When one calculate the real invetment return (incorporating inflation), one often replaces the nominal return R by R - i
or, more correctly, by (R - i) / (1+i) ... where i is the annual inflation rate.

For example, a nominal return of 8% means R = 0.08.
Incorporating a 3% inflation rate means i = 0.03 so, the real return (in terms of buying power) can be estimated by 0.08 - 0.03 = 0.05 or 5% or (more correctly) (0.08 - 0.03) / (1+0.03) = 0.0485 or 4.85%.

When calculating the value of your portfolio with annual withdrawals increasing with inflation, we can do this:

  1. Let P1, P2, P3, etc. be the portfolio values after year 1, 2, 3, etc. with a starting portfolio of p0.
  2. Let R1, R2, R3, ... and I1, I2, I3, ... be the nominal returns and inflation rates during years 1, 2, 3, ...
  3. Then the withdrawals at the end of years 1, 2, 3 ... are W0(1+I1), W0(1+I1)(1+I2), W0(1+I1)(1+I2)(1+I3)  where W0 is the initial withdrawal.
  4. At the end of years 1, 2, 3, ... the portfolio values are:
  • [1]   P1 = p0(1+R1) - W0(1+I1)   so P1/(1+I1) = p0(1+r1) - W0   where (1+r1) = (1+R1) / (1+I1)
      Let p1 = P1/(1+I1) which is then the first year portfolio "in today's dollars".
      Note too, that we can solve (1+r1) = (1+R1) / (1+I1) to get the "inflation reduced" or real return as r1 = (R1 - I1)/(1+R1).
      Hence, [1] can be rewritten:
    [1A]   p1 = p0(1+r1) - W0

  • [2]   P2 = P1(1+R2) - W0(1+I1)(1+I2)   so P2/(1+I1)(1+I2) = p1(1+r2) - W0   where (1+r2) = (1+R2) / (1+I2)
      Letting p2 = P2/(1+I1)(1+I2)   again in "in today's dollars", we rewrite [2] as:
    [2A]   p2 = p1(1+r2) - W0

  • [2]   P3 = P2(1+R3) - W0(1+I1)(1+I2)(1+I3)   so ...
>Yeah, I get the idea. So?
The point I'm making is that, in determining the value of your portfolio in today's dollars, you can use the real, inflation reduced return and assume a constant withdrawal amount.

>Constant withdrawal amount? That's W0, right?
Right. So, if you're picking random nominal returns R, just change each random selection
to (R - i) / (1+i) and withdraw a constant amount.

>And do you have a spreadsheet for that?
No, of course not! However, here's an example of what I mean
The 20 years of annual returns are selected from a lognormal distribution with Mean Return of 8% and Standard Deviation 15% and an assumed annual inflation of 3% and the initial withdrawal rate is 4%. When one portfolio is reduced to $0, so is the other