Paradoxes and Confusion

Consider reversion to the mean. We can identify it easily when we use the term correctly. We can eliminate it entirely if we use alternative definitions or apply it to individual companies.

Gummy has posted tutorials based on standard definitions applied to individual companies.
Gummy's (Peter Ponzo's) web site
Gummy's Tutorial on Mean Regression

Reversion to the mean DOES exist. It is important. With care, we can come up with suitable, broadly based definitions. But if we fail to use care, we can end up stuck at a meaningless dead-end.

Behind the term reversion to the mean is the notion that stock prices are related to earnings. This is often a loose relationship. It even varies with time. What is important is the mechanism, the cause-and-effect relationship between prices and earnings.

Having a mean that varies with time (as well as other factors) causes problems. It should not. I am sure that television reporting adds to the confusion. It is NOT the MEAN that is important. It is the tendency for prices to fall within a range of price-to-earnings ratios that is important. Prices may drift to extremes, but the probabilities favor their returning to a reasonable range. If prices are exceedingly high, the likelihood of a further price increase is diminished and the likelihood of a price decrease is increased.

Viewed from this perspective, it makes sense that the standard deviation of S&P500 stock returns falls faster than the standard (1/[square root of the number of years N] ) rate that applies when the year-to-year probabilities are independent. Raddr’s precise definition in terms of the standard deviation after a selected number of years makes sense. It is helpful. It improves the realism of models.

Confusion about reversion to the mean is innocent. Confusion about other factors is not always innocent.

Repeatedly, I see confusion caused by the manner in which the conventional Safe Withdrawal Rate methodology was set up. The conventional methodology tried to make a simple presentation and it tried to avoid statistical issues. There are many underlying statistical issues and addressing them all in detail can be a continual distraction. But the attempt to avoid confusion has failed. The methodology is not analytically valid. It leads to hopeless confusion.

Only after we introduce the concepts of probability and statistics can we unravel a variety of paradoxes.

Not only does this resolve the underlying confusion, it allows us to reach much more useful and powerful results.

The conventional methodology presents the Historical Surviving Withdrawal Rates for a specified number of years. It assumes that the cause of each individual HSWR cannot be known. In its basic form, it assumes that the smallest Historical Surviving Withdrawal Rate of the past is safe, going forward.

Immediately, we spot a problem of how the Historical Surviving Withdrawal Rate varies as more and more years are included with the data. The smallest rate can never increase. It either remains the same or decreases. Even a simple selection of percentiles would overcome this difficulty. Typically, we do not see the expenditure of even this minimal amount of effort.

We can recognize immediately that we have arrived at a standard mathematical problem with a simple, standard, approximate solution. Instead of relying on the single, lowest Historical Surviving Withdrawal Rate, most people would approximate the probability distribution (such as the Gaussian, bell shaped, normal distribution) so as to discern low probability effects.

If this is not done, a paradox pops up.

A withdrawal rate was safe last year but stock prices fell. This tells me that I can withdraw the same number of dollars as before and still be safe, except for a single year. If 4% was safe for all 30-year periods, then a new rate, using the same number of dollars but with a smaller initial balance, is safe for all 29-year periods. If the market fell by 50% last year, a 29-year withdrawal rate of 8% is safe today.

The underlying fallacy is the claim that the 4% withdrawal rate was safe. It never was entirely safe. It may have been safe enough. But the claim to have started with 100% safety is absurd.

Once we introduce the concepts of probability and statistics, we can start looking for factors that influence Historical Surviving Withdrawal Rates. Purchase prices certainly have an effect. By introducing a suitable measure of valuation, we can narrow down the range of likely outcomes. Of course, purchase prices are under our control.

It turns out that valuations allow us to snap down the range of uncertainty in our calculations of withdrawal rates to plus and minus 1.0% to 1.5%. This is a powerful advantage. It also turns out that today’s prices are among the most extreme. Using a 4% withdrawal rate today based on the conventional methodology is a reckless mistake. A withdrawal rate of 4% with today’s stock prices makes the odds of portfolio survival, looking forward, almost a coin toss. The probability of survival is in the neighborhood of 60%. Such a portfolio will not necessarily fail. But it is far from safe.

Have fun.

John Walter Russell
I posted this initially on 4-28-05.