A disbursement for each year is planned at the beginning of that year. This disbursement is the same as the disbursement that was planned for the prior year in real terms as long as that disbursement does not exceed a limit. This limit is calculated each year based on the then current value of the portfolio. This limit is the largest constant disbursement that could be obtained over the remainder of the disbursement period if the funds are invested at a constant rate of return. This rate of return is a decision variable that when it is increased reduces the risk of declines in the disbursements at the front of the disbursement period, and increases the risk of declines at the back of the period. This rate of return is called the Retrenchment Discount Rate, or RDR. The page, RDR, uses simulations to determine the best value to use for the RDR for different conditions, depending on the tradeoff of risk between the front and back of the period that it causes.
For the simulations, the value of the portfolio at the beginning of each year is obtained by applying the return on the portfolio over the prior year to the difference between the portfolio value at the beginning of the prior year and the planned disbursement for that year. Suppose the option in the model is being utilized that includes a small chance of an extra disbursement each year to cover an emergency. If such an emergency occurs for the prior year, that extra disbursement is deducted from what would otherwise be the portfolio value to get the portfolio value for the current year.
The first step to get the return on the portfolio over the prior year is to make a drawing from the normal probability distribution that has been specified for the annual real return of the intermediates. To get the annual real return on stocks, the first step is to multiply the return that was obtained for the intermediates by the assumed correlation coefficient. Added to this value is an independent drawing from a normal distribution that has been specified so as to give the assumed annual real return of stocks after taking into account the return of the intermediates, and assumed correlation. The return on the stocks so obtained is normally distributed since it is a linear combination of normally distributed variables. Applying the assumed allocation for the simulation to the returns obtained for the intermediates and stocks for the prior year gives the return on the portfolio.
Values of the disbursements in successive years are calculated in successive rows of a spreadsheet starting with a given initial disbursement. The value of the portfolio is obtained based on the amount invested at the beginning of the prior year, the return on the portfolio over the prior year, and less any disbursement for an emergency if there is one. The value of the portfolio is then used to calculate the value of the limit for the current year. The planned disbursement for the current year is the smaller of the limit for the current year or the planned disbursement for the prior year in real terms. The initial disbursement is given, and the initial portfolio value is generally set equal to 100 so that future disbursements and portfolio values are expressed as percentages of initial portfolio value. When running simulations, the disbursements are assumed to equal their planned values less the disbursement for emergencies, if any. In practice, however, disbursements may otherwise differ from their planned values. In this event, planned disbursements for the current year are based on the planned disbursement for the prior year, but the limit is calculated based on the actual value of the portfolio taking into account the actual disbursement for the prior year.
To illustrate procedures, three sets of expected returns are assumed. These are called the Pre 2008 Returns, the Lower Returns, and the Mid Returns. For each set of expected returns, the real returns on intermediates and stocks are assumed to be independently distributed over time with the same joint normal distribution and correlation each year. There is an exception at page, PAD, however, when the stock returns are assumed to depend on their prior average.
For the Pre 2008 Returns, the expected annual real return for the intermediates is assumed to be .03, and the standard deviation to be .07. These are about the average and the sample standard deviation for the historical returns from 1960 through 2007 reported in the SBBI Yearbook for intermediate term U. S. Governments. For the Lower Returns, the expected annual real return of the intermediates is assumed to be zero, and the standard deviation to be .04. These are about the average and sample standard deviation for this historical series from 2009 through 2021. Moreover, after the Federal Reserve increased interest rates in the early 2020s, deducting annual measures of inflation from the yields on intermediates has generally been less than half of .03. The Mid Returns assume an expected annual real return and standard deviation midway between those of the Pre 2008 Returns and the Lower Returns.
For stocks, the SBBI Yearbook reports that the average annual real return on the S&P 500 from l960 through 2007 was .073. The Pre 2008 Returns assume that the expected equity premium for stocks is .04 making the expected return of stocks equal to .07 compared to .03 for intermediates. From 2009 through 2022, the average annual real return on the S&P 500 was .114. One of the reasons for the very strong performance of equites was, of course, the decline in interest rates. As there is little likelihood that the very strong returns on stocks since 2009 will be repeated, the Low Returns and the Mid Returns assume the same equity premium of .04 as for the Pre 2008 Returns. The standard deviation of the stock returns is assumed to be .18 for the Pre 2008 Returns, .14 for the Lower Returns, and .16 for the Mid Returns.
The correlation coefficient for the annual return of the stocks on the annual return of the intermediates is assumed to be 0.4. Over the first half of the period since 1960 this coefficient was strongly positive as the dominant influence on financial markets was inflation. Over the second half of this period, however, the coefficient was strongly negative as the dominant influences on financial markets were periods of speculative excess. When the annual returns for stocks are regressed on the annual returns for intermediates for the period from 1960 through 2020, the coefficient is close to 0.4.
Basic Model
To specify the model formally, let V(t) be the value of the portfolio in real terms at the beginning of year t, before the disbursement for that year is made. Let C(t) be the planned value of the disbursement in real terms for year t 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 RDR 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)=a*r(t)+[1−a]*i(t) t = 1, 2, …, T
C*(t)=PMT(RDR,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 equal to RDR without any residual.
Suppose the Excel spreadsheet is augmented with @Risk or XLRisk 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 distributed as discussed earlier. In particular, each year r(t)=.4*i(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 Returns 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 Returns, the values of i(t) and r(t) are the following:
i(t)=RiskNormal(.03,.07) t = 1, 2, …,T
r(t)=RiskNormal(.058+.4*i(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+.4*i(t),.1391) t = 1,2, …,T
For the Mid Returns, the mean and standard deviation of r(t) are respectively .055 and .16. In this case, the mean and standard deviation of the independent variable must be set equal to .049 and .15848. Thus, for the Mid Returns, the values of i(t) and r(t) are the following:
i(t)=RiskNormal(.015,055) t = 1, 2, …,T
r(t)=RiskNormal(.049+.4*i(t),.15848) t = 1, 2, …,T
To start a simulation, V(1) can be set equal to 100 so that the disbursements are expressed as a percentage of the initial value of the portfolio. The initial value C(1) is given, and for sustainable disbursements is generally well below C*(1). The residual value of the portfolio is given by V(T+1).
Emergencies
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 to calculate the outlay for the emergency, 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,.2*V(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 October 2024
Model
Model
A disbursement for each year is planned at the beginning of that year. This disbursement is the same as the disbursement that was planned for the prior year in real terms as long as that disbursement does not exceed a limit. This limit is calculated each year based on the then current value of the portfolio. This limit is the largest constant disbursement that could be obtained over the remainder of the disbursement period if the funds are invested at a constant rate of return. This rate of return is a decision variable that when it is increased reduces the risk of declines in the disbursements at the front of the disbursement period, and increases the risk of declines at the back of the period. This rate of return is called the Retrenchment Discount Rate, or RDR. The page, RDR, uses simulations to determine the best value to use for the RDR for different conditions, depending on the tradeoff of risk between the front and back of the period that it causes.
For the simulations, the value of the portfolio at the beginning of each year is obtained by applying the return on the portfolio over the prior year to the difference between the portfolio value at the beginning of the prior year and the planned disbursement for that year. Suppose the option in the model is being utilized that includes a small chance of an extra disbursement each year to cover an emergency. If such an emergency occurs for the prior year, that extra disbursement is deducted from what would otherwise be the portfolio value to get the portfolio value for the current year.
The first step to get the return on the portfolio over the prior year is to make a drawing from the normal probability distribution that has been specified for the annual real return of the intermediates. To get the annual real return on stocks, the first step is to multiply the return that was obtained for the intermediates by the assumed correlation coefficient. Added to this value is an independent drawing from a normal distribution that has been specified so as to give the assumed annual real return of stocks after taking into account the return of the intermediates, and assumed correlation. The return on the stocks so obtained is normally distributed since it is a linear combination of normally distributed variables. Applying the assumed allocation for the simulation to the returns obtained for the intermediates and stocks for the prior year gives the return on the portfolio.
Values of the disbursements in successive years are calculated in successive rows of a spreadsheet starting with a given initial disbursement. The value of the portfolio is obtained based on the amount invested at the beginning of the prior year, the return on the portfolio over the prior year, and less any disbursement for an emergency if there is one. The value of the portfolio is then used to calculate the value of the limit for the current year. The planned disbursement for the current year is the smaller of the limit for the current year or the planned disbursement for the prior year in real terms. The initial disbursement is given, and the initial portfolio value is generally set equal to 100 so that future disbursements and portfolio values are expressed as percentages of initial portfolio value. When running simulations, the disbursements are assumed to equal their planned values less the disbursement for emergencies, if any. In practice, however, disbursements may otherwise differ from their planned values. In this event, planned disbursements for the current year are based on the planned disbursement for the prior year, but the limit is calculated based on the actual value of the portfolio taking into account the actual disbursement for the prior year.
To illustrate procedures, three sets of expected returns are assumed. These are called the Pre 2008 Returns, the Lower Returns, and the Mid Returns. For each set of expected returns, the real returns on intermediates and stocks are assumed to be independently distributed over time with the same joint normal distribution and correlation each year. There is an exception at page, PAD, however, when the stock returns are assumed to depend on their prior average.
For the Pre 2008 Returns, the expected annual real return for the intermediates is assumed to be .03, and the standard deviation to be .07. These are about the average and the sample standard deviation for the historical returns from 1960 through 2007 reported in the SBBI Yearbook for intermediate term U. S. Governments. For the Lower Returns, the expected annual real return of the intermediates is assumed to be zero, and the standard deviation to be .04. These are about the average and sample standard deviation for this historical series from 2009 through 2021. Moreover, after the Federal Reserve increased interest rates in the early 2020s, deducting annual measures of inflation from the yields on intermediates has generally been less than half of .03. The Mid Returns assume an expected annual real return and standard deviation midway between those of the Pre 2008 Returns and the Lower Returns.
For stocks, the SBBI Yearbook reports that the average annual real return on the S&P 500 from l960 through 2007 was .073. The Pre 2008 Returns assume that the expected equity premium for stocks is .04 making the expected return of stocks equal to .07 compared to .03 for intermediates. From 2009 through 2022, the average annual real return on the S&P 500 was .114. One of the reasons for the very strong performance of equites was, of course, the decline in interest rates. As there is little likelihood that the very strong returns on stocks since 2009 will be repeated, the Low Returns and the Mid Returns assume the same equity premium of .04 as for the Pre 2008 Returns. The standard deviation of the stock returns is assumed to be .18 for the Pre 2008 Returns, .14 for the Lower Returns, and .16 for the Mid Returns.
The correlation coefficient for the annual return of the stocks on the annual return of the intermediates is assumed to be 0.4. Over the first half of the period since 1960 this coefficient was strongly positive as the dominant influence on financial markets was inflation. Over the second half of this period, however, the coefficient was strongly negative as the dominant influences on financial markets were periods of speculative excess. When the annual returns for stocks are regressed on the annual returns for intermediates for the period from 1960 through 2020, the coefficient is close to 0.4.
Basic Model
To specify the model formally, let V(t) be the value of the portfolio in real terms at the beginning of year t, before the disbursement for that year is made. Let C(t) be the planned value of the disbursement in real terms for year t 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 RDR 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)=a*r(t)+[1−a]*i(t) t = 1, 2, …, T
C*(t)=PMT(RDR,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 equal to RDR without any residual.
Suppose the Excel spreadsheet is augmented with @Risk or XLRisk 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 distributed as discussed earlier. In particular, each year r(t)=.4*i(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 Returns 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 Returns, the values of i(t) and r(t) are the following:
i(t)=RiskNormal(.03,.07) t = 1, 2, …,T
r(t)=RiskNormal(.058+.4*i(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+.4*i(t),.1391) t = 1,2, …,T
For the Mid Returns, the mean and standard deviation of r(t) are respectively .055 and .16. In this case, the mean and standard deviation of the independent variable must be set equal to .049 and .15848. Thus, for the Mid Returns, the values of i(t) and r(t) are the following:
i(t)=RiskNormal(.015,055) t = 1, 2, …,T
r(t)=RiskNormal(.049+.4*i(t),.15848) t = 1, 2, …,T
To start a simulation, V(1) can be set equal to 100 so that the disbursements are expressed as a percentage of the initial value of the portfolio. The initial value C(1) is given, and for sustainable disbursements is generally well below C*(1). The residual value of the portfolio is given by V(T+1).
Emergencies
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 to calculate the outlay for the emergency, 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,.2*V(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 October 2024