The stock allocation at the beginning of each year is reset to the initial allocation. For a given stock allocation, simulations are used to find the cumulative probability distributions of the disbursements at years 12, 24, and 36. The best stock allocation is found by successively increasing the stock allocation by 10 percentage points until there is doubt that the last allocation has improved the distributions of the disbursements. When there is doubt, the prior stock allocation is the best allocation for the conditions that have been assumed. For sustainable disbursements, the best stock allocation is a moderate 30% for either the investment returns in the pre 2008 illustration, or the hypothetical lower returns. For larger disbursements, however, the best stock allocation increases much more for the lower returns. For the pre 2008 returns, the best stock allocation increases from 30% to 40% when the initial disbursement is increased from 3.5% to 4.5%, the equity premium is increased from .04 to .06, or the standard deviation for stocks is reduced from .18 to .15.
Pre 2008 Returns
Suppose simulations are made using the basic model in Model assuming initially that as many as 36 annual disbursements may be needed. The investment returns are those for the pre 2008 illustration. The RDR is set equal to the expected return of the portfolio, but similar results for the stock allocation are obtained if the RDR is increased to .02 or .04 above the expected return. Set the initial disbursement equal to 3.5% of portfolio value, which has good sustainability. Now successively increase the stock allocation by 10 percentage points each time. Graphs for the cumulative probability distributions of the disbursements at years 12, 24, and 36 for stock allocations of 10% and 20% are shown in Charts A.1, A.2, and A.3. For all years, the increase in the allocation from 10% to 20% provides significant improvement. The curves for 20% are always below or equal to those for 10% indicating that the chances for lower disbursements for 20% are always less than or equal to the chances for 10%. The chances of declines by the respective years are given by the probabilities on the vertical axis for the tops of the curves. Note that an allocation of 20% has good sustainability as the chances of a decline by year 12 are only .04, and .12 by the end of the period. Note also that the improvement provided by 20% is just about the same in each of the charts indicating that it is sufficient to look just at year 24 to determine whether an increase in the allocation is an improvement.
Suppose next that the stock allocation is increased from 20% to 30%. The curves in this case at year 24 are shown in Chart A.4. There is a small improvement, but nowhere near what occurs when increasing the allocation from 10% to 20%. The chances of declines by year 24 are reduced by only a small amount as shown by the difference on the vertical axis of the tops of the curves. Now suppose the stock allocation is further increased from 30% to 40%. The results in this case at year 24 are shown in Chart A.5. The curve for 40% is dashed because it is an increase, but it is now slightly above the curve for 30%, and therefore not an improvement. The best stock allocation for a 3.5% initial disbursement for the pre 2008 illustration is therefore 30%. There is not much difference, however, for stock allocations of 20%, 30%, and 40%. Chart A.6, however, shows that a more appreciable difference develops when the stock allocation is further increased to 50%.
Larger Disbursements
Suppose next that the initial disbursement is increased from 3.5% to 4.5%. Chart A.7 shows that increasing the stock allocation from 30% to 40% is now an improvement. The curve for 40% is now below the curve for 30%. When the allocation is further increased to 50%, however, Chart A.8 shows that this further increase is not an improvement. The curve for 50% is somewhat below the curve for 40% for large disbursements, but slightly above the curve for 40% for lower disbursements. Thus, 50% has the benefit of some reduction in small declines, but that benefit comes at the cost of 50% having somewhat larger chances of large declines. Thus, there appears to be doubt that 50% has a net benefit, and provides an improvement. The best stock allocation for the 4.5% initial disbursement is therefore 40%. Increasing the stock allocation increases expected return at the cost of more volatility. When funding larger disbursements it is desirable to accept more volatility to get higher expected returns.
Lower Returns
Suppose now that the investment returns are the hypothetical lower returns that are specified in Model to gauge the possible effect on results of lower returns. These lower returns reduce the expected annual real returns of both stocks and intermediates by three percentage points. The volatilities are also reduced, but the correlation between the returns each year remains the same. For these lower returns, the sustainability of a 2.5% initial disbursement is just about the same as a 3.5% initial disbursement for the higher returns. In either case, the chances of declines by years 12, 24, and 36 are just about the same. Charts A.9 and A.10 show that the best stock allocation for a sustainable disbursement of 2.5% with the lower returns is 30%, which is the same as for the higher returns.
Suppose next that the initial disbursement for the lower returns is increased from 2.5% to 3.5% of portfolio value. The curve for an allocation of 40% is now below the curve for 30%. Also, the curve for 50% is below the curve for 40% as shown in Chart A.11. Moreover, the curve for 60% is also below the curve for 50% as shown in Chart A.12. In Chart A.13, however, the curve for 70% is sufficiently above the curve for 60% for larger declines to cause doubt that 70% is an improvement. Thus, for the lower returns, the best allocation is 60% when the initial disbursement increases from 2.5% to 3.5%. For the higher returns, the best allocation increases only from 30% to 40% when the initial disbursement increases from 3.5% to 4.5%. Funding larger disbursements makes it desirable to accept more volatility and reach for higher expected returns. For the lower returns it is desirable to reach much further.
Better Stock Returns
For the investment returns in the pre 2008 illustration, increasing the initial disbursement from a reasonably sustainable level of 3.5% of portfolio value to 4.5% increases the best stock allocation from 30% to 40%. Suppose instead, however, that the initial disbursement remains at 3.5%, but the expected return of the stocks is increased, or the volatility of the stock returns is reduced. How do these improvements in the returns of stocks affect the best stock allocation? In particular, suppose that the expected real return on stocks is increased from .07 to .09 raising the equity premium from .04 to .06, or suppose the standard deviation of the stock returns is reduced from .18 to .15. It turns out that either of these improvements in the returns on stocks increases the best stock allocation from 30% to 40%, which is the same as increasing the initial disbursement from 3.5% to 4.5%.
Chart A.14 assumes the returns for the pre 2008 illustration except that the equity premium has been increased from .04 to .06. With the initial disbursement remaining at 3.5%, the curves show the effect of stock allocations of 20%, 30%, and 40%. Although not shown in the chart, the curve for 50% is above that for 40% showing that 40% is the best stock allocation for the higher equity premium. Similarly, Chart A.15 assumes that instead of increasing the equity premium that the standard deviation of the stock returns has been reduced from .18 to .15. With the initial disbursement remaining at 3.5%, the curves show the effect of stock allocations of 20%, 30%, and 40%. Although not shown in the chart, the curve for 50% is above that for 40% showing that 40% is also the best stock allocation for the reduced volatility of stock returns. Note that either for significantly larger disbursements or significant improvements in stock returns that the best stock allocation does not exceed a modest 40%. A more conventional stock allocation of 60% is obtained only by significantly increasing the disbursements for the lower returns. The problem for stocks in making disbursements appears to be the high volatility of stock returns. In contrast, Accumulation shows that much higher stock allocations tend to be desirable for accumulating funds.
Shorter Horizons
As the disbursements proceed, the number of remaining disbursements to be made is declining. The simulations assume that the allocation remains the same over the disbursement period. But is this a reasonable assumption? In theory, the best stock allocation might be expected to decline as the horizon becomes shorter. The reason is that with a shorter horizon there are fewer drawings to average out the higher volatility for the stock returns so as to reduce risk and provide the benefit of the higher expected return. Perhaps in practice, however, this benefit is sufficiently small that it can be ignored as an approximation. To find out, the simulations can be rerun with a shorter horizon.
Charts A.16 and A.17 assume that 18 more years of disbursements remain to be made. There has been no need to reduce the disbursement, and it is still equal to 3.5% of initial portfolio value. The value of the portfolio, however, is now equal to 60% of its initial value. With this lower portfolio value, the chances of a decline two-thirds of the way through the remaining disbursement period are the same as initially for a 30% stock allocation in Chart A.4. Two-thirds of the way through the remaining disbursement period, however, is now year 12 instead of year 24. Instead of being below the curve for 20%, the curve for 30% is now just about the same as that for 20%. These curves, however, are still meaningfully below the curve for 10%. Chart A.17 shows that the curve for 30% is now somewhat further below that for 40%. Thus, as expected, the simulations show that the best stock allocation declines as the horizon shortens. The effect is small, however, and assuming the allocation is fixed over the disbursement period appears to be a reasonable approximation. Nevertheless, in practice, reducing the stock allocation somewhat over the last few years seems advisable.
Conclusions
The best stock allocation is found by successively increasing the stock allocation by 10 percentage points until there is doubt that the latest increase is an improvement. Using this procedure, the best stock allocation for sustainable disbursements that may last as long as 36 years is 30% for either the investment returns in the pre 2008 illustration or the hypothetical lower returns. For larger than sustainable disbursements, it is desirable to accept more volatility and increase the expected return by increasing the allocation to stocks. For the lower returns, however, the increase in the best stock allocation is far larger than that for the higher returns for the same percentage point increase in the initial disbursement. For the returns in the pre 2008 illustration, the best stock allocation increases from 30% to 40% when the initial disbursement increases from 3.5% to 4.5%, the equity premium increases from .04 to .06, or the standard deviation of the returns on the stocks is reduced from .18 to .15. When the horizon declines as the disbursements proceed the best stock allocation declines. This decline is small enough, however, that assuming a fixed allocation over the disbursement period appears to nevertheless be a reasonable approximation.
Posted November, 2018 Revised, July, 2022
Addendum
For single period portfolio analysis, the major focus is on the correlation of the returns of the investments in the portfolio. These correlations determine how the portfolio should be allocated to achieve a desired balance between the expected value of the return on the portfolio and its volatility. For the multiperiod analysis at this site, the correlation between the returns is an important factor determining risk, but only one of many factors of interest in determining the results. A significant reduction in the correlation between the returns on stocks and intermediates reduces risk, as expected. But does a significant reduction in correlation affect the best allocation? A reduction in the correlation of their returns makes either the stocks or intermediates more valuable. But does one become more valuable relative to the other changing the best allocation? Simulations show that changes in the correlation of the returns on the stocks and intermediates have little effect on the best allocation.
Model assumes that the return on stocks each year is equal to a coefficient of .4 times the return on intermediates plus an independently and normally distributed random variable. The mean and standard deviation of the independent random variable are set so as to make the expected value and standard deviation of the returns on the stocks equal to specified values. The coefficient of .4 for the return on the intermediates is the coefficient obtained when the annual stock returns are regressed on the annual returns of the intermediates for a long period starting in 1960. For the period from 1960 to 1990, however, the major factor affecting investment returns was inflation. Increases and decreases in inflation have the same effect on the real returns of both stocks and intermediates. Thus, when inflation was the dominant influence for investment returns in the U. S. financial markets from 1960 to 1990 there was a stronger positive correlation between the real returns on stocks and intermediates than the .4 coefficient assumed in Model. In contrast, from 1990 up until the start of the pandemic in 2020 the dominant influences on U. S. financial markets were periods of speculative excess. First, there was the Dot Com Bubble of the late 1990s, and then the Credit Crunch of 2008. For periods of speculative excess fixed incomes of good quality provided a good hedge for stocks, and the correlation between the real returns on stocks and intermediates of good quality was strongly negative.
Suppose instead of .4 as assumed in Model that the correlation coefficient between the real returns on stocks and intermediates is either .8 or -.5. In these cases, the relation for simulating stock returns in Model must be changed to keep the expected value and standard deviation of the annual stock returns equal to the same specified values. The changes in these relations that do so are discussed in a note. As considered earlier in the text, suppose that the disbursements are sustainable and start at 3.5%. Consider cases in which the correlation coefficient of stock returns on the intermediates is either .8 or -.5 instead of .4 as assumed in Model. The results at year 24 for stock allocations of 20%, 30%, and 40% are shown in Chart A.18. As expected, there is much less risk when the correlation coefficient is -.5. The change in the correlation coefficient, however, does not affect the best stock allocation. For either of the coefficients the best stock allocation is 30% as found earlier for a coefficient of .4 in Charts A.4 and A.5 in the text. Looking to the future, disbursements will tend to be riskier if inflation is the major problem affecting financial markets. The best allocation, however, is little affected by whether the major problem is inflation or periods of speculative excess.
Note
The illustration based on pre 2008 returns assumes the annual real return on intermediates is normally distributed with an expected return of .03 and a standard deviation of .07. The annual return on stocks is assumed to be normally distributed with an expected return of .07 and a standard deviation of .18. When the annual return on stocks is assumed to be .4 of the annual return on intermediates plus an independently distributed normal variable, the independent variable must have an expected return of .058, and a standard deviation of .1778, to give the specified annual return on stocks. Thus, as given in Model:
r(t) = RiskNormal( .058 + .4i(t), .1778 )
When the annual real return on stocks is .8 of the annual real return on intermediates the expected return of the independent variable becomes .046 and the standard deviation equal to .17107. Thus, in this case r(t) is given by:
r(t) = RiskNormal( .046 + .8i(t), .17107 )
When the annual real return on stocks is -.5 of the annual real return on intermediates the expected return of the independent variable becomes .085 and the standard deviation is equal to .17656. In this case, r(t) is given by:
r(t) = RiskNormal( .085 – .5i(t), .17656 )
Posted August 2023
Addendum
Raising the initial disbursement increases the risk of the disbursements. The text of Allocation shows that it is possible to reduce this increase in risk by increasing the allocation to stocks. The risk of a given initial disbursement in the future depends on the value of the portfolio at that time. If investment returns in the future turn out to be less than expected the value of the portfolio will also be less than expected, and the value of the disbursement will increase relative to the value of the portfolio. The risk of the disbursements will increase just as the risk of the disbursements increases when the initial disbursement is raised relative to the initial value of the portfolio. When the value of the disbursement increases relative to the value of the portfolio in the future it should therefore be possible to reduce the risk of the disbursements by increasing the allocation to stocks just as occurs initially.
In particular, suppose the initial disbursement is 4.5% of portfolio value, and the stock allocation is 40% giving an RDR of .066, which is .02 above expected return for the pre 2008 returns. The distributions of the disbursements at years 12 and 24 are shown by the solid curves in Charts A.19 and A.20. Now suppose in the future that the stock allocation is increased from 40% to 60% and the RDR from .066 to .074 whenever the prior disbursement is equal to or greater than 6.5% of the current value of the portfolio. If subsequently the prior disbursement no longer meets this condition, the stock allocation is reduced back to 40% and the RDR to .066. The modifications of Model to incorporate these changes are discussed in a note. The results when these changes are made are shown by the dotted curves in Charts A.19 and A.20. There is a very small increase in the chances of large declines at year 24. Overall, however, increasing the stock allocation reduces risk when the risk of the disbursements increases in the future due to lower than expected portfolio values.
The results in Charts A.19 and A.20 assume the pre 2008 returns specified in Model. The text of Allocation shows for the lower returns specified in Model that the best allocation for a sustainable disbursement is the same as for the pre 2008 returns. When risk is increased by increasing the initial disbursement above a sustainable level, however, significantly higher stock allocations are desirable for the lower returns. Thus, for the lower returns, increasing the stock allocation in the future when risk increases might be expected to provide more improvement than for the pre 2008 returns. Increase shows for the lower returns that the Best Initial Disbursement (BID) declines by one percentage point from 4.5% to 3.5% of initial portfolio value. The text of Allocation shows for the lower returns that 60% is the best stock allocation for an initial disbursement of 3.5% and that the RDR is .044. Suppose for the lower returns that the stock allocation is increased from 60% to 80% whenever the prior disbursement is greater than or equal to 5.5% of current portfolio value, and the RDR is increased from .044 to .052. When this condition ceases to be satisfied the stock allocation and RDR return to their former values. The results for the lower returns at years 12 and 24 in Charts A.21 and A.22 for these conditions show that there is more improvement, as expected.
Model assumes that the allocation of the portfolio remains the same over the disbursement period. As the horizon shortens, however, there is less opportunity for averaging of the stock returns to reduce volatility, and make their higher expected return more valuable. Thus, a lower stock allocation might be expected to be desirable over time, and the text of Allocation shows that some decline is desirable. Instead of a decline, this addendum shows that some increase in the stock allocation is desirable if in the future the disbursement increases relative to the value of the portfolio indicating some increase in risk for the disbursements. Neither of these effects appears to be so strong as to disallow the use of a constant allocation as an approximation to gauge general effects. In practice, however, some reduction in the stock allocation is desirable as the end of the disbursement period approaches, and some increase in the stock allocation is desirable when the disbursement becomes much larger than earlier relative to portfolio value.
Note
To increase the stock allocation from 40% to 60% and the RDR from .066 to .074 if and only if the prior disbursement is greater than or equal to 6.5% of current portfolio value two columns are added to the spreadsheet for Model. For one of these columns, the first row is equal to .4 and subsequent rows equal to:
= If ( C(t-1) >= .065V(t), .6, .4 ) t = 2, 3, …, T
The values in each row of this column replace “a” in each row of the relation in Model for the return each period:
R(t) = ar(t) + [1-a]i(t) t = 1, 2, …, T
For the other column, the first row is equal to .066, and subsequent rows are equal to:
= If ( C(t-1) >= .065V(t), .074, .066 ) t = 2, 3, …, T
The values in each row of this column replace “m” in each row of the relation in Model for the limit each period:
C*(t) = PMT( m, T-t+1, V(t), 0, 1 ) t = 2, 3, …, T
Similar modifications are made for the lower returns.
Allocation
Best Allocations
The stock allocation at the beginning of each year is reset to the initial allocation. For a given stock allocation, simulations are used to find the cumulative probability distributions of the disbursements at years 12, 24, and 36. The best stock allocation is found by successively increasing the stock allocation by 10 percentage points until there is doubt that the last allocation has improved the distributions of the disbursements. When there is doubt, the prior stock allocation is the best allocation for the conditions that have been assumed. For sustainable disbursements, the best stock allocation is a moderate 30% for either the investment returns in the pre 2008 illustration, or the hypothetical lower returns. For larger disbursements, however, the best stock allocation increases much more for the lower returns. For the pre 2008 returns, the best stock allocation increases from 30% to 40% when the initial disbursement is increased from 3.5% to 4.5%, the equity premium is increased from .04 to .06, or the standard deviation for stocks is reduced from .18 to .15.
Pre 2008 Returns
Suppose simulations are made using the basic model in Model assuming initially that as many as 36 annual disbursements may be needed. The investment returns are those for the pre 2008 illustration. The RDR is set equal to the expected return of the portfolio, but similar results for the stock allocation are obtained if the RDR is increased to .02 or .04 above the expected return. Set the initial disbursement equal to 3.5% of portfolio value, which has good sustainability. Now successively increase the stock allocation by 10 percentage points each time. Graphs for the cumulative probability distributions of the disbursements at years 12, 24, and 36 for stock allocations of 10% and 20% are shown in Charts A.1, A.2, and A.3. For all years, the increase in the allocation from 10% to 20% provides significant improvement. The curves for 20% are always below or equal to those for 10% indicating that the chances for lower disbursements for 20% are always less than or equal to the chances for 10%. The chances of declines by the respective years are given by the probabilities on the vertical axis for the tops of the curves. Note that an allocation of 20% has good sustainability as the chances of a decline by year 12 are only .04, and .12 by the end of the period. Note also that the improvement provided by 20% is just about the same in each of the charts indicating that it is sufficient to look just at year 24 to determine whether an increase in the allocation is an improvement.
Suppose next that the stock allocation is increased from 20% to 30%. The curves in this case at year 24 are shown in Chart A.4. There is a small improvement, but nowhere near what occurs when increasing the allocation from 10% to 20%. The chances of declines by year 24 are reduced by only a small amount as shown by the difference on the vertical axis of the tops of the curves. Now suppose the stock allocation is further increased from 30% to 40%. The results in this case at year 24 are shown in Chart A.5. The curve for 40% is dashed because it is an increase, but it is now slightly above the curve for 30%, and therefore not an improvement. The best stock allocation for a 3.5% initial disbursement for the pre 2008 illustration is therefore 30%. There is not much difference, however, for stock allocations of 20%, 30%, and 40%. Chart A.6, however, shows that a more appreciable difference develops when the stock allocation is further increased to 50%.
Larger Disbursements
Suppose next that the initial disbursement is increased from 3.5% to 4.5%. Chart A.7 shows that increasing the stock allocation from 30% to 40% is now an improvement. The curve for 40% is now below the curve for 30%. When the allocation is further increased to 50%, however, Chart A.8 shows that this further increase is not an improvement. The curve for 50% is somewhat below the curve for 40% for large disbursements, but slightly above the curve for 40% for lower disbursements. Thus, 50% has the benefit of some reduction in small declines, but that benefit comes at the cost of 50% having somewhat larger chances of large declines. Thus, there appears to be doubt that 50% has a net benefit, and provides an improvement. The best stock allocation for the 4.5% initial disbursement is therefore 40%. Increasing the stock allocation increases expected return at the cost of more volatility. When funding larger disbursements it is desirable to accept more volatility to get higher expected returns.
Lower Returns
Suppose now that the investment returns are the hypothetical lower returns that are specified in Model to gauge the possible effect on results of lower returns. These lower returns reduce the expected annual real returns of both stocks and intermediates by three percentage points. The volatilities are also reduced, but the correlation between the returns each year remains the same. For these lower returns, the sustainability of a 2.5% initial disbursement is just about the same as a 3.5% initial disbursement for the higher returns. In either case, the chances of declines by years 12, 24, and 36 are just about the same. Charts A.9 and A.10 show that the best stock allocation for a sustainable disbursement of 2.5% with the lower returns is 30%, which is the same as for the higher returns.
Suppose next that the initial disbursement for the lower returns is increased from 2.5% to 3.5% of portfolio value. The curve for an allocation of 40% is now below the curve for 30%. Also, the curve for 50% is below the curve for 40% as shown in Chart A.11. Moreover, the curve for 60% is also below the curve for 50% as shown in Chart A.12. In Chart A.13, however, the curve for 70% is sufficiently above the curve for 60% for larger declines to cause doubt that 70% is an improvement. Thus, for the lower returns, the best allocation is 60% when the initial disbursement increases from 2.5% to 3.5%. For the higher returns, the best allocation increases only from 30% to 40% when the initial disbursement increases from 3.5% to 4.5%. Funding larger disbursements makes it desirable to accept more volatility and reach for higher expected returns. For the lower returns it is desirable to reach much further.
Better Stock Returns
For the investment returns in the pre 2008 illustration, increasing the initial disbursement from a reasonably sustainable level of 3.5% of portfolio value to 4.5% increases the best stock allocation from 30% to 40%. Suppose instead, however, that the initial disbursement remains at 3.5%, but the expected return of the stocks is increased, or the volatility of the stock returns is reduced. How do these improvements in the returns of stocks affect the best stock allocation? In particular, suppose that the expected real return on stocks is increased from .07 to .09 raising the equity premium from .04 to .06, or suppose the standard deviation of the stock returns is reduced from .18 to .15. It turns out that either of these improvements in the returns on stocks increases the best stock allocation from 30% to 40%, which is the same as increasing the initial disbursement from 3.5% to 4.5%.
Chart A.14 assumes the returns for the pre 2008 illustration except that the equity premium has been increased from .04 to .06. With the initial disbursement remaining at 3.5%, the curves show the effect of stock allocations of 20%, 30%, and 40%. Although not shown in the chart, the curve for 50% is above that for 40% showing that 40% is the best stock allocation for the higher equity premium. Similarly, Chart A.15 assumes that instead of increasing the equity premium that the standard deviation of the stock returns has been reduced from .18 to .15. With the initial disbursement remaining at 3.5%, the curves show the effect of stock allocations of 20%, 30%, and 40%. Although not shown in the chart, the curve for 50% is above that for 40% showing that 40% is also the best stock allocation for the reduced volatility of stock returns. Note that either for significantly larger disbursements or significant improvements in stock returns that the best stock allocation does not exceed a modest 40%. A more conventional stock allocation of 60% is obtained only by significantly increasing the disbursements for the lower returns. The problem for stocks in making disbursements appears to be the high volatility of stock returns. In contrast, Accumulation shows that much higher stock allocations tend to be desirable for accumulating funds.
Shorter Horizons
As the disbursements proceed, the number of remaining disbursements to be made is declining. The simulations assume that the allocation remains the same over the disbursement period. But is this a reasonable assumption? In theory, the best stock allocation might be expected to decline as the horizon becomes shorter. The reason is that with a shorter horizon there are fewer drawings to average out the higher volatility for the stock returns so as to reduce risk and provide the benefit of the higher expected return. Perhaps in practice, however, this benefit is sufficiently small that it can be ignored as an approximation. To find out, the simulations can be rerun with a shorter horizon.
Charts A.16 and A.17 assume that 18 more years of disbursements remain to be made. There has been no need to reduce the disbursement, and it is still equal to 3.5% of initial portfolio value. The value of the portfolio, however, is now equal to 60% of its initial value. With this lower portfolio value, the chances of a decline two-thirds of the way through the remaining disbursement period are the same as initially for a 30% stock allocation in Chart A.4. Two-thirds of the way through the remaining disbursement period, however, is now year 12 instead of year 24. Instead of being below the curve for 20%, the curve for 30% is now just about the same as that for 20%. These curves, however, are still meaningfully below the curve for 10%. Chart A.17 shows that the curve for 30% is now somewhat further below that for 40%. Thus, as expected, the simulations show that the best stock allocation declines as the horizon shortens. The effect is small, however, and assuming the allocation is fixed over the disbursement period appears to be a reasonable approximation. Nevertheless, in practice, reducing the stock allocation somewhat over the last few years seems advisable.
Conclusions
The best stock allocation is found by successively increasing the stock allocation by 10 percentage points until there is doubt that the latest increase is an improvement. Using this procedure, the best stock allocation for sustainable disbursements that may last as long as 36 years is 30% for either the investment returns in the pre 2008 illustration or the hypothetical lower returns. For larger than sustainable disbursements, it is desirable to accept more volatility and increase the expected return by increasing the allocation to stocks. For the lower returns, however, the increase in the best stock allocation is far larger than that for the higher returns for the same percentage point increase in the initial disbursement. For the returns in the pre 2008 illustration, the best stock allocation increases from 30% to 40% when the initial disbursement increases from 3.5% to 4.5%, the equity premium increases from .04 to .06, or the standard deviation of the returns on the stocks is reduced from .18 to .15. When the horizon declines as the disbursements proceed the best stock allocation declines. This decline is small enough, however, that assuming a fixed allocation over the disbursement period appears to nevertheless be a reasonable approximation.
Posted November, 2018 Revised, July, 2022
Addendum
For single period portfolio analysis, the major focus is on the correlation of the returns of the investments in the portfolio. These correlations determine how the portfolio should be allocated to achieve a desired balance between the expected value of the return on the portfolio and its volatility. For the multiperiod analysis at this site, the correlation between the returns is an important factor determining risk, but only one of many factors of interest in determining the results. A significant reduction in the correlation between the returns on stocks and intermediates reduces risk, as expected. But does a significant reduction in correlation affect the best allocation? A reduction in the correlation of their returns makes either the stocks or intermediates more valuable. But does one become more valuable relative to the other changing the best allocation? Simulations show that changes in the correlation of the returns on the stocks and intermediates have little effect on the best allocation.
Model assumes that the return on stocks each year is equal to a coefficient of .4 times the return on intermediates plus an independently and normally distributed random variable. The mean and standard deviation of the independent random variable are set so as to make the expected value and standard deviation of the returns on the stocks equal to specified values. The coefficient of .4 for the return on the intermediates is the coefficient obtained when the annual stock returns are regressed on the annual returns of the intermediates for a long period starting in 1960. For the period from 1960 to 1990, however, the major factor affecting investment returns was inflation. Increases and decreases in inflation have the same effect on the real returns of both stocks and intermediates. Thus, when inflation was the dominant influence for investment returns in the U. S. financial markets from 1960 to 1990 there was a stronger positive correlation between the real returns on stocks and intermediates than the .4 coefficient assumed in Model. In contrast, from 1990 up until the start of the pandemic in 2020 the dominant influences on U. S. financial markets were periods of speculative excess. First, there was the Dot Com Bubble of the late 1990s, and then the Credit Crunch of 2008. For periods of speculative excess fixed incomes of good quality provided a good hedge for stocks, and the correlation between the real returns on stocks and intermediates of good quality was strongly negative.
Suppose instead of .4 as assumed in Model that the correlation coefficient between the real returns on stocks and intermediates is either .8 or -.5. In these cases, the relation for simulating stock returns in Model must be changed to keep the expected value and standard deviation of the annual stock returns equal to the same specified values. The changes in these relations that do so are discussed in a note. As considered earlier in the text, suppose that the disbursements are sustainable and start at 3.5%. Consider cases in which the correlation coefficient of stock returns on the intermediates is either .8 or -.5 instead of .4 as assumed in Model. The results at year 24 for stock allocations of 20%, 30%, and 40% are shown in Chart A.18. As expected, there is much less risk when the correlation coefficient is -.5. The change in the correlation coefficient, however, does not affect the best stock allocation. For either of the coefficients the best stock allocation is 30% as found earlier for a coefficient of .4 in Charts A.4 and A.5 in the text. Looking to the future, disbursements will tend to be riskier if inflation is the major problem affecting financial markets. The best allocation, however, is little affected by whether the major problem is inflation or periods of speculative excess.
Note
The illustration based on pre 2008 returns assumes the annual real return on intermediates is normally distributed with an expected return of .03 and a standard deviation of .07. The annual return on stocks is assumed to be normally distributed with an expected return of .07 and a standard deviation of .18. When the annual return on stocks is assumed to be .4 of the annual return on intermediates plus an independently distributed normal variable, the independent variable must have an expected return of .058, and a standard deviation of .1778, to give the specified annual return on stocks. Thus, as given in Model:
r(t) = RiskNormal( .058 + .4i(t), .1778 )
When the annual real return on stocks is .8 of the annual real return on intermediates the expected return of the independent variable becomes .046 and the standard deviation equal to .17107. Thus, in this case r(t) is given by:
r(t) = RiskNormal( .046 + .8i(t), .17107 )
When the annual real return on stocks is -.5 of the annual real return on intermediates the expected return of the independent variable becomes .085 and the standard deviation is equal to .17656. In this case, r(t) is given by:
r(t) = RiskNormal( .085 – .5i(t), .17656 )
Posted August 2023
Addendum
Raising the initial disbursement increases the risk of the disbursements. The text of Allocation shows that it is possible to reduce this increase in risk by increasing the allocation to stocks. The risk of a given initial disbursement in the future depends on the value of the portfolio at that time. If investment returns in the future turn out to be less than expected the value of the portfolio will also be less than expected, and the value of the disbursement will increase relative to the value of the portfolio. The risk of the disbursements will increase just as the risk of the disbursements increases when the initial disbursement is raised relative to the initial value of the portfolio. When the value of the disbursement increases relative to the value of the portfolio in the future it should therefore be possible to reduce the risk of the disbursements by increasing the allocation to stocks just as occurs initially.
In particular, suppose the initial disbursement is 4.5% of portfolio value, and the stock allocation is 40% giving an RDR of .066, which is .02 above expected return for the pre 2008 returns. The distributions of the disbursements at years 12 and 24 are shown by the solid curves in Charts A.19 and A.20. Now suppose in the future that the stock allocation is increased from 40% to 60% and the RDR from .066 to .074 whenever the prior disbursement is equal to or greater than 6.5% of the current value of the portfolio. If subsequently the prior disbursement no longer meets this condition, the stock allocation is reduced back to 40% and the RDR to .066. The modifications of Model to incorporate these changes are discussed in a note. The results when these changes are made are shown by the dotted curves in Charts A.19 and A.20. There is a very small increase in the chances of large declines at year 24. Overall, however, increasing the stock allocation reduces risk when the risk of the disbursements increases in the future due to lower than expected portfolio values.
The results in Charts A.19 and A.20 assume the pre 2008 returns specified in Model. The text of Allocation shows for the lower returns specified in Model that the best allocation for a sustainable disbursement is the same as for the pre 2008 returns. When risk is increased by increasing the initial disbursement above a sustainable level, however, significantly higher stock allocations are desirable for the lower returns. Thus, for the lower returns, increasing the stock allocation in the future when risk increases might be expected to provide more improvement than for the pre 2008 returns. Increase shows for the lower returns that the Best Initial Disbursement (BID) declines by one percentage point from 4.5% to 3.5% of initial portfolio value. The text of Allocation shows for the lower returns that 60% is the best stock allocation for an initial disbursement of 3.5% and that the RDR is .044. Suppose for the lower returns that the stock allocation is increased from 60% to 80% whenever the prior disbursement is greater than or equal to 5.5% of current portfolio value, and the RDR is increased from .044 to .052. When this condition ceases to be satisfied the stock allocation and RDR return to their former values. The results for the lower returns at years 12 and 24 in Charts A.21 and A.22 for these conditions show that there is more improvement, as expected.
Model assumes that the allocation of the portfolio remains the same over the disbursement period. As the horizon shortens, however, there is less opportunity for averaging of the stock returns to reduce volatility, and make their higher expected return more valuable. Thus, a lower stock allocation might be expected to be desirable over time, and the text of Allocation shows that some decline is desirable. Instead of a decline, this addendum shows that some increase in the stock allocation is desirable if in the future the disbursement increases relative to the value of the portfolio indicating some increase in risk for the disbursements. Neither of these effects appears to be so strong as to disallow the use of a constant allocation as an approximation to gauge general effects. In practice, however, some reduction in the stock allocation is desirable as the end of the disbursement period approaches, and some increase in the stock allocation is desirable when the disbursement becomes much larger than earlier relative to portfolio value.
Note
To increase the stock allocation from 40% to 60% and the RDR from .066 to .074 if and only if the prior disbursement is greater than or equal to 6.5% of current portfolio value two columns are added to the spreadsheet for Model. For one of these columns, the first row is equal to .4 and subsequent rows equal to:
= If ( C(t-1) >= .065V(t), .6, .4 ) t = 2, 3, …, T
The values in each row of this column replace “a” in each row of the relation in Model for the return each period:
R(t) = ar(t) + [1-a]i(t) t = 1, 2, …, T
For the other column, the first row is equal to .066, and subsequent rows are equal to:
= If ( C(t-1) >= .065V(t), .074, .066 ) t = 2, 3, …, T
The values in each row of this column replace “m” in each row of the relation in Model for the limit each period:
C*(t) = PMT( m, T-t+1, V(t), 0, 1 ) t = 2, 3, …, T
Similar modifications are made for the lower returns.
Posted August, 2023