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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.

R. Monte
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3 Answers3

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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.

D Stanley
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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.

Community
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TTT
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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.

Xalorous
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