Question

Ashlyn made contributions to a Roth IRA over the course of 34 working years. His contributions averaged $3,100 annually. Ashlyn was in the 26% tax bracket during his working years. The average annual rate of return on the account was 6%. Upon retirement, Ashlyn stopped working and making Roth IRA contributions. Instead, he started living on withdrawals from the retirement account. At this point, Ashlyn dropped into the 12% tax bracket. Factoring in taxes, what is the effective value of Ashlyn’s Roth IRA at retirement? Assume annual compounding.

Answer 1

When, oh when, will question writers understand that an AVERAGE donation is just not enough information?!

1000 1000 1000 -- 3-year average is 1000
3000 0 0 -- 3-Year average is 1000
0 0 3000 -- 3-year average is 1000


We also don't know HOW those contributions were made. End of year, beginning, spread throughout?

Ignoring those glaring errors in the problem statement...

$3,100 annually
RoR 6%
Tax Bracket 26%
34 years

0.06*(1 - 0.26) = 0.0444

What's the accumulation over 34 years of 3100 annually at 4.44%? You tell me what your assumptions are as far as payment timing.

Answer 2

3100(1+4.44%)^34 ?

Answer 3

No, that's the accumulation of a single payment. You need 34 payments.

Payments at the beginning of the year.

3100(1+4.44%)^34
3100(1+4.44%)^33
3100(1+4.44%)^32
3100(1+4.44%)^31
3100(1+4.44%)^30
...
3100(1+4.44%)^1

Add all those up.

Answer 4

for the 1st one I got 13578

Answer 5

That is good, but you cannot do them one at a time. You need a better plan.

Some algebra

3100*1.0444 + 3100*1.0444^2 + ... + 3100*1.0444^34 =

3100*(1.0444 + 1.0444^2 + ... + 1.0444^34) =

Do you recognize that thing in the parentheses?

Answer 6

the dots ?

Answer 7

No, the hole thing. You should recognize the Geometric Series. Do you?

Answer 8

Oh yes I do

Answer 9

a = 1.0444
r = 1.0444
n = 34

What's the sum?

Answer 10

A(1+R)^(n) ?

Answer 11

No, that's just the one payment again. You need to be able to add up the geometric series. You may wish to track down an algebra book and brush up on this skill. It will serve you wall in this course.

(1.0444 + 1.0444^2 + ... + 1.0444^34) = \(\dfrac{1.0444 - 1.0444^{35}}{1 - 1.0444}\)

Do that arithmetic and let's see what you get.

Answer 12

Some guy messaged me saying the answer is $347,232.31

Answer 13

I might be able to help you with this. But I need to know what kind of algebra you are in, because there are different equations for figuring out compounding interest, depending on how exact you need the answer to be. Can you tell me what the equation is that you have learned for this problem?

Answer 14

It's Financial Algebra

Answer 15

So does this formula look familiar? A=P(1+r/n)^nt
A is the final amount; P is starting principal; r=interest rate; n=number of times compounded; t=time in years.

Answer 16

@Endo T=12 ?

Answer 17

Well since he invested for 34 years, t would be 34. The n would be 12 if it were compounded monthly, but since it's compounded yearly, just use 1 for n.

Answer 18

1.0444=P(1+1.0444/12)^(12*34)

Answer 19

or A=1.0444(1+1.0444/12)^(12*34)

Answer 20

I'm approaching this problem differently than tkhunny did. I'm suggesting we figure out what the average amount is that he invested after paying 26% taxes, so use 3100 - (3100*.26) = 2294.

Answer 21

And plug that into the compound interest equation: A=2294(1+0.06/1)^1*34

Answer 22

The use of the 1s in the compound interest equation is just for clarity in case you have to figure out a question where it is compounded monthly or daily instead of yearly. Does that make sense? So if you had to figure out interest compounded monthly you'd use A=P(1+0.06/12)^12*34

Answer 23

http://www.wolframalpha.com/input/?i=A%3D2294(1%2B0.06%2F1)%5E1*34&t=crmtb01&f=rc

Answer 24

I might be approaching this wrong. Did you use a formula in the class for this question? And why would they give you the tax rate after retirement of 12% when you don't have to pay taxes on IRAs after retirement? Is that just to see if you know about Roth IRAs?

Answer 25

I'm using this formula

Answer 26

Oh duh. I know where we're wrong. We need to do what tkhunny said, because this isn't just one payment left in the account for 34 years.

Answer 27

how we do that o.O

Answer 28

lol. Good question.

Answer 29

what a= ? r=? equal to

Answer 30

a = the initial payment and r = the interest rate

Answer 31

Let me see if I get the same answer for doing this tkhunny's way and the way I was thinking.

Answer 32

If you don't worry about the 26% interest thing, just figure out the compounded interest, you will use this formula: 3100((1.06^34-1)/(1.06-1)

Answer 33

So the equation above gives an answer of 322,969.64 ... But that's not the answer because you didn't give 3100 each year. Instead, you gave 2294, which is 3100 minus the 26% taxes. So plug in the same equation with 2294 as the payment—that's 238,997.53

Answer 34

http://www.wolframalpha.com/input/?i=3100((1.06%5E34-1)%2F(1.06-1)&t=crmtb01&f=rc

Answer 35

Yeah that's what I got.

Answer 36

But change 3100 to 2294

Answer 37

The way tkhunny was suggesting is to figure out a lower interest rate instead of changing the 3100 payment. That method might be more sophisticated. I don't know what your teacher wants you to do.

Answer 38

http://www.wolframalpha.com/input/?i=2294%28%281.06%5E34-1%29%2F%281.06-1%29

Answer 39

Yeah that's what I got. I hope you're not more confused. Maybe somebody more experienced will chip in to give us a better answer.

Answer 40

The question is pretty badly written anyway. Why give tax rates before and after if you don't need to worry about tax after retirement?

Answer 41

$295,565.64

$308,213.64

$347,232.31

$310,321.64

Answer 42

I'm going to go with $308,213.64

Answer 43

OK. So those are your four choices?

Answer 44

yea

Answer 45

FVOA=C*((1+i)^(n*t)-1))/(i)

Answer 46

@Endo

C=3,100
n= 1
t= 34

Answer 47

http://www.wolframalpha.com/input/?i=3%2C100*%28%281%2B0.26%29%5E%281*34%29-1%29%29%2F%280.26%29

Answer 48

http://www.wolframalpha.com/input/?i=0.3698283+-+3.08190250+

Answer 49

I times 0.12 * 3.08190250

Answer 50

then subtracted 0.3698283 - 3.08190250

Answer 51

At 4.44% $246,468.43
At 6.00% $342,347.82 - Whoops! Forget to pay taxes.