compound interest

Let's suppose we borrow 1000$. And we will have to give each mounth 100$ back along with an interest of 12% per year. How much will we give back in 1000/100 = 10 mounth?

1. Simple interest

The borrowed 1000$ is called the principal. Let's write P = 1000. If the (i.e. each) year interest rate is "r", then once the related year is finished, we will have to pay back
T = P + (the interest of P) = P + I = P + P x r = P(1 + r).
If we expect to pay back P in the coming "n" years, then the interest rate will be n x r and we will give back
P + nrP = P(1 + nr).

1.1. Each month

First mounth: 100$ + 100$ x (12% /12) = 100 ( 1 + 1%) $ = 101 $
Second mounth: 100 (1 + 2%) $ = 102 $
...
nth nouth : 100 ( 1 + n%) $
n = 1000/100 = 10

10th mounth: 100 (1 + 10%) $ = 110$
The total to pay back is : 101 + 102 + ... + 110 = 10 x 100 + ( 1 + 2 + ... + 10) =
1000 + (10 x 11 /2) = 1055 $ . Then 55$ extra.

1.2. At the end of the period

Here the period is equal to 10 month.

10th mounth: 1000 (1 + 10%) $ = 1100$ Then 100$ extra.

2. Compound interest

For each mounth and RIGHT NOW: 100$ + 100$ x (12% /12) = 100 ( 1 + 1%) $ = 101 $
First mounth: 100 (1 + 1%) $ = 101 $
If the seconf amount (101$) of the second mounth is not payed NOW, it will be charged its iterest of 1% for the next month. Then, for the next mounth: 100 (1 + 1%)² $ = 102.01 $
...
For the 10th mounth: 100 (1 + 1%)¹⁰

Let's write: r = 1 + 1%
The total to pay back is : 100 (r + r² + r³ + ... + r¹⁰)
T = 100 r (1 + r + r² + r³ + ... + r⁹ ) = 100 r [(1 - r¹⁰)/(1 - r)] = 100 r [(r¹⁰ - 1)/(r - 1)]

We have:
r - 1 = 1%

1. First approximation:
(r¹⁰ - 1 = (1 + 1%)¹⁰ - 1 = (1 + 10%) - 1 = 10%
Therefore:
T = 100 (1 + 1%) (10%) /1% =
100 x 1.01 x 10 = 1010 $

2. Second approximation:
(1 + 1%)¹⁰ = 1 + 10% + 10 x 9 x 1% x 1%/2 + ...
= 1 + 10% + 45%% + ... = 1.10 + 0.0045 = 1.1045 + ...

(r¹⁰ - 1 = 0.1045 = 10.45%
T = 100 (1 + 1%) (10.45%) /1% =
100 x 1.01 x 10.45 = 1055.45 $

3. Third approximation:
(1 + 1%)¹⁰ = 1 + 10% + 10 x 9 x 1% x 1%/2 + ... = 1 + 10% + 45%% + ...
= 1.10 + 0.0045 = 1.1045 + ...
(r¹⁰ - 1 = 0.1045 = 10.45%

T = 100 (1 + 1%) (10.45%) /1% =
100 x 1.01 x 10.45 = 1055.45 $

4. With Calculator:
(r¹⁰ - 1 = 0.104622125 = 10.4622125%
T = 100 (1 + 1%) (10.4622125%) /1% =
100 x 1.01 x 10.4622125 = 1056.68 $

3. nominal and effective rates

The nominal or stated rate is the unit rate: The interest rate "r" for a related period "n" is called the effective rate. Then the nominal rate is "r/n".
The interest becomes I = P x n x (r/n), and the repayment is R = P + I = P(1 + r) for the simple interest and R = P [1 + (r/n)]ⁿ for the compound interest.
The relationship between the two rates is:
P(1 + effective) = P [1 + (nominal)]ⁿ
Therefore:
effective = [1 + (nominal)]ⁿ - 1

effective rate =
(1 + nominal rate)^period - 1

Annex:

1. arithmetic series:
1 = 2 + 3 + ... + n = n(n + 1)/2, indeed:
Sn = 1 + 2 + 3 + ... + n
Sn = n + n-1 + n-2 + ... +1
2Sn = (1+n) + (1+n) + ... + (1+n) , n times. Therefore:
2Sn = n (1+n). Then:
Sn = n(n+1)/2

More generally:
First term: t₁ = a
Second term: t₂ = a + r
Third term: t₃ = a + r + r = a + 2r
...
nth term: t_n = a + r + r + r + ... + r = a + (n - 1)r

The sum is: S_n = n x a + r [1 + 2 + 3 + ... + (n - 1)] =
n x a + r x (n - 1) x n/2

S_n = n [t₁ + r(n - 1)/2]

Example:

For n = 10, t_n = 1, and r = 1, we have:
S_n = 10 [1 + 9/2] = 5 x 11 = 55.

2. Maclaurin polynomial:
(1 + x)ⁿ =
1 + n x + n(n - 1)x²/2! + n(n - 1)(n - 2)x³/3! + ... +

3. geometric series:
(1 + r + r² + r³ + ... + rⁿ) = [1 - rⁿ⁺¹]/(1 - r) =
[rⁿ⁺¹ - 1]/(r - 1)
1 + r + r² + r³ + ... + rⁿ = (rⁿ⁺¹ - 1)/(r - 1)

More generally:
First factor: f₁ = a
Second factor: f₂ = a x r
Third factor: f₃ = a x r x r = a x r²
...
nth factor: f_n = a x r x r x r x ... x r = a x r^{n - 1}

The sum is: S_n = a x r [1 + r + ... + r^{n - 2}] =
a x r x (1 - r^{n - 2})/(1 - r)

S_n = f₁ r (1 - r^n-1)/(1-r)

Example:

For n = 10, f_n = 1, and r = 2, we have:
S_n = 1 x 2 (1 - 2⁹)/(1 - 2) = 2 (2⁹ - 1)/(2 - 1) = 2 (2⁹ - 1) (2⁹ - 1) = 1022 ≈ 2¹⁰ = 1024.

2. Maclaurin polynomial:
(1 + x)ⁿ =
1 + n x + n(n - 1)x²/2! + n(n - 1)(n - 2)x³/3! + ... +

Application compound interest:
p x r
p x r²
p x r³
...
p x rⁿ

Therefore, the total is:
T = p x r (1 + r + r² + r³ + ... + rⁿ)
= p x r x (rⁿ⁺¹ - 1)/(r - 1)

Interesr rate = P r n
P: Principal, r: interest rate, n: priod

Simple interest: P → P ( 1 + n r)
Compound interest: P → P ( 1 + r)ⁿ

effective rate =
(1 + nominal rate)^period - 1

arithmetic series: t_n = a + (n - 1)r
S_n = n [t₁ + r(n - 1)/2]

geometric series: f_n = a x r^{n - 1}
S_n = f₁ r (1 - r^{n - 1})/(1-r)