 While the 30-30 Withdrawal Strategy is a nice clean method for withdrawing money from your portfolio, it does require you to keep track of a buffer which drives the calculation for your withdrawal amount every year. I felt like there should be a way to calculate the withdrawal in terms of only your current withdrawal amount and your current portfolio value thus alleviating the need to keep track of a separate spending buffer. After playing with the math, it turns out there is a pretty simple formula that achieves this goal.

## A Bit of Math

The amount that is initially spent from your portfolio is determined directly by multiplying your withdrawal rate by your portfolio value. Using the same variables as the Safe Withdrawal Rates for Vampires article,

``````W = Pw
``````

where,

``````W = annual withdrawal amount
w = annual withdrawal rate
P = portfolio value
k = buffer size in years
``````

And if we have a buffer of `k` years, then the total buffer size initially is,

``````initial buffer size = kW
``````

After a year of spending from the buffer, it is reduced by the withdrawal amount, `W`. So we are left with,

`````` buffer after 1 year of spending = kW-W = W(k-1)
``````

At this point, we add back an amount equal to the new portfolio value times the withdrawal rate to arrive at the new buffer size,

`````` updated buffer value after 1 year = W(k-1) + Pw
``````

And to compute the new withdrawal amount, we just divide this amount by the size of the buffer in years (`k`),

``````new annual withdrawal amount = W(k-1)/k + Pw/k
``````

Because I prefer to think of my withdrawals as monthly, let’s rewrite this in terms of monthly withdrawals,

``````new monthly withdrawal amount = (W/12)(k-1)/k + Pw/12k
``````

Given that `k` and `w` are constant values, we now have a formula for computing our new withdrawal amount from the current monthly withdrawal (`W/12`), and the current portfolio value (`P`).

## Example Using 30-30 Numbers

Let’s put some numbers to this to make it less abstract. Start with a portfolio of `P = \$720,000` and go with the 30-30 Withdrawal Strategy numbers for buffer size (`k = 2.5 years`), and withdrawal rate (`w = 3.33%`). For these values, the initial monthly withdrawal is simply,

``````Pw/12 = \$720000*0.0333/12 = \$2000/month
``````

If you want, you can set aside your buffer of 2.5 years (\$60,000) in a safe place. But the beauty of this strategy is that you don’t have to do so. You can also just assume that your buffer is fully invested within the bond portion of your portfolio; it’s up to you. This is why I call it a “virtual” buffer because you get the advantages the buffer gives you without having to keep track of it.

Now, let’s assume a year has passed and it has been a good year and the portfolio has gone up to \$765,000. Now your monthly withdrawal amount is computed as,

``````(W/12)(k-1)/k + Pw/12k = \$2000(1.5/2.5) + \$765000*.03333/(12*2.5))
= \$2000 * 0.6 + \$765000/900
= \$1200 + \$850
= \$2050
``````

So even though the portfolio increased by 6.25% (`765000/720000`), the monthly spending only went up 2.5% (`2050/2000`) due to the virtual 2.5-year buffering.

Also, look how simple the calculation is every year. In this example, multiplying your current income by 0.6, divide your portfolio by 900, and add those 2 numbers together. Pretty simple right?

## Making it Even Simpler

As easy as this calculation is, I couldn’t help but noticing that dividing by 900 is a little awkward. What if we changed the buffer size so that we ended up dividing by 1000? If we can divide by 1000 then the calculation can almost be done in your head. To state this mathematically, we want,

``````12k/w = 1000
``````

If the annual withdrawal rate is fixed, then we solve for the buffer size to get,

``````k = 1000w/12
``````

Substituting this back into our withdrawal equation,

``````new monthly withdrawal amount = (W/12)(1-0.012/w) + P/1000
``````

Revisiting the first example with a portfolio of \$720,000 and a withdrawal rate of 3.333%, the first year withdrawal is still \$2000/monthly because the withdrawal rate is the same. After the first year with a new portfolio value of \$765,000, the new monthly withdrawal now becomes,

``````(W/12)(1-0.012/w) + P/1000 = \$2000(1-0.012/0.03333) + \$765000/1000
= \$2000 * 0.64 + \$765000/1000
= \$1280 + \$765
= \$2045
``````

So, you can see that the slightly modified equation only decreased the income by \$5/month. This decrease is due to a slightly larger buffer (2.778 years vs 2.5 years) which means that the market gyrations are smoothed a little more than before.

## The Super-Simple Virtual-Buffer Withdrawal Formula

To summarize all the above math, in the first year of withdrawals, you withdrawal from your portfolio by simply multiplying your portfolio value by your withdrawal rate.

``````initial monthly withdrawal = Pw/12
``````

Then, after each year you compute your new withdrawal amount with this super-simple formula.

``````new monthly withdrawal = M * beta + P/1000
``````

where,

``````M = previous monthly withdrawal
beta = monthly withdrawal multiplier = 1 - 1.2%/w
w = annual withdrawal rate
P = current portfolio value
``````

The graph below shows the monthly withdrawal multiplier as a function of annual withdrawal rate. If you hover over the chart you will see other information including a “Sample Allocation” which shows how you might need to allocate more to stocks as your withdrawal rate goes up.

## Sample Portfolios

Here are a few possible portfolio types you could choose from depending on your risk tolerance. The Step-By-Step section that follows walks through a simple example using the “moderate” strategy numbers.

annual withdrawal rate monthly withdrawal multiplier virtual buffer size
conservative 2.40% 0.5 2.00 yrs
moderate 3.00% 0.6 2.50 yrs
moderately aggressive 3.33% 0.64 2.78 yrs
aggressive 4.00% 0.7 3.33 yrs

## Step-By-Step

Implementing the Super-Simple Virtual-Buffer Withdrawal strategy is as follows:

1. Compute your initial monthly withdrawal from the equation `Pw/12`. For example, with a \$400,000 portfolio (`P`) and a 3% withdrawal rate (`w`), your monthly withdrawal is,

`\$400,000*0.03/12 = \$1000`

2. Withdrawal from the portfolio for a full year using the last computed value.

3. At the end of each year, compute your new monthly withdrawal rate using the equation `M * beta + P/1000`. For example, with a 3% withdrawal rate (`w`), your beta multiplier is `1 - 1.2%/3% = 0.6`. If your portfolio (`P`) has increased to \$420,000 after the first year, then your new withdrawal is,

`\$1000*0.6 + \$420000/1000 = \$600 + \$420 = \$1020`

4. Go back to step 2 and repeat.

Some readers may have noticed that the formula for this withdrawal strategy looks an awful lot like an exponential smoothed average. If we rewrite the withdrawal formula like this,

``````M(t) = (1-alpha) * M(t-1) + alpha * Pw/12
``````

where

``````alpha = 1.2%/w
``````

then you can see that it is indeed an exponential moving average of the term `Pw/12`. That term represents the annual withdrawal rate projected down to the monthly level.

For exponential smoothed averages, a smaller `alpha` represents an average that has a larger smoothing effect. In the case of this withdrawal strategy, the smaller `alpha` corresponds to larger withdrawal rates which means that larger withdrawal rates will have larger smoothing effects.

## Parting Thoughts

Albert Einstein has been attributed with the saying that “everything should be made as simple as possible, but not simpler”. The super-simple virtual-buffer withdrawal strategy is simple and requires no bookkeeping other than checking your portfolio value once a year. Is there a simpler strategy? Yes, you could simply take a constant value out without checking your portfolio balance. But such a strategy could be risky since it ignores the possibility of a declining portfolio balance.

The super-simple strategy also has flexibility which comes from the fact that the buffer is “virtual”. Whether or not you keep an actual buffer is up to you. Generally it’s a good idea to keep a couple years or so of spending outside of your investments so that you know it will be there when you need it. But this strategy doesn’t require you do so, it’s up to you.