Let's say I want to buy a house that costs X, and that I have 20% of X to put as a down payment.

But I could put only 5% of X down and I would have to pay a 3.26% insurance premium on the mortgage amount because that's the law. So I would have an extra cost of X * 0.95 * 0.0326 but, on the other hand I would have 0.15 * X to invest and earn interest on.

The question is: How can I calculate how much the interest rate must be so that it will be more advantageous to put just the 5% down instead of putting 20% down?

Edit: To make it clearer, I am looking for the formula that finds what's the interest rate I would have to make on my investments to break-even with the insurance premium.

I forgot to mention something important, the insurance premium will be added to the mortgage balance.

  • What are you assuming for rate of return on the investments? – NL - Apologize to Monica Feb 23 '17 at 21:44
  • Your question is "how much should the interest rate on the mortgage be"? – NL - Apologize to Monica Feb 23 '17 at 21:48
  • @NathanL - I don't think this question is a dup. This question asks for a mathematical formula for breaking even given certain conditions. – TTT Feb 23 '17 at 22:51
  • In this case the percentages are really important. If you are talking a conventional loan, the 5% strategy will require PMI making it a silly proposition. A business case could be made for doing 20% as opposed to 35% down, but not 5 and 20. – Pete B. Feb 24 '17 at 12:48
  • @PeteB. OP already seems to be taking into account "a 3.26% insurance premium". – a CVn Feb 24 '17 at 14:23

I believe the following formula provides a reasonable approximation. You need to fill in the following variables:

  • MP5 (Monthly payment with 5% down)
  • MP20 (Monthly payment with 20% down)
  • PP (Purchase Price)
  • Y (Number of years you'll keep this mortgage. i.e. until you refi, sell, pay it off, etc.)

The average annual return you need on investing the 15% =

(((MP5 - MP20) * 12) + (.0326 * .95 * PP / Y)) / (PP *.15)

Example assuming an interest rate of 4% on a 100K home:

  • MP5 = $453.54/month (financing 95K)
  • MP20 = $381.93/month (financing 80K)
  • PP = $100,000
  • Y = 5 years

If you invest the $15K you'll break even if you make a 9.86% return per year on average. Here's the breakdown per year using these example numbers:

Years   Avg Return Needed
1       26.38%
2       16.05%
3       12.61%
4       10.89%
5       9.86%
6       9.17%
7       8.68%
8       8.31%
9       8.02%
10      7.79%
11      7.61%
12      7.45%
13      7.32%
14      7.20%
15      7.11%

Note this does not consider taxes.

  • Your response is the one that got closer to what I'm looking for. But I'm interested on breaking-even only on the insurance premium and it seems to me that you are considering also the interest on the balance on your formula. Also (and that's my fault, I forgot to mention it) the insurance premium would be added to the mortgage - so I would pay interest on it. – R. Monte Feb 24 '17 at 18:20
  • @R.Monte - The interest is automatically handled by the monthly payment difference. Adding the premium to the mortgage amount makes it more complicated because the amortization schedule would then become relevant on the extra amount. But, I think it will still be pretty close to these numbers, since your payments will be slightly higher, but you'll also slowly eat away at the principle on the extra mortgage amount each year. (I'd guess around 1% higher each year than the numbers you see here.) – TTT Feb 27 '17 at 18:55

You are comparing a risk-free cost with a risky return. If you can tolerate that level of risk (the ups and downs of the investment) for the chance that you'll come out ahead in the long-run, then sure, you could do that.

So the parameters to your equation would be:

  • What average rate of return on the risky investments do I want
  • What is the risk (variance) of the risky investments
  • What probability that I'll earn less than my risk-free cost can I tolerate?

If you assume that the risky returns are normally distributed, then you can use normal probability tables to determine what risk level you can tolerate.

To put some real numbers to it, take the average S&P 500 return of 10% and standard deviation of 18%. Using standard normal functions, we can calculate the probability that you earn more than various interest rates:

mean    10%
std dev 18%

Interest   Win %
0%         71.07%
1%         69.15%
2%         67.16%
3%         65.13%
4%         63.06%
5%         60.94%
6%         58.79%
7%         56.62%
8%         54.42%
9%         52.22%
10%        50.00%

so even with a low 3% interest rate, there's roughly a 1 in 3 chance that you'll actually be worse off (the gains on your investments will be less than the interest you pay). In any case there's a 3 in 10 chance that your investments will lose money.

  • Although interesting, this doesn't really answer the question. The question is "What return do I need in order to overcome the price of the insurance premium?" You are answering "What return can I expect to get?" – TTT Feb 23 '17 at 23:03
  • Indeed it doesn't really answer my question but it's an interesting insight, thank you! – R. Monte Feb 24 '17 at 18:27
  • @R.Monte The point of my answer is that there's not a return that guarantees that you'll come out ahead. The level of expected return and the level of risk determine the probability that you'll come out ahead but it's not guaranteed. If you pick an expected return that equals your interest rate + insurance premium you'll only come out ahead half of the time. – D Stanley Feb 24 '17 at 19:57

I wouldn't call it apples and oranges. This is literally an opportunity cost calculation. You can safely assume S&P500 will perform at least 11% over any 10 year period. Since failing companies are delisted and replaced with new growing companies, the market should continue to grow. No, it's not guaranteed. Lets use an aggressive number for inflation, 4%, leaving a 7% ROR estimate for S&P500. I assume OP has better credit than me, assume a rate around 3.5%. So it looks like net 3.5% ROR. The PMI erases that. You have to continue paying it until you pay off the loan.

Put 20% down, get a 15 year fixed at lowest rate. Pay it off quicker.

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