<p>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.</p>

<p>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). </p>

CC BY-SA 4.0
2020-04-06T12:05:57.747Z
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
CC BY-SA 4.0
2020-04-06T12:10:01.727Z
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
CC BY-SA 4.0
2020-04-05T20:47:07.773Z
2020-04-05T20:47:07.773Z
Brian Romanchuk
<p>Let <span class="math-container">$L$</span> be the loan amount, then <span class="math-container">$L=P\frac{1-(1+i)^{-n}}{i}$</span></p>

<p>The balance at time <span class="math-container">$t$</span> is defined as the present value of the remaining payments, </p>

<p><span class="math-container">$B_t=P\frac{1-(1+i)^{-(n-t)}}{i}$</span></p>

<p>By using the following equations</p>

<p><span class="math-container">$I_k+C_k=P$</span> and <span class="math-container">$I_k=iB_{k-1}$</span> where <span class="math-container">$I_k$</span> is the interest paid at time <span class="math-container">$k$</span> we derive the following equation</p>

<p><span class="math-container">$C_k=\frac{P}{(1+i)^{n-k+1}}$</span> and after a simple rearrangement we arrive at the formula you presented.</p>

CC BY-SA 4.0
2020-04-07T01:43:14.510Z
2020-04-07T01:43:14.510Z
actuarialboi9
<p>I have found this formula on wikipedia :
<span class="math-container">$$P=C_k(1+i)^ {(n-k+1)}$$</span></p>

<p>which describe the constant payment that has to be paid every year (<span class="math-container">$C_k$</span> is the part of the initial loan that is extinguished with the <span class="math-container">$k$</span> payment) in order to pay off a loan after a period of <span class="math-container">$ n $</span> years , all of them at the constant annual interest rate <span class="math-container">$ i $</span> .</p>

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

<mathematical-economics><finance><financial-economics><bonds>
Clarification on amortization of constant payments
Tortar
# Clarification on amortization of constant payments

(Asked by: username_0 on 2020-04-05)

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?

**Tags:** mathematical-economics, finance, financial-economics, bonds


---

## 2 answers

### Answer by username_2 on 2020-04-07. Score: 2 | Accepted answer

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.


---

### Answer by username_1 on 2020-04-05. Score: 1

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).

> **Comments:**
> **username_0**: 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  
> **username_1**: 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.  

---

