Portfolio fraction averaging

In the last couple of days, Michael James has been exploring variations of the 4% rule, and has proposed his own modified version, which simply states that you should spend 4% of your portfolio per year, and allow the resulting income to fluctuate.

Though this strategy as stated leaves some important questions unanswered, it does have a fundamental advantage over William Bengen's original 4% rule, which advises a constant withdrawal (in real terms) in dollars every year.

Bengen's advice amounts to dollar cost averaging in reverse: rather than periodically buying a fixed dollar value of assets, you periodically sell a fixed dollar value of assets.

The name "dollar cost averaging" refers to the fact that the average price you're paying for assets is determined by spending a fixed dollar cost for each purchase. The average price paid for the assets equals the total paid divided by the total units purchased, and in the case of dollar cost averaging, this works out to the harmonic mean of the purchase prices (though I'll spare you the mathematical details). This kind of mean makes DCA attractive while buying, and much less attractive while selling.

For example, consider just two purchases using dollar cost averaging, buying $1 of assets each time. For the first purchase, the asset cost 20 cents per unit, and for the second purchase, it cost 5 cents per unit. The first purchase gets 5 units, and the second gets 20, for a total of 25 units costing $2, which is 8 cents each.

Michael's plan, on the other hand, advises you to sell a fixed fraction of your portfolio each time period. This "portfolio fraction averaging" is, to a first approximation, much like selling a fixed number of asset units each time. (The diminishing number of units held is offset by new units purchased by reinvested dividends, so let's ignore both of these things for now.) The average price for each unit in such a scheme is the arithmetic mean of the individual sale prices.

Using the same example asset as before, consider just two sales using portfolio fraction averaging, where 4% of your portfolio amounts to one unit. The first sale would earn 20 cents, and the second sale would earn 5 cents. The two units together netted 25 cents, for an average of 12.5 cents each. Note that this is more than the 8 cent average we paid for these units. The arithmetic mean is always greater than the harmonic mean!

So, when you sell your assets, would you prefer to earn an average of 8 cents per unit, or 12.5 cents?

I thought so.