# Compound Interest Formula

#### Explain this!

##### Junior Member
How does (P + Pi) - P become P(1 + i)^n - 1? I know that factoring is involved, but I not sure what the steps are.

#### Subhotosh Khan

##### Super Moderator
Staff member
How does (P + Pi) - P become P(1 + i)^n - 1? I know that factoring is involved, but I not sure what the steps are.
Where did you see that "happening"? I think you are misquoting!

#### Explain this!

##### Junior Member
Where did you see that "happening"? I think you are misquoting!
I've seen this or something similar to it in finance books. Think of it in another way. Where does P(1 + i)^n - 1 come from? Isn't this the compound interest formula?

#### Subhotosh Khan

##### Super Moderator
Staff member
I've seen this or something similar to it in finance books. Think of it in another way. Where does P(1 + i)^n - 1 come from? Isn't this the compound interest formula?
However, you have NOT seen that

Most probably you have seen

Compound interest after n period = P[(1+i)^n -1]

Those [] are super-important. Now that you know the correct expression - google around a bit, you can find the derivation.

#### Explain this!

##### Junior Member
However, you have NOT seen that

Most probably you have seen

Compound interest after n period = P[(1+i)^n -1]

Those [] are super-important. Now that you know the correct expression - google around a bit, you can find the derivation.
Thanks for the reply! I'll search for an answer, but I thought that is what you "experts" are for -- to answer questions. Is there a simple reason why the brackets are important?

#### Subhotosh Khan

##### Super Moderator
Staff member
Thanks for the reply! I'll search for an answer, but I thought that is what you "experts" are for -- to answer questions. Is there a simple reason why the brackets are important?
"...I thought that is what you "experts" are for -- to answer questions"

Nope ... we are volunteers - we "choose" our actions - we GUIDE.

And we refuse to spoon-feed.....

#### Subhotosh Khan

##### Super Moderator
Staff member
Is there a simple reason why the brackets are important?
Yes . without parentheses - you'll get a different numerical answer.

#### firemath

##### Full Member
Is there a simple reason why the brackets are important?
The very simple reason that you will get it wrong if you do not follow the order of operations.

#### Jomo

##### Elite Member
(P + Pi) - P = Pi as the P cancels out to 0.

P(1 + i)^n - 1 means to multiply P and(1 + i)^n and then subtract 1

P[(1 + i)^n - 1] means to multiply P and (1 + i)^n - 1

P[(1 + i)^n - 1] = P(1 + i)^n - P which is different from P(1 + i)^n - 1

#### Explain this!

##### Junior Member
"...I thought that is what you "experts" are for -- to answer questions"

Nope ... we are volunteers - we "choose" our actions - we GUIDE.

And we refuse to spoon-feed.....
I'm certainly glad that I do not have you as an instructor, if that is your occupation!

#### Subhotosh Khan

##### Super Moderator
Staff member
I'm certainly glad that I do not have you as an instructor, if that is your occupation!

#### JeffM

##### Elite Member
How does (P + Pi) - P become P(1 + i)^n - 1? I know that factoring is involved, but I not sure what the steps are.
It doesn't. You are mixing up different formulas.

$$\displaystyle (P + Pi) - P = Pi$$.

That is the formula for calculating the interest paid on amount P for one period without compounding at an interest rate of 100 * i percent per period.

30000 dollars for one month at 1/4 % per month. What's i?

$$\displaystyle 100 * i = \dfrac{1}{4} = 0.25 \implies i = \dfrac{0.25}{100} = 0.0025.$$

So the interest paid is $$\displaystyle 30000 * 0.0025 = 75.$$

The other formula, which you got wrong, is

$$\displaystyle P\{(1 + i)^n - 1\}.$$

That is the formula for interest paid after n periods when interest at a rate of 100 * i percent per period is compounded rather than paid before maturity.

30,000 for 3 months at an interest rate of 1/4 % per month compounded monthly will involve payment of interest after 3 months of

$$\displaystyle 30000\{(1 + 0.0025)^3 - 1\} = 225.76 > 3 * 75.$$

#### Explain this!

##### Junior Member
It doesn't. You are mixing up different formulas.

$$\displaystyle (P + Pi) - P = Pi$$.

That is the formula for calculating the interest paid on amount P for one period without compounding at an interest rate of 100 * i percent per period.

30000 dollars for one month at 1/4 % per month. What's i?

$$\displaystyle 100 * i = \dfrac{1}{4} = 0.25 \implies i = \dfrac{0.25}{100} = 0.0025.$$

So the interest paid is $$\displaystyle 30000 * 0.0025 = 75.$$

The other formula, which you got wrong, is

$$\displaystyle P\{(1 + i)^n - 1\}.$$

That is the formula for interest paid after n periods when interest at a rate of 100 * i percent per period is compounded rather than paid before maturity.

30,000 for 3 months at an interest rate of 1/4 % per month compounded monthly will involve payment of interest after 3 months of

$$\displaystyle 30000\{(1 + 0.0025)^3 - 1\} = 225.76 > 3 * 75.$$

My question is what is the original formula that becomes P{(1 + i)^n - 1} and how does it become P{(1 + i)^n - 1}?

#### JeffM

##### Elite Member
I can give you an intuitive answer or a proof by weak mathematical induction.

#### Explain this!

##### Junior Member
I can give you an intuitive answer or a proof by weak mathematical induction.

Does P + Pr = P(1 + r)? If yes, then can you show me the steps?

#### JeffM

##### Elite Member
I can show you steps if you understand weak mathematical induction. You do realize that n can be any of an infinite number of values. I am not going to show you detailed steps for when n = 365.

I can show you an answer for n = 2, or n = 3, but that is not a general answer. The way to give a general answer is through weak mathematical induction.

#### Explain this!

##### Junior Member
I can show you steps if you understand weak mathematical induction. You do realize that n can be any of an infinite number of values. I am not going to show you detailed steps for when n = 365.

I can show you an answer for n = 2, or n = 3, but that is not a general answer. The way to give a general answer is through weak mathematical induction.
I'm trying to keep this simple:
First of all P + Pr has a common factor of P, so it can be expressed as P(1 + r).

I want to know how the "P" gets in front of (1 +r) to become P(1 + r). If I divide P + Pr by P, I get (1 + r), so how does the P get in front of (1 + r) as in P(1 + r)?

#### JeffM

##### Elite Member
3 * 6 + 3 * 8 = 18 + 24 = 42

3 * (6 + 8) = 3 * 14 = 42.

Now you can try experimenting with other numbers until you have convinced yourself of the validity of this basic law of arithmetic:

$$\displaystyle a(b + c) \equiv ab + ac.$$

It's called the distributive law of multiplication over multiplication.

There is a proof using induction and the Peano Postulates, but I have long since forgotten it.

#### Jomo

##### Elite Member
P=P*1

P + Pr = P*1 + Pr = P(1+r). The P was factored out.