Question

Lauren will make annual contributions in the amount of $4,120, on average, to a 401(k) over the next 32 years. Her employer will match 85% of her contributions. She is currently being taxed at 33%, but anticipates being taxed at 15% upon retirement. If her account grows at an average rate of 4.9% annually, what is the value of Lauren’s 401(k) upon retirement?

Answer 1

1) Why do we care about the tax rate AFTER retirement? It is smoke to confuse and not needed to answer the problem statement.

2) Average contributions and average earnings are just not good enough to provide a particularly good answer. I really do not understand the propensity to write such questions. It requires us to make broad, sweeping assumptions that are unlikely to have ANYTHING to do with reality.

3) Why do we care about the tax rate during accumulation? 401(k) contributions are BEFORE taxes and the tax rate is of no consequence. It's possible there is a rule that I'm overlooking. Maybe the 1st $2000 is tax free and anything in excess is subject to tax? You'll have to sort that out.

So, then, let's assume her $4,120 goes in at the END of each year.
The employer's contribution is $4,120*0.85 = $3,502
Total Contribution at the end fo each year is $4,120 + $3,502 = $7,622
Those are pretty hefty contributions.

Now, some definitions:

i = 0.049 -- Annual interest rate
r = (1+i) = 1.049 -- Annual accumulation factor

That's it. We're ready to build from BASIC principles.

Starting from the last payment, on the day of retirement.

\(7622 + 7622r + 7622r^{2} + 7622r^{3}+...+7622r^{31} = 7622(1 + r + r^{2} + ... + r^{31})\)

The sum of the expression in the parentheses is \(\dfrac{1-r^{32}}{1-r} = \dfrac{r^{32}-1}{r-1} = \dfrac{r^{32}-1}{i}\)

I get somewhere in the neighborhood of $½ Million.