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One of the key decisions a retiree must make is how much to withdrawal each year. When faced with this decision, it would be helpful to know the maximum amount that can be taken each year without running out of money.

Well, the good news is that there is an exact formula for the Maximum Withdrawal Rate. All you need to know is the annual total annual real return for each year of the withdrawal period. The bad news is that the future annual returns for most investments are not known.

Maximum Withdrawal Rate (MWR) Defined
DThe Maximum Withdrawal Rate is the initial rate of withdrawal that depletes the portfolio balance to zero at the end of the period. The first year’s withdrawal start at the MWR and are increased each year with inflation to maintain constant real consumption. The process is similar to a 30-year fixed mortgage payment calculation, except that the rate is not fixed.

If your withdrawal rate is greater than the MWR,  there will be a shortage at the end. In other words, you will run out of money prematurely. If your withdrawal rate is less the the MWR, there will be a surplus at the end.

The Magic Sum
The formula to calculate the MWR makes use of a magic sum. This sum is a remarkable figure that turns up in many calculations. The magic sum needs two things: all the annual inflation rates and all the annual returns. Here the terms used in the magic sum are defined:

• i is an annual Inflation Rate. (Example: if inflation is 3%, then i = 0.03)
• I = 1 + i is an Inflation Factor. (Example: 1.03)
• r an Annual Return for the portfolio. (Example: Annual Return is 8.9% so r = 0.089)
• g = 1 + r is the corresponding Gain Factor. (Example: g=1.089 means a $1.00 investment grows to $1.089 over the year)
• For years 1, 2, 3, … up to year N, we’ll call the inflation rates
  i₁, i₂, … i_N and similarly for the Annual Returns and Factors etc.
• G_n is the cumulative Gain Factor over n successive years, so G_n is the product of a bunch of annual Gain Factors, like: G_n = g₁g₂g₃…g_n

Finally, here is the formula for calculating the Magic Sum:

Formula from Magic Sum on gummy-stuff.org

Then the MWR is simply the inverse of the Magic Sum:

From Sensible Withdrawal Rates article on gummy-stuff.org

This formula is not as bad as it looks and is actually easy to program into an Excel spreadsheet. You can see how the MWR only depends on the total annual returns and the annual inflation rates.

[Notice that whether or not the returns come from dividends, interest or capital gains is irrelevant to calculating the maximum withdrawal rate. See previous article titled Eating Seed Corn.]

What Can be Learned From This Equation
First of all notice that there is a term for each year. So with each additional year another term is added and the sum is larger larger. Since the MWR is the inverse, the inference is that the MWR goes down with more years.

Second notice that the first year’s return is in every sum. The last year’s return is only in the last sum. We can infer that the early years have the most weight and are the most important. Further out years become less important.

So What Good Is It?
As mentioned at the top, each annual return must be known to calculate the MWR. We know this for past years, so we can calculate historical MWR for various portfolios.

This has some utility for comparing how various portfolios fared in past markets. For instance, we can see how various mixes of stocks and bonds would have served in the past. We may also be able to draw some general conclusions about constructing retirement portfolios by looking at MWR’s over past periods.

Back To The Future
We are still left with our original problem, which is how much to start withdrawing today? Future returns are unknown so you can’t just use this formual directly to find the maximum withdrawal rate. However, it is possible to play “What If?” and generate possible future scenarios. For instance, annual returns can be generated from a probability distribution and the MWR calculated for some hypothetical sequence of returns.

I’m going to be using the formula in future articles to compare various portfolios and withdrawal strategies.

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