To avoid even larger reductions in the future, and possibly running out of funds, the disbursements from a portfolio must be limited depending on the remaining value of the portfolio, and the remaining number of disbursements to be made. The limit used in the model at this site is to calculate the largest constant stream of disbursements over the remaining period that could be obtained if the remaining value of the portfolio is invested at a constant rate of return. This constant rate of return is called the Retrenchment Discount Rate, or RDR, and is a decision variable that governs how the risk of declines in the disbursements is distributed over the disbursement period. Increasing the RDR increases the limit and reduces the risk of declines at the front of the period, but increases the risk of declines at the back of the period. With a larger limit there is less protection for the value of the portfolio early in the period, and an increased chance of deterioration that will require lower disbursements in the future.
The best RDR to use can be found by gradually increasing the RDR and using simulations to find how much the risk of declines is reduced at the front of the period, and increased at the back of the period. The best RDR is found when the cost of a further increase in terms of the increased risk of declines at the back of the period exceeds the benefit in terms of the reduced declines at the front of the period. To make such tests, the portfolio is first assumed to be entirely allocated to intermediates with the returns assumed in the pre 2008 illustration. For a reasonably sustainable initial disbursement, cumulative probability distributions are obtained at years 12, 24, and 36 when the RDR is set equal to the expected return, or to .02 or .04 more. These tests are repeated when the initial disbursement is increased, and when half of the portfolio is allocated to stocks. In all cases, setting the RDR equal to .02 above the expected return of the portfolio provides significant improvement in the disbursements at the front of the period at the cost of significant deterioration at the back. Ignoring the difference in timing, the improvement at the front, although significant, is clearly not worth the deterioration at the back. Nevertheless, those making disbursements may be willing to discount results occurring far in the future sufficiently to obtain improvement in the near term.
Sustainable Disbursements
Suppose the initial annual disbursement is 3.5% of portfolio value and the portfolio is entirely allocated to intermediates with the returns specified in the pre 2008 illustration. The RDR is set equal to the expected return of the portfolio of .03, and to .05 and .07. The cumulative probability distributions of the disbursements at years 12, 24, and 36 are shown in Charts L.1, L.2, and L.3. Chart L.1 shows that increasing the RDR from .03 to .05 provides significant improvement at year 12. The chances of declines of up to 1.0 percentage points in initial portfolio value in size are reduced from .16 to .04. Chart L.2 shows at year 24 that the chances of declines of up to 1.0 percentage points in size are reduced from .25 to .15. The cost of this improvement at the front of the period is significant deterioration at the back. At year 36, the increase from .03 to .05 does provide some reduction in the chances of small declines, but this small improvement is significantly outweighed by a significant increase in the chances of large declines. In particular, there is a .15 chance of larger declines ranging in size from 0.5 to 2.5 percentage points of initial portfolio value. The increase in these declines is given by the distance on the horizontal axis between the curves for .05 and .03 at given probabilities on the vertical axis. On average, raising the RDR from .03 to .05 increases the size of these declines by about 0.35 percentage points of initial portfolio value.
Ignoring the difference in timing, the increase in the declines at year 36 is not worth accepting to get the reduction in the chances of declines earlier in the period. Those making disbursements, however, have good reason for discounting this later deterioration when comparing it to the earlier improvement. The declining chances of survival over the period mean that the results late in the period may never be applicable. Moreover, everyone wants to defer problems as long as doing so does not make the problems too much worse. Those making disbursements may decide that the increase in declines in Chart L.3 are sufficiently remote to accept so as to obtain the significant short term improvements in Charts L.1 and L.2 of increasing the RDR from .03 to .05. A further increase in the RDR from .05 to .07, however, does not look attractive. The reduction in the chances of declines at the front of the period is now substantially less. At year 12, for instance, the decline is now only from .04 to .01 instead of from .16 to .04, and at the back of the period the average increase in the size of the declines is now slightly larger.
Larger Disbursements
Suppose next that the initial disbursement is increased from 3.5% to 4.25%, which causes substantial deterioration in sustainability. The curves in this case are shown in Charts L.4, L.5, and L.6. At first glance, these graphs look very similar to the earlier graphs. Note, however, that the top of the scale on the vertical axis has been increased from .30 to 1.00. Thus, the chances that the declines are reduced at the front of the period are now substantially larger, as are the chances that the declines are increased at the back of the period. There is only a small increase, however, in the size of the declines that are reduced at the front of the period, and in the size of the declines that are increased at the back, and in the average size of the increase. At year 12 in Chart L.4, increasing the RDR from .03 to .05 now reduces the chances of declines from .68 to .25 instead of from .16 to .04. At year 36 in Chart L.6, the chances that the declines will be larger has increased from .15 to .45. As the chances of both the costs and benefit of an increase in the RDR from .03 to .05 have increased significantly, those making disbursements may see little reason for changing the decision from that for the smaller disbursements. If the increased chances of larger declines at the back of the period seemed sufficiently remote to discount in favor of significant reductions in earlier declines for the smaller disbursements that will continue to be true for the larger disbursements. Moreover, there continues to be little net benefit for a further increase in the RDR from .05 to .07.
Include Stocks
So far, the portfolio has been allocated entirely to intermediates. Suppose now that 50% of the portfolio is allocated to stocks with the returns specified in the pre 2008 illustration. As the assumed expected annual real return of the stocks is .07, a stock allocation of 50% increases the expected annual real return of the portfolio from .03 to .05. The question now is whether it is desirable to increase the RDR from .05 to .07 or .09? Also, allocating 50% to stocks allows the initial disbursement to be increased from 3.5% to 4.0% with about the same sustainability. The cumulative probability distributions of the disbursements in years 12, 24, and 36 under these conditions are shown in Charts L.7, L.8. and L.9. Not surprisingly, adding the stocks to the portfolio increases somewhat the size of the largest possible declines at both the front and back of the period. Also, the chances of declines are now somewhat less due to the improved efficiency of the portfolio. With less risk to reduce, increasing the RDR to .02 above the expected return reduces the chances of declines in Chart L.7 from .13 to .05 instead of from .16 to .04 in Chart L.1. In Chart L.9 the chances of increases in the declines are .13 instead of .15. Nevertheless, the charts with the 50% stock allocation look very similar to those with the portfolio allocated entirely to intermediates. Those willing to discount the increased chances of large declines at the back of the period to obtain a reduction in declines in the shorter term seem likely to continue doing so with stocks included in the portfolio.
Shorter Horizons
Another factor that could affect the desirability of increasing the RDR is the shortening horizon as the disbursements proceed. To test the effect of a shorter horizon, suppose that the initial disbursement was 3.5%, and that 18 of these disbursements have been made with the portfolio entirely allocated to intermediates. It is now the beginning of the last 18 years, and the portfolio is now equal to 58% of its initial value. A simulation shows that there is a .16 chance of a decline by the sixth year, which is a third of the way through the remaining period. That is the same chance of a decline as there was initially at the 12th year, which was a third of the way through the initial disbursement period. The question is does the shorter horizon change the desirability of increasing the RDR?
Charts L.10, L.11, and L.12 compare the results at years 6, 12, and 18 of increasing the RDR from .03 to .05 and .07. Increasing the RDR reduces the chances of declines in Charts L.10 and L.11 just about the same as in Charts L.1 and L.2 for the initial horizon. Moreover, the chances for larger declines at the end of the period in Charts L.12 and L.13 are just about the same when the RDR increases. On the other hand, the size of the declines that are being reduced are smaller with the shorter horizon. But the size of the declines that are being increased at the end of the period are also smaller as are the size of the increases. Thus, the shorter horizon appears to cause comparable reductions in both the costs and benefits of increasing the RDR. The increase in declines at the end of the period, however, is now closer, and more difficult to discount as being remote. Nevertheless, the shortening horizon as the disbursements proceed appears at most to have only a small effect on the desirability of increasing the RDR.
Conclusions
To avoid running out of funds disbursements from a portfolio must be limited depending on the value of the portfolio, and the remaining number of disbursements that may be necessary. The limit used at this site is the largest constant stream of disbursements that could be obtained when the portfolio is invested at a constant return equal to RDR over the remaining period. The value of RDR is a decision variable that when increased shifts the risk of declines in the disbursements from the front to the back of the disbursement period. Simulations show that those making disbursements will likely discount the future sufficiently to set the RDR equal to about two percentage points more than the expected return of the portfolio. That is true irrespective of the size of the disbursements, the allocation of the portfolio, or the remaining length of the horizon.
Posted November, 2018 Revised July, 2022
Addendum
Retrench discusses the dilemma that arises when a disbursement reaches its limit. The seriousness of this dilemma may not be fully recognized when viewing the chances of future declines when disbursements start well below their limit as in the simulations given in the text of RDR. The seriousness of the dilemma, however, does become evident when the chances of future declines are simulated assuming a disbursement has reached its limit in the future as is done in Retrench. In the simulations in Retrench there is the possibility of a large annual decline in the near future that will be disruptive. There are also much higher chances of very low disbursements in the more distant future than are evident when the disbursements start well below their limit. When a disbursement is at its limit the situation appears sufficiently serious that Retrench considers the possibility of undertaking a major retrenchment. This major retrenchment is instead of resetting overlimit disbursements to the limit as in Model.
Such a major retrenchment has a high certain cost as important habitual activities will have to be curtailed. The only near term benefit is to sharply reduce the chances of a large annual decline. Many of the important habitual activities that will have to be curtained may otherwise not need to be eliminated for many years, if ever. Thus, after weighing the costs and benefits, a decision may be made to reset an overlimit to the limit as in Model. Nevertheless, the cost of undertaking a major retrenchment is well worth considering when a disbursement reaches its limit in the future. Increasing the RDR to avoid a disbursement reaching its limit in the future may therefore be of more value than may be readily apparent when a disbursement starts well blow its limit as in the simulations in the text of RDR.
RDR
Best Limits
To avoid even larger reductions in the future, and possibly running out of funds, the disbursements from a portfolio must be limited depending on the remaining value of the portfolio, and the remaining number of disbursements to be made. The limit used in the model at this site is to calculate the largest constant stream of disbursements over the remaining period that could be obtained if the remaining value of the portfolio is invested at a constant rate of return. This constant rate of return is called the Retrenchment Discount Rate, or RDR, and is a decision variable that governs how the risk of declines in the disbursements is distributed over the disbursement period. Increasing the RDR increases the limit and reduces the risk of declines at the front of the period, but increases the risk of declines at the back of the period. With a larger limit there is less protection for the value of the portfolio early in the period, and an increased chance of deterioration that will require lower disbursements in the future.
The best RDR to use can be found by gradually increasing the RDR and using simulations to find how much the risk of declines is reduced at the front of the period, and increased at the back of the period. The best RDR is found when the cost of a further increase in terms of the increased risk of declines at the back of the period exceeds the benefit in terms of the reduced declines at the front of the period. To make such tests, the portfolio is first assumed to be entirely allocated to intermediates with the returns assumed in the pre 2008 illustration. For a reasonably sustainable initial disbursement, cumulative probability distributions are obtained at years 12, 24, and 36 when the RDR is set equal to the expected return, or to .02 or .04 more. These tests are repeated when the initial disbursement is increased, and when half of the portfolio is allocated to stocks. In all cases, setting the RDR equal to .02 above the expected return of the portfolio provides significant improvement in the disbursements at the front of the period at the cost of significant deterioration at the back. Ignoring the difference in timing, the improvement at the front, although significant, is clearly not worth the deterioration at the back. Nevertheless, those making disbursements may be willing to discount results occurring far in the future sufficiently to obtain improvement in the near term.
Sustainable Disbursements
Suppose the initial annual disbursement is 3.5% of portfolio value and the portfolio is entirely allocated to intermediates with the returns specified in the pre 2008 illustration. The RDR is set equal to the expected return of the portfolio of .03, and to .05 and .07. The cumulative probability distributions of the disbursements at years 12, 24, and 36 are shown in Charts L.1, L.2, and L.3. Chart L.1 shows that increasing the RDR from .03 to .05 provides significant improvement at year 12. The chances of declines of up to 1.0 percentage points in initial portfolio value in size are reduced from .16 to .04. Chart L.2 shows at year 24 that the chances of declines of up to 1.0 percentage points in size are reduced from .25 to .15. The cost of this improvement at the front of the period is significant deterioration at the back. At year 36, the increase from .03 to .05 does provide some reduction in the chances of small declines, but this small improvement is significantly outweighed by a significant increase in the chances of large declines. In particular, there is a .15 chance of larger declines ranging in size from 0.5 to 2.5 percentage points of initial portfolio value. The increase in these declines is given by the distance on the horizontal axis between the curves for .05 and .03 at given probabilities on the vertical axis. On average, raising the RDR from .03 to .05 increases the size of these declines by about 0.35 percentage points of initial portfolio value.
Ignoring the difference in timing, the increase in the declines at year 36 is not worth accepting to get the reduction in the chances of declines earlier in the period. Those making disbursements, however, have good reason for discounting this later deterioration when comparing it to the earlier improvement. The declining chances of survival over the period mean that the results late in the period may never be applicable. Moreover, everyone wants to defer problems as long as doing so does not make the problems too much worse. Those making disbursements may decide that the increase in declines in Chart L.3 are sufficiently remote to accept so as to obtain the significant short term improvements in Charts L.1 and L.2 of increasing the RDR from .03 to .05. A further increase in the RDR from .05 to .07, however, does not look attractive. The reduction in the chances of declines at the front of the period is now substantially less. At year 12, for instance, the decline is now only from .04 to .01 instead of from .16 to .04, and at the back of the period the average increase in the size of the declines is now slightly larger.
Larger Disbursements
Suppose next that the initial disbursement is increased from 3.5% to 4.25%, which causes substantial deterioration in sustainability. The curves in this case are shown in Charts L.4, L.5, and L.6. At first glance, these graphs look very similar to the earlier graphs. Note, however, that the top of the scale on the vertical axis has been increased from .30 to 1.00. Thus, the chances that the declines are reduced at the front of the period are now substantially larger, as are the chances that the declines are increased at the back of the period. There is only a small increase, however, in the size of the declines that are reduced at the front of the period, and in the size of the declines that are increased at the back, and in the average size of the increase. At year 12 in Chart L.4, increasing the RDR from .03 to .05 now reduces the chances of declines from .68 to .25 instead of from .16 to .04. At year 36 in Chart L.6, the chances that the declines will be larger has increased from .15 to .45. As the chances of both the costs and benefit of an increase in the RDR from .03 to .05 have increased significantly, those making disbursements may see little reason for changing the decision from that for the smaller disbursements. If the increased chances of larger declines at the back of the period seemed sufficiently remote to discount in favor of significant reductions in earlier declines for the smaller disbursements that will continue to be true for the larger disbursements. Moreover, there continues to be little net benefit for a further increase in the RDR from .05 to .07.
Include Stocks
So far, the portfolio has been allocated entirely to intermediates. Suppose now that 50% of the portfolio is allocated to stocks with the returns specified in the pre 2008 illustration. As the assumed expected annual real return of the stocks is .07, a stock allocation of 50% increases the expected annual real return of the portfolio from .03 to .05. The question now is whether it is desirable to increase the RDR from .05 to .07 or .09? Also, allocating 50% to stocks allows the initial disbursement to be increased from 3.5% to 4.0% with about the same sustainability. The cumulative probability distributions of the disbursements in years 12, 24, and 36 under these conditions are shown in Charts L.7, L.8. and L.9. Not surprisingly, adding the stocks to the portfolio increases somewhat the size of the largest possible declines at both the front and back of the period. Also, the chances of declines are now somewhat less due to the improved efficiency of the portfolio. With less risk to reduce, increasing the RDR to .02 above the expected return reduces the chances of declines in Chart L.7 from .13 to .05 instead of from .16 to .04 in Chart L.1. In Chart L.9 the chances of increases in the declines are .13 instead of .15. Nevertheless, the charts with the 50% stock allocation look very similar to those with the portfolio allocated entirely to intermediates. Those willing to discount the increased chances of large declines at the back of the period to obtain a reduction in declines in the shorter term seem likely to continue doing so with stocks included in the portfolio.
Shorter Horizons
Another factor that could affect the desirability of increasing the RDR is the shortening horizon as the disbursements proceed. To test the effect of a shorter horizon, suppose that the initial disbursement was 3.5%, and that 18 of these disbursements have been made with the portfolio entirely allocated to intermediates. It is now the beginning of the last 18 years, and the portfolio is now equal to 58% of its initial value. A simulation shows that there is a .16 chance of a decline by the sixth year, which is a third of the way through the remaining period. That is the same chance of a decline as there was initially at the 12th year, which was a third of the way through the initial disbursement period. The question is does the shorter horizon change the desirability of increasing the RDR?
Charts L.10, L.11, and L.12 compare the results at years 6, 12, and 18 of increasing the RDR from .03 to .05 and .07. Increasing the RDR reduces the chances of declines in Charts L.10 and L.11 just about the same as in Charts L.1 and L.2 for the initial horizon. Moreover, the chances for larger declines at the end of the period in Charts L.12 and L.13 are just about the same when the RDR increases. On the other hand, the size of the declines that are being reduced are smaller with the shorter horizon. But the size of the declines that are being increased at the end of the period are also smaller as are the size of the increases. Thus, the shorter horizon appears to cause comparable reductions in both the costs and benefits of increasing the RDR. The increase in declines at the end of the period, however, is now closer, and more difficult to discount as being remote. Nevertheless, the shortening horizon as the disbursements proceed appears at most to have only a small effect on the desirability of increasing the RDR.
Conclusions
To avoid running out of funds disbursements from a portfolio must be limited depending on the value of the portfolio, and the remaining number of disbursements that may be necessary. The limit used at this site is the largest constant stream of disbursements that could be obtained when the portfolio is invested at a constant return equal to RDR over the remaining period. The value of RDR is a decision variable that when increased shifts the risk of declines in the disbursements from the front to the back of the disbursement period. Simulations show that those making disbursements will likely discount the future sufficiently to set the RDR equal to about two percentage points more than the expected return of the portfolio. That is true irrespective of the size of the disbursements, the allocation of the portfolio, or the remaining length of the horizon.
Posted November, 2018 Revised July, 2022
Addendum
Retrench discusses the dilemma that arises when a disbursement reaches its limit. The seriousness of this dilemma may not be fully recognized when viewing the chances of future declines when disbursements start well below their limit as in the simulations given in the text of RDR. The seriousness of the dilemma, however, does become evident when the chances of future declines are simulated assuming a disbursement has reached its limit in the future as is done in Retrench. In the simulations in Retrench there is the possibility of a large annual decline in the near future that will be disruptive. There are also much higher chances of very low disbursements in the more distant future than are evident when the disbursements start well below their limit. When a disbursement is at its limit the situation appears sufficiently serious that Retrench considers the possibility of undertaking a major retrenchment. This major retrenchment is instead of resetting overlimit disbursements to the limit as in Model.
Such a major retrenchment has a high certain cost as important habitual activities will have to be curtailed. The only near term benefit is to sharply reduce the chances of a large annual decline. Many of the important habitual activities that will have to be curtained may otherwise not need to be eliminated for many years, if ever. Thus, after weighing the costs and benefits, a decision may be made to reset an overlimit to the limit as in Model. Nevertheless, the cost of undertaking a major retrenchment is well worth considering when a disbursement reaches its limit in the future. Increasing the RDR to avoid a disbursement reaching its limit in the future may therefore be of more value than may be readily apparent when a disbursement starts well blow its limit as in the simulations in the text of RDR.
Posted August, 2023