From Earnings Yield
Posted: Thu Apr 15, 2004 4:49 pm
From Earnings Yield
A straight line provides an excellent fit for a scatter plot of Historical Database Rates versus Earnings Yield (as measured by the inverse of P/E10). This allows us to calculate Safe Withdrawal Rates as a function of valuations.
The Portfolios
I have used the Retire Early Safe Withdrawal Calculator, Version 1.61, November 07, 2002 (with my modifications to make data reduction easier) to calculate Historical Database Rates for two portfolios. HDBR50 consists of 50% stocks and 50% commercial paper. HDBR80 consists of 80% stocks and 20% commercial paper.
Both portfolios were re-balanced annually. In both cases the Historical Database Rates were for a 30-year duration. In both cases the rates were determined with a precision of 0.1%. The rates are based upon a percentage of a portfolio's initial balance. A portfolio would have survived for the full 30 years at the Historical Database Rate, but it would have failed (i.e., the balance would have fallen to zero or become negative) at a withdrawal rate that was 0.1% higher. Withdrawal amounts were varied to match inflation (as measured by CPI-U). Expenses were set to 0.20% of the portfolio's current balance. I set the initial balance to $100K to minimize the effects of round off errors. Other calculator settings were left at their default values.
I have posted these Historical Database Rates. See the New HDBR Tables from Sunday, Jan 11, 2004 at 2:41 pm CST.
http://nofeeboards.com/boards/viewtopic.php?t=1962
I have restricted my investigations to portfolios beginning in the years 1921-1980. In fact, I have excluded the years 1921 and 1922 in most cases to get a better curve fit. Valuations were exceptionally low during those years. In the years prior to 1921, there is a qualitative difference in how Historical Database Rates and P/E10 behaved. Some of this can be traced to how commercial paper performed. In the late 1800s, for example, commercial paper by itself generally supported withdrawal rates of 6% or more.
The Plots
I have used Excel to make scatter plots of Historical Database Rates versus the Earnings Yield percentage 100/[P/E10], to calculate the best curve fits using the straight lines and to calculate R squared (i.e., the square of the correlation coefficient).
I used the equations for the straight lines to calculate standard deviations. I put the values of earning yield into the equations (of the straight lines), subtracted each Historical Database Rate for the same year (and the same earnings yield) and squared the difference. I added all of these values together and divided by the effective number of degrees of freedom. Finally, I took the square root.
I used the 90% confidence intervals appropriate for a Gaussian (or normal) distribution.
With the 50% stock portfolio, the Historical Database Rate (HDBR50) equation is HDBR50 = 0.3979x+2.6434%, where x = 100*(E10/P) or 100/[P/E10] = the earnings yield in percent and R squared equals 0.6975. When using this equation, the standard deviation of HDBR50 is 0.6178. The 90% confidence limits are plus and minus 1.01% of the calculated value.
With the 80% stock portfolio, the Historical Database Rate (HDBR80) equation is HDBR80 = 0.6685x+1.6424%, where x = 100*(E10/P) or 100/[P/E10] = the earnings yield in percent and R squared equals 0.7274. The standard deviation of HDBR80 using this formula is 0.9649. The 90% confidence limits are plus and minus 1.58%.
Statistical Considerations
I used 58 as the effective number of degrees of freedom (or 56 when data from 1921 and 1922 were excluded). The curve fit requires two degrees of freedom, one for the slope and the other for the intercept. (If only a mean were extracted from the data, the number of degrees of freedom would have been 59.)
Since there are 60 data points (or years for starting a retirement from 1921-1980), it is clear that I have treated each year as statistically independent. They are, in fact, very close to being independent. The fact that the sequences overlap strongly is only a secondary consideration. The reason is that the randomness comes from the percentage earnings yield, 100/[P/E10]. The earnings component of P/E10 is relatively stable because E10 is the average of ten years of (trailing, real) earnings. The randomness comes almost entirely from price fluctuations. In the very short-term, price fluctuations are very close to being entirely independent. It is only over longer periods that mean reversion (as properly defined and quantified by raddr) reduces the randomness.
It is necessary to use a confidence level lower than 100% whenever there is an element of chance. If I recall correctly, William Bernstein advocates a level around 80% to 85% for a variety of practical reasons. 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. But whenever you look at confidence levels greater than 90% to 95% (i.e., 1.64 sigma or 2 sigma), the statistical assumptions fall apart.
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 the answer is likely to be good enough for our purposes.
Applications
These predictions contain an element of uncertainty, which I identify by using 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 only a 5% chance of ending with as much as a zero balance (and a 95% chance of running out of money before the 30 years are up).
We can use the equations directly. I have to introduce two new terms, the Zero Balance Rate and the Unsafe Withdrawal Rate. Zero Balance Rates are what the straight-line equations calculate. The Unsafe Withdrawal Rate corresponds to the higher of the 90% confidence limits. The more familiar term, the Safe Withdrawal Rate, corresponds to the lower of the 90% confidence limits. [Notice that the term Safe Withdrawal Rate is reserved for a calculated value derived from the historical data and presented in a statistical context.]
For example, with the 50% stock portfolio, the 90% confidence limits are plus and minus 1.01% of the calculated value. When the earnings yield is 2.5%, the Zero Balance Rate is 3.64% plus and minus 1.01%. The Safe Withdrawal Rate (at a 95% level of safety) is 2.63%. The Unsafe Withdrawal Rate (with a level of safety no greater than 5%) is 4.65%. A portfolio that withdraws at the Safe Withdrawal Rate or less has a 95% chance of having a balance of zero or greater after 30 years. A portfolio that withdraws at the Unsafe Withdrawal Rate or more has a 95% chance of running out of money before 30 years have ended.
An earnings yield of 2.5% corresponds to a P/E10 of 40. At the peak of the bubble P/E10 exceeded 40.
In 1929 the P/E10 was 27.0 and the calculated Zero Balance Rate for a 50% stock portfolio is 4.12% [or 4.1171037%] plus and minus 1.01%. The Safe Withdrawal Rate is 3.11%. The Unsafe Withdrawal Rate is 5.13%.
With the 80% stock portfolio, the 90% confidence limits are plus and minus 1.58%.
For 1929 with P/E10 equal to 27.0, the calculated Zero Balance Rate for an 80% stock portfolio is 4.12% [or 4.1183259%] plus and minus 1.58%. The Safe Withdrawal Rate is 2.54%. The Unsafe Withdrawal Rate is 5.70%.
Notice that the Safe Withdrawal Rate for 1929 was lower with the 80% stock allocation than with the 50% stock allocation. The Zero Balance Rates were almost identical. The Unsafe Withdrawal Rate was higher with 80% stocks. The actual 1929 Historical Database Rates of 4.4% (for 80% stocks) and 4.5% (for 50% stocks) fell within the confidence intervals.
Using the 90% confidence limits to isolate intervals, we can break the data into two or three distinct sections. Referring again to HDBR50 with 50% stocks and 50% commercial paper, when the earnings yield is 5%, then the Zero Balance Rate is 4.63% plus and minus 1.01% according to the curve. When the earnings yield is 10%, then the Zero Balance Rate is 6.62% plus and minus 1.01%. Those are two distinct sections. (P/E10 is 20 when the earnings yield is 5% and it is 10 when the earnings yield is 10%.)
When the earnings yield is 2.5%, the Zero Balance Rate (as projected) is 3.64% plus and minus 1.01%. When the earnings yield is 7.5%, the Zero Balance Rate is 5.63% plus and minus 1.01%. When the earnings yield is 12.5%, the Zero Balance Rate is 7.62% plus and minus 1.01%. (P/E10 is 40 when the earnings yield is 2.5%. P/E10 is 13.3 when the earnings yield is 7.5%. P/E10 is 8 when the earnings yield is 12.5%.)
These intervals can be applied in the opposite direction, going from Zero Balance Rates to their corresponding earnings yields. Breaking the Zero Balance Rates into two sections, a Zero Balance Rate of 4% requires that the earnings yield be constrained to the interval of 3.41% plus and minus 2.55%. A Zero Balance Rate of 6% requires that the earnings yield be constrained to 8.44% plus and minus 2.55%. (P/E10 is 29.3 when the earnings yield is 3.41%. P/E10 is 11.9 when the earnings yield is 8.44%.)
A Zero Balance Rate of 3% requires that the earnings yield be constrained to the interval of 0.90% plus and minus 2.55%. (The actual lower limit is zero.) A Zero Balance Rate of 5% requires that the earnings yield be constrained to the interval of 5.92% plus and minus 2.55%. A Zero Balance Rate of 7% requires that the earnings yield be constrained to the interval of 10.95% plus and minus 2.55%. (P/E10 is 111.1 when the earnings yield is 0.90%. P/E10 is 16.9 when the earnings yield is 5.92%. P/E10 is 9.1 when the earnings yield is 10.95%.)
These numbers can be used in design. For example, if I were planning to withdraw at a 5% rate, I should be talking about a P/E10 close to 16.9. If the earnings yield were 8.47% (= 5.92% + 2.55%) and P/E10 were 11.8, I would be highly conservative. If the earnings yield were 3.37% (= 5.92% - 2.55%) and the P/E10 were 33.4, I would be hopelessly reckless. Another way of saying almost the same thing is that you can withdraw 5% when valuations are normal. You can withdraw even more when P/E10 falls below 12. Closing your eyes and gritting your teeth won't hack it should you decide to withdraw 5% at today's valuations.
Have fun.
John R.
A straight line provides an excellent fit for a scatter plot of Historical Database Rates versus Earnings Yield (as measured by the inverse of P/E10). This allows us to calculate Safe Withdrawal Rates as a function of valuations.
The Portfolios
I have used the Retire Early Safe Withdrawal Calculator, Version 1.61, November 07, 2002 (with my modifications to make data reduction easier) to calculate Historical Database Rates for two portfolios. HDBR50 consists of 50% stocks and 50% commercial paper. HDBR80 consists of 80% stocks and 20% commercial paper.
Both portfolios were re-balanced annually. In both cases the Historical Database Rates were for a 30-year duration. In both cases the rates were determined with a precision of 0.1%. The rates are based upon a percentage of a portfolio's initial balance. A portfolio would have survived for the full 30 years at the Historical Database Rate, but it would have failed (i.e., the balance would have fallen to zero or become negative) at a withdrawal rate that was 0.1% higher. Withdrawal amounts were varied to match inflation (as measured by CPI-U). Expenses were set to 0.20% of the portfolio's current balance. I set the initial balance to $100K to minimize the effects of round off errors. Other calculator settings were left at their default values.
I have posted these Historical Database Rates. See the New HDBR Tables from Sunday, Jan 11, 2004 at 2:41 pm CST.
http://nofeeboards.com/boards/viewtopic.php?t=1962
I have restricted my investigations to portfolios beginning in the years 1921-1980. In fact, I have excluded the years 1921 and 1922 in most cases to get a better curve fit. Valuations were exceptionally low during those years. In the years prior to 1921, there is a qualitative difference in how Historical Database Rates and P/E10 behaved. Some of this can be traced to how commercial paper performed. In the late 1800s, for example, commercial paper by itself generally supported withdrawal rates of 6% or more.
The Plots
I have used Excel to make scatter plots of Historical Database Rates versus the Earnings Yield percentage 100/[P/E10], to calculate the best curve fits using the straight lines and to calculate R squared (i.e., the square of the correlation coefficient).
I used the equations for the straight lines to calculate standard deviations. I put the values of earning yield into the equations (of the straight lines), subtracted each Historical Database Rate for the same year (and the same earnings yield) and squared the difference. I added all of these values together and divided by the effective number of degrees of freedom. Finally, I took the square root.
I used the 90% confidence intervals appropriate for a Gaussian (or normal) distribution.
With the 50% stock portfolio, the Historical Database Rate (HDBR50) equation is HDBR50 = 0.3979x+2.6434%, where x = 100*(E10/P) or 100/[P/E10] = the earnings yield in percent and R squared equals 0.6975. When using this equation, the standard deviation of HDBR50 is 0.6178. The 90% confidence limits are plus and minus 1.01% of the calculated value.
With the 80% stock portfolio, the Historical Database Rate (HDBR80) equation is HDBR80 = 0.6685x+1.6424%, where x = 100*(E10/P) or 100/[P/E10] = the earnings yield in percent and R squared equals 0.7274. The standard deviation of HDBR80 using this formula is 0.9649. The 90% confidence limits are plus and minus 1.58%.
Statistical Considerations
I used 58 as the effective number of degrees of freedom (or 56 when data from 1921 and 1922 were excluded). The curve fit requires two degrees of freedom, one for the slope and the other for the intercept. (If only a mean were extracted from the data, the number of degrees of freedom would have been 59.)
Since there are 60 data points (or years for starting a retirement from 1921-1980), it is clear that I have treated each year as statistically independent. They are, in fact, very close to being independent. The fact that the sequences overlap strongly is only a secondary consideration. The reason is that the randomness comes from the percentage earnings yield, 100/[P/E10]. The earnings component of P/E10 is relatively stable because E10 is the average of ten years of (trailing, real) earnings. The randomness comes almost entirely from price fluctuations. In the very short-term, price fluctuations are very close to being entirely independent. It is only over longer periods that mean reversion (as properly defined and quantified by raddr) reduces the randomness.
It is necessary to use a confidence level lower than 100% whenever there is an element of chance. If I recall correctly, William Bernstein advocates a level around 80% to 85% for a variety of practical reasons. 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. But whenever you look at confidence levels greater than 90% to 95% (i.e., 1.64 sigma or 2 sigma), the statistical assumptions fall apart.
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 the answer is likely to be good enough for our purposes.
Applications
These predictions contain an element of uncertainty, which I identify by using 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 only a 5% chance of ending with as much as a zero balance (and a 95% chance of running out of money before the 30 years are up).
We can use the equations directly. I have to introduce two new terms, the Zero Balance Rate and the Unsafe Withdrawal Rate. Zero Balance Rates are what the straight-line equations calculate. The Unsafe Withdrawal Rate corresponds to the higher of the 90% confidence limits. The more familiar term, the Safe Withdrawal Rate, corresponds to the lower of the 90% confidence limits. [Notice that the term Safe Withdrawal Rate is reserved for a calculated value derived from the historical data and presented in a statistical context.]
For example, with the 50% stock portfolio, the 90% confidence limits are plus and minus 1.01% of the calculated value. When the earnings yield is 2.5%, the Zero Balance Rate is 3.64% plus and minus 1.01%. The Safe Withdrawal Rate (at a 95% level of safety) is 2.63%. The Unsafe Withdrawal Rate (with a level of safety no greater than 5%) is 4.65%. A portfolio that withdraws at the Safe Withdrawal Rate or less has a 95% chance of having a balance of zero or greater after 30 years. A portfolio that withdraws at the Unsafe Withdrawal Rate or more has a 95% chance of running out of money before 30 years have ended.
An earnings yield of 2.5% corresponds to a P/E10 of 40. At the peak of the bubble P/E10 exceeded 40.
In 1929 the P/E10 was 27.0 and the calculated Zero Balance Rate for a 50% stock portfolio is 4.12% [or 4.1171037%] plus and minus 1.01%. The Safe Withdrawal Rate is 3.11%. The Unsafe Withdrawal Rate is 5.13%.
With the 80% stock portfolio, the 90% confidence limits are plus and minus 1.58%.
For 1929 with P/E10 equal to 27.0, the calculated Zero Balance Rate for an 80% stock portfolio is 4.12% [or 4.1183259%] plus and minus 1.58%. The Safe Withdrawal Rate is 2.54%. The Unsafe Withdrawal Rate is 5.70%.
Notice that the Safe Withdrawal Rate for 1929 was lower with the 80% stock allocation than with the 50% stock allocation. The Zero Balance Rates were almost identical. The Unsafe Withdrawal Rate was higher with 80% stocks. The actual 1929 Historical Database Rates of 4.4% (for 80% stocks) and 4.5% (for 50% stocks) fell within the confidence intervals.
Using the 90% confidence limits to isolate intervals, we can break the data into two or three distinct sections. Referring again to HDBR50 with 50% stocks and 50% commercial paper, when the earnings yield is 5%, then the Zero Balance Rate is 4.63% plus and minus 1.01% according to the curve. When the earnings yield is 10%, then the Zero Balance Rate is 6.62% plus and minus 1.01%. Those are two distinct sections. (P/E10 is 20 when the earnings yield is 5% and it is 10 when the earnings yield is 10%.)
When the earnings yield is 2.5%, the Zero Balance Rate (as projected) is 3.64% plus and minus 1.01%. When the earnings yield is 7.5%, the Zero Balance Rate is 5.63% plus and minus 1.01%. When the earnings yield is 12.5%, the Zero Balance Rate is 7.62% plus and minus 1.01%. (P/E10 is 40 when the earnings yield is 2.5%. P/E10 is 13.3 when the earnings yield is 7.5%. P/E10 is 8 when the earnings yield is 12.5%.)
These intervals can be applied in the opposite direction, going from Zero Balance Rates to their corresponding earnings yields. Breaking the Zero Balance Rates into two sections, a Zero Balance Rate of 4% requires that the earnings yield be constrained to the interval of 3.41% plus and minus 2.55%. A Zero Balance Rate of 6% requires that the earnings yield be constrained to 8.44% plus and minus 2.55%. (P/E10 is 29.3 when the earnings yield is 3.41%. P/E10 is 11.9 when the earnings yield is 8.44%.)
A Zero Balance Rate of 3% requires that the earnings yield be constrained to the interval of 0.90% plus and minus 2.55%. (The actual lower limit is zero.) A Zero Balance Rate of 5% requires that the earnings yield be constrained to the interval of 5.92% plus and minus 2.55%. A Zero Balance Rate of 7% requires that the earnings yield be constrained to the interval of 10.95% plus and minus 2.55%. (P/E10 is 111.1 when the earnings yield is 0.90%. P/E10 is 16.9 when the earnings yield is 5.92%. P/E10 is 9.1 when the earnings yield is 10.95%.)
These numbers can be used in design. For example, if I were planning to withdraw at a 5% rate, I should be talking about a P/E10 close to 16.9. If the earnings yield were 8.47% (= 5.92% + 2.55%) and P/E10 were 11.8, I would be highly conservative. If the earnings yield were 3.37% (= 5.92% - 2.55%) and the P/E10 were 33.4, I would be hopelessly reckless. Another way of saying almost the same thing is that you can withdraw 5% when valuations are normal. You can withdraw even more when P/E10 falls below 12. Closing your eyes and gritting your teeth won't hack it should you decide to withdraw 5% at today's valuations.
Have fun.
John R.