $41k in cash that I want to invest. If I put it into the mortgage, eliminating the PMI payment will save me $966 per year. That's a 2.35% return on my investment. Then I'm also no longer paying the 4.125% interest on that $41k.

The overall loan amount, payment terms, etc don't actually matter for these calculations.
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Nate79
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Re: Mortgage prepay math with PMI

Post by Nate79 »

allanj37 wrote:
Braje wrote:Could depend on the lender, but I thought you were require to keep the PMI for a minimum amount of time (I think 2 years) regardless of how much you pay down.
Probably does depend on the lender. We went with a local credit union, and this was something we specifically asked about before getting the loan. If we pay up to 20% equity after selling our old house, they'll cancel the PMI, and they'll even re-amortize the loan for free.
I think most lenders require an appraisal unless you get to 78% LTV to automatically remove PMI. Your credit union may be different. After a few years of paying PMI I realized the waste and decided to lump sum down to 78% LTV and my lender removed PMI automatically (this confirmed ahead of time with them how it worked). If I had only paid down to 80% I would have needed to pay for an appraisal which to me was also a waste.
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jimb_fromATL
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Re: Mortgage prepay math with PMI

Post by jimb_fromATL »

allanj37 wrote: So my quick estimation was that (80.51 * 12) / 41,180 = 2.346% yearly interest. Added on to our loan's interest, the 41k prepayment would be "earning" 6.471% (for the first 10 years 94 months we're scheduled to pay PMI; then back to 4.125% for the remainder of the life of our loan).

Does that sound correct? Or at least ballpark range?
It's in the ballpark, but you're not allowing for the fact that it's compound interest.

I think the method poster #cruncher shows is closer:

To approach it another way, let's look at the rate you'd need for the $41K to give you the same net result after 94 months in some other investment ... and verify some of the other math that some folks have mentioned, too.

Let's just use the spreadsheet library functions to do the math for us. The numbers will be slightly different because your balance will be lower by the time you make the lump sum payment; because of rouding in calculations; and because the PMI may be rounded up to an extra month.

By the way, have you verified that your lender will indeed drop PMI immediately if you pay down the balance? The low PMI rate suggests that it may be a credit union or some other loan that will allow it. But some lenders may require a minimum of a couple of years or more before they'll allow it; and they may require a new appraisal at your expense before they will allow it before the original amortization scheduled time.

If you owe $261,100 at 4.125% with 360 months remaining, the payment is $1265.42 per month for P&I.
=PMT(4.125%/12,360,-261100) ... returns $1265.42

If the 80% LTV is $219,920 then the purchase price must have been $274,900.

It will take =NPER(4.125%/12,1265.42, -261100, 219920) = 94.87 months to reduce the balance to $219,920.

As a double-check, In 94.87 months with the 1265.42 payment you will owe
=FV(4.125%/12, 94.866, 1265.42, -261100) = $219,920

If you pay an extra $41,180 the the balance of $219,920 at 4.125% with the $1265.42 payment will have a balance of $162,895 after 94.87 months.
=FV(4.125%/12, 94.87, 1265.42, -219920.00) … returns $162,895

The extra lump sum of $41,180 gives you a gain in net worth of $57,025 in 94.87 months while also giving you a reduction of $80.51 per month in payments.

In order for your $41K to give you $80 less per month in payments and be worth $57K at the end of 94 months:

=RATE(94.87, -80.51 , 41180, -57025)*12 … returns 6.14% ...same as #cruncher's result

To verify:
=FV(6.14%/12, 94.87, 80.51, -41180) … returns $57,025

Also bear in mind that in some other investment, you'd be paying tax on the interest/dividends from year to year, and cap gains tax on any increase in value of the funds. So your equivalent rate of return for paying down the mortgage is actually higher than the rate for the mortgage and PMI alone.

Plus, continuing to pay the extra $80 per month will be giving you the rate of the mortgage plus tax on earnings too.

jimb
Topic Author
allanj37
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Joined: Mon Feb 23, 2009 10:29 am

Re: Mortgage prepay math with PMI

Post by allanj37 »

Thanks cruncher and jimb -- that's the full math that I wasn't quite prepared to figure out (hence my simple interest ballpark method).

So it sounds like this will be a great investment for the first 94 months (6.14% with zero risk). After that it's just the 4.15% of the loan, which I guess is still decent considering there's no risk. But it's certainly not stock market returns.

Anyone want to calculate what the overall rate of return would be for this assuming we keep the loan for the full 30 years? (we haven't actually made our first payment yet)

We're planning on recasting/re-amortizing the loan if we go this route so that we'll have a lower monthly payment and still keep the loan term at 30 years. I'm assuming this doesn't change the calculations of the interest rate? (other than the fact that not recasting would effectively shorten the term of the loan)

For the questions about our lender, the credit union loan officer said that as long as we bump the equity up to 20% within the first year, they won't require an appraisal or anything to cancel the PMI. And they'll recast for free.

Thanks for the help everyone. This still won't be an easy decision, but I just wanted to verify the basic math before trying to actually make the choice.
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