# 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.

• Are you sure it is 62 and not 66? 1.15^30 is ~66. 75% of 66 is ~50 – amdopt Jul 19 '19 at 14:35
• @amdopt Yes very sure, added screenshot from the video as well with two other cases. – user13892 Jul 19 '19 at 15:29
• @amdopt added the link to video snippet as well. – user13892 Jul 19 '19 at 17:10
• The video provide the formula for calculating the difference but I can't figure out how to get the pre and after tax values. – user13892 Jul 19 '19 at 18:31

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. • If you apply the same logic for the case of 20 year investment then$0.75*{1.1}^{20}=\$5$ but he shows the value as $\$4\$. – user13892 Jul 19 '19 at 20:45
• Yes if you can show the derivation of the increment formula that would be helpful. – user13892 Jul 19 '19 at 20:47
• That 4 is the only one that does not match right? – Magic is in the chain Jul 19 '19 at 21:27
• And have added some thoughts around the derivation. Why is it called advanced investment btw? – Magic is in the chain Jul 20 '19 at 11:22
• Advanced probably because this course will be covering pricing in general including derivative pricing using risk neutral valuation and other complex stuff. Unfortunately I got stuck in the most basic stuff 😥. I am hoping he will present the ideas more intuitively since I never was able to explain complex ideas in plain English and my professors were mathematicians not economist or finance professionals so butchered the financial intuition. Hopefully I will have more intuitive ideas about instruments and their use after this course. While I expect this course will be butchering mathematics. – user13892 Jul 20 '19 at 14:13