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I am trying to wrap my head around the proper way to do this. I would like to simulate the portfolio value adjusted for inflation with a fixed withdrawal rate.

To simulate withdrawal rate, I will need to adjust my portfolio nominal return series to real return using CPI. After getting real return series, assume that my fixed withdrawal amount is 5% of initial equity. If my initial equity is, say, $1, then my withdrawal rate is 0.05. Since my simulation is real return based, do I need to adjust my fixed withdrawal rate for inflation or do I just keep it fixed at 0.05 and withdrawal this amount each period?

If I do adjust it, I am using the following equation: if inflation is 1% then current withdrawal rate = 0.05 + (0.05 * 0.01) = 0.0505

Which is correct?

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Let real wealth at time $t$ be defined as $W_{t}^{R}\equiv\frac{W_{t}^{N}}{P_{t}}$ where $W_{t}^{N}$ is nominal wealth and $P_{t}$ is the price level indexed to one at the initial period. You want to withdraw a $x_{t}$ percent of real wealth. This would give $$x_{t}W_{t}^{R}=x_{t}\frac{W_{t}^{N}}{P_{t}}$$.You could then consider a withdrawal rate in nominal terms $y_{t}\equiv\frac{x_{t}}{P_{t}}$ (ie. that you multiply by the nominal wealth) that would effectively mimic the real withdrawal rate. When wealth grows faster than inflation, the nominal withdrawal rate should decline and vice-versa.

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