0

There is a formula for mortgage month payment calculation: $$ A=P\cdot {\frac {r(1+r)^{n}}{(1+r)^{n}-1}} $$ where:

  • ${\displaystyle A}$ is the periodic amortization payment;
  • ${\displaystyle P}$ is the principal amount borrowed;
  • ${\displaystyle r}$ is the rate of interest expressed as a fraction (per month);
  • ${\displaystyle n}$ is the number of payments;

What exactly does the coefficient after ${\displaystyle P}$ represent? Why don't banks use a much simpler formula: $A=P\cdot {\frac {(1+r)^{n}}{n}}$ ?

gavrilikhin.d
  • 103

2 Answers2

0

About the "why":

Simple interests are not viable in front of compound ones. If compound interests did not exist, everyone would withdraw their deposits after one year and re- deposit the capital + interest on the next day, so achieving compound interests.

0

Consider the $k$th payment of $A$ in the future. That payment would correspond to a loan of $A(1+r)^{-k}$ in the present.

Summing the present values of the $n$ future payments,

$$\begin{align*} P &= \sum _{k = 1}^n A(1+r)^{-k}\\ &= \frac A{1+r} \sum_{k=0}^{n-1} (1+r)^{-k}\\ &= \frac A{1+r}\cdot \frac{1-(1+r)^{-n}}{1-(1+r)^{-1}}\\ &= A\cdot\frac{1-(1+r)^{-n}}{r}\\ &= A\cdot \frac{(1+r)^n - 1}{r(1+r)^n}\\ A &= P\cdot \frac{r(1+r)^n}{(1+r)^n - 1}\\ \end{align*}$$

For your proposed formula $A = \dfrac{P(1+r)^n}{n}$, consider an alternative payment schedule:

  • to borrow $P$ now,
  • and to pay $P(1+r)^n$ in full after $n$ months.

The total payment amount is the same as in the annuity payment schedule, yet the alternative allows a later payment.

This shows that either the annuity amount is not valued correctly, or the lender may be using the formula to promote a lower interest rate than the effective rate.

peterwhy
  • 22,256
  • Don't quite understand why $k$th payment correspond to that loan. Aren't all the payments the same? – gavrilikhin.d Nov 29 '20 at 19:43
  • It might be also important that from now on I specify $r$ as interest per month for every formula, to avoid confusion. – gavrilikhin.d Nov 29 '20 at 19:51
  • @gavrilikhin.d All the $n$ payment amounts are the same at each time of payment, but since they happen in different time in future, their present values are different. For example the first payment, to repay $A$ in 1 month the borrower would borrow $A(1+r)^{-1}$ now, but for the 10th payment, to repay the same amount $A$ in 10 months from now the borrower can only borrow $A(1+r)^{-10}$. – peterwhy Nov 29 '20 at 19:56
  • As far as I can understand, you are speaking about PV and FV, right? I might need to spend some time researching on this information to fully comprehend you answer. – gavrilikhin.d Nov 29 '20 at 20:08
  • @gavrilikhin.d Yes, and all these are built from the basic idea: to borrow $$1$ now means to repay $$(1+r)^k$ after $k$ months (in one lump); choosing to pay later means having to pay more (for the same borrowed amount). Or conversely, to repay $$1$ after $k$ months means to borrow $$(1+r)^{-k}$ now; choosing to pay later means getting less upfront (for the same payment amount). – peterwhy Nov 29 '20 at 20:57
  • Thanks, now I understand. – gavrilikhin.d Nov 30 '20 at 05:17