A single payment immediate lifetime annuity provides the beneficiary with a lifetime stream of benefits that generally is fixed initially in nominal terms. The payout rates available on the market for such annuities depend on the life expectancy of the beneficiary based on age and sex, and the expected nominal return on long term investments of good quality at the time. For beneficiaries in good health, the benefits from such an annuity are generally larger than the disbursements that can be obtained by investing the same amount because those who live longer than expected benefit from the annuities purchased by those who do not. Model can be modified to show how much the disbursements can be increased if some of the funds that would otherwise be in the portfolio are used to purchase a lifetime annuity. There are costs, however, to this improvement. The funds available in the portfolio to deal with possible emergencies are reduced. And if not needed for emergencies these funds will be available for making bequests. Also, being fixed in nominal terms, the benefits of the annuity are vulnerable to an increase in inflation. An increase in inflation may also reduce the real returns on portfolio investments, but these reductions are likely to be temporary as future nominal returns will likely increase to reflect the increase in inflation.
Ameriks, Veres, and Warshawasky (2001) have shown the benefits of annuitizing in a model, but one that differs significantly from the one developed at this site. Davidoff, Brown, and Diamond (2005) have provided very strong theoretical evidence as to why retirees in good health should annuitize a significant portion of their investable funds. They contrast this evidence with the very small portion of retirees who actually do voluntarily annuitize. Perhaps, however, they underestimate the reluctance of retirees to part with funds that they intuitively feel may be needed to deal with future problems whose size and nature are as yet unknown, and if not needed for that purpose will add to their bequests.
Whether to annuitize depends on whether doing so increases the Best Initial Disbursement (BID) by more than enough to compensate for the reduction in portfolio value. Two cases are considered here in which half of the initial investible funds are annuitized. In one, the returns on the portfolio are the assumptions given in Model called the Pre 2008 Returns, and the payout for the annuity is one that could have been purchased in the first half of 2007. In the other, the returns on the portfolio are the assumptions given in Model called the Lower Returns, and the payout for the annuity is one that could have been purchased in the fall of 2022. In either case, inflation is simulated by a simple recursive random process in which inflation in the long run is expected to be 3.0%. Prior inflation for the annuity purchased in the fall of 2022, however, is assumed to be higher than in 2007. In either case, it turns out that annuitizing half of the funds increases the Best Initial Disbursement (BID) by about half a percentage point of initial portfolio value. A decision must be made as to whether that increase is worth the reduction in portfolio value due to the annuitizing shown by the simulations.
Modifications
To include purchase of a lifetime annuity in Model a relation must be added that simulates the inflation that occurs each year over the disbursement period so that the nominal benefit that is received each year can be put on a real basis. Using the simulated inflation obtained from this relation each year in the past in an iteration of a simulation, an annual nominal annuity benefit when received is deflated and added to the portfolio. To determine the limit on the disbursement each year an amount must also be added to the value of the portfolio at that time to reflect the future stream of annuity benefits that is expected to be received. The future nominal benefits are discounted for the annual inflation that is expected before they are received as well as for the annual inflation that has already occurred in the past for the current iteration of the simulation. The future annuity benefits are also discounted for each year in the future before they will be received by the RDR because the RDR is the assumed rate of return at which funds are being invested. The specific relations that are added to the model to make these modifications are discussed in a note.
Annual inflation in the U. S. has exhibited strong serial correlation as there have been long periods over which annual inflation has been either persistently higher or lower than average. For instance, annual inflation was persistently high in the 1970s, and persistently low after 2007 up through 2020. Suppose annual inflation is regressed on its prior value over the period from 1960 through 2020. The coefficient for prior year inflation is about .75, which shows strong positive serial correlation as expected. In the model inflation is simulated by a simple recursive random process using .75 as the coefficient for inflation of the prior year. The constant term for this recursive relation is .25 of the annual inflation expected in the long run as this is the limit obtained when the expected inflation each year for this recursive relation is successively iterated into the future. Average annual inflation from 1960 through 2020 was 3.7%. In view of the Federal Reserve’s target for inflation of 2.0%, however, inflation in the long run is here assumed to be equal to 3.0%. The prior inflation for the recursive relation is assumed to be 3.0% for the annuity purchased in the first half of 2007, and 5.0% for the annuity purchased in the fall of 2022. The difference between simulated inflation each year and the expected inflation given by the recursive relation is assumed to be normally distributed with a zero mean and a standard deviation of 1.5%. The cumulative probability distributions of the inflation discounts obtained at years 12, 24, and 36 when starting with 3.0% are shown in Chart 12.1. The difference at year 12 when prior inflation is 5.0% instead of 3.0% is shown in Chart 12.2.
Pre 2008 Returns
Assume the Pre 2008 investment returns and the payout rate for a lifetime annuity of 7.50% that might have been purchased in the first half of 2007 by a 65 year old woman. To be consistent with the model the benefits for this annuity are assumed to be received at the beginning of each year including the initial year. As annuity benefits are typically paid monthly an approximation for a payout at the beginning of a year for a 65 year old woman was obtained by a quote for monthly benefits beginning in six months for a woman who had just reached 64. The payout of 7.50% was obtained for a much earlier study done back in 2007 when the yield on a 10 year Treasury was about .0475. Suppose that half of the initial portfolio is used to purchase this annuity, and that inflation for the year prior to the initial year is assumed to be 3.0%. To assess how much this annuitizing increases the Best Initial Disbursement (BID), the first steps are to see whether it affects the RDR or the stock allocation that should be used.
After the annuitizing, similar tests to those in RDR show that setting the RDR equal to .02 more than the expected return of the portfolio continues to give the best results. After the annuitizing an initial disbursement with the high sustainability of 3.5% without the annuitizing increases by half a percentage point to 4.0%. Tests show that the best stock allocation for this highly sustainable disbursement continues to be 30%. Moreover, for either initial disbursement with a 30% stock allocation and an RDR set at .02 above the expected return of the portfolio, there is only a very small chance of a decline by year 24. When the initial disbursement is increased above this highly sustainable level, however, the best stock allocation increases more after the annuitizing, but the difference is small. Without the annuitizing Allocation shows that the best stock allocation increases by 10 percentage points when the initial disbursement is increased by one percentage point above the sustainable level of 3.5%. After the annuitizing Chart 12.3 shows that the best stock allocation increases by 20 percentage points when the initial disbursement is increased by one percentage point above the highly sustainable level of 4.0%. That result will be used to determine the stock allocation after the annuitizing depending on how much the initial disbursement is increased.
Suppose after the annuitizing that disbursements starting at 4.0% and 4.5% are simulated. The cumulative probability distributions of these disbursements at years 12 and 24 are shown in Charts 12.4 and 12.5. These charts look virtually the same as the charts for starting at 3.5% and 4.0% without the annuitizing, except that the plots are shifted to the right by half a percentage point. Thus, the annuitizing has increased the disbursements with the same sustainability risk by about half a percentqge point of initial portfolio value. The cumulative probability distributions of disbursements starting at 3.5% and 4.0% at years 12 and 24 without the annuitizing are shown in Charts 5.1 and 5.2 at page BID.
The cost of the increase from 4.0% to 4.5% obtained with the annuitizing is the reduction in the value of the portfolio available to cover emergencies, or if not needed for that purpose to increase bequests. The initial reduction in portfolio value is, of course, the half of the portfolio that has been annuitized. The reduction in portfolio value at years 12, 24, and 36 can be obtained as outputs from the simulations. These comparisons are shown in Charts 12.6, 12.7, and 12.8. Initially, the portfolio values are two vertical lines at 50% and 100% of initial asset value. Over time the values become much more diverse and somewhat closer together. Based on averages, the portfolio is about 40% less at year 12 after the annuitizing, 30% less at year 24, and 20% less at year 36. Annuitizing is desirable if the half percentage point increase in the BID is deemed to more than compensate for the reductions in portfolio value.
Lower Returns
Now instead of assuming that the investment returns for the portfolio are those for the Pre 2008 Returns in Model assume the Lower Returns given in Model where the expected returns of both the intermediates and stocks are reduced by three percentage points, and there is also some reduction in volatility. The annuity payout for the lower returns is assumed to be 6.80%. This payout is based on a quote obtained in the fall of 2022 when the yield on a 10 year Treasury was about .0375, and the age and sex of the beneficiary were the same as for the 7.50% payout used for the Pre 2008 returns. It continues to be assumed that expected inflation in the long run is 3.0%, but prior inflation for this annuity is assumed to be 5.0% instead of 3.0%, given the higher inflation in 2022. Without the annuitizing, but with the lower returns, the simulations in Allocation and BID found that a reduction in the initial disbursement of one percentage point from 3.5% to 2.5% was highly sustainable, and that the best stock allocation continued to be 30%. Now suppose with the lower returns that half of the investible funds are annuitized with the 6.80% payout annuity. Using the inflation assumptions for this annuity, the initial disbursement that is highly sustainable increases by half a percentage point from 2.5% to 3.0%, and the best stock allocation continues to be 30%.
To find the BID after the annuitizing, the initial disbursement must be increased above the highly sustainable level of 3.0%, and the best allocations for these increases are required. In Allocation the Lower Returns cause stronger increases in the best stock allocation when the initial disbursement is increased above a sustainable level, and the same happens after the annuitizing. Chart 12.3 shows that the annuitizing increases the stock allocation from 30% to 50% when the initial disbursement is increased one percentage point above a highly sustainable level for the Pre 2008 Returns. Chart 12.9 shows that the best stock allocation increases from 30% to 70% when the initial disbursement is increased one percentage point above a highly sustainable level for the Lower Returns. The stock allocations for the Lower Returns will be based on this result when the initial disbursement is increased above a highly sustainable level depending on the size of the increase.
Suppose after the annuitizing that disbursements starting at 3.0% and 3.5% are simulated. The cumulative probaability distributions of these disbursments at years 12 and 24 are shown in Charts 12.10 and 12.11. These charts look virtually the same as the charts for starting at 2.5% and 3.0% without the annuitizing except that the plots are shifted to the right by half a percentage point. Thus, the annuitizing has increased the disbursements with the same sustainability risk by about half a percentage point. The cumulative probability distributions of the disbursements starting at 2.5% and 3.0% without the annuitizing are shown in Charts 5.4 and 5.5 at page BID. The cost of the half percentage point improvement from the annuitizing for the Lower Returns is a similar reduction in portfolio values as for the Pre 2008 Returns.
Conclusions
Annuitizing a portion of the initial portfolio offers the possibility of increasing the Best Initial Disbursement (BID) at the cost of reducing the value of the portfolio available to cover emergencies, or if not needed for that purpose to make bequests. Annuitizing is desirable if the increase in the BID is judged to be sufficient to more than compensate for the reduction in portfolio value. Cases have been considered in which the returns on the portfolio are either those assumed in Model for the Pre 2008 Returns or the Lower Returns. The annuity payout rate assumed for the Pre 2008 Returns is one that might have been purchased in the first half of 2007, whereas for the Lower Returns it is one that might have been purchased in the fall of 2022. As the payouts are in nominal terms, future benefits must be deflated for inflation to incorporate them in the model. To do so, annual inflation is assumed to follow a recursive random process with expected inflation in the long run equal to 3.0%. Prior inflation for the Pre 2008 Returns is assumed to be 3.0%, and for the Lower Returns to be 5.0%. In either case, when half of the initial portfolio is annuitized, the BID increases by about a half percentage point of initial portfolio value. The initial portfolio reduction is, of course, 50%. On average, however, simulations show that it is about 40% after 12 years, 30% after 24 years, and 20% at the end of the period. The annuitizing is desirable if the half percentage point increase in the BID is more than enough to compensate for these lower portfolio values.
Note
To incorporate annuity benefits into the model relations in four new columns are added to the spreadsheet, and the real value of the annuity benefit is added to the value of the portfolio at the beginning of each year. Also, the real present value of the future stream of annuity benefits is added to the value of the portfolio at any time to determine the limit on the disbursement. One of the new columns has the relations that determine the value of inflation each year. Inflation in year t is denoted by j(t) and given by:
j(t)=h*j(t-1)+(1-h)*j^+k(t) t = 1, 2, …, T-1
where j^ is the expected value of inflation in the long run, and j(0) is the assumed inflation for the year prior to the start of a simulation. For the reported simulations the value of j^ is assumed to be .03, and j(0) is .03 for the annuity payout from 2007, and .05 for the payout from 2022. The value of h is .75. The term, k(t), is the random component of inflation in year t, which is determined in another new column and given by:
k(t)=RiskNormal(0,s) t = 1, 2, …, T-1
where for the reported simulations the standard deviation, s, is assumed to be .015. The cumulated discount rate for inflation at the beginning of t is denoted by J(t) and given in another new column by:
J(t+1)=J(t)/(1+j(t)) t = 1, 2, …, T-1
where J(1)=1. The fourth and last new column gives the present value denoted by A(t) of the stream of annuity benefits to be received as of the beginning of t discounted in future years for expected inflation and the RDR:
A(t)=PV((1+ j^)*(1+m)–1,T–t+1,-g,0,1) t = 1, 2, …, T
where g is the annual annuity benefit in nominal terms at the beginning of each year, including the first year. Suppose the initial value of the assets before the purchase of the annuity is set equal to 100. The value of V(1) is then the percentage of initial asset value that is not annuitized, and g is the annual annuity payout as a percentage of initial asset value. The value of the disbursement, C(t), and portfolio value, V(t), are then expressed as percentages of initial asset value. PV() is the Excel function that calculates the present value of the stream of g received at the beginning of each of T-t+1 years, including the first year, discounted at an annual rate of (1+j^)*(1+m)-1 for each year in the future to account for expected inflation and the RDR. To add the annuity benefit to the portfolio each year the relation for V(t+1) is rewritten to be:
V(t+1)=(V(t)+g*J(t)–C(t))*(1+R(t)) t = 1, 2, …, T.
To add the values of the stream of annuity benefits to the value of the portfolio when determining the limit on the disbursement at t, the relation for C*(t) is rewritten to be:
C*(t)=PMT(m,T-t+1,-V(t)-J(t)*A(t),0,1) t = 1, 2, …, T.
References
Ameriks, John, Robert Veres, and Mark J. Warshawsky. 2001. “Making Retirement Income Last a Lifetime.” Journal of Financial Planning 14, 12 (December):62-76.
Davidoff, Thomas, Jeffery R. Brown, and Peter A. Diamond. 2005. “Annuities and Individual Welfare. “ American Economic Review 95, (December):1573-1590.
Annuitize
Lifetime Annuities
A single payment immediate lifetime annuity provides the beneficiary with a lifetime stream of benefits that generally is fixed initially in nominal terms. The payout rates available on the market for such annuities depend on the life expectancy of the beneficiary based on age and sex, and the expected nominal return on long term investments of good quality at the time. For beneficiaries in good health, the benefits from such an annuity are generally larger than the disbursements that can be obtained by investing the same amount because those who live longer than expected benefit from the annuities purchased by those who do not. Model can be modified to show how much the disbursements can be increased if some of the funds that would otherwise be in the portfolio are used to purchase a lifetime annuity. There are costs, however, to this improvement. The funds available in the portfolio to deal with possible emergencies are reduced. And if not needed for emergencies these funds will be available for making bequests. Also, being fixed in nominal terms, the benefits of the annuity are vulnerable to an increase in inflation. An increase in inflation may also reduce the real returns on portfolio investments, but these reductions are likely to be temporary as future nominal returns will likely increase to reflect the increase in inflation.
Ameriks, Veres, and Warshawasky (2001) have shown the benefits of annuitizing in a model, but one that differs significantly from the one developed at this site. Davidoff, Brown, and Diamond (2005) have provided very strong theoretical evidence as to why retirees in good health should annuitize a significant portion of their investable funds. They contrast this evidence with the very small portion of retirees who actually do voluntarily annuitize. Perhaps, however, they underestimate the reluctance of retirees to part with funds that they intuitively feel may be needed to deal with future problems whose size and nature are as yet unknown, and if not needed for that purpose will add to their bequests.
Whether to annuitize depends on whether doing so increases the Best Initial Disbursement (BID) by more than enough to compensate for the reduction in portfolio value. Two cases are considered here in which half of the initial investible funds are annuitized. In one, the returns on the portfolio are the assumptions given in Model called the Pre 2008 Returns, and the payout for the annuity is one that could have been purchased in the first half of 2007. In the other, the returns on the portfolio are the assumptions given in Model called the Lower Returns, and the payout for the annuity is one that could have been purchased in the fall of 2022. In either case, inflation is simulated by a simple recursive random process in which inflation in the long run is expected to be 3.0%. Prior inflation for the annuity purchased in the fall of 2022, however, is assumed to be higher than in 2007. In either case, it turns out that annuitizing half of the funds increases the Best Initial Disbursement (BID) by about half a percentage point of initial portfolio value. A decision must be made as to whether that increase is worth the reduction in portfolio value due to the annuitizing shown by the simulations.
Modifications
To include purchase of a lifetime annuity in Model a relation must be added that simulates the inflation that occurs each year over the disbursement period so that the nominal benefit that is received each year can be put on a real basis. Using the simulated inflation obtained from this relation each year in the past in an iteration of a simulation, an annual nominal annuity benefit when received is deflated and added to the portfolio. To determine the limit on the disbursement each year an amount must also be added to the value of the portfolio at that time to reflect the future stream of annuity benefits that is expected to be received. The future nominal benefits are discounted for the annual inflation that is expected before they are received as well as for the annual inflation that has already occurred in the past for the current iteration of the simulation. The future annuity benefits are also discounted for each year in the future before they will be received by the RDR because the RDR is the assumed rate of return at which funds are being invested. The specific relations that are added to the model to make these modifications are discussed in a note.
Annual inflation in the U. S. has exhibited strong serial correlation as there have been long periods over which annual inflation has been either persistently higher or lower than average. For instance, annual inflation was persistently high in the 1970s, and persistently low after 2007 up through 2020. Suppose annual inflation is regressed on its prior value over the period from 1960 through 2020. The coefficient for prior year inflation is about .75, which shows strong positive serial correlation as expected. In the model inflation is simulated by a simple recursive random process using .75 as the coefficient for inflation of the prior year. The constant term for this recursive relation is .25 of the annual inflation expected in the long run as this is the limit obtained when the expected inflation each year for this recursive relation is successively iterated into the future. Average annual inflation from 1960 through 2020 was 3.7%. In view of the Federal Reserve’s target for inflation of 2.0%, however, inflation in the long run is here assumed to be equal to 3.0%. The prior inflation for the recursive relation is assumed to be 3.0% for the annuity purchased in the first half of 2007, and 5.0% for the annuity purchased in the fall of 2022. The difference between simulated inflation each year and the expected inflation given by the recursive relation is assumed to be normally distributed with a zero mean and a standard deviation of 1.5%. The cumulative probability distributions of the inflation discounts obtained at years 12, 24, and 36 when starting with 3.0% are shown in Chart 12.1. The difference at year 12 when prior inflation is 5.0% instead of 3.0% is shown in Chart 12.2.
Pre 2008 Returns
Assume the Pre 2008 investment returns and the payout rate for a lifetime annuity of 7.50% that might have been purchased in the first half of 2007 by a 65 year old woman. To be consistent with the model the benefits for this annuity are assumed to be received at the beginning of each year including the initial year. As annuity benefits are typically paid monthly an approximation for a payout at the beginning of a year for a 65 year old woman was obtained by a quote for monthly benefits beginning in six months for a woman who had just reached 64. The payout of 7.50% was obtained for a much earlier study done back in 2007 when the yield on a 10 year Treasury was about .0475. Suppose that half of the initial portfolio is used to purchase this annuity, and that inflation for the year prior to the initial year is assumed to be 3.0%. To assess how much this annuitizing increases the Best Initial Disbursement (BID), the first steps are to see whether it affects the RDR or the stock allocation that should be used.
After the annuitizing, similar tests to those in RDR show that setting the RDR equal to .02 more than the expected return of the portfolio continues to give the best results. After the annuitizing an initial disbursement with the high sustainability of 3.5% without the annuitizing increases by half a percentage point to 4.0%. Tests show that the best stock allocation for this highly sustainable disbursement continues to be 30%. Moreover, for either initial disbursement with a 30% stock allocation and an RDR set at .02 above the expected return of the portfolio, there is only a very small chance of a decline by year 24. When the initial disbursement is increased above this highly sustainable level, however, the best stock allocation increases more after the annuitizing, but the difference is small. Without the annuitizing Allocation shows that the best stock allocation increases by 10 percentage points when the initial disbursement is increased by one percentage point above the sustainable level of 3.5%. After the annuitizing Chart 12.3 shows that the best stock allocation increases by 20 percentage points when the initial disbursement is increased by one percentage point above the highly sustainable level of 4.0%. That result will be used to determine the stock allocation after the annuitizing depending on how much the initial disbursement is increased.
Suppose after the annuitizing that disbursements starting at 4.0% and 4.5% are simulated. The cumulative probability distributions of these disbursements at years 12 and 24 are shown in Charts 12.4 and 12.5. These charts look virtually the same as the charts for starting at 3.5% and 4.0% without the annuitizing, except that the plots are shifted to the right by half a percentage point. Thus, the annuitizing has increased the disbursements with the same sustainability risk by about half a percentqge point of initial portfolio value. The cumulative probability distributions of disbursements starting at 3.5% and 4.0% at years 12 and 24 without the annuitizing are shown in Charts 5.1 and 5.2 at page BID.
The cost of the increase from 4.0% to 4.5% obtained with the annuitizing is the reduction in the value of the portfolio available to cover emergencies, or if not needed for that purpose to increase bequests. The initial reduction in portfolio value is, of course, the half of the portfolio that has been annuitized. The reduction in portfolio value at years 12, 24, and 36 can be obtained as outputs from the simulations. These comparisons are shown in Charts 12.6, 12.7, and 12.8. Initially, the portfolio values are two vertical lines at 50% and 100% of initial asset value. Over time the values become much more diverse and somewhat closer together. Based on averages, the portfolio is about 40% less at year 12 after the annuitizing, 30% less at year 24, and 20% less at year 36. Annuitizing is desirable if the half percentage point increase in the BID is deemed to more than compensate for the reductions in portfolio value.
Lower Returns
Now instead of assuming that the investment returns for the portfolio are those for the Pre 2008 Returns in Model assume the Lower Returns given in Model where the expected returns of both the intermediates and stocks are reduced by three percentage points, and there is also some reduction in volatility. The annuity payout for the lower returns is assumed to be 6.80%. This payout is based on a quote obtained in the fall of 2022 when the yield on a 10 year Treasury was about .0375, and the age and sex of the beneficiary were the same as for the 7.50% payout used for the Pre 2008 returns. It continues to be assumed that expected inflation in the long run is 3.0%, but prior inflation for this annuity is assumed to be 5.0% instead of 3.0%, given the higher inflation in 2022. Without the annuitizing, but with the lower returns, the simulations in Allocation and BID found that a reduction in the initial disbursement of one percentage point from 3.5% to 2.5% was highly sustainable, and that the best stock allocation continued to be 30%. Now suppose with the lower returns that half of the investible funds are annuitized with the 6.80% payout annuity. Using the inflation assumptions for this annuity, the initial disbursement that is highly sustainable increases by half a percentage point from 2.5% to 3.0%, and the best stock allocation continues to be 30%.
To find the BID after the annuitizing, the initial disbursement must be increased above the highly sustainable level of 3.0%, and the best allocations for these increases are required. In Allocation the Lower Returns cause stronger increases in the best stock allocation when the initial disbursement is increased above a sustainable level, and the same happens after the annuitizing. Chart 12.3 shows that the annuitizing increases the stock allocation from 30% to 50% when the initial disbursement is increased one percentage point above a highly sustainable level for the Pre 2008 Returns. Chart 12.9 shows that the best stock allocation increases from 30% to 70% when the initial disbursement is increased one percentage point above a highly sustainable level for the Lower Returns. The stock allocations for the Lower Returns will be based on this result when the initial disbursement is increased above a highly sustainable level depending on the size of the increase.
Suppose after the annuitizing that disbursements starting at 3.0% and 3.5% are simulated. The cumulative probaability distributions of these disbursments at years 12 and 24 are shown in Charts 12.10 and 12.11. These charts look virtually the same as the charts for starting at 2.5% and 3.0% without the annuitizing except that the plots are shifted to the right by half a percentage point. Thus, the annuitizing has increased the disbursements with the same sustainability risk by about half a percentage point. The cumulative probability distributions of the disbursements starting at 2.5% and 3.0% without the annuitizing are shown in Charts 5.4 and 5.5 at page BID. The cost of the half percentage point improvement from the annuitizing for the Lower Returns is a similar reduction in portfolio values as for the Pre 2008 Returns.
Conclusions
Annuitizing a portion of the initial portfolio offers the possibility of increasing the Best Initial Disbursement (BID) at the cost of reducing the value of the portfolio available to cover emergencies, or if not needed for that purpose to make bequests. Annuitizing is desirable if the increase in the BID is judged to be sufficient to more than compensate for the reduction in portfolio value. Cases have been considered in which the returns on the portfolio are either those assumed in Model for the Pre 2008 Returns or the Lower Returns. The annuity payout rate assumed for the Pre 2008 Returns is one that might have been purchased in the first half of 2007, whereas for the Lower Returns it is one that might have been purchased in the fall of 2022. As the payouts are in nominal terms, future benefits must be deflated for inflation to incorporate them in the model. To do so, annual inflation is assumed to follow a recursive random process with expected inflation in the long run equal to 3.0%. Prior inflation for the Pre 2008 Returns is assumed to be 3.0%, and for the Lower Returns to be 5.0%. In either case, when half of the initial portfolio is annuitized, the BID increases by about a half percentage point of initial portfolio value. The initial portfolio reduction is, of course, 50%. On average, however, simulations show that it is about 40% after 12 years, 30% after 24 years, and 20% at the end of the period. The annuitizing is desirable if the half percentage point increase in the BID is more than enough to compensate for these lower portfolio values.
Note
To incorporate annuity benefits into the model relations in four new columns are added to the spreadsheet, and the real value of the annuity benefit is added to the value of the portfolio at the beginning of each year. Also, the real present value of the future stream of annuity benefits is added to the value of the portfolio at any time to determine the limit on the disbursement. One of the new columns has the relations that determine the value of inflation each year. Inflation in year t is denoted by j(t) and given by:
j(t)=h*j(t-1)+(1-h)*j^+k(t) t = 1, 2, …, T-1
where j^ is the expected value of inflation in the long run, and j(0) is the assumed inflation for the year prior to the start of a simulation. For the reported simulations the value of j^ is assumed to be .03, and j(0) is .03 for the annuity payout from 2007, and .05 for the payout from 2022. The value of h is .75. The term, k(t), is the random component of inflation in year t, which is determined in another new column and given by:
k(t)=RiskNormal(0,s) t = 1, 2, …, T-1
where for the reported simulations the standard deviation, s, is assumed to be .015. The cumulated discount rate for inflation at the beginning of t is denoted by J(t) and given in another new column by:
J(t+1)=J(t)/(1+j(t)) t = 1, 2, …, T-1
where J(1)=1. The fourth and last new column gives the present value denoted by A(t) of the stream of annuity benefits to be received as of the beginning of t discounted in future years for expected inflation and the RDR:
A(t)=PV((1+ j^)*(1+m)–1,T–t+1,-g,0,1) t = 1, 2, …, T
where g is the annual annuity benefit in nominal terms at the beginning of each year, including the first year. Suppose the initial value of the assets before the purchase of the annuity is set equal to 100. The value of V(1) is then the percentage of initial asset value that is not annuitized, and g is the annual annuity payout as a percentage of initial asset value. The value of the disbursement, C(t), and portfolio value, V(t), are then expressed as percentages of initial asset value. PV() is the Excel function that calculates the present value of the stream of g received at the beginning of each of T-t+1 years, including the first year, discounted at an annual rate of (1+j^)*(1+m)-1 for each year in the future to account for expected inflation and the RDR. To add the annuity benefit to the portfolio each year the relation for V(t+1) is rewritten to be:
V(t+1)=(V(t)+g*J(t)–C(t))*(1+R(t)) t = 1, 2, …, T.
To add the values of the stream of annuity benefits to the value of the portfolio when determining the limit on the disbursement at t, the relation for C*(t) is rewritten to be:
C*(t)=PMT(m,T-t+1,-V(t)-J(t)*A(t),0,1) t = 1, 2, …, T.
References
Ameriks, John, Robert Veres, and Mark J. Warshawsky. 2001. “Making Retirement Income Last a Lifetime.” Journal of Financial Planning 14, 12 (December):62-76.
Davidoff, Thomas, Jeffery R. Brown, and Peter A. Diamond. 2005. “Annuities and Individual Welfare. “ American Economic Review 95, (December):1573-1590.
Posted 2020 Revised October 2022 May 2025