The stock allocation at the beginning of each year is reset to the initial allocation. For a given stock allocation, simulations are used to find the cumulative probability distributions of the disbursements at years 12, 24, and 36. The best stock allocation is found by successively increasing the stock allocation by 10 percentage points until there is doubt that the last allocation has improved the distributions of the disbursements. When there is doubt, the prior stock allocation is the best allocation for the conditions that have been assumed. For reasonably sustainable disbursements, the best stock allocations are moderate being 30% for the Pre 2008 Returns, and 40% for the Lower Returns. For larger disbursements, the best stock allocations increase, but much more for the Lower Returns. When the risk for the disbursements increases it is desirable to accept more risk in funding them, and to reach for higher expected returns. When the expected returns are poor it is desirable to reach even further.
Pre 2008 Returns
Suppose simulations are made using the basic model and assuming initially that as many as 36 annual disbursements may be needed. The investment returns are those for the Pre 2008 Returns. The RDR is set equal to two percentage points above the expected return of the portfolio, as was found to be best at page, RDR. Set the initial disbursement equal to 4.0% of portfolio value, which has reasonably good sustainability. Now successively increase the stock allocation by 10 percentage points each time. Graphs for the cumulative probability distributions of the disbursements at years 12, 24, and 36 for stock allocations of 10% and 20% are shown in Charts 4.1, 4.2, and 4.3. For all years, the increase in the allocation from 10% to 20% provides significant improvement. The curves for 20% are always below or equal to those for 10% indicating that the chances for lower disbursements for 20% are always less than or equal to the chances for 10%. The chances of declines by the respective years are given by the probabilities on the vertical axis for the tops of the curves. Note that an allocation of 20% has reasonably good sustainability as the chances of a decline by year 12 are only .05, and .175 by the end of the period. Note also that the improvement provided by 20% is just about the same in each of the charts indicating that it is sufficient to look just at year 24 to determine whether an increase in the allocation is an improvement.
Suppose next that the stock allocation is increased from 20% to 30%. The curves in this case at year 24 are shown in Chart 4.4. There is an improvement, but not as much as occurs when increasing the allocation from 10% to 20%. The chances of declines by year 24 are reduced by only a small amount as shown by the difference on the vertical axis of the tops of the curves. Now suppose the stock allocation is further increased from 30% to 40%. The results in this case at year 24 are shown in Chart 4.5. There is very little difference, but there is clearly doubt that 40% is an improvement. In fact, for disbursements less than 3.0 on the horizontal axis, the curve for 40% is slightly above that for 30% indicating that 40% has a very slight chance of larger declines than 30%. Thus, there is certainly doubt that 40% is an improvement, and 30% is therefore the best allocation under these conditions. Chart 4.6 shows that a further increase to 50% causes more significant deterioration.
Larger Disbursements
Suppose next that the initial disbursement is increased from 4.0% to 5.0%. Chart 4.7 shows that increasing the stock allocation from 30% to 40% is now an improvement. When the allocation is further increased to 50%, however, there is doubt that 50% is an improvement. There is little difference, but 50% does reduce somewhat the chances of small declines. On the other hand, 50% also slightly increases the chances of large declines. The increase in the chances of large declines raises doubt that the increase to 50% is an improvement. An allocation of 40% is therefore assumed to be the best allocation when the initial disbursement is increased from 4.0% to 5.0%. Even if the best allocation were instead assumed to be 50%, the difference has little effect when at page, BID, different initial disbursements are tested to find the best initial disbursement. In any case, when funding larger disbursements it is desirable to increase the stock allocation, and accept more volatility to get higher expected returns.
Lower Returns
Suppose now instead of assuming the Pre 2008 Returns that the Lower Returns provided at the page, Model, are used. The Lower Returns assume that the expected annual real return on the intermediates each year is zero instead of .03. The expected equity premium for stocks continues to be .04. To keep the disbursements reasonably sustainable for the Lower Returns, the initial disbursement is reduced from 4.0% to 3.0%. The RDR continues to be set two percentage points above the expected return of the portfolio. The results for stock allocations of 20% and 30% are shown in Chart 4.9. There is no doubt that 30% is an improvement over 20%. When 40% is compared to 30%, Chart 4.10 shows that 40% also appears to be an improvement although 40% has a slightly higher risk of large declines. Chart 4.11 shows that a further increase to 50% is not an improvement as the small reduction in smaller declines is outweighed by an increase in larger declines. Thus, for the Lower Returns, the best allocation increases from 30% to 40%. Accepting somewhat more risk becomes desirable to fund the disbursements.
Suppose now that the initial disbursement is increased from 3.0% to 4.0%. Chart 4.12 shows that accepting additional risk has become desirable. A stock allocation of 50% is now without a doubt an improvement over 40%. Moreover, Chart 4.13 shows that the stock allocation should be further increased to 70%. For 70% there is a very slight increase in the risk of large declines, but this deterioration appears to be outweighed by more significant reductions in the chances of smaller declines. Thus, for the Lower Returns when the initial disbursement is increased from 3.0% to 4.0%, the best allocation increases from 40% to 70%. This compares with an increase from 30% to 40% for the Pre 2008 Returns when the initial disbursement is increased from 4.0% to 5.0%. For lower returns, it is desirable to reach further for higher expected returns when the risk of the disbursements increases.
Shorter Horizons
As the disbursements proceed, the number of remaining disbursements to be made is declining. The simulations assume that the allocation remains the same over the disbursement period. But is this a reasonable assumption? In theory, the best stock allocation might be expected to decline as the horizon becomes shorter. The reason is that with a shorter horizon there are fewer drawings to average out the higher volatility for the stock returns so as to reduce risk and provide the benefit of the higher expected return. Perhaps in practice, however, this benefit is sufficiently small that it can be ignored as an approximation. To find out, the simulations can be rerun with a shorter horizon.
Consider again the conditions for Chart 4.4 with the Pre 2008 Returns, and an initial disbursement of 4.0%. Supposer over the first 18 years that the disbursement has not reached its limit and is still equal to 4.0% of initial portfolio value. Now, however, there are 18 instead of 36 disbursements that remain to be made. Also, the value of the portfolio is now equal to 63.5% of its initial value. Suppose that the same simulations are made as originally, but now with a portfolio value of 63.5% instead of 100% of the original initial value, and with 18 instead of 36 disbursements that may need to be made. The disbursement continues to be 4.0% of the initial value of the portfolio. Looking ahead two-thirds of the way through the disbursement period is now year 12 instead of year 24.
The results at year 12 for stock allocations of 10%, 20% and 30% are shown in Chart 4.14. Using a 20% allocation still shows significant improvement over 10%, but 30% now shows significantly less improvement versus 20% than it did in Chart 4.4 with the initial 36 year horizon. As expected, a shorter horizon does make stocks less attractive, and the best allocation does decline as the horizon shortens. Nevertheless, continuing to use the same allocation as the horizon shortens appears to be a tolerable approximation for gauging general effects. In practice, however, reducing the stock allocation when the remaining number of disbursements becomes small appears advisable.
Riskier Future Disbursements
Raising the initial disbursement increases the risk of the disbursements. Earlier results showed that it is possible to reduce this increase in risk by increasing the allocation to stocks. The risk of a given initial disbursement in the future depends on the value of the portfolio at that time. If investment returns in the future turn out to be less than expected the value of the portfolio will also be less than expected, and the value of the disbursement will increase relative to the value of the portfolio. The risk of the disbursements will increase just as the risk of the disbursements increases when the initial disbursement is raised relative to the initial value of the portfolio. When the value of the disbursement increases relative to the value of the portfolio in the future it should therefore be possible to reduce the risk of the disbursements by increasing the allocation to stocks just as occurs initially.
The earlier analysis showed that the best stock allocation for a 4.0% initial disbursement was 30% for the Pre 2008 returns. When the initial disbursement increased from 4.0% to 5.0% the best allocation increased from 30% to 40%. Suppose when starting at 4.0% that investment returns in the future turn out to be sufficiently worse than expected that the disbursement becomes 5.0% or more than the value of the portfolio. It should then be desirable to increase the allocation from 30% to 40 % to reduce risk, just as a 40% allocation was desirable when starting with a 5.0% disbursement. Moreover, the RDR should be increased in accord with the higher allocation. Suppose subsequently investment returns improve and the disbursement falls back below 5.0% of the value of the portfolio. The allocation and the RDR are then reduced back to their earlier values.
The results of using such a variable allocation are compared to a fixed allocation of 30% in Chart 4.15. The modification of the model to provide this variable allocation are discussed in a note. The results using the variable allocation are designated as vlb1, and clearly show significant improvement. Given this improvement, suppose that the allocation is further increased to 50% if the disbursement is greater than or equal to 6.0% of portfolio value. The results in this case are shown in Chart 4.16. Little further improvement is evident, however, for this further increase in the stock allocation when the value of the disbursement further increases relative to the value of the portfolio.
Suppose now that the Lower Returns are being used instead of the Pre 2008 Returns. Earlier results showed that lower returns make higher stock allocations desirable especially when the initial disbursement is above a reasonably sustainable level. For lower returns the results should therefore be more sensitive to increasing the stock allocation in the future when the initial disbursement increases relative to the value of the portfolio. For the Lower Returns, the earlier analysis showed that the initial disbursement had to be reduced from 4.0% to 3.0% to get a reasonably sustainable disbursement, and that the best stock allocation increased from 30% to 40% for a reasonably sustainable disbursement. When the initial disbursement increased from 3.0% to 4.0%, the allocation increased from 40% to 70%. When starting at 3.0% for the Lower Returns, suppose that the allocation is increased from 40% to 70% if the disbursement becomes equal to 4.0% or more of the value of the portfolio, and is reduced back to 40% if the disbursement falls back below 4.0% of portfolio value. The RDR is also adjusted in line with the change in the stock allocation.
The results for this case are shown in Chart 4.17. As expected, for the lower returns, the results are more sensitive to increasing the stock allocation when the disbursement rises relative to portfolio value. In this case, however, as well as significant reductions in the chances of smaller declines, there are also appreciable increases in the chances of larger declines. Nevertheless, there does appear to be a net benefit to increasing the stock allocation when the disbursement rises relative to portfolio value. The net benefit, however, might be somewhat less than the benefit of increasing the stock allocation for the higher returns.
Correlation
For single period portfolio analysis, the major focus is on the correlation of the returns of the investments in the portfolio. These correlations determine how the portfolio should be allocated to achieve a desired balance between the expected value of the return on the portfolio and its volatility. For multiperiod analysis, the correlation between the returns is an important factor determining risk, but only one of many factors of interest in determining the results. A significant reduction in the correlation between the returns on stocks and intermediates reduces risk, as expected. But does a significant reduction in correlation affect the best allocation? A reduction in the correlation of their returns makes either the stocks or intermediates more valuable. But does one become more valuable relative to the other changing the best allocation? Simulations show that changes in the correlation of the returns on the stocks and intermediates have little effect on the best allocation.
Generally, the model assumes that the return on stocks each year is equal to a coefficient of .4 times the return on intermediates plus an independently and normally distributed random variable. The mean and standard deviation of the independent random variable are set so as to make the expected value and standard deviation of the returns on the stocks equal to specified values. The coefficient of .4 for the return on the intermediates is close to the coefficient obtained when the annual stock returns are regressed on the annual returns of the intermediates for the period from 1960 to 2020. For the period from 1960 to 1990, however, the major factor affecting investment returns was inflation. Increases and decreases in inflation have the same effect on the real returns of both stocks and intermediates. Thus, when inflation was the dominant influence for investment returns in the U. S. financial markets from 1960 to 1990 there was a stronger positive correlation between the real returns on stocks and intermediates than the .4 coefficient generally assumed in the model. In contrast, from 1990 up until the start of the pandemic in 2020 the dominant influences on U. S. financial markets were periods of speculative excess. First, there was the Speculative Bubble of the late 1990s, and then the Credit Crunch of 2008. For periods of speculative excess fixed incomes of good quality provided a good hedge for stocks, and the correlation between the real returns on stocks and intermediates of good quality was strongly negative.
Suppose instead of .4 as generally assumed in the model that the correlation coefficient between the real returns on stocks and intermediates is either .8 or -.5. In these cases, the relation for simulating stock returns must be changed to keep the expected value and standard deviation of the annual stock returns equal to the same specified values. The changes in these relations that do so are discussed in a note. As considered earlier, suppose that the disbursements are reasonably sustainable and start at 4.0%. Consider cases in which the correlation coefficient of stock returns on the intermediates is either .8 or -.5 instead of .4 as generally assumed in the model. The results at year 24 for stock allocations of 20%, 30%, and 40% are shown in Chart 4.18. As expected, there is much less risk when the correlation coefficient is -.5. The change in the correlation coefficient, however, does not affect the best stock allocation. For either of the coefficients the best stock allocation is 30% as found earlier for a coefficient of .4 in Charts 4.4 and 4.5. Looking to the future, disbursements will tend to be riskier if inflation is the major problem affecting financial markets. The best allocation, however, is little affected by whether the major problem is inflation or periods of speculative excess.
Conclusions
Best stock allocations are found by successively increasing the stock allocation by 10 percentage points until there is doubt that the latest increase is an improvement, in which case the prior allocation is the best for the conditions that have been assumed. In general, the best stock allocation increases when disbursements become riskier, and for poorer investment returns. For disbursements that are reasonably sustainable initially the best stock allocations are moderate being 30% for the Pre 2008 Returns and 40% for the Lower Returns. When the initial disbursement is increased the best stock allocation increases, but much more for the Lower Returns. The best stock allocation also increases in the future if investment returns are less than expected causing the disbursement to increase relative to the value of the portfolio. As there are fewer drawings to average out risk, the best stock allocation declines as the horizon shortens. This reduction has been ignored as an approximation for gauging general effects, and the portfolio has been rebalanced each year in the model. In practice, however, stock allocations should be reduced when few disbursements remain to be made. Correlation between the returns on stocks and intermediates appears to have little effect on the best allocation. If inflation is expected to be a major influence on financial markets, correlation will be more positive than assumed in the simulations, and disbursements will have more risk.
Notes
To include vlb1 in the model, two columns are added to the spreadsheet. The rows of the first column give the allocation each year, and the rows of the second column give the RDR. In the relation for R(t) in the model a is replaced by a(t), which is the allocation for each year given by the rows of the first column. The values of a(t) are given by:
a(t)=If(C(t-1)/V(t)>=.05,.4,.3) t=2,3,…T a(1)=.3
In the relation for C*(t) in the model RDR is replaced by RDR(t), which is the RDR for each year given by the rows of the second column. The values of RDR(t) are given by:
The illustration based on the Pre 2008 Returns assumes that the annual real return on intermediates is normally distributed with an expected return of .03 and a standard deviation of .07. The annual return on stocks is assumed to be normally distributed with an expected return of .07 and a standard deviation of .18. The annual return on stocks is assumed to be .4 of the annual return on intermediates plus an independently distributed normal variable. This independent variable must have an expected return of .058, and a standard deviation of .1778, to give the specified annual return on stocks. Thus, as given in Model:
r(t)=RiskNormal(.058+.4*i(t),.1778)
When the annual real return on stocks is .8 of the annual real return on intermediates the expected return of the independent variable becomes .046 and the standard deviation equal to .17107. Thus, in this case r(t) is given by:
r(t)=RiskNormal(.046+.8*i(t),.17107)
When the annual real return on stocks is -.5 of the annual real return on intermediates the expected return of the independent variable becomes .085 and the standard deviation is equal to .17656. In this case, r(t) is given by:
r(t)=RiskNormal(.085-.5i(t),.17656)
Posted November 2018 Revised July 2022 October 2024
Allocation
Best Allocations
The stock allocation at the beginning of each year is reset to the initial allocation. For a given stock allocation, simulations are used to find the cumulative probability distributions of the disbursements at years 12, 24, and 36. The best stock allocation is found by successively increasing the stock allocation by 10 percentage points until there is doubt that the last allocation has improved the distributions of the disbursements. When there is doubt, the prior stock allocation is the best allocation for the conditions that have been assumed. For reasonably sustainable disbursements, the best stock allocations are moderate being 30% for the Pre 2008 Returns, and 40% for the Lower Returns. For larger disbursements, the best stock allocations increase, but much more for the Lower Returns. When the risk for the disbursements increases it is desirable to accept more risk in funding them, and to reach for higher expected returns. When the expected returns are poor it is desirable to reach even further.
Pre 2008 Returns
Suppose simulations are made using the basic model and assuming initially that as many as 36 annual disbursements may be needed. The investment returns are those for the Pre 2008 Returns. The RDR is set equal to two percentage points above the expected return of the portfolio, as was found to be best at page, RDR. Set the initial disbursement equal to 4.0% of portfolio value, which has reasonably good sustainability. Now successively increase the stock allocation by 10 percentage points each time. Graphs for the cumulative probability distributions of the disbursements at years 12, 24, and 36 for stock allocations of 10% and 20% are shown in Charts 4.1, 4.2, and 4.3. For all years, the increase in the allocation from 10% to 20% provides significant improvement. The curves for 20% are always below or equal to those for 10% indicating that the chances for lower disbursements for 20% are always less than or equal to the chances for 10%. The chances of declines by the respective years are given by the probabilities on the vertical axis for the tops of the curves. Note that an allocation of 20% has reasonably good sustainability as the chances of a decline by year 12 are only .05, and .175 by the end of the period. Note also that the improvement provided by 20% is just about the same in each of the charts indicating that it is sufficient to look just at year 24 to determine whether an increase in the allocation is an improvement.
Suppose next that the stock allocation is increased from 20% to 30%. The curves in this case at year 24 are shown in Chart 4.4. There is an improvement, but not as much as occurs when increasing the allocation from 10% to 20%. The chances of declines by year 24 are reduced by only a small amount as shown by the difference on the vertical axis of the tops of the curves. Now suppose the stock allocation is further increased from 30% to 40%. The results in this case at year 24 are shown in Chart 4.5. There is very little difference, but there is clearly doubt that 40% is an improvement. In fact, for disbursements less than 3.0 on the horizontal axis, the curve for 40% is slightly above that for 30% indicating that 40% has a very slight chance of larger declines than 30%. Thus, there is certainly doubt that 40% is an improvement, and 30% is therefore the best allocation under these conditions. Chart 4.6 shows that a further increase to 50% causes more significant deterioration.
Larger Disbursements
Suppose next that the initial disbursement is increased from 4.0% to 5.0%. Chart 4.7 shows that increasing the stock allocation from 30% to 40% is now an improvement. When the allocation is further increased to 50%, however, there is doubt that 50% is an improvement. There is little difference, but 50% does reduce somewhat the chances of small declines. On the other hand, 50% also slightly increases the chances of large declines. The increase in the chances of large declines raises doubt that the increase to 50% is an improvement. An allocation of 40% is therefore assumed to be the best allocation when the initial disbursement is increased from 4.0% to 5.0%. Even if the best allocation were instead assumed to be 50%, the difference has little effect when at page, BID, different initial disbursements are tested to find the best initial disbursement. In any case, when funding larger disbursements it is desirable to increase the stock allocation, and accept more volatility to get higher expected returns.
Lower Returns
Suppose now instead of assuming the Pre 2008 Returns that the Lower Returns provided at the page, Model, are used. The Lower Returns assume that the expected annual real return on the intermediates each year is zero instead of .03. The expected equity premium for stocks continues to be .04. To keep the disbursements reasonably sustainable for the Lower Returns, the initial disbursement is reduced from 4.0% to 3.0%. The RDR continues to be set two percentage points above the expected return of the portfolio. The results for stock allocations of 20% and 30% are shown in Chart 4.9. There is no doubt that 30% is an improvement over 20%. When 40% is compared to 30%, Chart 4.10 shows that 40% also appears to be an improvement although 40% has a slightly higher risk of large declines. Chart 4.11 shows that a further increase to 50% is not an improvement as the small reduction in smaller declines is outweighed by an increase in larger declines. Thus, for the Lower Returns, the best allocation increases from 30% to 40%. Accepting somewhat more risk becomes desirable to fund the disbursements.
Suppose now that the initial disbursement is increased from 3.0% to 4.0%. Chart 4.12 shows that accepting additional risk has become desirable. A stock allocation of 50% is now without a doubt an improvement over 40%. Moreover, Chart 4.13 shows that the stock allocation should be further increased to 70%. For 70% there is a very slight increase in the risk of large declines, but this deterioration appears to be outweighed by more significant reductions in the chances of smaller declines. Thus, for the Lower Returns when the initial disbursement is increased from 3.0% to 4.0%, the best allocation increases from 40% to 70%. This compares with an increase from 30% to 40% for the Pre 2008 Returns when the initial disbursement is increased from 4.0% to 5.0%. For lower returns, it is desirable to reach further for higher expected returns when the risk of the disbursements increases.
Shorter Horizons
As the disbursements proceed, the number of remaining disbursements to be made is declining. The simulations assume that the allocation remains the same over the disbursement period. But is this a reasonable assumption? In theory, the best stock allocation might be expected to decline as the horizon becomes shorter. The reason is that with a shorter horizon there are fewer drawings to average out the higher volatility for the stock returns so as to reduce risk and provide the benefit of the higher expected return. Perhaps in practice, however, this benefit is sufficiently small that it can be ignored as an approximation. To find out, the simulations can be rerun with a shorter horizon.
Consider again the conditions for Chart 4.4 with the Pre 2008 Returns, and an initial disbursement of 4.0%. Supposer over the first 18 years that the disbursement has not reached its limit and is still equal to 4.0% of initial portfolio value. Now, however, there are 18 instead of 36 disbursements that remain to be made. Also, the value of the portfolio is now equal to 63.5% of its initial value. Suppose that the same simulations are made as originally, but now with a portfolio value of 63.5% instead of 100% of the original initial value, and with 18 instead of 36 disbursements that may need to be made. The disbursement continues to be 4.0% of the initial value of the portfolio. Looking ahead two-thirds of the way through the disbursement period is now year 12 instead of year 24.
The results at year 12 for stock allocations of 10%, 20% and 30% are shown in Chart 4.14. Using a 20% allocation still shows significant improvement over 10%, but 30% now shows significantly less improvement versus 20% than it did in Chart 4.4 with the initial 36 year horizon. As expected, a shorter horizon does make stocks less attractive, and the best allocation does decline as the horizon shortens. Nevertheless, continuing to use the same allocation as the horizon shortens appears to be a tolerable approximation for gauging general effects. In practice, however, reducing the stock allocation when the remaining number of disbursements becomes small appears advisable.
Riskier Future Disbursements
Raising the initial disbursement increases the risk of the disbursements. Earlier results showed that it is possible to reduce this increase in risk by increasing the allocation to stocks. The risk of a given initial disbursement in the future depends on the value of the portfolio at that time. If investment returns in the future turn out to be less than expected the value of the portfolio will also be less than expected, and the value of the disbursement will increase relative to the value of the portfolio. The risk of the disbursements will increase just as the risk of the disbursements increases when the initial disbursement is raised relative to the initial value of the portfolio. When the value of the disbursement increases relative to the value of the portfolio in the future it should therefore be possible to reduce the risk of the disbursements by increasing the allocation to stocks just as occurs initially.
The earlier analysis showed that the best stock allocation for a 4.0% initial disbursement was 30% for the Pre 2008 returns. When the initial disbursement increased from 4.0% to 5.0% the best allocation increased from 30% to 40%. Suppose when starting at 4.0% that investment returns in the future turn out to be sufficiently worse than expected that the disbursement becomes 5.0% or more than the value of the portfolio. It should then be desirable to increase the allocation from 30% to 40 % to reduce risk, just as a 40% allocation was desirable when starting with a 5.0% disbursement. Moreover, the RDR should be increased in accord with the higher allocation. Suppose subsequently investment returns improve and the disbursement falls back below 5.0% of the value of the portfolio. The allocation and the RDR are then reduced back to their earlier values.
The results of using such a variable allocation are compared to a fixed allocation of 30% in Chart 4.15. The modification of the model to provide this variable allocation are discussed in a note. The results using the variable allocation are designated as vlb1, and clearly show significant improvement. Given this improvement, suppose that the allocation is further increased to 50% if the disbursement is greater than or equal to 6.0% of portfolio value. The results in this case are shown in Chart 4.16. Little further improvement is evident, however, for this further increase in the stock allocation when the value of the disbursement further increases relative to the value of the portfolio.
Suppose now that the Lower Returns are being used instead of the Pre 2008 Returns. Earlier results showed that lower returns make higher stock allocations desirable especially when the initial disbursement is above a reasonably sustainable level. For lower returns the results should therefore be more sensitive to increasing the stock allocation in the future when the initial disbursement increases relative to the value of the portfolio. For the Lower Returns, the earlier analysis showed that the initial disbursement had to be reduced from 4.0% to 3.0% to get a reasonably sustainable disbursement, and that the best stock allocation increased from 30% to 40% for a reasonably sustainable disbursement. When the initial disbursement increased from 3.0% to 4.0%, the allocation increased from 40% to 70%. When starting at 3.0% for the Lower Returns, suppose that the allocation is increased from 40% to 70% if the disbursement becomes equal to 4.0% or more of the value of the portfolio, and is reduced back to 40% if the disbursement falls back below 4.0% of portfolio value. The RDR is also adjusted in line with the change in the stock allocation.
The results for this case are shown in Chart 4.17. As expected, for the lower returns, the results are more sensitive to increasing the stock allocation when the disbursement rises relative to portfolio value. In this case, however, as well as significant reductions in the chances of smaller declines, there are also appreciable increases in the chances of larger declines. Nevertheless, there does appear to be a net benefit to increasing the stock allocation when the disbursement rises relative to portfolio value. The net benefit, however, might be somewhat less than the benefit of increasing the stock allocation for the higher returns.
Correlation
For single period portfolio analysis, the major focus is on the correlation of the returns of the investments in the portfolio. These correlations determine how the portfolio should be allocated to achieve a desired balance between the expected value of the return on the portfolio and its volatility. For multiperiod analysis, the correlation between the returns is an important factor determining risk, but only one of many factors of interest in determining the results. A significant reduction in the correlation between the returns on stocks and intermediates reduces risk, as expected. But does a significant reduction in correlation affect the best allocation? A reduction in the correlation of their returns makes either the stocks or intermediates more valuable. But does one become more valuable relative to the other changing the best allocation? Simulations show that changes in the correlation of the returns on the stocks and intermediates have little effect on the best allocation.
Generally, the model assumes that the return on stocks each year is equal to a coefficient of .4 times the return on intermediates plus an independently and normally distributed random variable. The mean and standard deviation of the independent random variable are set so as to make the expected value and standard deviation of the returns on the stocks equal to specified values. The coefficient of .4 for the return on the intermediates is close to the coefficient obtained when the annual stock returns are regressed on the annual returns of the intermediates for the period from 1960 to 2020. For the period from 1960 to 1990, however, the major factor affecting investment returns was inflation. Increases and decreases in inflation have the same effect on the real returns of both stocks and intermediates. Thus, when inflation was the dominant influence for investment returns in the U. S. financial markets from 1960 to 1990 there was a stronger positive correlation between the real returns on stocks and intermediates than the .4 coefficient generally assumed in the model. In contrast, from 1990 up until the start of the pandemic in 2020 the dominant influences on U. S. financial markets were periods of speculative excess. First, there was the Speculative Bubble of the late 1990s, and then the Credit Crunch of 2008. For periods of speculative excess fixed incomes of good quality provided a good hedge for stocks, and the correlation between the real returns on stocks and intermediates of good quality was strongly negative.
Suppose instead of .4 as generally assumed in the model that the correlation coefficient between the real returns on stocks and intermediates is either .8 or -.5. In these cases, the relation for simulating stock returns must be changed to keep the expected value and standard deviation of the annual stock returns equal to the same specified values. The changes in these relations that do so are discussed in a note. As considered earlier, suppose that the disbursements are reasonably sustainable and start at 4.0%. Consider cases in which the correlation coefficient of stock returns on the intermediates is either .8 or -.5 instead of .4 as generally assumed in the model. The results at year 24 for stock allocations of 20%, 30%, and 40% are shown in Chart 4.18. As expected, there is much less risk when the correlation coefficient is -.5. The change in the correlation coefficient, however, does not affect the best stock allocation. For either of the coefficients the best stock allocation is 30% as found earlier for a coefficient of .4 in Charts 4.4 and 4.5. Looking to the future, disbursements will tend to be riskier if inflation is the major problem affecting financial markets. The best allocation, however, is little affected by whether the major problem is inflation or periods of speculative excess.
Conclusions
Best stock allocations are found by successively increasing the stock allocation by 10 percentage points until there is doubt that the latest increase is an improvement, in which case the prior allocation is the best for the conditions that have been assumed. In general, the best stock allocation increases when disbursements become riskier, and for poorer investment returns. For disbursements that are reasonably sustainable initially the best stock allocations are moderate being 30% for the Pre 2008 Returns and 40% for the Lower Returns. When the initial disbursement is increased the best stock allocation increases, but much more for the Lower Returns. The best stock allocation also increases in the future if investment returns are less than expected causing the disbursement to increase relative to the value of the portfolio. As there are fewer drawings to average out risk, the best stock allocation declines as the horizon shortens. This reduction has been ignored as an approximation for gauging general effects, and the portfolio has been rebalanced each year in the model. In practice, however, stock allocations should be reduced when few disbursements remain to be made. Correlation between the returns on stocks and intermediates appears to have little effect on the best allocation. If inflation is expected to be a major influence on financial markets, correlation will be more positive than assumed in the simulations, and disbursements will have more risk.
Notes
To include vlb1 in the model, two columns are added to the spreadsheet. The rows of the first column give the allocation each year, and the rows of the second column give the RDR. In the relation for R(t) in the model a is replaced by a(t), which is the allocation for each year given by the rows of the first column. The values of a(t) are given by:
a(t)=If(C(t-1)/V(t)>=.05,.4,.3) t=2,3,…T a(1)=.3
In the relation for C*(t) in the model RDR is replaced by RDR(t), which is the RDR for each year given by the rows of the second column. The values of RDR(t) are given by:
RDR(t)=If(C(t-1)/V(t)>=.05,.066,.062) t=2,3…T RDR(1)=.062
To include vlb2, the relation for a(t) is replaced by:
a(t)=If(C(t-1)/V(t)>=.05,If(C(t-1)/V(t)>=.06,.5,.4),.3) t=2,3,…T a(1)=.3
and the relation for RDR(t) is replaced by:
RDR=If(C(t-1)/V(t)>=.05,If(C(t-1)/V(t)>=.06,.070,.066),.062) t=2,3,…T RDR(1)=.062
The relations for vlb3 are similar to vlb1.
The illustration based on the Pre 2008 Returns assumes that the annual real return on intermediates is normally distributed with an expected return of .03 and a standard deviation of .07. The annual return on stocks is assumed to be normally distributed with an expected return of .07 and a standard deviation of .18. The annual return on stocks is assumed to be .4 of the annual return on intermediates plus an independently distributed normal variable. This independent variable must have an expected return of .058, and a standard deviation of .1778, to give the specified annual return on stocks. Thus, as given in Model:
r(t)=RiskNormal(.058+.4*i(t),.1778)
When the annual real return on stocks is .8 of the annual real return on intermediates the expected return of the independent variable becomes .046 and the standard deviation equal to .17107. Thus, in this case r(t) is given by:
r(t)=RiskNormal(.046+.8*i(t),.17107)
When the annual real return on stocks is -.5 of the annual real return on intermediates the expected return of the independent variable becomes .085 and the standard deviation is equal to .17656. In this case, r(t) is given by:
r(t)=RiskNormal(.085-.5i(t),.17656)
Posted November 2018 Revised July 2022 October 2024