harhit makes regular deposits of $375 each month into a savings plan with a fixed APR of 4.2% compounded monthly. What is the balance of his account after 15 years?

Guest Jun 12, 2014

#6**+5 **

Actually we are all equally correct I think. The question does say when the first deposit is made!

If it is made at the begining of the first month then CPhill is correct (Present value of an annuity due question)

If it is made at the end of the month then I am correct. (present value of an ordinary annuity problem)

CPhill answer is probably preferable.

Thanks Chris and Alan for making me think about it.

Melody
Jun 13, 2014

#1**+5 **

I'll attempt to derive a "formula" for calculating this......

First, let's assume that all deposits are made at the first of the month (an "annuity-due")

Let "d" be the monthly deposit and let r be the geometric difference between successive terms of the sums in the annulty calculation. So, the sum of this annuity is given by:

S = [dr + dr^2 + dr^3 + ..... + dr^(n-1) + dr^(n) ] where n is the number of months over the life of the annuity

Now, multiply this sum by (1/r) =

(1/r)S = (d + dr + dr^2 + .... + dr^(n-2) + dr^(n-1) ]

Subtracting the second sum from the first, we have

(1- 1/r)S = [-d + dr^(n)]

(1 - 1/r)S = d [r^(n) - 1]

S = d [ r^(n) -1 ] / [ 1 - 1/r ]

S = d*(r) [ r^(n) -1 ] / [r - 1]

So.....d = 375 and r = (1 + .042/12) = 1.0035 and n = 180 ... So we have

S = 375(1.0035)[ 1.0035^(180) - 1 ] / [.0035] ≈ $94,136.88

That's my best attempt......!!!!!!

CPhill
Jun 12, 2014

#2**+5 **

This is the future value of an ordinary annuity problem

$$FV=R\times\frac{(1+i)^n-1}{i}$$

where

R=$375

i=0.042/12=0.0035

n=15*12=180

$${\frac{{\mathtt{375}}{\mathtt{\,\times\,}}\left({{\mathtt{1.003\: \!5}}}^{{\mathtt{180}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{0.003\: \!5}}}} = {\mathtt{93\,808.549\: \!983\: \!472\: \!642\: \!600\: \!2}}$$

$93808.55

That's my answer anyway.

Melody
Jun 13, 2014

#3**+5 **

The difference between Chris and Melody is just because of when they assume interest is added. See:

Alan
Jun 13, 2014

#5**+5 **

I was only talking about when the deposit was made....at the first of the month is called an "annuity due" - if I remember correctly. Deposits made at the end of the month are known as an "ordinary" annuity. I was assuming the first situation....but the second could be assumed also...... it's not stated in the problem....

CPhill
Jun 13, 2014

#6**+5 **

Best Answer

Actually we are all equally correct I think. The question does say when the first deposit is made!

If it is made at the begining of the first month then CPhill is correct (Present value of an annuity due question)

If it is made at the end of the month then I am correct. (present value of an ordinary annuity problem)

CPhill answer is probably preferable.

Thanks Chris and Alan for making me think about it.

Melody
Jun 13, 2014

#7**0 **

If you save money in a bank, the bank uses Melody's method to decide how much to pay you back. If the bank loans you money it uses Chris's method to decide how much you must pay back.

No, ignore me; I'm just being cynical!!

Alan
Jun 14, 2014

#8**0 **

If you are repaying a loan then your first payment will be due at the end of the first payment period.

This is referred to as a "A present value of an ordinary annuity"

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If you are putting money in the bank at regular intervals then you would normally put the money in the bank at the beginning of the first time period.

what it grows to is referred to a "The future value of an annuity due"

CPHill used the annuity due method - his answer is a better. (Although the question did not state when the first deposit was made)

Melody
Jun 14, 2014