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Lesson 45 · Loans and honest rates
What borrowing actually costs
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What borrowing actually costs

A loan is an annuity pointed the other way: you hold the balance, and equal payments grind it to exactly zero on the last one. The level payment is M=P i1−(1+i)−nM = \dfrac{P\,i}{1 - (1+i)^{-n}} — M equals P i over, one minus, one plus i to the minus n. PP is the principal borrowed in dollars, ii is the rate per PAYMENT period as a decimal (an annual rate divided by 12 for monthly payments), nn is the number of payments (years times 12), and MM is the payment itself. Both ii and nn must speak the same period or the formula answers a question nobody asked.

Then the line the advertisement never prints: I=M n−PI = M\,n - P, the total interest — everything handed over across the whole term, minus the amount actually borrowed. On a 30-year mortgage it routinely exceeds the price of the house. That is not a scandal; it is a rental fee for a very large sum over a very long time. But it deserves to be computed rather than assumed.

Finally, the only fair way to compare two quoted rates: EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \dfrac{r}{m}\right)^{m} - 1, the effective annual rate, where rr is the nominal annual rate as a decimal and mm is the number of compoundings per year. 12% compounded monthly is 12.68% effective, so it loses to 12.5% compounded annually. The bigger headline was the cheaper loan, and only the EAR could have told you.

PMni

M=P i1−(1+i)−nM = \frac{P\,i}{1 - (1+i)^{-n}}

  • MM= Payment per period (money)
  • PP= Principal borrowed (money)
  • ii= Interest rate per period
  • nn= Number of payments
Loan Payment (Amortized Loan or Mortgage) solver →
PIM n

I=M n−PI = M\,n - P

  • II= Total interest paid (money)
  • MM= Payment per period (money)
  • nn= Number of payments
  • PP= Principal borrowed (money)
Total Interest Paid Over a Loan solver →
rmEAR

EAR=(1+rm)m−1\mathit{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1

  • EAR\mathit{EAR}= Effective annual rate
  • rr= Nominal annual rate
  • mm= Compounds per year
Effective Annual Rate from a Nominal Rate solver →