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What’s New? A Major Revision
This posting makes an important revision in the procedure for finding the Best Initial Disbursement, or BID. To describe this revision, the former page, Increase, has been rewritten and renamed, BID. The first step in the revision is to use simulations to find the largest highly sustainable initial disbursement. This highly sustainable initial disbursement is the largest that simulations show has very little chance of needing to be reduced in the future under worst case conditions. These worst case conditions are the worst investment returns that seem reasonably possible, and assuming that there is a small chance each year that a large extra disbursement may be necessary to cover an emergency. For a 36-year initial horizon, this highly sustainable initial disbursement turns out to equal about 2.5%. Any highly sustainable funds from other sources, such as Social Security, are then added to this highly sustainable disbursement from the portfolio.
The next step is to find the highest priority activities that can be covered with these highly sustainable funds, and in descending order of priority the activities that cannot be covered. In a few cases, it may be possible to cover all existing activities with highly sustainable funding. In that case, the only question is whether to increase spending, including perhaps with some increase in sustainability risk. In most cases, however, there will be important habitual activities that cannot be covered from highly sustainable funding, and that will be very painful to curtail. In these cases, the next step is to increase the initial disbursement by accepting risks that seem reasonable to avoid retrenchment. These include using better returns in the simulations than the worst that seem reasonably possible, accepting chances of retrenchment in the future to avoid initial retrenchment, and not including the possibility of extra disbursements to cover emergencies in the simulations.
By accepting risks that seem reasonable to avoid painful initial retrenchment the initial disbursement can be increased from 2.5% to as much as 4.0% to cover activities that cannot be covered with highly sustainable funding. Thus, a New 4% Rule is formulated. This New 4% Rule does not pose any risk of running out of funds, but it does have sustainability risks that appear acceptable to avoid initial retrenchment. Many, however, will have important habitual activities that cannot be covered with a 4.0% initial disbursement. They will want to consider how much additional retrenchment risk in the future will have to be accepted to avoid initially curtailing these activities. Simulations show the costs and benefits of further increases in the initial disbursement from 4.0% to 5.0% or even 6.0%. In general, the initial disbursement is increased until the marginal cost in terms of increased sustainability risk exceeds the marginal benefit of the additional activities being funded by the additional funds provided.
Besides revising the determination of the BID, this revision also makes some improvements at other pages. These include Model, RDR, Allocation, Retrench, Dampen, and, Horizons. Also, a new page, Increases, has been added. This page looks at the possibility of increasing the disbursement in the future when the cushion protecting the disbursement below its limit becomes larger than the initial percentage. Making these increases, however, substantially increases the risk of covering possible emergencies. Thus, beneficiaries may prefer to allow these surpluses to accumulate, knowing that if not needed for emergencies they will add to bequests.
The revised procedure for determining the BID does not affect the determination of the annual disbursement in the basic model. The budgeted disbursement for the current year is the same as the year before in real terms unless this disbursement exceeds the calculated value of a limit. This limit is the largest constant annuity that could be obtained each year for the longest that the disbursements might be needed. The value of this annuity is calculated assuming that the then current value of the portfolio is invested at a constant rate of return called the Retrenchment Discount Rate, or RDR.
The value to use for the RDR is determined by simulating the use of an increasing series of values to see which gives the best results. These simulations are made assuming that the portfolio is some mix of stocks with returns resembling those on the S&P 500 and intermediate term fixed incomes with returns resembling those on U.S. government issues. For any simulation, the allocation of the portfolio is generally assumed to be reset to be the same at the beginning of each year over the disbursement period, but exceptions are investigated. Increasing the RDR improves the disbursements at the front of the disbursement period relative to the back. In the limit, when the RDR becomes infinite, the initial disbursement is always sustained until the portfolio runs out of funds. Simulations at page, RDR, show that a good balance between the disbursements at the front and back of the period is obtained by setting the RDR equal to somewhat more than the expected return of the portfolio.
As the RDR depends on the expected return of the portfolio it depends on its allocation. To determine the optimal allocation, disbursements are simulated as the allocation to stocks is gradually increased. The best allocation is found when the future disbursements stop improving and begin to deteriorate. The optimal stock allocation increases when the initial disbursement increases, and when less favorable returns are assumed for the investments in the portfolio. These issues are discussed at the page, Allocation.
As discussed at the page, Model, simulations are based on three sets of assumptions about future investment returns. For intermediates, these are based on best and worst cases for historic returns before and after 2008, and a case midway between. The expected return of stocks over intermediates is assumed to be the same in all three cases, and equal to the .04 observed from 1960 up to 2008. The much stronger returns on stocks since 2008 are presumably at least in part due to the lowering of interest rates, and in any case seem unlikely to be repeated.
An important factor affecting the size of an initial disbursement is the length of the initial horizon. Most illustrations at this site assume initially that the disbursements may be needed initially for as long as 36 years. For someone just reaching age 65 that would allow a last disbursement upon reaching age 100. Those with health issues will surely want to plan on a much shorter horizon, and reducing the horizon by a half can increase the initial disbursement with the same sustainability by more than half. Others may be starting sooner and wanting to plan that disbursements may be needed longer. Increasing the horizon by a quarter reduces the initial disbursement with the same sustainability by an eighth. When disbursements are being made by an organization or a trust with a specified objective the disbursements may be considered as perpetual. In this case, there are limits on the chances of declines in the far distant future. Further discussion of the effect of the horizon is at the page, Horizons.
In general, ongoing disbursements are not increased in real terms because recipients are assumed to be more sensitive to the pain of decreases than the gain from increases. Increases in real terms in ongoing disbursements are therefore not desirable because they make later reductions more likely and larger if they occur. In the future, however, an emergency may arise requiring a significant increase in spending that cannot be avoided, and that is not fully covered by insurance. Suppose, for instance, that a serious deterioration in health requires significant expense for assisted living for a long time. In effect, the simulations include self-insurance for such a problem because they assume beforehand that the disbursements will continue for as long as they might possibly be needed. A serious deterioration in health reduces the longest that the disbursements might be needed, and increases the limit on disbursements. Simulations that reduce how long the disbursements might be needed at some future point show how much the disbursements might be increased to cover increased spending on health care.
Increased spending for an emergency that is unavoidable, however, is not confined to health issues. There are many possibilities the nature and size of which initially are unknown, and some may not even be dimly recognized. One possibility that might be recognized is the possible need to provide aid for a child. Another is water damage to a residence that is only partially covered by insurance, if at all. The chances of a large outlay for an emergency in any one year are very small, but over a long period the chances become significant and could have a comparable effect on the portfolio as large as a bear market for stocks. Such a possibility can be included in the simulations to assess the effect on the sustainability of the disbursements available for normal living expenses. Including the possibility of emergencies is an option in the model and further discussed at the page, Model.
A disbursement initially is generally well below its limit, but when investment returns are less than expected the limit declines and in the future the disbursement can exceed its limit. In such cases the basic model resets the disbursement to the limit. Doing so, however, has the risk of very large annual declines in the disbursements, which will be disruptive. Moreover, there is a risk of very low disbursements in the future. To deal with these issues, the alternative is to reset an overlimit disbursement to far below its limit. A dilemma arises because of the very large costs and benefits of such a major retrenchment. There are significant reductions in the chances of low disbursements in the future, and in the risk of annual declines. These benefits, however, come at the cost of a significant curtailment of important habitual activities. Moreover, otherwise, many of these activities might never have to be curtailed for many years, and some might never have to be eliminated. To help in making a decision, simulations show the chances of future low disbursements with and without the major retrenchment. Simulations also show the risk of large annual declines if a major retrenchment is not undertaken. These issues are discussed at the page, Retrench.
An alternative way of reducing the disruption caused by resetting overlimits to the limit is to reset the disbursement only part way to the limit. Dampening these adjustments, however, can result in running out of funds, unless a large reduction is made in a disbursement if it gets too far above its limit. Simulations show, however, that is unlikely to be necessary if an overlimit is reduced at least half way to the limit. Moreover, such dampening is sufficient to significantly reduce such disruptions. Dampening the adjustment of overlimits is discussed at the page, Dampen.
The assumptions generally used at this site to illustrate procedures assume that investment returns are independently and identically distributed over time. Robert Shiller, however, famously argues that due to emotion stock prices in bull and bear markets tend to overshoot the levels warranted by future dividends and earnings. If stock prices overshoot, a period of much lower than normal returns in the past should be associated with higher than normal returns in the future. Reductions in the disbursements that otherwise might be needed will therefore tend to be less, and sustainability improved. These favorable hedging benefits tend to increase the optimal stock allocation. High prior average returns before the disbursements begin, however, could offset these favorable hedging benefits. There is evidence that stock returns are negatively related to very long moving averages of prior returns. Using Prior Average Dependent (PAD) stock returns in the simulations shows the extent to which negative PAD stock returns can improve sustainability. These issues are discussed at the page, PAD.
Instead of investing all of the funds available initially in a portfolio of stocks and fixed income issues, retirees in good health could use some of the funds to purchase an immediate lifetime annuity. To evaluate this possibility the determination of the limit must be modified to add the annual annuity benefit each year in the future to the portfolio, and to add the present value of these future benefits in real terms to the value of the portfolio in determining the current limit for a disbursement. Also, since almost all available annuities pay benefits that are fixed in nominal terms, provision must be made to adjust the future nominal benefits for inflation.
As lifetime annuity benefits have some similarities to fixed income issues, purchasing an annuity might affect the optimal stock allocation of the remaining funds in the investment portfolio. Simulations show that annuitizing does increase somewhat the best stock allocation of the remaining funds. The key issue is does annuitizing increase the Best Initial Disbursement (BID) enough to compensate for the loss of portfolio value to cover emergencies, or if not needed for that purpose to increase bequests? One test assumes the pre 2008 returns and an annuity rate that might have been obtained in the first half of 2007. Inflation follows a simple recursive random process with initial inflation and inflation expected in the long run both equal to 3%. Annuitizing half the portfolio increases the BID by about half a percentage point of initial portfolio value. By the back of the disbursement period on average the portfolio value reduction declines from 50% to about 30%. The annuitizing is desirable if the half percentage point increase in the BID is deemed to more than compensate for the loss of portfolio value. A similar tradeoff is obtained using the lower return assumptions in Model and an annuity rate available in the fall of 2022. For this test initial inflation is increased from 3% to 5%, but expected inflation in the long run remains at 3%. These issues are discussed at the page, Annuitize.
In addition to investigating how best to make periodic disbursements from a portfolio of risky assets, this site also looks at the best way to accumulate that portfolio. These issues are discussed at the page, Accumulation. For accumulation the assets in the portfolio are the same as when making the disbursements. Some key questions to answer: How should the portfolio be allocated over time when accumulating? How much more savings are required for the lower assumptions about investment return?
For the assumptions at this site, the optimal stock allocation for making sustainable disbursements is a moderate 30%. Very large stock allocations are optimal only when making large initial disbursements with the low assumptions about investment returns. In contrast, very large stock allocations are desirable early in the period when accumulating over a long period with either the low or higher investment return assumptions. Moreover, very large stock allocations can be desirable over much shorter periods when sizable additional contributions are being made to the portfolio in the future. Large stock allocations are desirable early in a long accumulation period due to the benefits of hedging over time. They are desirable when there will be sizable contributions in the future because these future contributions absorb some of the volatility. In either case, the higher volatility of the stocks tends to be suppressed making the higher expected return of the stocks more desirable.
Large stock allocations for accumulating are desirable even when the downside risk for accumulated value is strictly limited. In particular, for the analysis at this site the stock allocation for any year is determined by finding the largest stock allocation for that year whose lowest 5 percentile for the accumulated value is not less than the lowest 5 percentile value when the portfolio is entirely allocated to intermediates. When there is only one more year for accumulation the stock allocation with this property can be determined analytically. When there is more than one year simulations must be used with a backwards recursive process in which the future allocations are those that have already been determined to have this property. Suppose there are not any future contributions and that the pre 2008 investment returns are assumed. Suppose the years remaining for accumulation are 20, 15, 10, 5, and 1. The stock allocations obtained are respectively: 84%, 76.5%, 64%, 49%, and 28%.
Now suppose that there are contributions to the portfolio at the beginning of future years that are equal in real terms and that might represent the addition of more savings. In this case, an addition is made to the allocation each year that is a multiple of the ratio of the annual contribution to the initial portfolio value for that year. The values of these multiples are determined using a backwards recursive process in which the values of the future multiples are those that have already been determined to have the desired property. The desired value of a multiple is the largest for which the lowest 5 percentile for the accumulated value is not less than the lowest 5 percentile value when the portfolio is entirely allocated to intermediates.
Sometimes, however, the downside risk for accumulated value with 100% allocated to stocks is not greater than with 100% allocated to intermediates. In such cases, the stock allocation is set at 100% because leverage is not allowed, and it is not possible to allocate enough to stocks to cause as much risk as when allocating entirely to intermediates. Such situations commonly arise in the early years when the future contributions are equal to the initial value of the portfolio. Later on, however, the value of the portfolio rises substantially reducing the ratio of the contributions to portfolio value. A declining series of stock allocations can then be found over the remainder of the period with comparable downside risk for accumulated value to investing entirely in intermediates.
Assuming equal contributions in real terms to the portfolio each year over a long period provides a useful model for seeing how low returns might affect the savings needed for retirement. For the low returns, new allocations and multiples for the ratios must be determined using the backwards regressive processes. With the allocations determined simulations show the cumulative probability distributions for the funds accumulated with a constant stream of savings over 20 years. The probability of the funds accumulated as a percentage of contributed value can then be compared for the low and higher assumptions for investment returns. This comparison shows that the central tendency is for the accumulated value to be about 40% higher for the higher returns. For the assumptions about disbursements at this site, about 40% more is also required in the initial portfolio with the low returns to get a sustainable stream of disbursements of the same size. As 40% more savings is required to get a portfolio that is 40% larger, these increases are compounded. Savings must be almost twice as much to get the same sustainable stream of disbursements in retirement.
Overview
Warning: Charts and tables are not printing at the backs of pages with some browsers. If charts and tables are not printing try a different browser. All charts and tables are showing on the screen, and two or three are printing at the front of pages.
What’s New? A Major Revision
This posting makes an important revision in the procedure for finding the Best Initial Disbursement, or BID. To describe this revision, the former page, Increase, has been rewritten and renamed, BID. The first step in the revision is to use simulations to find the largest highly sustainable initial disbursement. This highly sustainable initial disbursement is the largest that simulations show has very little chance of needing to be reduced in the future under worst case conditions. These worst case conditions are the worst investment returns that seem reasonably possible, and assuming that there is a small chance each year that a large extra disbursement may be necessary to cover an emergency. For a 36-year initial horizon, this highly sustainable initial disbursement turns out to equal about 2.5%. Any highly sustainable funds from other sources, such as Social Security, are then added to this highly sustainable disbursement from the portfolio.
The next step is to find the highest priority activities that can be covered with these highly sustainable funds, and in descending order of priority the activities that cannot be covered. In a few cases, it may be possible to cover all existing activities with highly sustainable funding. In that case, the only question is whether to increase spending, including perhaps with some increase in sustainability risk. In most cases, however, there will be important habitual activities that cannot be covered from highly sustainable funding, and that will be very painful to curtail. In these cases, the next step is to increase the initial disbursement by accepting risks that seem reasonable to avoid retrenchment. These include using better returns in the simulations than the worst that seem reasonably possible, accepting chances of retrenchment in the future to avoid initial retrenchment, and not including the possibility of extra disbursements to cover emergencies in the simulations.
By accepting risks that seem reasonable to avoid painful initial retrenchment the initial disbursement can be increased from 2.5% to as much as 4.0% to cover activities that cannot be covered with highly sustainable funding. Thus, a New 4% Rule is formulated. This New 4% Rule does not pose any risk of running out of funds, but it does have sustainability risks that appear acceptable to avoid initial retrenchment. Many, however, will have important habitual activities that cannot be covered with a 4.0% initial disbursement. They will want to consider how much additional retrenchment risk in the future will have to be accepted to avoid initially curtailing these activities. Simulations show the costs and benefits of further increases in the initial disbursement from 4.0% to 5.0% or even 6.0%. In general, the initial disbursement is increased until the marginal cost in terms of increased sustainability risk exceeds the marginal benefit of the additional activities being funded by the additional funds provided.
Besides revising the determination of the BID, this revision also makes some improvements at other pages. These include Model, RDR, Allocation, Retrench, Dampen, and, Horizons. Also, a new page, Increases, has been added. This page looks at the possibility of increasing the disbursement in the future when the cushion protecting the disbursement below its limit becomes larger than the initial percentage. Making these increases, however, substantially increases the risk of covering possible emergencies. Thus, beneficiaries may prefer to allow these surpluses to accumulate, knowing that if not needed for emergencies they will add to bequests.
The revised procedure for determining the BID does not affect the determination of the annual disbursement in the basic model. The budgeted disbursement for the current year is the same as the year before in real terms unless this disbursement exceeds the calculated value of a limit. This limit is the largest constant annuity that could be obtained each year for the longest that the disbursements might be needed. The value of this annuity is calculated assuming that the then current value of the portfolio is invested at a constant rate of return called the Retrenchment Discount Rate, or RDR.
The value to use for the RDR is determined by simulating the use of an increasing series of values to see which gives the best results. These simulations are made assuming that the portfolio is some mix of stocks with returns resembling those on the S&P 500 and intermediate term fixed incomes with returns resembling those on U.S. government issues. For any simulation, the allocation of the portfolio is generally assumed to be reset to be the same at the beginning of each year over the disbursement period, but exceptions are investigated. Increasing the RDR improves the disbursements at the front of the disbursement period relative to the back. In the limit, when the RDR becomes infinite, the initial disbursement is always sustained until the portfolio runs out of funds. Simulations at page, RDR, show that a good balance between the disbursements at the front and back of the period is obtained by setting the RDR equal to somewhat more than the expected return of the portfolio.
As the RDR depends on the expected return of the portfolio it depends on its allocation. To determine the optimal allocation, disbursements are simulated as the allocation to stocks is gradually increased. The best allocation is found when the future disbursements stop improving and begin to deteriorate. The optimal stock allocation increases when the initial disbursement increases, and when less favorable returns are assumed for the investments in the portfolio. These issues are discussed at the page, Allocation.
As discussed at the page, Model, simulations are based on three sets of assumptions about future investment returns. For intermediates, these are based on best and worst cases for historic returns before and after 2008, and a case midway between. The expected return of stocks over intermediates is assumed to be the same in all three cases, and equal to the .04 observed from 1960 up to 2008. The much stronger returns on stocks since 2008 are presumably at least in part due to the lowering of interest rates, and in any case seem unlikely to be repeated.
An important factor affecting the size of an initial disbursement is the length of the initial horizon. Most illustrations at this site assume initially that the disbursements may be needed initially for as long as 36 years. For someone just reaching age 65 that would allow a last disbursement upon reaching age 100. Those with health issues will surely want to plan on a much shorter horizon, and reducing the horizon by a half can increase the initial disbursement with the same sustainability by more than half. Others may be starting sooner and wanting to plan that disbursements may be needed longer. Increasing the horizon by a quarter reduces the initial disbursement with the same sustainability by an eighth. When disbursements are being made by an organization or a trust with a specified objective the disbursements may be considered as perpetual. In this case, there are limits on the chances of declines in the far distant future. Further discussion of the effect of the horizon is at the page, Horizons.
In general, ongoing disbursements are not increased in real terms because recipients are assumed to be more sensitive to the pain of decreases than the gain from increases. Increases in real terms in ongoing disbursements are therefore not desirable because they make later reductions more likely and larger if they occur. In the future, however, an emergency may arise requiring a significant increase in spending that cannot be avoided, and that is not fully covered by insurance. Suppose, for instance, that a serious deterioration in health requires significant expense for assisted living for a long time. In effect, the simulations include self-insurance for such a problem because they assume beforehand that the disbursements will continue for as long as they might possibly be needed. A serious deterioration in health reduces the longest that the disbursements might be needed, and increases the limit on disbursements. Simulations that reduce how long the disbursements might be needed at some future point show how much the disbursements might be increased to cover increased spending on health care.
Increased spending for an emergency that is unavoidable, however, is not confined to health issues. There are many possibilities the nature and size of which initially are unknown, and some may not even be dimly recognized. One possibility that might be recognized is the possible need to provide aid for a child. Another is water damage to a residence that is only partially covered by insurance, if at all. The chances of a large outlay for an emergency in any one year are very small, but over a long period the chances become significant and could have a comparable effect on the portfolio as large as a bear market for stocks. Such a possibility can be included in the simulations to assess the effect on the sustainability of the disbursements available for normal living expenses. Including the possibility of emergencies is an option in the model and further discussed at the page, Model.
A disbursement initially is generally well below its limit, but when investment returns are less than expected the limit declines and in the future the disbursement can exceed its limit. In such cases the basic model resets the disbursement to the limit. Doing so, however, has the risk of very large annual declines in the disbursements, which will be disruptive. Moreover, there is a risk of very low disbursements in the future. To deal with these issues, the alternative is to reset an overlimit disbursement to far below its limit. A dilemma arises because of the very large costs and benefits of such a major retrenchment. There are significant reductions in the chances of low disbursements in the future, and in the risk of annual declines. These benefits, however, come at the cost of a significant curtailment of important habitual activities. Moreover, otherwise, many of these activities might never have to be curtailed for many years, and some might never have to be eliminated. To help in making a decision, simulations show the chances of future low disbursements with and without the major retrenchment. Simulations also show the risk of large annual declines if a major retrenchment is not undertaken. These issues are discussed at the page, Retrench.
An alternative way of reducing the disruption caused by resetting overlimits to the limit is to reset the disbursement only part way to the limit. Dampening these adjustments, however, can result in running out of funds, unless a large reduction is made in a disbursement if it gets too far above its limit. Simulations show, however, that is unlikely to be necessary if an overlimit is reduced at least half way to the limit. Moreover, such dampening is sufficient to significantly reduce such disruptions. Dampening the adjustment of overlimits is discussed at the page, Dampen.
The assumptions generally used at this site to illustrate procedures assume that investment returns are independently and identically distributed over time. Robert Shiller, however, famously argues that due to emotion stock prices in bull and bear markets tend to overshoot the levels warranted by future dividends and earnings. If stock prices overshoot, a period of much lower than normal returns in the past should be associated with higher than normal returns in the future. Reductions in the disbursements that otherwise might be needed will therefore tend to be less, and sustainability improved. These favorable hedging benefits tend to increase the optimal stock allocation. High prior average returns before the disbursements begin, however, could offset these favorable hedging benefits. There is evidence that stock returns are negatively related to very long moving averages of prior returns. Using Prior Average Dependent (PAD) stock returns in the simulations shows the extent to which negative PAD stock returns can improve sustainability. These issues are discussed at the page, PAD.
Instead of investing all of the funds available initially in a portfolio of stocks and fixed income issues, retirees in good health could use some of the funds to purchase an immediate lifetime annuity. To evaluate this possibility the determination of the limit must be modified to add the annual annuity benefit each year in the future to the portfolio, and to add the present value of these future benefits in real terms to the value of the portfolio in determining the current limit for a disbursement. Also, since almost all available annuities pay benefits that are fixed in nominal terms, provision must be made to adjust the future nominal benefits for inflation.
As lifetime annuity benefits have some similarities to fixed income issues, purchasing an annuity might affect the optimal stock allocation of the remaining funds in the investment portfolio. Simulations show that annuitizing does increase somewhat the best stock allocation of the remaining funds. The key issue is does annuitizing increase the Best Initial Disbursement (BID) enough to compensate for the loss of portfolio value to cover emergencies, or if not needed for that purpose to increase bequests? One test assumes the pre 2008 returns and an annuity rate that might have been obtained in the first half of 2007. Inflation follows a simple recursive random process with initial inflation and inflation expected in the long run both equal to 3%. Annuitizing half the portfolio increases the BID by about half a percentage point of initial portfolio value. By the back of the disbursement period on average the portfolio value reduction declines from 50% to about 30%. The annuitizing is desirable if the half percentage point increase in the BID is deemed to more than compensate for the loss of portfolio value. A similar tradeoff is obtained using the lower return assumptions in Model and an annuity rate available in the fall of 2022. For this test initial inflation is increased from 3% to 5%, but expected inflation in the long run remains at 3%. These issues are discussed at the page, Annuitize.
In addition to investigating how best to make periodic disbursements from a portfolio of risky assets, this site also looks at the best way to accumulate that portfolio. These issues are discussed at the page, Accumulation. For accumulation the assets in the portfolio are the same as when making the disbursements. Some key questions to answer: How should the portfolio be allocated over time when accumulating? How much more savings are required for the lower assumptions about investment return?
For the assumptions at this site, the optimal stock allocation for making sustainable disbursements is a moderate 30%. Very large stock allocations are optimal only when making large initial disbursements with the low assumptions about investment returns. In contrast, very large stock allocations are desirable early in the period when accumulating over a long period with either the low or higher investment return assumptions. Moreover, very large stock allocations can be desirable over much shorter periods when sizable additional contributions are being made to the portfolio in the future. Large stock allocations are desirable early in a long accumulation period due to the benefits of hedging over time. They are desirable when there will be sizable contributions in the future because these future contributions absorb some of the volatility. In either case, the higher volatility of the stocks tends to be suppressed making the higher expected return of the stocks more desirable.
Large stock allocations for accumulating are desirable even when the downside risk for accumulated value is strictly limited. In particular, for the analysis at this site the stock allocation for any year is determined by finding the largest stock allocation for that year whose lowest 5 percentile for the accumulated value is not less than the lowest 5 percentile value when the portfolio is entirely allocated to intermediates. When there is only one more year for accumulation the stock allocation with this property can be determined analytically. When there is more than one year simulations must be used with a backwards recursive process in which the future allocations are those that have already been determined to have this property. Suppose there are not any future contributions and that the pre 2008 investment returns are assumed. Suppose the years remaining for accumulation are 20, 15, 10, 5, and 1. The stock allocations obtained are respectively: 84%, 76.5%, 64%, 49%, and 28%.
Now suppose that there are contributions to the portfolio at the beginning of future years that are equal in real terms and that might represent the addition of more savings. In this case, an addition is made to the allocation each year that is a multiple of the ratio of the annual contribution to the initial portfolio value for that year. The values of these multiples are determined using a backwards recursive process in which the values of the future multiples are those that have already been determined to have the desired property. The desired value of a multiple is the largest for which the lowest 5 percentile for the accumulated value is not less than the lowest 5 percentile value when the portfolio is entirely allocated to intermediates.
Sometimes, however, the downside risk for accumulated value with 100% allocated to stocks is not greater than with 100% allocated to intermediates. In such cases, the stock allocation is set at 100% because leverage is not allowed, and it is not possible to allocate enough to stocks to cause as much risk as when allocating entirely to intermediates. Such situations commonly arise in the early years when the future contributions are equal to the initial value of the portfolio. Later on, however, the value of the portfolio rises substantially reducing the ratio of the contributions to portfolio value. A declining series of stock allocations can then be found over the remainder of the period with comparable downside risk for accumulated value to investing entirely in intermediates.
Assuming equal contributions in real terms to the portfolio each year over a long period provides a useful model for seeing how low returns might affect the savings needed for retirement. For the low returns, new allocations and multiples for the ratios must be determined using the backwards regressive processes. With the allocations determined simulations show the cumulative probability distributions for the funds accumulated with a constant stream of savings over 20 years. The probability of the funds accumulated as a percentage of contributed value can then be compared for the low and higher assumptions for investment returns. This comparison shows that the central tendency is for the accumulated value to be about 40% higher for the higher returns. For the assumptions about disbursements at this site, about 40% more is also required in the initial portfolio with the low returns to get a sustainable stream of disbursements of the same size. As 40% more savings is required to get a portfolio that is 40% larger, these increases are compounded. Savings must be almost twice as much to get the same sustainable stream of disbursements in retirement.
Posted November, 2020 Revised July, 2022 February, 2023 November, 2024