FOr the formula y= a(1+n/m)^mt What numbers so tyyou use when an investment is daily, yearly , quarterly and semi anuulay, Please provide example

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quirozd's profile pic

quirozd | High School Teacher | (Level 3) Adjunct Educator

Posted on

Given `y=a(1+n/m)^(mt)` , what does each variable represent in terms of compounding interest?

This is an exponential equation known as the compound interest formula. Money is put in an account and earnings are made by applying the same percent to the previous amount. 

t is time, but what unit is it in? Think about the units themselves too, If you put dollars in, you would expect to get dollars out (not dollars^year for example).

So, the trick to figuring these variables out is by thinking about "what would happen if," and make sense of the units

I just put my money in, what is the time? We start with no time elapsed, which means that t=0, so we would have `a(1+n/m)^0` . Anything raised to the zero power is 1. so `y=a` .

That must mean that y is the earnings, and a is the initial deposit.

What would happen if interest was zero? If you put your money in a zero-interest account, you would expect your money to stay the same, no matter how long you left it in there. Look at the `(1+n/m)` quantity. 1 raised to any power is still 1. The only way we can make `(1+n/m) = 1` is by letting `n = 0` . Thus, n must be the interest rate. Interest rates are usually given in percent and in terms of one year. In other words 5% per year `5%/text(year)` . This also means that the 1 in the equation really represents 100%. So 5% would mean your money would be 105% of its original value after 1 time period.

Since n is interest rate per year, that means m has to have units of something per year as well (so that the units cancel). Notice that when m is multiplied by t, that the units cancel as well. For the answer, we look at the terms left over: annually, quarterly, etc. These terms tell us how many times per year we are applying the percentage and reinvesting. So m is the number of times per year you are compounding.

To recap:

y = earnings

a = initial deposit

t = time in years

m = number of times per year compounded

n = interest rate per year

Note that if t is in a different unit of time, then your other numbers should be in the same unit of time.

*Often the equation looks like `P=C(1+r/n)^(nt)`

Sources:
pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

y= a(1+n/m)^mt

Given formula `y=a(1+n/m)^(mt)`

This is exponential formula which we use in general in finance. When we invest or hold some commodity or using credit card etc.

y - it is compound amount either increase or decrease.

a - iniatial investment /holding

n - rate of increase or decrease in percent i.e ( n/100)

m- conversion period number of time it will compound

m=365 /366  in case of daily converstion in case of using credit cards .

m=1   yearly conversion  national GDP etc.

m=2   semi annually  in banking sector ( Indian Bank's)

m=4   quarterly    growth factor of industries etc.

t - no of years.

 

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