Compound interest

• Jul 4th 2012, 05:09 AM
daigo
Compound interest
Quote:

A deposit of $25 is made at the beginning of the 1st month, and successive monthly deposits after that is$25 more than the previous month (2nd month is a $50 deposit, 3rd month is a$75, etc.). At the beginning of the next year (after 12 months), the deposit cycle is reset back to $25 the first month, etc. and this pattern continues for 5 years. The account pays 5% compounded interest monthly at the end of each month. What is the balance of the account after 5 years? So I'm trying to find out the formula by writing out the expanded version first and simplifying it, but I'm not sure how to write this in terms of exponents. Month 1:$\displaystyle 25 + 25(.05)$Let x =$\displaystyle 25 + 25(.05)$Month 2:$\displaystyle (x + 50)(.05) + (x + 50)$Month 3:$\displaystyle ((x + 50)(.05) + (x + 50) + 75)(.05) + ((x + 50)(.05) + (x + 50) + 75)$Month 4:$\displaystyle (((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) + (x + 50) + 100))(.05) + ((x + 50)(.05) + (x + 50) + 100)(.05) + ((x + 50)(.05) + (x + 50) + 100)$But then I remembered the formula will probably change after 12 months since the deposits start over, but the balance is different...so I'm not sure how else to really approach this. • Jul 4th 2012, 08:39 AM daigo Re: Compound interest So far, this is what I got for the first year:$\displaystyle (25m)(1 + \frac{.05}{12})(\frac{1 - (1 + \frac{.05}{12})^{12}}{{1 - (1 + \frac{.05}{12})}})$where m = the month... Not sure if this is correct, but do I need to make a new formula for each year to make up for the "new" principal in the account (the total from the preceding year[s])? • Jul 4th 2012, 09:23 AM Wilmer Re: Compound interest Daigo, I'll give you the account balance after 5 years: 11,012.44 Deposited is$1,950 per year, total $9,750 : 11012.44 - 9750 = 1262.44 is interest. The stuff you've shown makes no sense (sorry!), so I'm not going to try and show you how to get this by formula...why? Because I don't know where you're at with annuities. Here's a deal:$25 is deposited monthly for 12 months.
The 1st deposit is at end of 1st month.
The rate is 5% annual compounded monthly.
What is the accont balance at end of 12th month?

Show me the solution, and HOW you got it.
Then I'll have a better idea of how to help. How's that?!
• Jul 4th 2012, 10:32 AM
daigo
Re: Compound interest

Month 1: $0 principal +$25 deposited
5% compounded interest of $25 =$26.25

Month 2: $26.25 principal +$25 deposited
5% compounded interest of $51.25 =$53.8125

Month 3: $53.8125 principal +$25 deposited
5% compounded interest of $78.8125 =$82.753125

Month 4: $82.753125 principal +$25 deposited
5% compounded interest of $107.753125 =$113.140781

Month 5: $113.140781 principal +$25 deposited
5% compounded interest of $138.140781 =$145.04782

Month 6: $145.04782 principal +$25 deposited
5% compounded interest of $170.04782 =$178.550211

Month 7: $178.550211 principal +$25 deposited
5% compounded interest of $203.550211 =$213.727722

Month 8: $213.727722 principal +$25 deposited
5% compounded interest of $238.727722 =$250.664108

Month 9: $250.664108 principal +$25 deposited
5% compounded interest of $275.664108 =$289.447313

Month 10: $289.447313 principal +$25 deposited
5% compounded interest of $314.447313 =$330.169679

Month 11: $330.169679 principal +$25 deposited
5% compounded interest of $355.169679 =$372.928163

Month 12: $372.928163 principal +$25 deposited
5% compounded interest of $397.928163 =$417.824571

So after 1 year the total is $417.824571 with 5% compounded monthly interest if you make a$25 deposit each month

I might as well do the 1st year for my question too:

Beginning of month 1: $0 principal +$25 deposited
End of month 1: 5% compounded interest of $25 =$26.25

Beginning of month 2: $26.25 principal +$50 deposited
End of month 2: 5% compounded interest of $76.25 =$80.0625

Beginning of month 3: $80.0625 principal +$75 deposited
End of month 3: 5% compounded interest of $155.0625 =$162.815625

Beginning of month 4: $162.815625 principal +$100 deposited
End of month 4: 5% compounded interest of $262.815625 =$275.956406

Beginning of month 5: $275.956406 principal +$125 deposited
End of month 5: 5% compounded interest of $400.956406 =$421.004226

Beginning of month 6: $421.004226 principal +$150 deposited
End of month 6: 5% compounded interest of $571.004226 =$599.554437

Beginning of month 7: $599.554437 principal +$175 deposited
End of month 7: 5% compounded interest of $774.554437 =$813.282159

Beginning of month 8: $813.282159 principal +$200 deposited
End of month 8: 5% compounded interest of $1,013.28216 =$1,063.94627

Beginning of month 9: $1,063.94627 principal +$225 deposited
End of month 9: 5% compounded interest of $1,288.94627 =$1,353.39358

Beginning of month 10: $1,353.39358 principal +$250 deposited
End of month 10: 5% compounded interest of $1,603.39358 =$1,683.56326

Beginning of month 11: $1,683.56326 principal +$275 deposited
End of month 11: 5% compounded interest of $1,958.56326 =$2,056.49142

Beginning of month 12: $2,056.49142 principal +$300 deposited
End of month 12: 5% compounded interest of $2,356.49142 =$2,474.31599

And then at the beginning of year 2 the deposits reset:

Beginning of month 1: $2,474.31599 principal +$25 deposited
End of month 1: 5% compounded interest of $2,499.31599 =$2,624.28179

etc.
• Jul 4th 2012, 12:19 PM
Wilmer
Re: Compound interest
Quote:

Originally Posted by daigo
Month 1: $0 principal +$25 deposited
5% compounded interest of $25 =$26.25
..........
Month 12: $372.928163 principal +$25 deposited
5% compounded interest of $397.928163 =$417.824571

So after 1 year the total is $417.824571 with 5% compounded monthly interest if you make a$25 deposit each month

You're somewhat on right track...but:
Month 1 is $25 (no interest). You have to calculate interest using .05/12, not .05 : .05 each month means a 60% annual rate! Works out like this: Code: M DEPOSIT INTEREST BALANCE 0 .00 1 25.00 .00 25.00 2 25.00 .10 50.10 3 25.00 .21 75.31 4 25.00 .31 100.62 ... 12 25.00 1.17 306.97 Formula: d = 25 n = 12 i = .05/12 F = ? F = d[(1+i)^n - 1] / i = 306.97 • Jul 4th 2012, 12:34 PM daigo Re: Compound interest Wouldn't a 60% annual rate be the interest for the entire year? i.e. ($25 deposit per month x 12 months) + ($25 deposit per month x 12 months)(.6 interest) But anyway, trying to do the way you showed it: Month 2:$25 principal + ($25 principal * (.5/12 interest)) +$25 deposit = $51.0416667 Month 3:$51.0416667 + ($51.0416667 * (.5/12)) +$25 deposit = $78.1684028 I'm still not getting the numbers you are getting...what am I doing wrong? • Jul 4th 2012, 01:25 PM Wilmer Re: Compound interest Quote: Originally Posted by daigo Month 2:$25 principal + ($25 principal * (.5/12 interest)) +$25 deposit = \$51.0416667

Please be careful! .05/12, not .5/12
• Jul 4th 2012, 01:36 PM
daigo
Re: Compound interest
Never mind, I think I finally get what "compounding" actually means now. I had been thinking differently before.

But how do I make each deposit increment by +25 from the previous month and add it to the current month?