need help

- sparrow2

need help

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- schrodinger

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- sparrow2

question is about compound interest. \[a=p(1+r/m)mt and a=p(1+r)t\]

- sparrow2

symbols are ordinary, why this two result arn't the same?

- sparrow2

a=p(1+r/m)^mt a=p(1+r)^t

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## More answers

- sparrow2

@Michele_Laino

- 0_0youscareme

@.Gjallarhorn.

- Michele_Laino

I'm sorry, I don't know your answer since I'm not good with financial mathematics

- sparrow2

@Abhisar

- sparrow2

@amistre64

- sparrow2

@empty

- anonymous

\[a=p \left( 1+r \right)^t\]
here interest is compounded annually
\[a=p \left( 1+\frac{ r }{ m } \right)^{mt}\]
here interest is compounded monthly.
or we can say interest of first month is also principal for next month and so on.

- sparrow2

i know that ,but they must be equal, i think so

- anonymous

no, for first year principal is same during whole year.
but in second case principal is different for every month.

- sparrow2

so if i have r as anual rate, and they tell me to fing future value of smt,but intereset must be monthly ,what i will do?

- sparrow2

@Palmo4ka esli mojesh pomogi :)

- anonymous

let us take the case principal=100
rate=12% annually
then amount after first month
\[a=100\left( 1+\frac{ .12 }{ 12 } \right)^1=101\]
amount after second month
\[a=101\left( 1+\frac{ .12 }{ 12 } \right)^1=101*1.01=102.01\]
..........
this becomes principal for third month
and so on.
we see principal increases every month.
but in the first case principal is same whole year.

- sparrow2

i understand that man, but why did you diveded anual rate by 12,is it ok?

- anonymous

@sparrow2 Do you speak Russian?

- anonymous

no

- sparrow2

kaneshna gavariu, no po angliski luchshe :)

- anonymous

а я, наоборот, говорю лучше по-русски:) @sparrow2

- anonymous

rate is annual ,but we are calculating every month so we divide by 12
if we have to calculate after every 6 months,we divide by 2
if we have to calculate every 3 months ,we divide by 4

- sparrow2

so the answers will be different when calculating anualy and monthly?

- anonymous

yes

- sparrow2

and dividing by periods is ok? so do they mean so when asking for periods(we know only annualy)

- sparrow2

@Palmo4ka potomushto ti iz rossi( no ia net)

- anonymous

i am sorry i don't understand your language fully,i am an indian.

- sparrow2

i'm also not native speaker :D

- sparrow2

so you divided it by 12 or by 6 and so on(anual rate) is it ok?

- sparrow2

so if i have anual rate like 12%, if they askme me to calcualte anualy i use 12%,but by monthly i will use 1% and the answers will be different

- anonymous

correct

- anonymous

but at the sametime multiply t by 12

- sparrow2

i thought that this was just 2 different ways,but the answers must be the same, i thougt so

- anonymous

no ,as you see principal goes on changing for compound interest.

- sparrow2

on earth :)

- sparrow2

okay thanks man :)

- anonymous

yw

- ganeshie8

It might be enlightening to see what happens if you calculate the interest every day, every hour, every second..

- sparrow2

@ganeshie it's true. you see, if you increase periods the amount is increasing too(it's more beneficila for banks to use more periods :D )

- sparrow2

pe^rt will be if its continous

- ganeshie8

Haha these days most banks calculate interest continuously(every nano second or so)
The interest wont increase much by increasing the periods, it saturates after 12 periods or so

- ganeshie8

Notice that we get that continuous formula by letting \(m\to \infty\) in the discrete version :
\[\large \lim\limits_{m\to\infty}~p\left(1+\frac{r}{m}\right)^{mt} = pe^{rt} \]

- sparrow2

yeah i see, it easy to prove :)

- ganeshie8

Letting \(p=r=t=1\), we get the definition of euler constant \(e\) :
\[\large \lim\limits_{m\to\infty}~\left(1+\frac{1}{m}\right)^{m} = e\]

- sparrow2

yeah that is cool way to define e :) creative

- sparrow2

okay thanks @ganeshie8

- ganeshie8

that is indeed one of the most useful definitions of \(e\)
https://en.wikipedia.org/wiki/E_(mathematical_constant)#History

- sparrow2

so when you have like 12% anualy can you always say that monthly will be like 1 % @ganeshie8

- sparrow2

oh i closed the question

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