We consider the financial aspects of putting money INSIDE a tax-deferred investment plan (wherein you don't pay taxes on the dollars invested, but pay when the money is withdrawn, at retirement), and investing OUTSIDE such a plan (where you invest with after-tax dollars and pay taxes on capital gains). In each case, we assume that at, retirement, you buy a fixed M-year annuity which lasts, hopefully, until you drop dead.

The variables are:

• A = annual investment (before taxes).
• R₁ = annual return on INSIDE investments.
• R₂ = annual return on OUTSIDE investments (after capital gain taxes, if any).
• N = numbers of years of investing.
• T₁ = tax rate while investing.
• T₂ = tax rate after retirement (while withdrawing).
• M = years of an annuity.
• r = capital gains tax-factor ... what fraction of T₂?

The magic formulas are (see the Math):

INSIDE after-tax income = A{(1+R₁)^N-1}/{1-(1+R₁)^-M}(1-T₂)

OUTSIDE after-tax income = A(1-T₁)[ {(1+R₂)^N-1}/{1-(1+R₂)^-M}(1-rT₂) + rT₂N/M ]

Annual Investment A = $ INSIDE return R1 = % OUTSIDE return R2 = % Years of investing N = Tax Rate while investing T1 = % (until retirement) Tax Rate while withdrawing T2 = % (after retirement) Years of annuity M = (years until annuity ceases) Capital gains tax-factor r = %     so capital gains tax rate is THIS percentage of T2 INSIDE after-tax income at retirement =     in today's dollars = OUTSIDE after-tax income at retirement =     in today's dollars = in today's dollars assumes a 3% inflation rate

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