Calculating Compound Interest: January 2010 Cash Balance Analysis

[Astro]

Homework Statement

January 1, 2010, opening cash balance is $20590.17.
January 31, 2010, closing cash balance is $20626.89.
Interest rate is 2.1%.
Interest paid is $36.72.
(There are 31 days in January 2010).

I=interest
A=amount (ie. P+I)
P=principal
i=interest rate per compounding period
n=number of compounding periods

Homework Equations

Show how the interest was calculated.

The Attempt at a Solution

I=P(1+i)^n-P
=$20590.17(1+0.21/365)^31-$20590.17
=$36.75555504
=$36.75

$36.75 not= $36.72 x_X Where have I gone wrong? :S

 

[Char. Limit]

With the two answers just .03 apart... you didn't really.

Although...

Should that .21 be a .021?

 

[Mark44]

Is the interest compounded daily? You seem to be using that in your formula, but didn't mention it in the problem statement.

You have (1 +0.21/365) as part of your interest calculation. 2.1% is 0.021, not 0.21. I think this was a typo that you didn't make when you did your calculation.

You used 365 days. I tried it with 365.25 days, and got $.02 closer, to $36.73. Still off one cent.

 

[jacksonpeeble]

I didn't notice any fundamental problems with your calculation besides the previously-mentioned error in your percentage, which you must have fixed anyhow.

A year is almost exactly 365 days, 5 hours, 48 minutes and 46 seconds (don't worry - I didn't have that memorized, I had to look it up); maybe that would help you get an even closer answer?

You're certainly within a good margin of error, though.

 

[Astro]

Thank you for all your answers. :3

You're right, I did a typo with the percentage when I wrote it up on the forum but not when I calculated it. That still doesn't explain the difference, minor though it is.

But! I did find the answer here: http://www.cardratings.com/creditcardblog/moneysavingstips/2005/10/calculate-interest-on-savings-account.html

The formula I was using was completely wrong for the bank senerio I was thinking of.
This is because at a bank, in a savaings account, they calculate it daily and pay it monthly. The formula I used assumes that the frequency at which interest is calculated & deposited equals the number of compounding periods. At the bank, the frequency at which interest is calculated and frequency at which interest is deposited are not equal.

The correct formula is as follows:

Let r = interest rate per annum. (Here, r=2.1%)
Let f = frequency at which interest calculations are made (but not deposited).
Let capital N = equal the frequency at which interested is deposited. (In this example, ever 31 days)
Let n = still equals the number of compounding periods (in this example, 1, brecause we're only looking at January.)
Let A = Principal + I
Let P= principal

Note: f & N must be in the same unites. (In this example, days).


I=PrfN
=$20590.17(0.021)(1/365)(31)
=$36.72383745...
=$36.72 :)


If anyone knows the correct mathematical notiation for that calculation, please let me know what the proper symbols for the variables should be. I had to invent them. I'd rather adhere to what the common usage is.

I've come up with this formula in the interm:

A₁=P+PrfN

A₂=(P+PrfN)+(P+PrfN)rfN
=P(1+rfN)+P(1+rfN)rfN
=P(1+rfN)(1+rfN)
=P(1+rfN)²

A₃=P(1+rfN)²+P(1+rfN)²rfn
=P(1+frN)²+(1+rfn)
=P(1+rfn)³

.
.
.
Aⁿ=P(1+rfN)^N

Thank you! ^^