Robert Shiller in his book, Irrational Exuberance, and elsewhere famously argues that due to emotion, stock prices in both bull and bear markets overshoot the prices warranted by future dividends and earnings. If such overshooting occurs, returns on stocks should be negatively Prior Average Dependent (PAD). In particular, lower than normal average returns in the past should be associated with higher than normal returns in the future. If so, negative PAD stock returns should improve the sustainability of disbursements. Persistently lower than expected returns that might otherwise require reductions in disbursements will tend to be offset by higher than originally expected returns in the future.
When historic real returns are regressed on their prior averages the relationship is negative, and becomes more strongly negative as the length of the average increases up to 30 years. Due to inherent bias in such regressions, however, such a negative relationship would be obtained even if the returns are Independently and Identically Distributed (IID) over time. Nevertheless, the regression coefficients obtained could be sufficiently negative to provide evidence that stock returns are negative PAD, and that the results obtained are not just due to bias in the regressions. Such evidence can be provided by simulating a very large number of time series of returns that are IID, have a similar annual probability distribution to the historic returns, and a similar length as the series for which the regressions have been made. Regressions can then be run on these simulated series to determine the likelihood of obtaining regression coefficients as negative as those for the historic series when returns are known to be IID. If the coefficients for the historic returns are sufficiently negative to be unlikely to be from IID returns, there is evidence that stock returns are negative PAD, despite the bias. These tests show that there is evidence that stock returns are negative PAD for a 30 year average.
Negative PAD returns affect disbursements both by a hedging benefit, and by the effect of lower or higher than normal average returns before the disbursements begin. To better understand the effect of PAD returns, it is desirable to consider each of these effects separately. The hedging benefit can be isolated by comparing the effect of disbursements from negative PAD stock returns to those from IID stock returns when the returns for the PAD returns before the first disbursement are the same as the expected IID return. To study the effect of the returns prior to the initial disbursement, the returns are set equal to 30 year segments of historical returns. Before doing so, however, the entire historical series is adjusted by the same amount each year so that the average of the historical returns is the same as the expected IID return.
The effect of a prior 30 year segment depends not only on its average return, but on the pattern of the returns over the segment. For instance, the 30 year segment from 1995 through 2024 has a very high average, but the impact of that high average is lessened because over the first five years the unusually strong bull market from 1995 through 1999 is coming out of the 30 year average. Thus, the negative effect of the high average prior returns when starting in 2025 is less than indicated by just looking at the average. It turns out that the worst results are obtained for the prior 30 year average ending in 1965 that followed the prolonged bullish stock market after WWII. The best results are obtained for returns prior to the mid 1980s that followed the prolonged bearish period in the 1970s, and as the strong returns of the 1950s were coming out of the 30 year average.
Negative PAD stock returns can affect the optimal allocation to stocks as PAD returns can make stocks more valuable relative to intermediates. When prior returns are equal to the expected IID return, the hedging benefit of PAD returns increases the optimal allocation to stocks by 10 to 20 percentage points depending on the reaction coefficient assumed for the expected return to the prior average. When the prior returns are the high values in the early or mid 1960s, however, it turns out that the adverse effect of these high prior returns offsets the hedging benefit of negative PAD returns. In this case, stocks are no longer more valuable relative to intermediates, and there is no change in the optimal stock allocation from IID stock returns. When the prior stock returns are abnormally low, there is a further increase in the optimal stock allocation beyond that due to the hedging benefit of PAD stock returns.
When initial prior returns are equal to the expected IID return, the hedging benefits of PAD returns provide a meaningful improvement in sustainability. The improvement, however, is not as large as that obtained by reducing the initial disbursement from 4.0% to 3.5% for IID returns. More improvement than reducing the initial disbursement from 4.0% to 3.5% can be obtained, however, when the initial prior returns are as low as have been observed historically. When the initial prior returns are as high as have been observed historically the deterioration in sustainability is serious, but not as bad as increasing the initial disbursement from 4.0% to 4.5%, for IID returns. Thus, PAD stock returns provide a meaningful benefit for normal initial prior returns. Much larger benefits occur when prior returns are as low as sometimes in the past. Serious deterioration in sustainability occurs when prior returns are as high as have sometimes been observed.
Evidence
The SBBI Yearbooks have a well-known series for the annual real returns for large capitalization stocks listed on the NYSE starting in 1926. These returns are identical to those for the S&P 500 starting in the late 1950s when the S&P 500 was introduced. Publication of the SBBI Yearbook, however, ceased in 2023, covering the historic returns up through 2022. Others, however, have been publishing on line comparable historical returns for the S&P 500, which include reinvestment of dividends, and adjustment for inflation based on the CPI. Returns from the site maintained by Aswath Damodaran were used to update the series from the SBBI Yearbooks for large capitalization stocks for years 2023 and 2024.
To use this series to provide a prior average as long as 40 years, the first year that can be used for a regression of these returns on their prior averages is 1966. The results of such regressions for the returns from 1966 to 2024 on their prior moving averages are shown in Table 11.1. The regression coefficients are negative and become more strongly negative as the length of the moving average increases to 30 years. As just noted, however, such regressions have a bias to give negative coefficients even if the returns in the series are not negative PAD, but IID. There is a negative bias because having a prior moving average above its own average over the sample period makes it more likely that subsequent returns in the sample period are below average, and this tendency is stronger the longer is the moving average. Thus, there is a negative relation between returns in a sample period and a prior moving average even if those returns are IID.
The negative bias can be confirmed and its size investigated by running regressions on simulated time series of returns that are IID. To do so, suppose that 10,000 time series of returns are simulated that are of the same length as the series of historic returns that were used to obtain the results in Table 11.1. Suppose that these returns are IID with a normal distribution with a mean and standard deviation equal to the average and sample standard deviation of the Large Cap Stocks from 1926 to 2024. Results from these regressions are shown in the two columns on the right of Table 11.1.
The chances that the regression coefficients are negative increase from .72 for a 10 year average to .87 for a 40 year average. The column on the far right shows that regression coefficients as negative as those obtained for the historic returns with shorter averages could easily be due to chance if the underlying returns are IID. Coefficients as negative as those obtained for averages of 25 or 30 years, however, are less likely to be due to chance. In particular, the coefficient for the 30 year average has only a .025 chance of occurring if the returns are IID, which is less than the .05 chance often used for establishing significance. Perhaps, however, the coefficient for the 30 year average is unlikely because the underlying stock returns have a different mean or standard deviation than assumed for the IID returns. When the mean or standard deviation is varied by .02, however, virtually the same results are obtained as in the two columns on the right of Table 11.1. It is of interest, but not necessarily of much consequence, that the lowest and highest 30 year averages over 1966 to 2024 occurred just prior to the unusually strong bull market of 1995/99, and the unusually severe bear market of 1973/74 as discussed in a note.
Modeling
Elsewhere, the assumptions about the future expected return of stocks in the simulations have been those specified in Model as either the Pre 2008, Lower, or Mid Returns. These sets of assumptions all assume that the return on stocks each year is a linear function of the return on intermediates for that year plus an independently and normally distributed random variable with a zero mean. To include the dependence of stock returns on the 30 year average of prior returns, the linear function is expanded to include the 30 year average of prior returns. The coefficient of the 30 year average is assumed to be -1, but the effect of using -2 is also tested. To determine the constant term of the linear relation the expected value of the 30 year average is assumed to be the expected return of stocks for the assumed set of returns. For instance, for the Mid Returns, the expected return for stocks is .055, and for intermediates is .015. To make the expected return of stocks equal to .055 when the coefficient for the 30 year average is -1 the constant for the linear relation is set equal to .104 as .104+.4(.015)-1(.055) =.055.
For these assumptions, the stock returns continue to be normally distributed as they are linear combinations of normally distributed variables. After the initial year, however, the volatility of stock returns is affected by the variability of prior returns in the moving average as additional returns in the average are being determined in the simulation as it proceeds. The effect of the returns in the moving average on volatility, however, is very small. For instance, the volatility for the second year is increased by (1/30) squared of the variance of the return for the first year when the coefficient is -1, which increases the standard deviation of the return by only .0001. In subsequent years the additional increase is even less. By year 36 simulations show that the total increase in the standard deviation is only about .0015. To reduce any impact of the higher volatility caused by the moving average a small reduction can be made in the volatility of the stock returns without the PAD dependence.
Consider two sets of returns, one of which is the same as the Mid Returns in Model. The other set differs by having stock returns that are PAD with a coefficient of -1 for the prior 30 year average. The expected stock return for this set is the same as for the Mid Returns when the prior 30 year average return is equal to the expected stock return for the Mid Returns assumption. The standard deviation of the annual stock returns for this set is assumed to be .001 less than for the Mid Returns. These two sets of returns will be used to illustrate the effect of PAD stock returns on sustainability risk. First, all 30 prior stock returns are assumed to equal the expected stock return for the Mid Returns. This case illustrates the effect of PAD stock returns when the effect of the initial 30 years of returns is neutral.
When the effect of the initial prior 30 annual returns is neutral, stocks that are PAD become more valuable relative to intermediates, and the best stock allocation increases. Sustainability also shows a meaningful improvement. The next step is to see how these results are affected when the 30 prior returns have some of the patterns that have been observed in the past, and what the effect might be of the prior 30 annual returns in 2025. To gauge the effect of the pattern of prior historical returns an adjustment is made for the difference in the overall average of the historical returns and the assumed expected stock return. All of the historic returns are reduced by this difference before the effect of any of the 30 year segments of the historic returns is tested. Thus, these tests attempt to show the effect of prior variability about the mean rather than differences in the mean itself. The changes in the spreadsheet needed to make these tests in the model are discussed in a note.
Results
For the IID Mid Returns viewed initially the expected stock return at the beginning of each year in the future is the same and equal to .055. For the PAD Mid Returns viewed initially, the expected stock return at the beginning of each year in the future is the larger, the lower has been the average of the 30 prior returns. To gauge the size of the hedging benefit, suppose first that the effect of the initial prior 30 annual returns is neutral because each of these prior returns is equal to the expected stock return for the Mid Returns of .055. Suppose also for the PAD Mid Returns that the coefficient for the 30 year average is -1, and that the linear relation for the stock returns is that described earlier for this case. Simulations can provide as an output the probability distribution of the trailing 30 year average each year as the future unfolds. Deducting these averages from .0110 gives the expected stock return as discussed earlier. Thus, using this procedure the probability distribution of expected stock returns at the beginning of future years can be obtained.
The percentiles for these cumulative probability distributions are shown in Chart 11.1. The upper curve for the 95th percentile gives the lower bound for the highest 5% of expected stock returns at the beginning of each year. These correspond to the upper bound for the lower 5% of 30 year prior averages that have been obtained for those years from the simulations. The chart shows by year 10 that the next expected stock return has increased from .055 to .080. Suppose the coefficient of the 30 year average is -2 instead of -1. In this case, the expected stock return at year 10 increases to .095 instead of .080. As expected, the hedge provided by the PAD stocks becomes stronger.
To consider the effect of their hedging benefit on sustainability, suppose the New 4% Rule is being used as discussed earlier at page, BID. The earlier results at years 12 and 24 are shown by the solid curves in Charts 11.2 and 11.3 labeled IID. Now suppose that the PAD returns just described are being used instead of the IID returns. As PAD makes stocks more valuable relative to intermediates the stock allocation increases from 50% to 60% or 70% depending on whether the coefficient of the 30 year average is -1 or -2. The results in this case are shown by the dashed curves labeled PAD. There is a meaningful improvement in sustainability, but it is well short of that obtained by reducing the initial disbursement from 4.0% to 3.5%.
Initial Prior Returns
So far, the initial prior returns have been set equal to the expected IID return. Now suppose that the prior returns are the most recent 30 annual real returns in the 1926 to 2024 historical series, or the 30 consecutive returns in that series that would have an unusually strong impact either positive or negative on sustainability. To have a strong impact on sustainability the prior average must be unusually low or high and more recent returns should also be unusually low or high as recent returns will remain in the average for a long time. Unusually high averages with more recent high returns occurred prior to the early and mid 1960s following the long bullish period in the stock market after WWII. In particular, the returns prior to 1966 will be considered as that is the first year for the regressions in Table 11.1. An unusually low average with more recent low returns occurred in the mid 1980s following the long bearish period in the stock market in the 1970s. The mid 1980s also reflects some of the unusually strong returns in the 1950s coming out of the 30 year average. In particular, the returns prior to 1985 will be considered. To isolate the effect of the pattern of the returns in the historical series from the difference between the average historical returns and the expected IID return, an adjustment is required. All of the returns in the historical series are adjusted by the difference between the average return of the historical series and the expected IID return.
For normal initial prior returns the hedging benefit of PAD returns makes stocks more valuable relative to intermediates, and increases the allocation to stocks by 10 or 20 percentage points. This advantage for stocks with PAD returns can be nullified, however, if the initial prior returns are too high. Suppose, for instance, that the initial prior stock returns are those in 1966, after the adjustment just described. When the prior returns for 1966 replace the neutral returns the optimal stock allocations for coefficients of -1 and -2 revert back to being the same as for the IID returns. When the prior returns for 1985 replace the IID returns the optimal allocation increases by another 10 percentage points. Change in the 30 year average return as the disbursement period unfolds might make some later change in allocation desirable, but doing so will not have much effect on sustainability, and is not considered.
For initial prior returns equal to the expected stock return for the Mid Returns, Charts 11.2 and 11.3 show the reduced chances of declines at years 12 and 24 for the PAD returns. Tables 11.2 and 11.3 compare these reductions to those obtained when the initial prior returns are those for the adjusted historical returns ending in 2024, 1985, and 1966. As expected, for the low returns prior to 1985, there are additional reductions. On the other hand, for the high returns prior to 1966, the sustainabiliy is worse than for the IID returns, and more so when the reaction coefficient is more negative. The surprising result is that the 30 year average ending in 2024 gives normal results despite the 30 year average ending in 2024 being higher than the 30 year average ending in 1966. Thus, at first glance, the 30 year average of past returns is showing that stocks are highly overvalued in early 2025 just like price earnings ratio measures of valuation. The explanation for the normal results in the tables is that the exceptionally strong bull market of 1995 through 1999 is about to come out of the 30 year average of past returns. Thus, whether the high 30 year average is showing that stocks are highly overvalued in early 2025 is questionable.
Conclusions
There is evidence that stock returns are negatively related to the trailing 30 year average of prior returns. This negative relationship provides a hedging benefit that reduces sustainability risk when making disbursements. Simulations show that the chances of future declines are reduced when the 30 annual returns before the disbursements begin are equal to the expected stock return. When the initial prior returns are as low as have been observed in the past, sustainability is further improved. When the prior returns are as high as observed in the past, sustainability is worse than without the negative PAD relationship. Surprisingly, however, the 30 prior annual returns up through 2024 have little effect on the PAD hedging benefit, despite the 30 year prior average being higher than in the past. The explanation is that the exceptionally strong bull market from 1995 through 1999 is about to come out of the 30 year average. Thus, in early 2025, the prior 30 year average indicates that stocks are highly overvalued just like price earnings ratios of market valuation. The high value of the prior 30 year average, however, is questionable as the strong bull market of the late 1990s is about to come out of the prior 30 year average.
Notes
1. Over 1966 to 2024, the strongest bull market was over 1995 to 1999 when the annual increase for those five years averaged 26% in real terms. The most severe bear market was over 1973 and 1974 when the annual decline averaged 28% in real terms for these two years. It is of interest to note that the trough and peak of the 30 year average over 1966 to 2024 occurred in the year just prior to each of these events. Of course, there was no way of knowing that was the peak or trough until long afterwards. Moreover, for many years beforehand the 30 year average was almost as low or as high as at the peak or trough. Thus, the 30 year average would not have been of much help in forecasting the timing of these events. Highly abnormal measures of stock market valuation are helpful in indicating that a highly abnormal move of the market is possible, not when it will occur.
2. To incorporate PAD stock returns into the model, two new columns must be added to the spreadsheet, and a revision made in the relation for the stock return, r(t). The first of the new columns has the 65 returns that are used in each iteration for a simulation of the disbursements. The first 30 of these returns starts with the most distant of the 30 given prior returns, and ends with the given return for the year prior to the beginning of the disbursements. The last 35 returns start with the drawing for the return for the first year in the iteration, and end with the drawing for the return for the 35th year in that iteration. The second column has the 36 successive moving averages of the 30 prior returns used to calculate the expected return for each successive year in an iteration. The first of these is the average of the 30 given prior returns, and the last is the 30 returns drawn in the simulation for the 6th through the 35th years of an iteration.
References
Aswath Damadoran, https://pages.stern.nyu.edu
Kroll, 2023 SBBI Yearbook, New York, 2023.
Robert J. Shiller, Irrational Exuberance, Princeton University Press, Princeton, NJ, 2000.
PAD
PAD Stock Returns
Robert Shiller in his book, Irrational Exuberance, and elsewhere famously argues that due to emotion, stock prices in both bull and bear markets overshoot the prices warranted by future dividends and earnings. If such overshooting occurs, returns on stocks should be negatively Prior Average Dependent (PAD). In particular, lower than normal average returns in the past should be associated with higher than normal returns in the future. If so, negative PAD stock returns should improve the sustainability of disbursements. Persistently lower than expected returns that might otherwise require reductions in disbursements will tend to be offset by higher than originally expected returns in the future.
When historic real returns are regressed on their prior averages the relationship is negative, and becomes more strongly negative as the length of the average increases up to 30 years. Due to inherent bias in such regressions, however, such a negative relationship would be obtained even if the returns are Independently and Identically Distributed (IID) over time. Nevertheless, the regression coefficients obtained could be sufficiently negative to provide evidence that stock returns are negative PAD, and that the results obtained are not just due to bias in the regressions. Such evidence can be provided by simulating a very large number of time series of returns that are IID, have a similar annual probability distribution to the historic returns, and a similar length as the series for which the regressions have been made. Regressions can then be run on these simulated series to determine the likelihood of obtaining regression coefficients as negative as those for the historic series when returns are known to be IID. If the coefficients for the historic returns are sufficiently negative to be unlikely to be from IID returns, there is evidence that stock returns are negative PAD, despite the bias. These tests show that there is evidence that stock returns are negative PAD for a 30 year average.
Negative PAD returns affect disbursements both by a hedging benefit, and by the effect of lower or higher than normal average returns before the disbursements begin. To better understand the effect of PAD returns, it is desirable to consider each of these effects separately. The hedging benefit can be isolated by comparing the effect of disbursements from negative PAD stock returns to those from IID stock returns when the returns for the PAD returns before the first disbursement are the same as the expected IID return. To study the effect of the returns prior to the initial disbursement, the returns are set equal to 30 year segments of historical returns. Before doing so, however, the entire historical series is adjusted by the same amount each year so that the average of the historical returns is the same as the expected IID return.
The effect of a prior 30 year segment depends not only on its average return, but on the pattern of the returns over the segment. For instance, the 30 year segment from 1995 through 2024 has a very high average, but the impact of that high average is lessened because over the first five years the unusually strong bull market from 1995 through 1999 is coming out of the 30 year average. Thus, the negative effect of the high average prior returns when starting in 2025 is less than indicated by just looking at the average. It turns out that the worst results are obtained for the prior 30 year average ending in 1965 that followed the prolonged bullish stock market after WWII. The best results are obtained for returns prior to the mid 1980s that followed the prolonged bearish period in the 1970s, and as the strong returns of the 1950s were coming out of the 30 year average.
Negative PAD stock returns can affect the optimal allocation to stocks as PAD returns can make stocks more valuable relative to intermediates. When prior returns are equal to the expected IID return, the hedging benefit of PAD returns increases the optimal allocation to stocks by 10 to 20 percentage points depending on the reaction coefficient assumed for the expected return to the prior average. When the prior returns are the high values in the early or mid 1960s, however, it turns out that the adverse effect of these high prior returns offsets the hedging benefit of negative PAD returns. In this case, stocks are no longer more valuable relative to intermediates, and there is no change in the optimal stock allocation from IID stock returns. When the prior stock returns are abnormally low, there is a further increase in the optimal stock allocation beyond that due to the hedging benefit of PAD stock returns.
When initial prior returns are equal to the expected IID return, the hedging benefits of PAD returns provide a meaningful improvement in sustainability. The improvement, however, is not as large as that obtained by reducing the initial disbursement from 4.0% to 3.5% for IID returns. More improvement than reducing the initial disbursement from 4.0% to 3.5% can be obtained, however, when the initial prior returns are as low as have been observed historically. When the initial prior returns are as high as have been observed historically the deterioration in sustainability is serious, but not as bad as increasing the initial disbursement from 4.0% to 4.5%, for IID returns. Thus, PAD stock returns provide a meaningful benefit for normal initial prior returns. Much larger benefits occur when prior returns are as low as sometimes in the past. Serious deterioration in sustainability occurs when prior returns are as high as have sometimes been observed.
Evidence
The SBBI Yearbooks have a well-known series for the annual real returns for large capitalization stocks listed on the NYSE starting in 1926. These returns are identical to those for the S&P 500 starting in the late 1950s when the S&P 500 was introduced. Publication of the SBBI Yearbook, however, ceased in 2023, covering the historic returns up through 2022. Others, however, have been publishing on line comparable historical returns for the S&P 500, which include reinvestment of dividends, and adjustment for inflation based on the CPI. Returns from the site maintained by Aswath Damodaran were used to update the series from the SBBI Yearbooks for large capitalization stocks for years 2023 and 2024.
To use this series to provide a prior average as long as 40 years, the first year that can be used for a regression of these returns on their prior averages is 1966. The results of such regressions for the returns from 1966 to 2024 on their prior moving averages are shown in Table 11.1. The regression coefficients are negative and become more strongly negative as the length of the moving average increases to 30 years. As just noted, however, such regressions have a bias to give negative coefficients even if the returns in the series are not negative PAD, but IID. There is a negative bias because having a prior moving average above its own average over the sample period makes it more likely that subsequent returns in the sample period are below average, and this tendency is stronger the longer is the moving average. Thus, there is a negative relation between returns in a sample period and a prior moving average even if those returns are IID.
The negative bias can be confirmed and its size investigated by running regressions on simulated time series of returns that are IID. To do so, suppose that 10,000 time series of returns are simulated that are of the same length as the series of historic returns that were used to obtain the results in Table 11.1. Suppose that these returns are IID with a normal distribution with a mean and standard deviation equal to the average and sample standard deviation of the Large Cap Stocks from 1926 to 2024. Results from these regressions are shown in the two columns on the right of Table 11.1.
The chances that the regression coefficients are negative increase from .72 for a 10 year average to .87 for a 40 year average. The column on the far right shows that regression coefficients as negative as those obtained for the historic returns with shorter averages could easily be due to chance if the underlying returns are IID. Coefficients as negative as those obtained for averages of 25 or 30 years, however, are less likely to be due to chance. In particular, the coefficient for the 30 year average has only a .025 chance of occurring if the returns are IID, which is less than the .05 chance often used for establishing significance. Perhaps, however, the coefficient for the 30 year average is unlikely because the underlying stock returns have a different mean or standard deviation than assumed for the IID returns. When the mean or standard deviation is varied by .02, however, virtually the same results are obtained as in the two columns on the right of Table 11.1. It is of interest, but not necessarily of much consequence, that the lowest and highest 30 year averages over 1966 to 2024 occurred just prior to the unusually strong bull market of 1995/99, and the unusually severe bear market of 1973/74 as discussed in a note.
Modeling
Elsewhere, the assumptions about the future expected return of stocks in the simulations have been those specified in Model as either the Pre 2008, Lower, or Mid Returns. These sets of assumptions all assume that the return on stocks each year is a linear function of the return on intermediates for that year plus an independently and normally distributed random variable with a zero mean. To include the dependence of stock returns on the 30 year average of prior returns, the linear function is expanded to include the 30 year average of prior returns. The coefficient of the 30 year average is assumed to be -1, but the effect of using -2 is also tested. To determine the constant term of the linear relation the expected value of the 30 year average is assumed to be the expected return of stocks for the assumed set of returns. For instance, for the Mid Returns, the expected return for stocks is .055, and for intermediates is .015. To make the expected return of stocks equal to .055 when the coefficient for the 30 year average is -1 the constant for the linear relation is set equal to .104 as .104+.4(.015)-1(.055) =.055.
For these assumptions, the stock returns continue to be normally distributed as they are linear combinations of normally distributed variables. After the initial year, however, the volatility of stock returns is affected by the variability of prior returns in the moving average as additional returns in the average are being determined in the simulation as it proceeds. The effect of the returns in the moving average on volatility, however, is very small. For instance, the volatility for the second year is increased by (1/30) squared of the variance of the return for the first year when the coefficient is -1, which increases the standard deviation of the return by only .0001. In subsequent years the additional increase is even less. By year 36 simulations show that the total increase in the standard deviation is only about .0015. To reduce any impact of the higher volatility caused by the moving average a small reduction can be made in the volatility of the stock returns without the PAD dependence.
Consider two sets of returns, one of which is the same as the Mid Returns in Model. The other set differs by having stock returns that are PAD with a coefficient of -1 for the prior 30 year average. The expected stock return for this set is the same as for the Mid Returns when the prior 30 year average return is equal to the expected stock return for the Mid Returns assumption. The standard deviation of the annual stock returns for this set is assumed to be .001 less than for the Mid Returns. These two sets of returns will be used to illustrate the effect of PAD stock returns on sustainability risk. First, all 30 prior stock returns are assumed to equal the expected stock return for the Mid Returns. This case illustrates the effect of PAD stock returns when the effect of the initial 30 years of returns is neutral.
When the effect of the initial prior 30 annual returns is neutral, stocks that are PAD become more valuable relative to intermediates, and the best stock allocation increases. Sustainability also shows a meaningful improvement. The next step is to see how these results are affected when the 30 prior returns have some of the patterns that have been observed in the past, and what the effect might be of the prior 30 annual returns in 2025. To gauge the effect of the pattern of prior historical returns an adjustment is made for the difference in the overall average of the historical returns and the assumed expected stock return. All of the historic returns are reduced by this difference before the effect of any of the 30 year segments of the historic returns is tested. Thus, these tests attempt to show the effect of prior variability about the mean rather than differences in the mean itself. The changes in the spreadsheet needed to make these tests in the model are discussed in a note.
Results
For the IID Mid Returns viewed initially the expected stock return at the beginning of each year in the future is the same and equal to .055. For the PAD Mid Returns viewed initially, the expected stock return at the beginning of each year in the future is the larger, the lower has been the average of the 30 prior returns. To gauge the size of the hedging benefit, suppose first that the effect of the initial prior 30 annual returns is neutral because each of these prior returns is equal to the expected stock return for the Mid Returns of .055. Suppose also for the PAD Mid Returns that the coefficient for the 30 year average is -1, and that the linear relation for the stock returns is that described earlier for this case. Simulations can provide as an output the probability distribution of the trailing 30 year average each year as the future unfolds. Deducting these averages from .0110 gives the expected stock return as discussed earlier. Thus, using this procedure the probability distribution of expected stock returns at the beginning of future years can be obtained.
The percentiles for these cumulative probability distributions are shown in Chart 11.1. The upper curve for the 95th percentile gives the lower bound for the highest 5% of expected stock returns at the beginning of each year. These correspond to the upper bound for the lower 5% of 30 year prior averages that have been obtained for those years from the simulations. The chart shows by year 10 that the next expected stock return has increased from .055 to .080. Suppose the coefficient of the 30 year average is -2 instead of -1. In this case, the expected stock return at year 10 increases to .095 instead of .080. As expected, the hedge provided by the PAD stocks becomes stronger.
To consider the effect of their hedging benefit on sustainability, suppose the New 4% Rule is being used as discussed earlier at page, BID. The earlier results at years 12 and 24 are shown by the solid curves in Charts 11.2 and 11.3 labeled IID. Now suppose that the PAD returns just described are being used instead of the IID returns. As PAD makes stocks more valuable relative to intermediates the stock allocation increases from 50% to 60% or 70% depending on whether the coefficient of the 30 year average is -1 or -2. The results in this case are shown by the dashed curves labeled PAD. There is a meaningful improvement in sustainability, but it is well short of that obtained by reducing the initial disbursement from 4.0% to 3.5%.
Initial Prior Returns
So far, the initial prior returns have been set equal to the expected IID return. Now suppose that the prior returns are the most recent 30 annual real returns in the 1926 to 2024 historical series, or the 30 consecutive returns in that series that would have an unusually strong impact either positive or negative on sustainability. To have a strong impact on sustainability the prior average must be unusually low or high and more recent returns should also be unusually low or high as recent returns will remain in the average for a long time. Unusually high averages with more recent high returns occurred prior to the early and mid 1960s following the long bullish period in the stock market after WWII. In particular, the returns prior to 1966 will be considered as that is the first year for the regressions in Table 11.1. An unusually low average with more recent low returns occurred in the mid 1980s following the long bearish period in the stock market in the 1970s. The mid 1980s also reflects some of the unusually strong returns in the 1950s coming out of the 30 year average. In particular, the returns prior to 1985 will be considered. To isolate the effect of the pattern of the returns in the historical series from the difference between the average historical returns and the expected IID return, an adjustment is required. All of the returns in the historical series are adjusted by the difference between the average return of the historical series and the expected IID return.
For normal initial prior returns the hedging benefit of PAD returns makes stocks more valuable relative to intermediates, and increases the allocation to stocks by 10 or 20 percentage points. This advantage for stocks with PAD returns can be nullified, however, if the initial prior returns are too high. Suppose, for instance, that the initial prior stock returns are those in 1966, after the adjustment just described. When the prior returns for 1966 replace the neutral returns the optimal stock allocations for coefficients of -1 and -2 revert back to being the same as for the IID returns. When the prior returns for 1985 replace the IID returns the optimal allocation increases by another 10 percentage points. Change in the 30 year average return as the disbursement period unfolds might make some later change in allocation desirable, but doing so will not have much effect on sustainability, and is not considered.
For initial prior returns equal to the expected stock return for the Mid Returns, Charts 11.2 and 11.3 show the reduced chances of declines at years 12 and 24 for the PAD returns. Tables 11.2 and 11.3 compare these reductions to those obtained when the initial prior returns are those for the adjusted historical returns ending in 2024, 1985, and 1966. As expected, for the low returns prior to 1985, there are additional reductions. On the other hand, for the high returns prior to 1966, the sustainabiliy is worse than for the IID returns, and more so when the reaction coefficient is more negative. The surprising result is that the 30 year average ending in 2024 gives normal results despite the 30 year average ending in 2024 being higher than the 30 year average ending in 1966. Thus, at first glance, the 30 year average of past returns is showing that stocks are highly overvalued in early 2025 just like price earnings ratio measures of valuation. The explanation for the normal results in the tables is that the exceptionally strong bull market of 1995 through 1999 is about to come out of the 30 year average of past returns. Thus, whether the high 30 year average is showing that stocks are highly overvalued in early 2025 is questionable.
Conclusions
There is evidence that stock returns are negatively related to the trailing 30 year average of prior returns. This negative relationship provides a hedging benefit that reduces sustainability risk when making disbursements. Simulations show that the chances of future declines are reduced when the 30 annual returns before the disbursements begin are equal to the expected stock return. When the initial prior returns are as low as have been observed in the past, sustainability is further improved. When the prior returns are as high as observed in the past, sustainability is worse than without the negative PAD relationship. Surprisingly, however, the 30 prior annual returns up through 2024 have little effect on the PAD hedging benefit, despite the 30 year prior average being higher than in the past. The explanation is that the exceptionally strong bull market from 1995 through 1999 is about to come out of the 30 year average. Thus, in early 2025, the prior 30 year average indicates that stocks are highly overvalued just like price earnings ratios of market valuation. The high value of the prior 30 year average, however, is questionable as the strong bull market of the late 1990s is about to come out of the prior 30 year average.
Notes
1. Over 1966 to 2024, the strongest bull market was over 1995 to 1999 when the annual increase for those five years averaged 26% in real terms. The most severe bear market was over 1973 and 1974 when the annual decline averaged 28% in real terms for these two years. It is of interest to note that the trough and peak of the 30 year average over 1966 to 2024 occurred in the year just prior to each of these events. Of course, there was no way of knowing that was the peak or trough until long afterwards. Moreover, for many years beforehand the 30 year average was almost as low or as high as at the peak or trough. Thus, the 30 year average would not have been of much help in forecasting the timing of these events. Highly abnormal measures of stock market valuation are helpful in indicating that a highly abnormal move of the market is possible, not when it will occur.
2. To incorporate PAD stock returns into the model, two new columns must be added to the spreadsheet, and a revision made in the relation for the stock return, r(t). The first of the new columns has the 65 returns that are used in each iteration for a simulation of the disbursements. The first 30 of these returns starts with the most distant of the 30 given prior returns, and ends with the given return for the year prior to the beginning of the disbursements. The last 35 returns start with the drawing for the return for the first year in the iteration, and end with the drawing for the return for the 35th year in that iteration. The second column has the 36 successive moving averages of the 30 prior returns used to calculate the expected return for each successive year in an iteration. The first of these is the average of the 30 given prior returns, and the last is the 30 returns drawn in the simulation for the 6th through the 35th years of an iteration.
References
Aswath Damadoran, https://pages.stern.nyu.edu
Kroll, 2023 SBBI Yearbook, New York, 2023.
Robert J. Shiller, Irrational Exuberance, Princeton University Press, Princeton, NJ, 2000.
Posted March, 2019 Revised January, 2025