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.

A Canadian's random thoughts on personal finance

## Dec 15, 2009

## Dec 11, 2009

### Is the 4% rule good advice?

If you've spent any time reading personal finance blogs, you must have come across the "4% rule". It is quoted with impressive regularity, yet it is rarely explained, and even more rarely questioned.

The 4% rule was first developed in a paper by William Bengen. Here is the 4% rule from the horse's mouth:

What is Bengen's advice for asset allocation?

Allan is retiring in March 2009. Barney retired in March 2008. At the time, Barney's nest egg was $1.3M. He dutifully withdrew $52k for the year.

What advice do we give these two men in March 2009?

We advise Allan to withdraw $40,000 (which is 4% of his portfolio), but we advise Barney to withdraw $53,500 (the previous withdrawal adjusted for inflation) despite being in precisely the same financial situation. To me, this is inherently irrational, but it's hard to offer better advice to poor Barney, who would be understandably upset at the prospect of decreasing his annual income by 20%.

To understand this inefficiency, first consider that if you kept your nest egg as cash under your mattress, you could safely withdraw 4% per year for 25 years with zero risk of running out of money during that time. Suddenly, investing in a mix of stocks and bonds to achieve a probable 33 years of withdrawals is not so impressive anymore.

Of course, your mattress doesn't compensate you for inflation, but this is easily fixed using inflation-protected government bonds. With a typical real return of 2%, Scott et al argue that the retiree could finance 30 years of guaranteed inflation-adjusted withdrawals for the price of just 22.4 years. This amounts to a withdrawal rate of 1/22.4 = 4.46%!

What gives? How could stogy old inflation-protected bonds offer a withdrawal rate 11% higher than that of the stocks-and-bonds portfolio?

The answer is that choosing the bonds over the market portfolio gives up both the potential upside and downside of the market. Because the market generally increases over time, the upside over a 30-year time frame is much higher than the downside risk, and the market values this at approximately 11% of the withdrawals per year.

The bond investor has also accepted the certainty that his money will be gone after year 30. The stocks-and-bonds investor likely still has money left to continue his withdrawals indefinitely, and extremely likely still has enough at least for years 31-33.

This is what the inflation-protected bond investor has given up in exchange for a 11% higher withdrawal rate and zero risk.

#### What is the 4% rule?

Many would summarize the 4% rule as follows:You can safely withdraw 4% of your retirement savings each year in retirement without risk of running out of money.This statement is not only vague; it's inaccurate. It doesn't say anything about how the retirement savings is invested, or how to cope with inflation, or just how much risk there is of running out of money.

The 4% rule was first developed in a paper by William Bengen. Here is the 4% rule from the horse's mouth:

For a client just beginning retirement, determine first the "safe" withdrawal rate. … For a client of age 60–65, this will usually be about 4 percent. The withdrawal dollar amount for the first year (calculated as the withdrawal percentage times the starting value of the portfolio), will be adjusted up or down for inflation every succeeding year. After the first year, the withdrawal rate is no longer used for computing the amount withdrawn; that will be computed instead from last year's withdrawal, plus an inflation factor.So this is not a withdrawal rate of 4% per year. It's a withdrawal rate of 4% in the first year, adjusted for inflation thereafter.

What is Bengen's advice for asset allocation?

Despite advice you may have heard to the contrary, the historical record supports an allocation of between 50-percent and 75-percent stocks as the best starting allocation for a client. For most clients, it can be maintained throughout retirement, or until their investing goals change. Stock allocations below 50 percent and above 75 percent are counterproductive.And just how safe is this strategy? Bengen back-tests the strategy using market performance and inflation numbers between 1926 and 1992. With a 50/50 split between stocks and bonds, Bengen concludes:

Assuming a minimum requirement of 30 years of portfolio longevity, a first-year withdrawal of 4 percent, followed by inflation-adjusted withdrawals in subsequent years, should be safe. In no past case has it caused a portfolio to be exhausted before 33 years, and in most cases it will lead to portfolio lives of 50 years or longer.Of course, the merits of conclusions based on back-testing can be questionable, but at least we know exactly where the 4% rule comes from.

#### Does this make sense?

The main problem I have with this advice is that it breaks what I consider a very reasonable soundness test: two people in identical situations should be given the same advice. However, consider Allan and Barney, both 65 years old with $1M in 50/50 stocks and bonds, and both with the same actuarial risks (both married, non-smokers, living in the same part of the country, etc.). Suppose inflation is 3%.Allan is retiring in March 2009. Barney retired in March 2008. At the time, Barney's nest egg was $1.3M. He dutifully withdrew $52k for the year.

What advice do we give these two men in March 2009?

We advise Allan to withdraw $40,000 (which is 4% of his portfolio), but we advise Barney to withdraw $53,500 (the previous withdrawal adjusted for inflation) despite being in precisely the same financial situation. To me, this is inherently irrational, but it's hard to offer better advice to poor Barney, who would be understandably upset at the prospect of decreasing his annual income by 20%.

#### Can we do better?

A subsequent paper by Scott, Sharpe, and Watson addresses the question of whether Bengen's strategy is the most cost-efficient way to ensure you get a reliable cash flow during retirement. They identify two sources of inefficiency: one is a relatively minor one that they address using option trading; I find that one fairly uninteresting because it only amounts to 2-4% of your savings, and recouping that requires some fairly esoteric financial maneuvering. However, the more interesting inefficiency is fairly straightforward and accounts for 10-20% of the portfolio's value.To understand this inefficiency, first consider that if you kept your nest egg as cash under your mattress, you could safely withdraw 4% per year for 25 years with zero risk of running out of money during that time. Suddenly, investing in a mix of stocks and bonds to achieve a probable 33 years of withdrawals is not so impressive anymore.

Of course, your mattress doesn't compensate you for inflation, but this is easily fixed using inflation-protected government bonds. With a typical real return of 2%, Scott et al argue that the retiree could finance 30 years of guaranteed inflation-adjusted withdrawals for the price of just 22.4 years. This amounts to a withdrawal rate of 1/22.4 = 4.46%!

What gives? How could stogy old inflation-protected bonds offer a withdrawal rate 11% higher than that of the stocks-and-bonds portfolio?

The answer is that choosing the bonds over the market portfolio gives up both the potential upside and downside of the market. Because the market generally increases over time, the upside over a 30-year time frame is much higher than the downside risk, and the market values this at approximately 11% of the withdrawals per year.

The bond investor has also accepted the certainty that his money will be gone after year 30. The stocks-and-bonds investor likely still has money left to continue his withdrawals indefinitely, and extremely likely still has enough at least for years 31-33.

This is what the inflation-protected bond investor has given up in exchange for a 11% higher withdrawal rate and zero risk.

#### So what should we do?

The best course of action may be a combination of the two. Buy inflation-protected bonds to fund 30 years of living expenses based on your budget, and use the remainder of your portfolio to invest in stocks and bonds, which could improve your standard of living, fund your expenses after 30 years, and leave an inheritance for your family.
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