Calculating the Compounding Benefit of Tax-Deferred Investing
Summary
The document explains an example comparing the future value of pre-tax and post-tax retirement contributions. It starts with a dollar contribution, a fixed tax rate, an annual return, and a retirement horizon. The post-tax amount is compounded directly; the pre-tax case is described as that compounded amount plus an adjustment for the value of deferring tax. The adjustment is derived by comparing the tax paid today with the same tax amount paid at the end of the investment period, then accounting for the after-tax portion.
The example illustrates how compounding can make tax timing affect accumulated wealth even when the stated investment return and risk are unchanged. Its calculation depends on simplifying assumptions: a constant tax rate and return, a single contribution, and deferred tax paid at the end of the horizon. It does not model account-specific rules, contribution limits, changing tax rates, withdrawals, or differences in when taxes are ultimately due, so the result is an illustration rather than a general retirement-account comparison.
Key ideas
- The example compares compounded values for pre-tax and post-tax contributions.
- The proposed tax-deferral adjustment is based on the future value of tax paid later rather than today.
- The calculation assumes a constant tax rate and investment return over the full horizon.
- Actual retirement-account outcomes can differ because tax rules and payment timing vary.
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Full text
# How is the pre-tax and post-tax return calculated?
# How is the pre-tax and post-tax return calculated?
I am looking at the lectures about advanced investments and in the first lecture of the series, the professor mentions,
To increase the return without bearing more risk one has to invest with pre-tax dollars via pension fund, 401k or individual retirement account.
- These accounts either let you to invest with salary before it is taxed or let you deduct your investment from your taxable income.
- You will be taxed eventually but always in the future. This is better than getting taxed now because you will have more of your money making money.
Then he shows example including,
If your tax rate is 25% and the annual return is 15% and you invest for 30 years before you retire then you will have \$62 for every dollar invested pre-tax rather than \$50 if you invest post-tax.
How do you calculate to get \$50 and \$62?
Here is a screenshot from the video with two other cases as well. The tax rate is 25% fixed.
## Answer by Magic is in the chain (score 2)
https://quant.stackexchange.com/a/46715
So this is what he has done:
Take pre-tax investment of $1. At 25% tax rate, 0.75 goes into the retirement account, which grows at 15% for 30 years:
$ 0.75*1.15^{30}=50 $
Then he applies the formula he has given just above the snapshot to compute the incremental amount:
$0.25*(1-0.25)*\left(1.15^{30}-1\right)=0.25×0.75×(1.15^{30}−1)=12$
And adding this 12 to 50 gives 62.
Re-derivation of the increment formula: $x*(1-x)*\left((1+R)^{T}-1\right)$, I am not familiar with the two types of 401k, but I would justify the formula as follows (let me know if I am missing something obvious!):
The difference on pre-tax 1 dollar in terms of the deferred taxation is: under one system you pay $1*x=x$ today, whereas under the other you pay $x$ at least after T years, which he assumes to be at T for simplification. So the difference between the two in terms of time value of money is $x(1+R)^T-x$. And you then multiply this by $1-x$ to calculate its after tax benefit:
$\mathrm{Value \,of \, the \,Benefit \, at \, time \,} T=(1-x)\left(x(1+R)^T-x\right)$
$=x(1-x)\left((1+R)^T-1\right)$
By the way this multiplication by $1-x$ seems to be the reason behind the 62 vs 66 that @amdopt mentioned in the comment.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.