What's the meaning behind annuity coefficient?

Question

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: $A=P\cdot {\frac {(1+r)^{n}}{n}}$ ?

Answer 1

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,

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

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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.

Answer 2

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.