Let V(t) be the value of the investment in real terms at the beginning of year t, before the disbursement for that year is made. Let C(t) be the value of the disbursement in real terms at the beginning of year t. And let C*(t) be the limit on its value. Let R(t) be the return on the investment in real terms over year t including the reinvestment of any income earned during the year. On the same basis, r(t) is the return earned on stocks and i(t) the return earned on the intermediate term fixed income issues. The proportion allocated to stocks at the beginning of t is a.
When the disbursements begin initially let T be the largest possible total number of annual disbursements. Let m be the annual real return used to calculate the value of the annuity that limits the disbursements. The calculations to be made for each year in the rows of an Excel spreadsheet are then as follows:
V(t+1) = [ V(t) – C(t) ][ 1 + R(t) ] t = 1, 2, …, T
R(t) = ar(t) + [ 1−a ]i(t) t = 1, 2, …, T
C*(t) = PMT( m, T−t+1, V(t), 0, 1 ) t = 2, 3, …, T
C(t) = Min( C(t-1), C*(t) ) t = 2, 3, …, T
In these expressions PMT is the Excel function that calculates the constant amount that can be obtained at the beginning of each of T−t+1 years. This is from an investment V(t), at an annual rate of interest m without any residual.
Suppose the Excel spreadsheet is augmented with @Risk software for making Monte Carlo simulations. The values of i(t) and r(t) are obtained from this software assuming that i(t) and r(t) are each independently distributed over time with its own normal distribution each year. The distribution of r(t) each year is assumed to be a linear function of i(t) plus an independently distributed normal variable the mean and standard deviation of which are set so as to make the mean and standard deviation of r(t) equal to specified values. The coefficient of i(t) for this linear relation in the reported simulations is assumed to be 0.4. This is the coefficient obtained when the annual real return of the S&P 500 is regressed on the annual real return of intermediate term U. S. government issues for the period from 1960 through 2018. Over the first half of that period, however, the correlation is strongly positive, and strongly negative over the second half.
For the period from 1960 through 2018, the average annual real return for intermediate term U. S. government issues was about .03. The sample standard deviation of those annual returns was about .07. If the equity premium for stocks is assumed to be .04, the expected annual real return of stocks is .07. Suppose the standard deviation for the annual real return for stocks is .18. These assumptions have been used in prior postings at this site to illustrate procedures, and will be called the pre 2008 illustration. Looking forward in the early 2020s, however, much lower expected real investment returns are possible. In fact, over the last 10 years of the 1960 to 2018 period, the average annual real return on intermediate term U. S. government issues was about zero, and the sample standard deviation was about .04. To help gauge the possible effect of much lower returns, an illustration using a hypothetical set of much lower returns will be considered. For these lower returns, the expected annual real return of intermediates is zero, and the standard deviation of the annual returns is .04. For stocks, the expected annual real return is .04, which keeps the equity premium at .04. The standard deviation of the annual returns is reduced from .18 to .14. The correlation for the returns each year remains the same.
As specified above, each year r(t) = .4i(t) plus a drawing from an independent and normally distributed variable. The mean and standard deviation of this independent variable are set so as to keep the mean and standard deviation of r(t) equal to its specified values. For the pre 2008 illustration these are respectively .07 and .18, and the mean and standard deviation of the independent variable must be set equal respectively to .058 and .1778. Thus, for the pre 2008 illustration, the values of i(t) and r(t) are the following:
For the lower returns, the mean and standard deviation of r(t) are respectively .04 and .14. In this case, the mean and standard deviation of the independent variable must be set equal respectively to .04 and .1391. Thus, for the lower returns, the values of i(t) and r(t) are the following:
To start a simulation, V(t) can be set equal to 100 so that the disbursements are expressed as a percentage of the initial value of the investment. The initial value C(1) is given, and for sustainable disbursements is generally well below C*(1). The residual value of the investment is given by V(T+1).
Suppose it is desired to add to the model each year a small chance that a large extra expenditure will be required for an emergency. In particular, suppose each year that there is an independent .05 chance that an extra disbursement will be required equal to 20% of the initial value of the portfolio. If an outlay that large would seriously deplete the funds needed for continuing activities in the future, however, the outlay is not allowed to exceed 25% of the current value of the portfolio. The possibility of such an outlay is incorporated into the model as follows. An additional column is added to the spreadsheet, and the relation for V(t) in each row is modified. Let K(t) be the amount required at t to cover the emergency if it occurs, and the outlay is not constrained. The value of K(1)=0, and for the other rows given by the following relation:
K(t) = If ( Rand() <= .05, .2V(1), 0 ) t = 2, 3, …, T
where Rand() is the Excel function that for each replication is equally likely to give any value between 0 and 1. If (x, y, z ) is an Excel function that is equal to y when x is true, and to z when x is false. The relation for V(t) is rewritten as follows:
V(t) = Max [ ( V(t-1) – C(t-1) )( 1 + R(t-1) ) – K(t), .75( V(t-1) – C(t-1))(1 + R(t-1)) ] t = 2, 3, …, T
where V(1) is a given value.
Posted November 2018 Revised July 2022
Addendum
Note that the disbursement in Model denoted by C(t) is the budgeted disbursement for year t at the beginning of t. In general, the actual disbursement over t may differ from the budgeted amount depending on the expenses that must be covered over t. For instance, as just discussed, a very large extra disbursement may be necessary to cover an emergency over t. More generally, much smaller differences may also exist between the actual disbursement over t and the budgeted amount given by C(t) due to the expenses that must be covered. In simulating future disbursements at this site actual disbursements are assumed to be the same as the budgeted amounts unless an emergency occurs.
The limit on C(t+1) given by C*(t+1) depends on the value of the portfolio at the beginning of t+1 denoted by V(t+1). This value incorporates a disbursement that might be required over t for an emergency. C(t+1) is limited by the value of C*(t+1) so determined, but otherwise is equal to the budgeted disbursement for t, which is C(t). When disbursements are actually being made in practice, however, there may be small differences between the budgeted and actual amounts due to the expenses over t that must be covered. When actually making disbursements in practice in such cases, the value of the portfolio that occurs as a result of the actual disbursement is used to calculate the limit, C*(t+1). The budgeted disbursement for t+1 is this limit or the prior budgeted disbursement, C(t), whichever is smaller.
Model
Model
Let V(t) be the value of the investment in real terms at the beginning of year t, before the disbursement for that year is made. Let C(t) be the value of the disbursement in real terms at the beginning of year t. And let C*(t) be the limit on its value. Let R(t) be the return on the investment in real terms over year t including the reinvestment of any income earned during the year. On the same basis, r(t) is the return earned on stocks and i(t) the return earned on the intermediate term fixed income issues. The proportion allocated to stocks at the beginning of t is a.
When the disbursements begin initially let T be the largest possible total number of annual disbursements. Let m be the annual real return used to calculate the value of the annuity that limits the disbursements. The calculations to be made for each year in the rows of an Excel spreadsheet are then as follows:
V(t+1) = [ V(t) – C(t) ][ 1 + R(t) ] t = 1, 2, …, T
R(t) = ar(t) + [ 1−a ]i(t) t = 1, 2, …, T
C*(t) = PMT( m, T−t+1, V(t), 0, 1 ) t = 2, 3, …, T
C(t) = Min( C(t-1), C*(t) ) t = 2, 3, …, T
In these expressions PMT is the Excel function that calculates the constant amount that can be obtained at the beginning of each of T−t+1 years. This is from an investment V(t), at an annual rate of interest m without any residual.
Suppose the Excel spreadsheet is augmented with @Risk software for making Monte Carlo simulations. The values of i(t) and r(t) are obtained from this software assuming that i(t) and r(t) are each independently distributed over time with its own normal distribution each year. The distribution of r(t) each year is assumed to be a linear function of i(t) plus an independently distributed normal variable the mean and standard deviation of which are set so as to make the mean and standard deviation of r(t) equal to specified values. The coefficient of i(t) for this linear relation in the reported simulations is assumed to be 0.4. This is the coefficient obtained when the annual real return of the S&P 500 is regressed on the annual real return of intermediate term U. S. government issues for the period from 1960 through 2018. Over the first half of that period, however, the correlation is strongly positive, and strongly negative over the second half.
For the period from 1960 through 2018, the average annual real return for intermediate term U. S. government issues was about .03. The sample standard deviation of those annual returns was about .07. If the equity premium for stocks is assumed to be .04, the expected annual real return of stocks is .07. Suppose the standard deviation for the annual real return for stocks is .18. These assumptions have been used in prior postings at this site to illustrate procedures, and will be called the pre 2008 illustration. Looking forward in the early 2020s, however, much lower expected real investment returns are possible. In fact, over the last 10 years of the 1960 to 2018 period, the average annual real return on intermediate term U. S. government issues was about zero, and the sample standard deviation was about .04. To help gauge the possible effect of much lower returns, an illustration using a hypothetical set of much lower returns will be considered. For these lower returns, the expected annual real return of intermediates is zero, and the standard deviation of the annual returns is .04. For stocks, the expected annual real return is .04, which keeps the equity premium at .04. The standard deviation of the annual returns is reduced from .18 to .14. The correlation for the returns each year remains the same.
As specified above, each year r(t) = .4i(t) plus a drawing from an independent and normally distributed variable. The mean and standard deviation of this independent variable are set so as to keep the mean and standard deviation of r(t) equal to its specified values. For the pre 2008 illustration these are respectively .07 and .18, and the mean and standard deviation of the independent variable must be set equal respectively to .058 and .1778. Thus, for the pre 2008 illustration, the values of i(t) and r(t) are the following:
i(t) = RiskNormal ( .03, .07 ) t = 1, 2, …,T
r(t) = RiskNormal ( .058 + .4i(t), .1778 ) t = 1,2, …,T
For the lower returns, the mean and standard deviation of r(t) are respectively .04 and .14. In this case, the mean and standard deviation of the independent variable must be set equal respectively to .04 and .1391. Thus, for the lower returns, the values of i(t) and r(t) are the following:
i(t) = RiskNormal ( .00, .04 ) t = 1, 2, …,T
r(t) = RiskNormal ( .04 + .4i(t), .1391 ) t = 1,2, …,T
To start a simulation, V(t) can be set equal to 100 so that the disbursements are expressed as a percentage of the initial value of the investment. The initial value C(1) is given, and for sustainable disbursements is generally well below C*(1). The residual value of the investment is given by V(T+1).
Suppose it is desired to add to the model each year a small chance that a large extra expenditure will be required for an emergency. In particular, suppose each year that there is an independent .05 chance that an extra disbursement will be required equal to 20% of the initial value of the portfolio. If an outlay that large would seriously deplete the funds needed for continuing activities in the future, however, the outlay is not allowed to exceed 25% of the current value of the portfolio. The possibility of such an outlay is incorporated into the model as follows. An additional column is added to the spreadsheet, and the relation for V(t) in each row is modified. Let K(t) be the amount required at t to cover the emergency if it occurs, and the outlay is not constrained. The value of K(1)=0, and for the other rows given by the following relation:
K(t) = If ( Rand() <= .05, .2V(1), 0 ) t = 2, 3, …, T
where Rand() is the Excel function that for each replication is equally likely to give any value between 0 and 1. If (x, y, z ) is an Excel function that is equal to y when x is true, and to z when x is false. The relation for V(t) is rewritten as follows:
V(t) = Max [ ( V(t-1) – C(t-1) )( 1 + R(t-1) ) – K(t), .75( V(t-1) – C(t-1))(1 + R(t-1)) ] t = 2, 3, …, T
where V(1) is a given value.
Posted November 2018 Revised July 2022
Addendum
Note that the disbursement in Model denoted by C(t) is the budgeted disbursement for year t at the beginning of t. In general, the actual disbursement over t may differ from the budgeted amount depending on the expenses that must be covered over t. For instance, as just discussed, a very large extra disbursement may be necessary to cover an emergency over t. More generally, much smaller differences may also exist between the actual disbursement over t and the budgeted amount given by C(t) due to the expenses that must be covered. In simulating future disbursements at this site actual disbursements are assumed to be the same as the budgeted amounts unless an emergency occurs.
The limit on C(t+1) given by C*(t+1) depends on the value of the portfolio at the beginning of t+1 denoted by V(t+1). This value incorporates a disbursement that might be required over t for an emergency. C(t+1) is limited by the value of C*(t+1) so determined, but otherwise is equal to the budgeted disbursement for t, which is C(t). When disbursements are actually being made in practice, however, there may be small differences between the budgeted and actual amounts due to the expenses over t that must be covered. When actually making disbursements in practice in such cases, the value of the portfolio that occurs as a result of the actual disbursement is used to calculate the limit, C*(t+1). The budgeted disbursement for t+1 is this limit or the prior budgeted disbursement, C(t), whichever is smaller.
Posted July 2023