Taxes in TEA and TDA: pre- and after-tax contributions?

Janet Lowell is a financial advisor meeting with a new client, Jim Harris. Harris is subject to the following tax rates: interest (25%), dividends (15%), short-term gains (25%), and long-term gains (20%). Short-term capital gains include realized capital gains on investments held for less than one year. Harris’s objective is to minimize taxes payable whenever possible.
Harris has $700,000 of cash that he recently inherited. Based on her recommended asset allocation with annual rebalancing, Lowell estimates that Harris could earn an average annual pre-tax return of 8% if the portfolio is invested in a balanced mix of stocks and bonds. Harris has an investment horizon of 20 years.
Lowell also addresses the tax implications of asset allocation. Because Harris has never contributed to a tax-deferred account (TDA), the tax laws in his jurisdiction permit him to contribute up to $450,000 of funds into a TDA immediately in the current year. An immediate tax deduction would be provided to Harris for the contribution to the TDA and he will reinvest the entire tax refund from his tax deduction in the TDA. Therefore, Harris is considering investing half of his $700,000 in a TDA and the other half in a taxable account. The relevant tax rates for both accounts are assumed to be 20%.
In the following year, Harris earns $120,000 of pre-tax employment income and is subject to a flat income tax rate of 20%. He is permitted by the tax laws to set aside one-third of the pre-tax income to invest in either a TDA or a tax-exempt account (TEA). Both accounts are estimated to have an average annual pre-tax return of 8%.

Calculate the amount that Harris will have in his investment account at the end of his investment horizon if in the current year he invests the entire $700,000 and reinvests his entire tax refund in the TDA.
TDA: ($350,000 / (1 – 0.20)] × (1 + 0.08)20(1 – 0.20) = $1,631,335
Taxable account: $350,000 × (1 + 0.08(1 – 0.20))20 = $1,210,321
Total: $1,631,335 + $1,210,321 = $2,841,656

Assuming that Harris sets aside one-third of his pre-tax employment income in the following year, he should most likely contribute his funds to:
A) the tax-deferred account (TDA).
B) the tax-exempt account (TEA).
C) either the TDA or the TEA.
Superficially, it appears that the TEA is superior to the TDA because there is no tax due on withdrawals. However, the fact that TEA contributions are made with after-tax funds and TDA contributions are made with pre-tax funds is ignored.
For example, taking into account the immediate tax deduction for the TDA, we can calculate the value of the contribution at the end of 20 years for the TDA as:
TDA: [$40,000 × (1 – 0.20) + ($40,000 × 0.20)] × [(1.08)20(1 – 0.20)] = $149,151
For the TEA, the $40,000 of pre-tax funds is subject to 20% tax and then everything else afterward is tax-free:
TEA: $40,000 × (1 – 0.20) × (1.08)20 = $149,151

TEA contributions are made with after-tax funds and TDA contributions are made with pre-tax funds - is this always the case?


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Now I’'m struggling with this… namely, OK, fully understood, TDA uses pre-tax funds/returns. But, in this case, I thought the 350k is the before-tax. Why do we need to do another “pre-tax” and divide by 0,80? Thanks a lot!