COMPOUND INTEREST
- Thread starter Saumyojit
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Good question.
I want to know how did the amount compounded annually formula came ?
A = P (1 + r/100)^n
n = time
r=rate of interest.
First, you made slight mistakes. The variable r is the rate of annual interest expressed as a percent. And n is the number of years.
The more general formula is:
[MATH]f = p(1 + i)^n, \text {where}[/MATH]
[MATH]f = \text {FUTURE value,}[/MATH]
[MATH]p = \text {PRESENT value;}[/MATH]
[MATH]i = \text {INTEREST rate for period expressed as a decimal; and}[/MATH]
[MATH]n = \text {NUMBER of periods.}[/MATH]
So the division by 100 in your formula is just to turn a percent into a decimal. Got it?
Now what do we mean by an interest rate? It is just the ratio of the increment repaid over the amount borrowed or invested. So if you must repay 101.00 for borrowing 100.00 after one month, the monthly interest rate is
[MATH]\dfrac{101.00 - 100.0}{100.00} = 0.01 \text { per month.}[/MATH]
Notice that I have replaced your A with f, your P with p, and your r with i. By definition,
[MATH]i = \dfrac{f - p}{p} \implies pi = f - p \implies f = p + ip = p(1 + i) = p(1 + i)^1.[/MATH]
That gives us f if n = 1. With me so far?
How about if n = 2.
Well after 1 period, we have [MATH]p(1 + i).[/MATH]
Thus at the end of the second period we have
[MATH]f = \{p(1 + i)^1\}(1 + i) = p(1 + )^2 \text { if n} = 2.[/MATH]
What do you reason f will be if n = 3?
I want to know how did the amount compounded annually formula came ?
A = P (1 + r/100)^n
n = time
r=rate of interest.
This formula is right. time in years and rate in percentage thats why divided by 100 common sense!
It is right only if r is an annual rate expressed as a percentage, compounding is annual, and n is the number of years. Otherwise it is wrong.
Why do you ask a question if you pay no attention to the answer?
Suppose Interest rate is 5 Percent.
Then 5/100 of principal.
What is this "It is just the ratio of the increment repaid over the amount borrowed or invested."
"Then 5/100 of principal" is not even a valid Englsh sentence. It is meaningless.
I suspect that what you mean is that the interest paid is 5/100 (or 0.05) of principal. If so, that is exactly correct. You repay the original principal plus an increment. The increment is called "interest." Divide the increment by the principal (find the ratio) to determine the interest rate.
As a practical matter, interest rates are described legally as annual percentage rates in the U.S., but financial professionals use periodic interest rates in decimals in their math. Now of course if the compounding period is a year, nothing more is involved to go from annual percentage rates to period decimal rates than to divide by 100. That will not do if the compounding period is a month. Then to go from an annual percentage rate to a period decimal rate, you must divide the percentage rate by 1200.
The increment is 0.05 . and principle is rs 200 suppose then my Rate of interest is 0.05/200 = ...?
No.
The principal is 200. The amount of the required repayment is 210.
Then the interest (increment) over the original principal is 210 - 200 = 10.
Therefore the ratio of increment to original payment is 10/200 = 0.05.
Steven G
Elite Member
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For simplicity let's use r for what you call r/100. As pointed out, r is the yearly interest rate and n is the number of years you have the account for (again this is only if you get interest once per year).
I want to know how did the amount compounded annually formula came ?
A = P (1 + r/100)^n
n = time
r=rate of interest.
Start: year 0: You have A = P (1 + r)^0 = P
year 1: A= P + rP = P(1+r) = P(1+r)^1
year 2: A = P(1+r)^1 + rP(1+r)^1 = P(1+r)(1+r)^1 = P(1+r)^2
year 3: A = P(1+r)^2 + rP(1+r)^2 = (1+r)P(1+r)^2 = P(1+r)^3
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year n: P(1+r)^n
if The increment is 0.05 . and principle is rs 200 then amount must be 200+0.05=200.05
Steven G
Elite Member
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- Dec 30, 2014
- Messages
- 14,483
It's only February and you are giving up! Just take a little break. Good night!