Confidence Limits
Posted: Sat Feb 14, 2004 10:21 am
I have misstated what our confidence limits mean. I have referred to Safe Withdrawal Rates plus and minus their statistical confidence limits.
The Safe Withdrawal Rate is the number at the lower confidence limit. The Unsafe Withdrawal Rate is the number at the upper confidence limit. I do not have a good term for the estimate itself. For the time being, I will refer to it as the Zero Balance Withdrawal Rate. I do not like that term since it excludes many approaches such as withdrawing a percentage of a portfolio's current balance.
Historical Database Rates are withdrawal rates that would have ended at a zero balance after a specified period of time, usually 30 years. We refer to those as actual outcomes. Based on information extracted from those outcomes, I make predictions of withdrawal rates that will end with a zero balance (after 30 years).
Those predictions contain an element of uncertainty, which I identify by using confidence limits. I have elected to use a 90% confidence level. There is a 10% chance that actual outcomes will be outside of the confidence limits. Each basic prediction has a 50% chance of being safe. At the lower confidence limit, at the calculated withdrawal rate a portfolio has a 95% chance of ending with a balance of zero or higher. At the upper confidence limit, at the calculated withdrawal rate a portfolio has no more than a 5% chance of ending with as much as a zero balance (and at least a 95% chance of running out of money before the 30 years are up).
Part of the uncertainty in my calculations is based upon our inability to extract information from the historical record. For example, if I had used Tobin's Q instead of P/E10, the uncertainty would be greater because my measure of valuation would not have been as good (for such calculations). We expect a degree of uncertainty to remain always because of such things as the sequence of returns. We expect the future to be similar to the past, but not identical to the past.
It is necessary to use a confidence level lower than 100% whenever there is an element of chance. If I am not mistaken, William Bernstein advocates using a level around 80% to 85% for a variety of practical reasons. In addition, whenever one looks at confidence levels greater than 90% to 95% (i.e., 1.64 sigma or 2 sigma), the statistical assumptions fall apart. (In practice, people start redefining terms, giving different names to various sources of randomness.) The bell curve (or the normal distribution or the Gaussian distribution) is an excellent approximation as long as you do not set the confidence levels too high. There are sound mathematical reasons for this. It is still useful around 3 sigma. It is meaningless when you try to extend it to 6 sigma. (As I mentioned previously, all sorts of special adjustments become necessary. When done right, something such as 6 sigma becomes meaningful again. But only after making the adjustments.)
If you talk to a professional statistician, he will tell you that using the bell curve is usually OK, but that the confidence levels of the actual distribution are almost certain to differ somewhat. When I calculate a 90% confidence level, the real number (in an idealized, theoretical sense and which I am unable to calculate) may be something like 82% or 93%. But my answer is likely to be good enough.
We can apply our estimates to the past. They tell us what would have been reasonable to expect under similar, but not identical, circumstances. We can compare these estimates to Historical Database Rates to determine how representative the past outcomes were. When we make our estimates, we are generally extracting information from the totality of the historical record, not just from one or two specific years.
Have fun.
John R.
The Safe Withdrawal Rate is the number at the lower confidence limit. The Unsafe Withdrawal Rate is the number at the upper confidence limit. I do not have a good term for the estimate itself. For the time being, I will refer to it as the Zero Balance Withdrawal Rate. I do not like that term since it excludes many approaches such as withdrawing a percentage of a portfolio's current balance.
Historical Database Rates are withdrawal rates that would have ended at a zero balance after a specified period of time, usually 30 years. We refer to those as actual outcomes. Based on information extracted from those outcomes, I make predictions of withdrawal rates that will end with a zero balance (after 30 years).
Those predictions contain an element of uncertainty, which I identify by using confidence limits. I have elected to use a 90% confidence level. There is a 10% chance that actual outcomes will be outside of the confidence limits. Each basic prediction has a 50% chance of being safe. At the lower confidence limit, at the calculated withdrawal rate a portfolio has a 95% chance of ending with a balance of zero or higher. At the upper confidence limit, at the calculated withdrawal rate a portfolio has no more than a 5% chance of ending with as much as a zero balance (and at least a 95% chance of running out of money before the 30 years are up).
Part of the uncertainty in my calculations is based upon our inability to extract information from the historical record. For example, if I had used Tobin's Q instead of P/E10, the uncertainty would be greater because my measure of valuation would not have been as good (for such calculations). We expect a degree of uncertainty to remain always because of such things as the sequence of returns. We expect the future to be similar to the past, but not identical to the past.
It is necessary to use a confidence level lower than 100% whenever there is an element of chance. If I am not mistaken, William Bernstein advocates using a level around 80% to 85% for a variety of practical reasons. In addition, whenever one looks at confidence levels greater than 90% to 95% (i.e., 1.64 sigma or 2 sigma), the statistical assumptions fall apart. (In practice, people start redefining terms, giving different names to various sources of randomness.) The bell curve (or the normal distribution or the Gaussian distribution) is an excellent approximation as long as you do not set the confidence levels too high. There are sound mathematical reasons for this. It is still useful around 3 sigma. It is meaningless when you try to extend it to 6 sigma. (As I mentioned previously, all sorts of special adjustments become necessary. When done right, something such as 6 sigma becomes meaningful again. But only after making the adjustments.)
If you talk to a professional statistician, he will tell you that using the bell curve is usually OK, but that the confidence levels of the actual distribution are almost certain to differ somewhat. When I calculate a 90% confidence level, the real number (in an idealized, theoretical sense and which I am unable to calculate) may be something like 82% or 93%. But my answer is likely to be good enough.
We can apply our estimates to the past. They tell us what would have been reasonable to expect under similar, but not identical, circumstances. We can compare these estimates to Historical Database Rates to determine how representative the past outcomes were. When we make our estimates, we are generally extracting information from the totality of the historical record, not just from one or two specific years.
Have fun.
John R.