# Clarification on amortization of constant payments

I have found this formula on wikipedia : $$P=C_k(1+i)^ {(n-k+1)}$$

which describe the constant payment that has to be paid every year ($$C_k$$ is the part of the initial loan that is extinguished with the $$k$$ payment) in order to pay off a loan after a period of $$n$$ years , all of them at the constant annual interest rate $$i$$ .

My problem is that $$1 + i$$ is elevated to $$n-k+1$$ while I expected $$k$$ because it's the numbers of years passed when the $$k$$ payment is done . It seems that the loan is paid inside out , namely starting to pay the interests at the time $$n$$ on the first year and so on…Am i right?

## 2 Answers

Let $$L$$ be the loan amount, then $$L=P\frac{1-(1+i)^{-n}}{i}$$

The balance at time $$t$$ is defined as the present value of the remaining payments,

$$B_t=P\frac{1-(1+i)^{-(n-t)}}{i}$$

By using the following equations

$$I_k+C_k=P$$ and $$I_k=iB_{k-1}$$ where $$I_k$$ is the interest paid at time $$k$$ we derive the following equation

$$C_k=\frac{P}{(1+i)^{n-k+1}}$$ and after a simple rearrangement we arrive at the formula you presented.

The simplest way to understand this is to write out the numbers for a particular loan. You can track the balance, and interest over time. Very easily done in a spreadsheet.

The amount of principal repaid in each payment is increasing as time passes, as the amount of interest paid falls (since the interest is proportional to a falling principal balance).

• My problem is about the fact that the interest and the principals are created through the debt left instead of the principal which is paid in that particular year...the total amount would'nt change if the calculation was made that way , just the order of the payments . It is strange to me that it's calculated by the debt left – Tortar Apr 6 '20 at 12:05
• The interest due in each period is the previous principal times the interest rate. Instead of trying to put a verbal interpretation on the equation, run actual numbers in a spreadsheet. – Brian Romanchuk Apr 6 '20 at 12:10