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I have a mortgage whose initial value is $\$615,000$ and interest rate is $6.63\%$ compounded monthly over a 30 year period. When I substitute these into the usual compound-interest formula, I get $$\$615000\left(1+\frac{.0663}{12}\right)^{12\times30}=\$4470018,$$ which then works out to a monthly repayment of $$\$12417,$$ which is absurdly high.

Am I doing something horribly wrong here? I realize that mortgages may not be exactly calculated using this formula, but I would think it would provide somewhat of a close estimate.

ryang
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mmmmmm
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    Yes, you are doing something horribly wrong. Your $(1+\frac{.0663}{12})^{12}\approx 1.06835226$ i.e. an effective annual interest rate of about $6.8%$. Compounding this over $30$ years gives about $7.238$ and multiplying $$615000$ by that gives your four million+. But it has nothing to do with monthly payments: it is what you would owe at the end if you made zero monthly payments. – Henry Mar 28 '25 at 01:46
  • If you wanted to estimate, at the start you have to pay interest of $615000\times \frac{.0663}{12} \approx 3398$ a month. If there were no interest, and you were paying back the capital debt in equal amounts, you would pay off $615000\times \frac{1}{12\times 30 } \approx 1708$. So your monthly payment will be somewhere between $3398$ and $3398+1708 =5106$. In fact you pay less interest and more capital at the end as by then you have reduced the debt, and so need to pay back less capital debt at the start in addition to the full interest, so monthly payments should be about $3937$ a month. – Henry Mar 28 '25 at 02:10
  • Upon further investigation, I see now that I should have been using the loan amortization formula rather than the compounding interest formula. Thanks for the feedback. – mmmmmm Mar 28 '25 at 02:25

1 Answers1

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Your computation (simply dividing the loan principal's end-of-tenure future value by the total number of repayments) grossly overestimates the required monthly repayment because it disregards the time value of money, failing to account for the fact that the final repayment $\$A$ has a smaller time value than the other repayments (especially the initial repayment) $\$A.$

For a $\$P$ loan with nominal interest rate $r\%$ per annum and repayment $\$A$ made $m$ times per year over $t$ years:

  • if compounded daily, $$A = \frac{P \left( \left(1 + \frac{r}{36500}\right)^{\frac{365}{m}} - 1 \right)}{1 - \frac{1}{\left(1 + \frac{r}{36500}\right)^{365t}}};\tag1$$

  • if compounded $m$ times per year, $$A = \frac{Pr}{100m\left(1 - \frac{1}{\left(1 + \frac{r}{100m}\right)^{tm}}\right)}.\tag2$$

(These formulae are derived as sums of geometric series.)

Since loans typically accrue interest on a daily basis (even as payments are required monthly), formula $\boldsymbol{(2)}$ typically underestimates the required monthly repayment, compared to formula $\boldsymbol{(1).}$

For your monthly-compounding example, the required monthly repayment $A$ is \$3940. However, if in fact your loan compounds interest daily, then paying this figure instead of the correct amount \$3947 will result in a growing shortfall (as well as interest accrued on it), leading to an outstanding loan balance at the end of the $30$-year term.

ryang
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