Future value of annuities

[hannah19142]

Hi there! I'm stuck on this problem about future values of annuities:

You are planning to make annual deposits of $4,680 into a retirement account that pays 8 percent interest compounded monthly. How large will your account balance be in 25 years?

I guess what I'm confused about is how you incorporate annual payments with monthly compounding. I tried altering my calculations to put everything in terms of months, without much luck. I made N=300 (12 months x 25 years), I= 8/12 (eight percent annual interest/12 months) and the payment amount = $4680/12 (annual payment/ 12 months). This gave me $370,900.29, which is wrong.

Is the payment amount correct? It seems like it should be a $4680 payment every year as opposed to a $390 payment every month, since the compounding will alter the total. I'm just not sure how to make the payment annual when everything else is in terms of months.

Thanks!

 

[tkhunny]

You will never be stuck if you understand "Basic Principles". No one said to make monthly payments. Why did you think that was a good idea?

If interest is 8% annually, compounded monthy, you have...

i = 0.08 -- Annual Nominal Interest Rate

j = 0.08/12 = 0.00666666... -- Monthly Interest Rate

(1+j) = 1.0066666.... -- Monthly Accumulation Factor

\(\displaystyle r\;=\;(1+j)^{12}\) = 1.0829995068... -- Annual Accumulation Factor

Now you build it.

\(\displaystyle P = 4680\) -- Your annual payment.

\(\displaystyle P\cdot (r^{25} + r^{24} + r^{23} +\;...\;+\;r)\) OR \(\displaystyle P\cdot (r^{25} + r^{24} + r^{23} +\;...\;+\;r + 1)\)

You'll have to decide what the problem statement means. Do you make a deposit at the end of the 25 years (the second one) or don't you (the first one)?

I'll just take the second, assuming there IS a final payment at the end.

\(\displaystyle P\cdot (r^{25} + r^{24} + r^{23} +\;...\;+\;r + 1)\;=\;P\cdot \frac{r^{26} - 1}{r - 1}\) -- You should be able to do this in your sleep. It is a primary defense mechanism against ever being stuck again.

Now, you show me the version WITHOUT the final payment.