Given an initial disbursement, simulations using the basic model in Model provide the cumulative probability distributions of the disbursements at years 12, 24, and 36. In Allocation, the best stock allocation for a given initial disbursement is found by successively increasing the stock allocation by 10 percentage points until doubt arises that the last increase is an improvement, in which case the prior allocation is the best stock allocation. The Best Initial Disbursement (BID) will now be found by a similar process. The stock allocation is set equal to the best allocation found in Allocation, and the RDR to .02 more than the expected return of the portfolio based on the results in RDR. The initial disbursement is successively increased by half a percentage point of initial portfolio value until there is doubt that the last increase is an improvement, in which case the prior initial disbursement is the Best Initial Disbursement (BID).
To assess whether an increase in the initial disbursement is an improvement a judgment is made as to whether the increased risk of future declines in the disbursements caused by the increase is worth accepting to obtain the higher disbursements provided to increase spending. Such decisions might be made by comparing metrics calculated from successive probability distributions at future years. Perhaps, however, such decisions can better be made by simply visually comparing graphs of the successive cumulative probability distributions at future years. In particular, comparisons can be made at year 12, which indicate the effect on risk at the front of the period, and at year 24, which indicate the effect at the back of the period.
It is important to note, however, that the BID is not necessarily the initial disbursement that will be made from the portfolio. The initial disbursement that will be made from the portfolio depends on the prior spending of the beneficiary, and any other funds that may be available to cover spending. The BID is used along with this additional information to determine the initial disbursement that will be made from the portfolio. If adding the BID to the other funds available provides more than enough to cover prior spending a decision must be made as to whether or not to increase spending, and by how much. On the other hand, suppose that adding the BID to the other funds available is insufficient to cover prior spending, and that important habitual activities will have to be curtailed. In this case, more than the BID may be disbursed from the portfolio initially because the tradeoff has changed. The BID is determined by deciding whether the increased risk of future declines caused by increasing the initial disbursement is worth accepting to increase spending. Now, however, the additional funds provided are not being used to increase spending, but to avoid retrenchment. In making disbursements, funds to avoid retrenchment are given higher priority than to increase spending. For instance, when disbursements are ongoing, funds from better than expected investment returns are allowed to accumulate as a cushion to avoid future retrenchment rather than being used to increase spending.
Using the investment returns in the pre 2008 illustration, Charts I.1 and I.2 show the cumulative probability distributions of the disbursements at years 12 and 24 for initial disbursements of 3.5% and 4.0%. There is no doubt that the increase from 3.5% to 4.0% is an improvement. At year 12 there is very little chance that the higher disbursement will not be sustained, and almost no chance that it will be less than the lower disbursement. By year 24 there is only a .15 chance that the increase will not be sustained. There is a .10 chance that the disbursement will have declined by more than the increase, and a .05 chance by more than twice the increase. Nevertheless, there appears to be no doubt that this small increased risk at the back of the period is worth accepting to obtain the very favorable results at the front, and that the increase from 3.5% to 4.0% is an improvement.
Next, increase the initial disbursement to 4.5% with appropriate increases in the stock allocation and RDR. Comparisons of starting at 4.5% and 4.0% at years 12 and 24 are shown in Charts I.3 and I.4. For 4.5%, the chances of a decline by year 12 increase from .05 to .15. And there is now a .10 chance of a decline larger than the increase, instead of almost no chance. There is, however, almost no chance of a decline larger than twice the increase. Thus, there has been an increase in the risk of declines at the front of the period, but this increase in risk is modest, and the increase to 4.5% at the front of the period appears favorable. At the back of the period, however, the increase in risk caused by 4.5% is of more concern. The chances that there has been a decline have increased from .15 to.30. There is now a better than .20 chance of a decline more than the increase instead of a .10 chance, and a .10 chance of a decline more than three times the increase, instead of a .025 chance. The back of the period, however, is far in the future, and with declining chances of survival, may never be applicable. Thus, those making disbursements may be willing to accept this increased risk as being sufficiently far in the future to discount so as to obtain the favorable results at the front of the period. On balance, there appears to be little doubt that the increase from 4.0% to 4.5% is an improvement to increase the spending of the beneficiary.
Now suppose the initial disbursement is further increased from 4.5% to 5.0% of portfolio value with appropriate increases in the stock allocation and RDR. The distributions of the disbursements at years 12 and 24 in this case are compared in Charts I.5 and I.6. The increased risk at the front of the period has now become a concern. The chances of a decline by year 12 have increased to .30. There is now a .20 chance of a decline more than the increase, instead of a .10 chance. Moreover, there is a .10 chance of a decline more than twice the increase instead of a .025 chance. Coming at the front of the period this increased risk can no longer be discounted as being far away. In addition, there is now significant increased risk at the back of the period. The chances of a decline have increased from .30 to.45. Moreover, there is now almost a .25 chance of a decline more than three times the increase instead of a .10 chance. There is almost a .10 chance that the disbursement will be reduced by more than a half. Thus, there is significant doubt that the increased risk of future declines caused by an increase from 4.5% to 5.0% is worth accepting to obtain larger disbursements to increase spending. The Best Initial Disbursement (BID) for the pre 2008 illustration is therefore 4.5%.
Avoiding Retrenchment
Suppose now, however, that instead of being used to increase spending that the larger disbursements are being used to avoid additional curtailment of important habitual activities. Now, the increased risk of future retrenchment is being accepted to avoid an increase in current retrenchment, not to increase spending. More increased risk of future declines and retrenchment is now acceptable, as avoiding retrenchment has higher priority than increasing spending. Some current retrenchment, however, is acceptable to avoid too high a risk of low disbursements in the future. The increased risk caused by an increase from 4.5% to 5.0% may be acceptable to reduce current retrenchment, but not the further increase in risk caused by the increase from 5.0% to 5.5% shown in Charts I.7 and I.8. The 5.5% initial disbursement will further reduce current retrenchment, but not by enough to compensate for the increased risk of future declines that it causes. (An addendum to Retrench argues, however, that an initial disbursement even as high as 7.5% can be acceptable to avoid cuts in important habitual activities, and is consistent with not undertaking a major retrenchment when a much more sustainable initial disbursement reaches its limit in the future.) It is important to note, however, that these results do not affect the BID. The BID remains at 4.5% because the BID depends on how much of an increase in the risk of future declines is acceptable to increase spending. When more than the BID is needed to cover prior spending, however, the initial disbursement may increase above the BID.
Lower Returns
The simulations to find the BID use the investment returns for the pre 2008 illustration in Model. Returns subsequent to 2020, however, may be substantially less, and a hypothetical set of lower returns is presented in Model that can be used to help gauge the possible effect of lower returns on the results. These lower returns reduce the expected annual real returns for both the intermediates and stocks by .03, and also make some reduction in the volatility of the returns for both. The effect of these lower returns on the best stock allocation is investigated in Allocation. For disbursements with good sustainability, the best stock allocation is the same for either the higher or lower returns, and equal to 30%. When the initial disbursement increases above a sustainable level, however, the best stock allocation for the lower returns increases much more. The increase is 15 percentage points for a half percentage point increase in the initial disbursement instead of the 5 percentage point increase for the higher returns.
For the pre 2008 illustration, an initial disbursement of 3.5% has no chance of a decline at the front of the period, and only a slight chance of a decline at the back. For the lower returns the same is true when the initial disbursement is one percentage point less and equal to 2.5%. For the pre 2008 returns, Charts I.3 and I.4 show the distributions of the disbursements at years 12 and 24 for initial disbursements of 4.0% and 4.5%. Rerun the simulations for these charts using the lower returns and reducing the initial disbursements by one percentage point. As these initial disbursements are a half and one percentage point above a sustainable level increase the stock allocation by 15 and 30 percentage points above the sustainable allocation of 30% to 45% and 60%. Using these allocations recalculate the RDRs to be .02 above the expected returns of the portfolios in these cases. The results are shown in Charts I.9 and I.10. The curves look very similar to those in Charts I.3 and I.4 except for being one percentage point less on the horizontal axis.
For the pre 2008 returns, Charts I.5 and I.6 show the distributions of the disbursements starting at 4.5% and 5.0%. Rerun these simulations in the same way using the lower returns and noting that the stock allocations now increase to 60% and 75%. The results are shown in Charts I.11 and I.12. Except for being one percentage point less on the horizontal axis, the curves look just about the same as those in Charts I.5 and I.6. For the pre 2008 returns, the BID is 4.5% because starting at 4.5% is an improvement over 4.0%, but there is doubt that 5.0% is an improvement over 4.5%. For the lower returns, the curves look the same when the values on the horizontal axis are one percentage point less. Thus, for the lower returns the BID is one percentage point less than the 4.5% for the higher returns, and equal to 3.5%.
Emergencies
Providing a constant stream of disbursements in real terms assumes the beneficiary needs such a stream to fund expenditures on activities that provide a steady stream of satisfaction. Sometimes, however, a very large unexpected expenditure may be necessary to maintain the current level of satisfaction. Such an expenditure might be to cover a loss that might be covered by insurance. There are limits, however, on what can be covered by insurance, and some losses are not insurable, or even recognized as possible beforehand. Some possible problems that might require large unexpected outlays include extensive water damage to a residence not covered by insurance, an urgent need for financial assistance by a child, or a personal medical problem for which insurance does not cover the best available solution. An extra disbursement from the portfolio to cover such an expenditure could have the same effect as a bear market in stocks. The loss from a bear market, however, may be recovered by a subsequent rebound, but not the expenditure to cover an emergency.
The chance that a large extra disbursement may be needed to cover an emergency can be included in the simulations. Suppose, for instance, each year that there is an independent .05 chance that an extra disbursement will be required equal to 20% of the initial value of the portfolio. If an outlay that large would seriously deplete the funds needed for continuing activities in the future, however, the outlay is limited. In particular, the outlay for the emergency is not allowed to exceed 25% of the current value of the portfolio. With a .05 chance of an emergency each year, over a 20 year period the expected number of emergencies is one, which seems reasonable. The modifications to incorporate the possibility of emergencies in the model are provided in Model.
Charts I.3 and I.4 show the distributions of the disbursements at years 12 and 24 for initial disbursements of 4.0% and 4.5% using the pre 2008 illustration. Suppose those simulations are rerun incorporating the possibility of the emergencies just described, and the initial disbursements are reduced by half a percentage point. The results are shown in Charts I.13 and I.14. The curves look just about the same except that they are half a percentage point less on the horizontal axis. Next, do the same for Charts I.5 and I.6 that show the distributions of the disbursements for initial disbursements of 4.5% and 5.0%. The results are shown in Charts I.15 and I.16. Again, the curves look just about the same except that they are half a percentage point less on the horizontal axis. Earlier, it was argued that 4.5% was the BID for those using the disbursements to increase spending because starting at 4.5% is an improvement over starting at 4.0%, but there is doubt that starting at 5.0% is better than starting at 4.5%. When the emergencies just described are included in the model with the pre 2008 returns the curves look the same except that the values on the horizontal axis are half a percentage point less. Thus, when the emergencies are included, the BID is half a percentage point less than the BID without the emergencies of 4.5%, and equal to 4.0%.
Conclusions
The Best Initial Disbursement (BID) is found by successively increasing the initial disbursement by half a percentage point of portfolio value until there is doubt that the latest increase is an improvement. Using this process, the BID for the investment returns in the pre 2008 illustration appears to be 4.5% when beneficiaries are using the disbursements to increase spending. For the hypothetical lower investment returns where the expected annual real returns are reduced by .03 and there is also some reduction in volatility, the BID declines one percentage point from 4.5% to 3.5%. When the possibility is introduced that one or more large additional disbursements may be required to cover emergencies, the BID is reduced by half a percentage point. Suppose the BID is insufficient to cover the ongoing stream of spending by the beneficiary, and that curtailment will be required in important habitual activities. In this case, more risk of future declines becomes acceptable because it is avoiding current retrenchment. The initial disbursement in this case may be larger than the BID, and increase to 5.0% or more.
Posted November, 2018 Revised July, 2022
Addendum
The text investigates how much lower returns or the possibility of emergencies may reduce the Best Initial Disbursement (BID). Another factor that may reduce the BID that is available for covering living expenses is federal income tax. Many retirees in the U. S. have portfolios that have largely been accumulated on a tax deferred basis. Tax will have to be paid, however, on any funds disbursed from the portfolio. Moreover, Required Minimum Distributions (RMD) may make taxable disbursements necessary in excess of those needed to cover living expenses and pay tax on those disbursements. Suppose initially that the entire portfolio has been accumulated on a tax deferred basis. No tax is payable on any income received on investments in the portfolio, or on transactions made within the portfolio. The tax paid on disbursements, however, has a comparable effect on the funds available to cover living expenses as the lower returns or possible emergencies considered in the text. Consider a single filer starting at age 65 under the tax brackets and other rules applicable in 2022. To get the same sustainability for the disbursements with the tax at year 24 the initial disbursement must be reduced from 4.5% to 3.9% for a $1 million initial portfolio, or 3.7% for a $4 million portfolio.
Major changes in Model are required to make the simulations to get these results. The Retrenchment Rule continues to apply each year to the disbursement made to cover living expenses. A withdrawal is also made to pay the tax owed based on the taxable income generated in the prior year. In addition, beginning at age 72 under the 2022 rules, a larger withdrawal may be necessary to meet the Required Minimum Distribution Requirement (RMD). If the RMD is larger it is assumed that the surplus is invested in a short term reserve that earns after tax a zero real return. The reserve is available in the future to cover withdrawals when they exceed the RMD or to provide funds if the tax deferred portfolio is exhausted. It turns out almost half the time that the RMD never exceeds the withdrawals that would otherwise be required to cover living expenses and taxes. A detailed description of the changes that must be made in Model to include taxes under these assumptions is given in a note.
The curve for No Tax in Chart I.17 is the same as that in the text for an initial disbursement of 4.5% of portfolio value in Chart I.6. With the tax assumptions, the other curves show how much the initial disbursement must be reduced to get the same chances of declines at year 24 for initial portfolios of $1 million or $ 4 million. A single filer is assumed to start at age 65 with the tax brackets and with RMD rules similar to those applicable in 2022. The tax brackets are indexed for inflation and remain the same in real terms in future years. Someone starting at year 1 at age 65 will be age 72 at year 8. The simulations assume the RMD at age 72 in the 2022 rules is applicable at the beginning of year 8 and successively in future years for the respective ages at these years. Taxable income each year includes any withdrawals from the portfolio plus an assumed $10,000 in real terms that covers taxable Social Security less the standard deduction for a single filer age 65 or more.
As just indicated, the simulations assume any funds withdrawn to cover the RMD that are not needed to cover living expenses or tax payments are deposited in a short term reserve where they earn a zero after tax real return. The value of the funds in this reserve can be obtained as an output of the simulations. The probability distributions of the value of the reserve in future years for the portfolio with $1 million of initial value and an initial disbursement of 3.9% are shown in Chart I.18. These distributions show the chances that the reserve is larger than the percent of initial portfolio value shown on the horizontal axis. Note at the back of the period that the chances are close to a half that there will not be any funds in the reserve. Thus, the chances are close to a half that the RMD will never require disbursements that are in excess of the funds needed to cover living expenses and the taxes that must be paid to obtain those funds from the portfolio. The chart also shows that there is little in the reserve at year 12 which corresponds to age 76 for someone starting at age 65. Thus, increasing the age at which the RMD begin has only a small effect on reducing taxes.
Chart I.18 shows that there is a small chance of accumulating large balances in the reserve at the back of the period. The marginal tax rate on these extra withdrawals is generally 22%. Nevertheless, if a large balance is accumulating in the reserve and these extra taxes are being paid, declines are not occurring in the disbursements to cover living expenses. The major concern posed by tax deferred portfolios is not the small chance that large extra tax payments may be necessary due to the RMD. The major concern is the one shown in Chart I.17. The tax causes a meaningful reduction in the disbursements available to cover living expenses just like lower returns or the possibility of emergencies.
Note
At the beginning of t, let Tx(t), TxInc(t), RMD(t), and Rsv(t) denote respectively the tax paid, taxable income, the RMD, and the funds in the reserve. Tax is calculated in dollars from taxable income in dollars, whereas C(t) and V(t) in Model are expressed as a percentage of initial portfolio value. For the tax calculations, C(t) and V(t) must be multiplied by a factor, s, to express them in dollars, and Tx(t) must be divided by s to express the tax payment as a percentage of initial portfolio value when it is deducted from V(t) in Model. For an initial portfolio of $ 1 million s is 10,000 so that a disbursement of 4.5% becomes $45,000. For an initial portfolio of $4 million, s is equal to 40,000 so that a 4.5% disbursement becomes $180,000. The beneficiary is age 65 at year 1 and becomes age 72 at year 8. The RMD are assumed to start at the beginning of year 8 with the factor for a single filer at age 72 in the 2022 rules of 27.4. Thus, RMD(8) = sV(8)/27.4 and subsequent RMD are given by the respective factors for later ages in the 2022 rules. TxInc(t) is calculated from the portfolio withdrawals at the beginning of t plus an assumed $10,000 in real terms each year to allow for taxable Social Security less the standard deduction for a single filer age 65 or older. Tx(t) is calculated from TxInc(t-1) based on a sum of conditional statements for each of the seven tax brackets in 2022. For example, the conditional statements for the first three brackets, which are the relevant range for a $1 million portfolio, are the following:
Prior to year 8 when the RMD begin the withdrawals from the portfolio to get taxable income are equal simply to sC(t) + Tx(t). At year 9, however, determining the taxable withdrawal from the portfolio is more complicated because there may be funds in the reserve in the prior year that cover at least part of sC(t) + Tx(t) if sC(t) + Tx(t) is larger than RMD(t). At year 9 and thereafter the withdrawal from the portfolio is given by the relation in (1):
Moreover, Rsv(t-1) is included with V(t) in the calculation of the limit on C(t) given by C*(t). As a result the withdrawal given by (1) could at some point exceed V(t). In such an event any funds in the portfolio are used to cover as much of the withdrawal as possible and the remainder is obtained from the reserve. Subsequent sC(t) + Tx(t) are covered from the reserve.
Taking into account the possibility that V(t) may not be larger than the withdrawal given
by (1), V(t+1) is given by:
If( V(t) > (1), ( V(t) – (1) )( 1 + R(t) ), 0 )
and TxInc(t) by:
If( V(t) > (1), s(1), sV(t) ) +10,000
Since V(t+1) = 0 when V(t) <= (1) note that RMD(t+1) is also zero in this case. Moreover, at t+1 (1) is also zero because (1) is equal to the RMD when the withdrawal is being funded by the reserve and RMD(t+1) = 0. TxInc(t+1) is therefore equal to 10,000 in this case as are subsequent values of taxable income.
RMD(t) adds to the reserve when it exceeds sC(t) + Tx(t). The reserve is reduced when RMD(t) is less than sC(t) + Tx(t), and there are funds in the reserve. Assuming V(t) > (1) the change in the reserve at the beginning of t is therefore given by (2):
If V(t) <= (1), V(t) is deducted from the portfolio and the remainder of the withdrawal at t comes from the reserve. Any further withdrawals in the future come from the reserve. The value of Rsv(t) is therefore given by the following relation:
Model assumes that the portfolio is rebalanced each year to the same allocation. There are transaction costs for doing so that are ignored, but they are likely to be small. For taxable portfolios, however, there are also taxes on capital gains that could be large. The question arises as to whether the benefits of annual rebalancing are sufficient to cover these costs, especially of taxable portfolios when there are large taxes on capital gains. The benefits of annual rebalancing can be determined by comparing the results that have been obtained for annual rebalancing to the results when the portfolio is not rebalanced. Such comparisons show the benefits of annual rebalancing to be surprisingly small. Thus, there is doubt as to whether annual rebalancing is desirable especially for taxable portfolios when there are significant taxes on capital gains to be paid.
Instead of being rebalanced each year a portfolio could be rebalanced after a given number of years, and in the limit might never be rebalanced. In the latter case, the initial values for the stocks and intermediates are determined by the given allocation and initial value of the portfolio. Depending on the allocation, a portion of the initial disbursement is deducted from the initial values of the stocks and intermediates. The returns on the stocks and intermediates over the initial year are then applied to the residuals of each giving the value of the stocks and intermediates at the beginning of the next year. The disbursement for the next year is determined based on a limit calculated using the sum of the value of the stocks and intermediates at the beginning of the next year. A portion of that disbursement is deducted from the value of the stocks and intermediates depending on the given allocation. The returns on the stocks and intermediates over the next year are then applied to the residuals giving the value of the stocks and intermediates at the beginning of the following year. This same process continues over the succeeding years until the end of the disbursement period. The disbursements obtained are those for the given allocation when there is not any rebalancing.
The first question is whether the best allocation is affected by not rebalancing the portfolio? To find out, rerun the simulations in Allocation when there is not any rebalancing. It turns out that the best allocations obtained are the same as for annual rebalancing. For instance, Charts I.19 and I.20 show the results for the same conditions as Charts A.4 and A.5 in Allocation except that there is not any rebalancing. These two sets of charts look just about the same. Thus, for a 3.5% initial disbursement with good sustainability, the best allocation is 30% irrespective of whether or not there is annual rebalancing. Suppose that the initial disbursement is increased from 3.5% to 4.5% causing significant deterioration is sustainability. The results in this case without annual rebalancing are shown in Charts I.21 and I.22. these charts are almost the same as A.6 and A.7 in Allocation for the same conditions except with annual rebalancing. With or without the rebalancing the best allocation increases from 30% to 40% when the initial disbursement increases from 3.5% to 4.5%.
Suppose now that the simulations earlier in Increase are rerun without annual rebalancing. The solid curves in Chart I.23 are the same as those in Chart I.4 that show the cumulative probability distributions of the disbursement at year 24 when starting at 4.0% and 4.5% with annual rebalancing. The results without the rebalancing are shown by the dotted curves. As expected, without the annual rebalancing, there is deterioration. The deterioration, however is reasonably small. Perhaps the benefits of annual rebalancing are sufficient to cover very small transaction costs. The benefits, however, do not seem sufficient to cover the cost of paying large capital gains on taxable portfolios.
Increase
Best Initial Disbursements
Given an initial disbursement, simulations using the basic model in Model provide the cumulative probability distributions of the disbursements at years 12, 24, and 36. In Allocation, the best stock allocation for a given initial disbursement is found by successively increasing the stock allocation by 10 percentage points until doubt arises that the last increase is an improvement, in which case the prior allocation is the best stock allocation. The Best Initial Disbursement (BID) will now be found by a similar process. The stock allocation is set equal to the best allocation found in Allocation, and the RDR to .02 more than the expected return of the portfolio based on the results in RDR. The initial disbursement is successively increased by half a percentage point of initial portfolio value until there is doubt that the last increase is an improvement, in which case the prior initial disbursement is the Best Initial Disbursement (BID).
To assess whether an increase in the initial disbursement is an improvement a judgment is made as to whether the increased risk of future declines in the disbursements caused by the increase is worth accepting to obtain the higher disbursements provided to increase spending. Such decisions might be made by comparing metrics calculated from successive probability distributions at future years. Perhaps, however, such decisions can better be made by simply visually comparing graphs of the successive cumulative probability distributions at future years. In particular, comparisons can be made at year 12, which indicate the effect on risk at the front of the period, and at year 24, which indicate the effect at the back of the period.
It is important to note, however, that the BID is not necessarily the initial disbursement that will be made from the portfolio. The initial disbursement that will be made from the portfolio depends on the prior spending of the beneficiary, and any other funds that may be available to cover spending. The BID is used along with this additional information to determine the initial disbursement that will be made from the portfolio. If adding the BID to the other funds available provides more than enough to cover prior spending a decision must be made as to whether or not to increase spending, and by how much. On the other hand, suppose that adding the BID to the other funds available is insufficient to cover prior spending, and that important habitual activities will have to be curtailed. In this case, more than the BID may be disbursed from the portfolio initially because the tradeoff has changed. The BID is determined by deciding whether the increased risk of future declines caused by increasing the initial disbursement is worth accepting to increase spending. Now, however, the additional funds provided are not being used to increase spending, but to avoid retrenchment. In making disbursements, funds to avoid retrenchment are given higher priority than to increase spending. For instance, when disbursements are ongoing, funds from better than expected investment returns are allowed to accumulate as a cushion to avoid future retrenchment rather than being used to increase spending.
Using the investment returns in the pre 2008 illustration, Charts I.1 and I.2 show the cumulative probability distributions of the disbursements at years 12 and 24 for initial disbursements of 3.5% and 4.0%. There is no doubt that the increase from 3.5% to 4.0% is an improvement. At year 12 there is very little chance that the higher disbursement will not be sustained, and almost no chance that it will be less than the lower disbursement. By year 24 there is only a .15 chance that the increase will not be sustained. There is a .10 chance that the disbursement will have declined by more than the increase, and a .05 chance by more than twice the increase. Nevertheless, there appears to be no doubt that this small increased risk at the back of the period is worth accepting to obtain the very favorable results at the front, and that the increase from 3.5% to 4.0% is an improvement.
Next, increase the initial disbursement to 4.5% with appropriate increases in the stock allocation and RDR. Comparisons of starting at 4.5% and 4.0% at years 12 and 24 are shown in Charts I.3 and I.4. For 4.5%, the chances of a decline by year 12 increase from .05 to .15. And there is now a .10 chance of a decline larger than the increase, instead of almost no chance. There is, however, almost no chance of a decline larger than twice the increase. Thus, there has been an increase in the risk of declines at the front of the period, but this increase in risk is modest, and the increase to 4.5% at the front of the period appears favorable. At the back of the period, however, the increase in risk caused by 4.5% is of more concern. The chances that there has been a decline have increased from .15 to.30. There is now a better than .20 chance of a decline more than the increase instead of a .10 chance, and a .10 chance of a decline more than three times the increase, instead of a .025 chance. The back of the period, however, is far in the future, and with declining chances of survival, may never be applicable. Thus, those making disbursements may be willing to accept this increased risk as being sufficiently far in the future to discount so as to obtain the favorable results at the front of the period. On balance, there appears to be little doubt that the increase from 4.0% to 4.5% is an improvement to increase the spending of the beneficiary.
Now suppose the initial disbursement is further increased from 4.5% to 5.0% of portfolio value with appropriate increases in the stock allocation and RDR. The distributions of the disbursements at years 12 and 24 in this case are compared in Charts I.5 and I.6. The increased risk at the front of the period has now become a concern. The chances of a decline by year 12 have increased to .30. There is now a .20 chance of a decline more than the increase, instead of a .10 chance. Moreover, there is a .10 chance of a decline more than twice the increase instead of a .025 chance. Coming at the front of the period this increased risk can no longer be discounted as being far away. In addition, there is now significant increased risk at the back of the period. The chances of a decline have increased from .30 to.45. Moreover, there is now almost a .25 chance of a decline more than three times the increase instead of a .10 chance. There is almost a .10 chance that the disbursement will be reduced by more than a half. Thus, there is significant doubt that the increased risk of future declines caused by an increase from 4.5% to 5.0% is worth accepting to obtain larger disbursements to increase spending. The Best Initial Disbursement (BID) for the pre 2008 illustration is therefore 4.5%.
Avoiding Retrenchment
Suppose now, however, that instead of being used to increase spending that the larger disbursements are being used to avoid additional curtailment of important habitual activities. Now, the increased risk of future retrenchment is being accepted to avoid an increase in current retrenchment, not to increase spending. More increased risk of future declines and retrenchment is now acceptable, as avoiding retrenchment has higher priority than increasing spending. Some current retrenchment, however, is acceptable to avoid too high a risk of low disbursements in the future. The increased risk caused by an increase from 4.5% to 5.0% may be acceptable to reduce current retrenchment, but not the further increase in risk caused by the increase from 5.0% to 5.5% shown in Charts I.7 and I.8. The 5.5% initial disbursement will further reduce current retrenchment, but not by enough to compensate for the increased risk of future declines that it causes. (An addendum to Retrench argues, however, that an initial disbursement even as high as 7.5% can be acceptable to avoid cuts in important habitual activities, and is consistent with not undertaking a major retrenchment when a much more sustainable initial disbursement reaches its limit in the future.) It is important to note, however, that these results do not affect the BID. The BID remains at 4.5% because the BID depends on how much of an increase in the risk of future declines is acceptable to increase spending. When more than the BID is needed to cover prior spending, however, the initial disbursement may increase above the BID.
Lower Returns
The simulations to find the BID use the investment returns for the pre 2008 illustration in Model. Returns subsequent to 2020, however, may be substantially less, and a hypothetical set of lower returns is presented in Model that can be used to help gauge the possible effect of lower returns on the results. These lower returns reduce the expected annual real returns for both the intermediates and stocks by .03, and also make some reduction in the volatility of the returns for both. The effect of these lower returns on the best stock allocation is investigated in Allocation. For disbursements with good sustainability, the best stock allocation is the same for either the higher or lower returns, and equal to 30%. When the initial disbursement increases above a sustainable level, however, the best stock allocation for the lower returns increases much more. The increase is 15 percentage points for a half percentage point increase in the initial disbursement instead of the 5 percentage point increase for the higher returns.
For the pre 2008 illustration, an initial disbursement of 3.5% has no chance of a decline at the front of the period, and only a slight chance of a decline at the back. For the lower returns the same is true when the initial disbursement is one percentage point less and equal to 2.5%. For the pre 2008 returns, Charts I.3 and I.4 show the distributions of the disbursements at years 12 and 24 for initial disbursements of 4.0% and 4.5%. Rerun the simulations for these charts using the lower returns and reducing the initial disbursements by one percentage point. As these initial disbursements are a half and one percentage point above a sustainable level increase the stock allocation by 15 and 30 percentage points above the sustainable allocation of 30% to 45% and 60%. Using these allocations recalculate the RDRs to be .02 above the expected returns of the portfolios in these cases. The results are shown in Charts I.9 and I.10. The curves look very similar to those in Charts I.3 and I.4 except for being one percentage point less on the horizontal axis.
For the pre 2008 returns, Charts I.5 and I.6 show the distributions of the disbursements starting at 4.5% and 5.0%. Rerun these simulations in the same way using the lower returns and noting that the stock allocations now increase to 60% and 75%. The results are shown in Charts I.11 and I.12. Except for being one percentage point less on the horizontal axis, the curves look just about the same as those in Charts I.5 and I.6. For the pre 2008 returns, the BID is 4.5% because starting at 4.5% is an improvement over 4.0%, but there is doubt that 5.0% is an improvement over 4.5%. For the lower returns, the curves look the same when the values on the horizontal axis are one percentage point less. Thus, for the lower returns the BID is one percentage point less than the 4.5% for the higher returns, and equal to 3.5%.
Emergencies
Providing a constant stream of disbursements in real terms assumes the beneficiary needs such a stream to fund expenditures on activities that provide a steady stream of satisfaction. Sometimes, however, a very large unexpected expenditure may be necessary to maintain the current level of satisfaction. Such an expenditure might be to cover a loss that might be covered by insurance. There are limits, however, on what can be covered by insurance, and some losses are not insurable, or even recognized as possible beforehand. Some possible problems that might require large unexpected outlays include extensive water damage to a residence not covered by insurance, an urgent need for financial assistance by a child, or a personal medical problem for which insurance does not cover the best available solution. An extra disbursement from the portfolio to cover such an expenditure could have the same effect as a bear market in stocks. The loss from a bear market, however, may be recovered by a subsequent rebound, but not the expenditure to cover an emergency.
The chance that a large extra disbursement may be needed to cover an emergency can be included in the simulations. Suppose, for instance, each year that there is an independent .05 chance that an extra disbursement will be required equal to 20% of the initial value of the portfolio. If an outlay that large would seriously deplete the funds needed for continuing activities in the future, however, the outlay is limited. In particular, the outlay for the emergency is not allowed to exceed 25% of the current value of the portfolio. With a .05 chance of an emergency each year, over a 20 year period the expected number of emergencies is one, which seems reasonable. The modifications to incorporate the possibility of emergencies in the model are provided in Model.
Charts I.3 and I.4 show the distributions of the disbursements at years 12 and 24 for initial disbursements of 4.0% and 4.5% using the pre 2008 illustration. Suppose those simulations are rerun incorporating the possibility of the emergencies just described, and the initial disbursements are reduced by half a percentage point. The results are shown in Charts I.13 and I.14. The curves look just about the same except that they are half a percentage point less on the horizontal axis. Next, do the same for Charts I.5 and I.6 that show the distributions of the disbursements for initial disbursements of 4.5% and 5.0%. The results are shown in Charts I.15 and I.16. Again, the curves look just about the same except that they are half a percentage point less on the horizontal axis. Earlier, it was argued that 4.5% was the BID for those using the disbursements to increase spending because starting at 4.5% is an improvement over starting at 4.0%, but there is doubt that starting at 5.0% is better than starting at 4.5%. When the emergencies just described are included in the model with the pre 2008 returns the curves look the same except that the values on the horizontal axis are half a percentage point less. Thus, when the emergencies are included, the BID is half a percentage point less than the BID without the emergencies of 4.5%, and equal to 4.0%.
Conclusions
The Best Initial Disbursement (BID) is found by successively increasing the initial disbursement by half a percentage point of portfolio value until there is doubt that the latest increase is an improvement. Using this process, the BID for the investment returns in the pre 2008 illustration appears to be 4.5% when beneficiaries are using the disbursements to increase spending. For the hypothetical lower investment returns where the expected annual real returns are reduced by .03 and there is also some reduction in volatility, the BID declines one percentage point from 4.5% to 3.5%. When the possibility is introduced that one or more large additional disbursements may be required to cover emergencies, the BID is reduced by half a percentage point. Suppose the BID is insufficient to cover the ongoing stream of spending by the beneficiary, and that curtailment will be required in important habitual activities. In this case, more risk of future declines becomes acceptable because it is avoiding current retrenchment. The initial disbursement in this case may be larger than the BID, and increase to 5.0% or more.
Posted November, 2018 Revised July, 2022
Addendum
The text investigates how much lower returns or the possibility of emergencies may reduce the Best Initial Disbursement (BID). Another factor that may reduce the BID that is available for covering living expenses is federal income tax. Many retirees in the U. S. have portfolios that have largely been accumulated on a tax deferred basis. Tax will have to be paid, however, on any funds disbursed from the portfolio. Moreover, Required Minimum Distributions (RMD) may make taxable disbursements necessary in excess of those needed to cover living expenses and pay tax on those disbursements. Suppose initially that the entire portfolio has been accumulated on a tax deferred basis. No tax is payable on any income received on investments in the portfolio, or on transactions made within the portfolio. The tax paid on disbursements, however, has a comparable effect on the funds available to cover living expenses as the lower returns or possible emergencies considered in the text. Consider a single filer starting at age 65 under the tax brackets and other rules applicable in 2022. To get the same sustainability for the disbursements with the tax at year 24 the initial disbursement must be reduced from 4.5% to 3.9% for a $1 million initial portfolio, or 3.7% for a $4 million portfolio.
Major changes in Model are required to make the simulations to get these results. The Retrenchment Rule continues to apply each year to the disbursement made to cover living expenses. A withdrawal is also made to pay the tax owed based on the taxable income generated in the prior year. In addition, beginning at age 72 under the 2022 rules, a larger withdrawal may be necessary to meet the Required Minimum Distribution Requirement (RMD). If the RMD is larger it is assumed that the surplus is invested in a short term reserve that earns after tax a zero real return. The reserve is available in the future to cover withdrawals when they exceed the RMD or to provide funds if the tax deferred portfolio is exhausted. It turns out almost half the time that the RMD never exceeds the withdrawals that would otherwise be required to cover living expenses and taxes. A detailed description of the changes that must be made in Model to include taxes under these assumptions is given in a note.
The curve for No Tax in Chart I.17 is the same as that in the text for an initial disbursement of 4.5% of portfolio value in Chart I.6. With the tax assumptions, the other curves show how much the initial disbursement must be reduced to get the same chances of declines at year 24 for initial portfolios of $1 million or $ 4 million. A single filer is assumed to start at age 65 with the tax brackets and with RMD rules similar to those applicable in 2022. The tax brackets are indexed for inflation and remain the same in real terms in future years. Someone starting at year 1 at age 65 will be age 72 at year 8. The simulations assume the RMD at age 72 in the 2022 rules is applicable at the beginning of year 8 and successively in future years for the respective ages at these years. Taxable income each year includes any withdrawals from the portfolio plus an assumed $10,000 in real terms that covers taxable Social Security less the standard deduction for a single filer age 65 or more.
As just indicated, the simulations assume any funds withdrawn to cover the RMD that are not needed to cover living expenses or tax payments are deposited in a short term reserve where they earn a zero after tax real return. The value of the funds in this reserve can be obtained as an output of the simulations. The probability distributions of the value of the reserve in future years for the portfolio with $1 million of initial value and an initial disbursement of 3.9% are shown in Chart I.18. These distributions show the chances that the reserve is larger than the percent of initial portfolio value shown on the horizontal axis. Note at the back of the period that the chances are close to a half that there will not be any funds in the reserve. Thus, the chances are close to a half that the RMD will never require disbursements that are in excess of the funds needed to cover living expenses and the taxes that must be paid to obtain those funds from the portfolio. The chart also shows that there is little in the reserve at year 12 which corresponds to age 76 for someone starting at age 65. Thus, increasing the age at which the RMD begin has only a small effect on reducing taxes.
Chart I.18 shows that there is a small chance of accumulating large balances in the reserve at the back of the period. The marginal tax rate on these extra withdrawals is generally 22%. Nevertheless, if a large balance is accumulating in the reserve and these extra taxes are being paid, declines are not occurring in the disbursements to cover living expenses. The major concern posed by tax deferred portfolios is not the small chance that large extra tax payments may be necessary due to the RMD. The major concern is the one shown in Chart I.17. The tax causes a meaningful reduction in the disbursements available to cover living expenses just like lower returns or the possibility of emergencies.
Note
At the beginning of t, let Tx(t), TxInc(t), RMD(t), and Rsv(t) denote respectively the tax paid, taxable income, the RMD, and the funds in the reserve. Tax is calculated in dollars from taxable income in dollars, whereas C(t) and V(t) in Model are expressed as a percentage of initial portfolio value. For the tax calculations, C(t) and V(t) must be multiplied by a factor, s, to express them in dollars, and Tx(t) must be divided by s to express the tax payment as a percentage of initial portfolio value when it is deducted from V(t) in Model. For an initial portfolio of $ 1 million s is 10,000 so that a disbursement of 4.5% becomes $45,000. For an initial portfolio of $4 million, s is equal to 40,000 so that a 4.5% disbursement becomes $180,000. The beneficiary is age 65 at year 1 and becomes age 72 at year 8. The RMD are assumed to start at the beginning of year 8 with the factor for a single filer at age 72 in the 2022 rules of 27.4. Thus, RMD(8) = sV(8)/27.4 and subsequent RMD are given by the respective factors for later ages in the 2022 rules. TxInc(t) is calculated from the portfolio withdrawals at the beginning of t plus an assumed $10,000 in real terms each year to allow for taxable Social Security less the standard deduction for a single filer age 65 or older. Tx(t) is calculated from TxInc(t-1) based on a sum of conditional statements for each of the seven tax brackets in 2022. For example, the conditional statements for the first three brackets, which are the relevant range for a $1 million portfolio, are the following:
If( And( TxInc(t-1) > 0, TxInc(t-1) <= 10,275 ), .10TxInc(t-1), 0 ) +
If( And( TxInc(t-1) > 10,275, TxInc(t-1) <= 41,775 ), 1,027.5 + .12( TxInc(t-1) – 10,275 ), 0 ) +
If( And( TxInc(t-1) > 41,775, TxInc(t-1) <= 89,075 ), 4,807.5 + .22( TxInc(t-1) – 41,775 ), 0 )
Prior to year 8 when the RMD begin the withdrawals from the portfolio to get taxable income are equal simply to sC(t) + Tx(t). At year 9, however, determining the taxable withdrawal from the portfolio is more complicated because there may be funds in the reserve in the prior year that cover at least part of sC(t) + Tx(t) if sC(t) + Tx(t) is larger than RMD(t). At year 9 and thereafter the withdrawal from the portfolio is given by the relation in (1):
(1) If( RMD(t)/s >= C(t) + Tx(t)/s, RMD(t)/s, If( Rsv(t-1) >= C(t) + Tx(t)/s – RMD(t)/s,
RMD(t)/s, If( Rsv(t-1) > 0, C(t) + Tx(t)/S – Rsv(t-1), C(t) + Tx(t)/s )))
Moreover, Rsv(t-1) is included with V(t) in the calculation of the limit on C(t) given by C*(t). As a result the withdrawal given by (1) could at some point exceed V(t). In such an event any funds in the portfolio are used to cover as much of the withdrawal as possible and the remainder is obtained from the reserve. Subsequent sC(t) + Tx(t) are covered from the reserve.
Taking into account the possibility that V(t) may not be larger than the withdrawal given
by (1), V(t+1) is given by:
If( V(t) > (1), ( V(t) – (1) )( 1 + R(t) ), 0 )
and TxInc(t) by:
If( V(t) > (1), s(1), sV(t) ) +10,000
Since V(t+1) = 0 when V(t) <= (1) note that RMD(t+1) is also zero in this case. Moreover, at t+1 (1) is also zero because (1) is equal to the RMD when the withdrawal is being funded by the reserve and RMD(t+1) = 0. TxInc(t+1) is therefore equal to 10,000 in this case as are subsequent values of taxable income.
RMD(t) adds to the reserve when it exceeds sC(t) + Tx(t). The reserve is reduced when RMD(t) is less than sC(t) + Tx(t), and there are funds in the reserve. Assuming V(t) > (1) the change in the reserve at the beginning of t is therefore given by (2):
(2) If( RMD(t) > sC(t) + Tx(t), RMD(t) – sC(t) – Tx(t), – If( Rsv((t-1) >= sC(t) + Tx(t) – RMD(t),
sC(t) + Tx(t) – RMD(t), If( Rsv(t-1) > 0, Rsv(t-1), 0 )))
If V(t) <= (1), V(t) is deducted from the portfolio and the remainder of the withdrawal at t comes from the reserve. Any further withdrawals in the future come from the reserve. The value of Rsv(t) is therefore given by the following relation:
Rsv(t-1) + If( V(t) > (1), (2), -If( V(t) > 0, sC(t) +Tx(t) – sV(t), sC(t) +Tx(t) ))
Posted February 2023 Revised March 2023
Addendum
Model assumes that the portfolio is rebalanced each year to the same allocation. There are transaction costs for doing so that are ignored, but they are likely to be small. For taxable portfolios, however, there are also taxes on capital gains that could be large. The question arises as to whether the benefits of annual rebalancing are sufficient to cover these costs, especially of taxable portfolios when there are large taxes on capital gains. The benefits of annual rebalancing can be determined by comparing the results that have been obtained for annual rebalancing to the results when the portfolio is not rebalanced. Such comparisons show the benefits of annual rebalancing to be surprisingly small. Thus, there is doubt as to whether annual rebalancing is desirable especially for taxable portfolios when there are significant taxes on capital gains to be paid.
Instead of being rebalanced each year a portfolio could be rebalanced after a given number of years, and in the limit might never be rebalanced. In the latter case, the initial values for the stocks and intermediates are determined by the given allocation and initial value of the portfolio. Depending on the allocation, a portion of the initial disbursement is deducted from the initial values of the stocks and intermediates. The returns on the stocks and intermediates over the initial year are then applied to the residuals of each giving the value of the stocks and intermediates at the beginning of the next year. The disbursement for the next year is determined based on a limit calculated using the sum of the value of the stocks and intermediates at the beginning of the next year. A portion of that disbursement is deducted from the value of the stocks and intermediates depending on the given allocation. The returns on the stocks and intermediates over the next year are then applied to the residuals giving the value of the stocks and intermediates at the beginning of the following year. This same process continues over the succeeding years until the end of the disbursement period. The disbursements obtained are those for the given allocation when there is not any rebalancing.
The first question is whether the best allocation is affected by not rebalancing the portfolio? To find out, rerun the simulations in Allocation when there is not any rebalancing. It turns out that the best allocations obtained are the same as for annual rebalancing. For instance, Charts I.19 and I.20 show the results for the same conditions as Charts A.4 and A.5 in Allocation except that there is not any rebalancing. These two sets of charts look just about the same. Thus, for a 3.5% initial disbursement with good sustainability, the best allocation is 30% irrespective of whether or not there is annual rebalancing. Suppose that the initial disbursement is increased from 3.5% to 4.5% causing significant deterioration is sustainability. The results in this case without annual rebalancing are shown in Charts I.21 and I.22. these charts are almost the same as A.6 and A.7 in Allocation for the same conditions except with annual rebalancing. With or without the rebalancing the best allocation increases from 30% to 40% when the initial disbursement increases from 3.5% to 4.5%.
Suppose now that the simulations earlier in Increase are rerun without annual rebalancing. The solid curves in Chart I.23 are the same as those in Chart I.4 that show the cumulative probability distributions of the disbursement at year 24 when starting at 4.0% and 4.5% with annual rebalancing. The results without the rebalancing are shown by the dotted curves. As expected, without the annual rebalancing, there is deterioration. The deterioration, however is reasonably small. Perhaps the benefits of annual rebalancing are sufficient to cover very small transaction costs. The benefits, however, do not seem sufficient to cover the cost of paying large capital gains on taxable portfolios.
Posted May 2023