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 here 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. When the RDR is infinite, there is no limit, and the initial disbursement continues until the portfolio runs out of funds.
Reaching the limit is undesirable not only because it results in a decline in the disbursement, but because it causes uncertainty about the possibility of future declines. The cushion that has been protecting the disbursement from lower than expected returns no longer exists. Lower than expected returns may now result in large unpredictable declines in the disbursement each year in the future, which will be disruptive. To avoid this risk, the page, Retrench, considers the possibility of undertaking a large retrenchment of say 30% to create a new cushion instead of resetting an overlimit to the limit as specified in the basic model. The page, Dampen, considers the possibility of dampening, and reducing any overlimit only part way to the limit. In any case, it is especially desirable to avoid such situations early in the disbursement period, which is a factor to consider in selecting the RDR.
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, simulations are used to find the cumulative probability distributions of the disbursements 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 made under a variety of different conditions. In all cases, the best RDR appears to be at least .02 more than the expected return of the portfolio.
Pre 2008 Returns
The first test assumes the Pre 2008 Returns and that the portfolio is entirely allocated to the intermediates. For the Pre 2008 Returns the page, Model, gives the assumed expected return on the intermediates as .03. Thus, with the portfolio entirely allocated to intermediates, the expected return of the portfolio is .03. The values of the RDR to be tested are .03, .05, and .07. Suppose the initial disbursement is set equal to 3.5%. The cumulative probability distributions of the disbursements for the three values of the RDR at years 12, 24, and 36 are shown in Charts 3.1, 3.2, and 3.3.
A quick glance at the charts shows that increasing the RDR has caused significant improvement at years 12 and 24, but deterioration at the end of the disbursement period as expected. Moreover, most of the improvement is due to the increase from .03 to .05. A further increase from .05 to .07 causes at least as much if not more deterioration at the end of the period as the increase from .03 to .05, and provides much less improvement earlier in the period. Suppose the beneficiary is willing to discount results in the distant future for declining life expectancy and a general desire to defer problems as long as possible as long as they do not become too much worse. There then appears to be good reason to increase the RDR to at least 05.
More specifically, note that the chances of reaching the limit are shown by the probability on the vertical axis for the tops of the curves. Thus, increasing the RDR reduces the chances of reaching the limit for each of the years, and the reduction is significantly more for the increase from .03 to .05 than for .05 to .07. Note also that the chances of a half percentage point decline or more are shown by the probability on the vertical axis for a curve when its value on the horizontal axis is 3.0. These chances decline at years 12 and 24 when the RDR is increased, but there is no change at year 36. The chances of a one percentage point decline or more are shown by the probability on the vertical axis for a curve when the value on the horizontal axis is 2.5. These chances do not change at years 12 and 24 when the RDR is increased. There is, however, a significant increase in these chances at year 36. There are also significant increases in the chances of declines of 1.5 or 2.0 percentage points or more.
Now suppose that instead of being entirely allocated to intermediates that half of the portfolio is allocated to stocks. Since the expected equity premium is .04 doing so increases the expected return of the portfolio from .03 to .05. The RDRs to be tested are now .05, .07, and .09 instead of .03, .05, and .07. Moreover, with the increase in expected return, the initial disbursement can now be increased from 3.5% to 4.0% with little change in the chances of reaching the limit at various years in the future. The results in this case are shown in Charts 3.4, 3.5, and 3.6. The general patterns are similar to those obtained before, and there is not any change in the conclusions. When the stocks are included, however, the curves are reduced somewhat less when the RDR is increased. Also, as to be expected with the stocks in the portfolio, there are somewhat higher chances for larger declines.
Lower Returns
Suppose next that the assumed returns are those called the Lower Returns in the page, Model, with the expected annual real return on the intermediates equal to zero instead of .03. In this case, when the portfolio is entirely allocated to intermediates, the RDRs to be tested are .00, 02, and .04. Also, to get comparable chances of reaching the limit in future years, the initial disbursement is reduced from 3.5% to 2.5%. The results for the Lower Returns are shown in Charts 3.7, 3.8, and 3.9. These charts show that increasing the RDR above the expected return of the portfolio is far more important for these lower returns than the higher returns just considered, even though the benefits for the higher returns were significant. In fact, for the lower returns, increasing the RDR from zero to .02 almost eliminates any chance of the disbursement reaching its limit over the first 12 years versus almost a .30 chance of doing so when the RDR is equal to the expected return of the portfolio of zero. Over the following 12 years with an RDR of .02 there is a better than .15 chance of reaching the limit, but this is still an improvement. When the RDR is zero the chances of having reached the limit by year 24 have further increased to .40
There is no doubt that these improvements of setting the RDR to .02 are well worth the cost of larger declines at the end of the period. Chart 3.9 shows at the end of the period that increasing the RDR from 0 to .02 increases the chances of a half percentage point decline or more from .09 to .16. And the chances of a one percentage point decline or more increase from 0 to.04. Any beneficiary, however, will presumably be willing to discount these costs at the end of the period sufficiently to prefer an increase in the RDR to .02. Whether a further increase from .02 to .04 is an improvement, however, is questionable. The further reduction in the chances of reaching the limit by year 12 are very small. The improvement at year 24 is more appreciable, but it is debatable whether it is worth the higher risk of declines at the end of the period when the RDR is increased from .02 to .04.
Now suppose with the lower returns that half of the portfolio is allocated to stocks. In this case, the RDRs to be tested increase from 0, .02, and .04 to .02, .04, and .06. For the increased expected returns with the stocks, the initial disbursement is also increased from 2.5% to 3.0% to keep the chances comparable in future years of reaching the limit. The results for this case are shown in Charts 3.10, 3.11, and 3.12. With stocks included in the portfolio for the Lower Returns, the improvement from increasing the RDR is not as remarkable, but still significant. Moreover, there is more improvement than for the 50% stock allocation with the Pre 2008 Returns. There appears to be no doubt that beneficiaries will prefer the results when the RDR is increased to .02 above the expected return of the portfolio. A further increase may also be desirable, but is more questionable.
Shorter Horizons
A factor that could affect the desirability of using a larger RDR is the shorter horizon as the disbursements proceed. Consider again the first case considered with the Pre 2008 Returns, and the portfolio entirely allocated to intermediates. Suppose over the first 18 years that the disbursement has not reached its limit, and that 18 more disbursements remain to made. The value of the portfolio, however, is now down to 58% of its initial value. Year 6 of the remaining 18 years is a third of the way through the remaining disbursement period, just as year 12 was a third of the way through the initial 36 years. With an RDR of .03, simulations show that the chances of reaching the limit at year 6 are about .15, which is the same as the chances were of reaching the limit by year 12 initially with an RDR of .03. The question is whether the shorter horizon has affected the desirability of increasing the RDR?
The shorter horizon makes the chances of declines of any size less likely. Simulations show, however, that it is still desirable to increase the RDR to reduce the chances of reaching the limit in the future just as it was initially. Charts 3.13, 3.14, and 3.15 show the chances of declines at years 6, 12, and 18 when the RDR is equal to .03, .05, and .07. In these charts the chances of declines of any size are less as expected. The desirability of increasing the RDR, however, appears to be about the same as for the longer horizon. The declining horizon as the disbursements proceed has little effect on the choice of 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 here is the largest constant stream of disbursements that could be obtained when the portfolio is invested at a constant return equal to the RDR over the remaining period. The value of the 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 at least two percentage points more than the expected return of the portfolio. That is true irrespective of the assumptions made about the returns, the allocation of the portfolio, or the remaining length of the horizon.
Posted November 2018 Revised July 2022 October 2024
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 here 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. When the RDR is infinite, there is no limit, and the initial disbursement continues until the portfolio runs out of funds.
Reaching the limit is undesirable not only because it results in a decline in the disbursement, but because it causes uncertainty about the possibility of future declines. The cushion that has been protecting the disbursement from lower than expected returns no longer exists. Lower than expected returns may now result in large unpredictable declines in the disbursement each year in the future, which will be disruptive. To avoid this risk, the page, Retrench, considers the possibility of undertaking a large retrenchment of say 30% to create a new cushion instead of resetting an overlimit to the limit as specified in the basic model. The page, Dampen, considers the possibility of dampening, and reducing any overlimit only part way to the limit. In any case, it is especially desirable to avoid such situations early in the disbursement period, which is a factor to consider in selecting the RDR.
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, simulations are used to find the cumulative probability distributions of the disbursements 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 made under a variety of different conditions. In all cases, the best RDR appears to be at least .02 more than the expected return of the portfolio.
Pre 2008 Returns
The first test assumes the Pre 2008 Returns and that the portfolio is entirely allocated to the intermediates. For the Pre 2008 Returns the page, Model, gives the assumed expected return on the intermediates as .03. Thus, with the portfolio entirely allocated to intermediates, the expected return of the portfolio is .03. The values of the RDR to be tested are .03, .05, and .07. Suppose the initial disbursement is set equal to 3.5%. The cumulative probability distributions of the disbursements for the three values of the RDR at years 12, 24, and 36 are shown in Charts 3.1, 3.2, and 3.3.
A quick glance at the charts shows that increasing the RDR has caused significant improvement at years 12 and 24, but deterioration at the end of the disbursement period as expected. Moreover, most of the improvement is due to the increase from .03 to .05. A further increase from .05 to .07 causes at least as much if not more deterioration at the end of the period as the increase from .03 to .05, and provides much less improvement earlier in the period. Suppose the beneficiary is willing to discount results in the distant future for declining life expectancy and a general desire to defer problems as long as possible as long as they do not become too much worse. There then appears to be good reason to increase the RDR to at least 05.
More specifically, note that the chances of reaching the limit are shown by the probability on the vertical axis for the tops of the curves. Thus, increasing the RDR reduces the chances of reaching the limit for each of the years, and the reduction is significantly more for the increase from .03 to .05 than for .05 to .07. Note also that the chances of a half percentage point decline or more are shown by the probability on the vertical axis for a curve when its value on the horizontal axis is 3.0. These chances decline at years 12 and 24 when the RDR is increased, but there is no change at year 36. The chances of a one percentage point decline or more are shown by the probability on the vertical axis for a curve when the value on the horizontal axis is 2.5. These chances do not change at years 12 and 24 when the RDR is increased. There is, however, a significant increase in these chances at year 36. There are also significant increases in the chances of declines of 1.5 or 2.0 percentage points or more.
Now suppose that instead of being entirely allocated to intermediates that half of the portfolio is allocated to stocks. Since the expected equity premium is .04 doing so increases the expected return of the portfolio from .03 to .05. The RDRs to be tested are now .05, .07, and .09 instead of .03, .05, and .07. Moreover, with the increase in expected return, the initial disbursement can now be increased from 3.5% to 4.0% with little change in the chances of reaching the limit at various years in the future. The results in this case are shown in Charts 3.4, 3.5, and 3.6. The general patterns are similar to those obtained before, and there is not any change in the conclusions. When the stocks are included, however, the curves are reduced somewhat less when the RDR is increased. Also, as to be expected with the stocks in the portfolio, there are somewhat higher chances for larger declines.
Lower Returns
Suppose next that the assumed returns are those called the Lower Returns in the page, Model, with the expected annual real return on the intermediates equal to zero instead of .03. In this case, when the portfolio is entirely allocated to intermediates, the RDRs to be tested are .00, 02, and .04. Also, to get comparable chances of reaching the limit in future years, the initial disbursement is reduced from 3.5% to 2.5%. The results for the Lower Returns are shown in Charts 3.7, 3.8, and 3.9. These charts show that increasing the RDR above the expected return of the portfolio is far more important for these lower returns than the higher returns just considered, even though the benefits for the higher returns were significant. In fact, for the lower returns, increasing the RDR from zero to .02 almost eliminates any chance of the disbursement reaching its limit over the first 12 years versus almost a .30 chance of doing so when the RDR is equal to the expected return of the portfolio of zero. Over the following 12 years with an RDR of .02 there is a better than .15 chance of reaching the limit, but this is still an improvement. When the RDR is zero the chances of having reached the limit by year 24 have further increased to .40
There is no doubt that these improvements of setting the RDR to .02 are well worth the cost of larger declines at the end of the period. Chart 3.9 shows at the end of the period that increasing the RDR from 0 to .02 increases the chances of a half percentage point decline or more from .09 to .16. And the chances of a one percentage point decline or more increase from 0 to.04. Any beneficiary, however, will presumably be willing to discount these costs at the end of the period sufficiently to prefer an increase in the RDR to .02. Whether a further increase from .02 to .04 is an improvement, however, is questionable. The further reduction in the chances of reaching the limit by year 12 are very small. The improvement at year 24 is more appreciable, but it is debatable whether it is worth the higher risk of declines at the end of the period when the RDR is increased from .02 to .04.
Now suppose with the lower returns that half of the portfolio is allocated to stocks. In this case, the RDRs to be tested increase from 0, .02, and .04 to .02, .04, and .06. For the increased expected returns with the stocks, the initial disbursement is also increased from 2.5% to 3.0% to keep the chances comparable in future years of reaching the limit. The results for this case are shown in Charts 3.10, 3.11, and 3.12. With stocks included in the portfolio for the Lower Returns, the improvement from increasing the RDR is not as remarkable, but still significant. Moreover, there is more improvement than for the 50% stock allocation with the Pre 2008 Returns. There appears to be no doubt that beneficiaries will prefer the results when the RDR is increased to .02 above the expected return of the portfolio. A further increase may also be desirable, but is more questionable.
Shorter Horizons
A factor that could affect the desirability of using a larger RDR is the shorter horizon as the disbursements proceed. Consider again the first case considered with the Pre 2008 Returns, and the portfolio entirely allocated to intermediates. Suppose over the first 18 years that the disbursement has not reached its limit, and that 18 more disbursements remain to made. The value of the portfolio, however, is now down to 58% of its initial value. Year 6 of the remaining 18 years is a third of the way through the remaining disbursement period, just as year 12 was a third of the way through the initial 36 years. With an RDR of .03, simulations show that the chances of reaching the limit at year 6 are about .15, which is the same as the chances were of reaching the limit by year 12 initially with an RDR of .03. The question is whether the shorter horizon has affected the desirability of increasing the RDR?
The shorter horizon makes the chances of declines of any size less likely. Simulations show, however, that it is still desirable to increase the RDR to reduce the chances of reaching the limit in the future just as it was initially. Charts 3.13, 3.14, and 3.15 show the chances of declines at years 6, 12, and 18 when the RDR is equal to .03, .05, and .07. In these charts the chances of declines of any size are less as expected. The desirability of increasing the RDR, however, appears to be about the same as for the longer horizon. The declining horizon as the disbursements proceed has little effect on the choice of 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 here is the largest constant stream of disbursements that could be obtained when the portfolio is invested at a constant return equal to the RDR over the remaining period. The value of the 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 at least two percentage points more than the expected return of the portfolio. That is true irrespective of the assumptions made about the returns, the allocation of the portfolio, or the remaining length of the horizon.
Posted November 2018 Revised July 2022 October 2024