When a disbursement reaches its limit, Retrench shows that there is a risk of large annual declines in the annual budget. One way of dealing with this risk is to undertake a major retrenchment, and reestablish a cushion to absorb worse than expected investment returns. A major retrenchment not only deals with the risk of annual budget declines, but also reduces the chances of lower disbursements later in the disbursement period. If a major retrenchment is not undertaken, however, there is another strategy to reduce the threat of unpredictable annual budget declines. Instead of reducing overlimit disbursements to the limit they can be reduced only part way.
Dampening the reduction of overlimit disbursements to the limit provides a way to reduce the threat of large annual reductions in the budget without undertaking a major retrenchment. Instead of reducing overlimits to the limit they can be reduced only a quarter or half the way. The problem with dampening, however, is that it runs the risk of running out of funds. The solution to that problem is to undertake a major retrenchment if the disbursement becomes too far above its limit. In this case, a major retrenchment is not voluntary, but necessary to avoid the possibility of running out of funds. This major retrenchment, however, causes the risk of much lower disbursements later in the disbursement period.
The likelihood that a major retrenchment will be needed when dampening, however, depends on the size of the reduction in the gap that is made with the dampening. If the reduction in the gap is sufficiently large it is possible that any need for a reset may be sufficiently small so as not to pose a risk of later declines. The possibility that a reset may cause the risk of declines later in the disbursement period can be tested with simulations. It turns out that reducing overlimits by only a quarter of the gap does cause a risk of declines later in the period. When the gap is reduced by a half when there is an overlimit, however, there is little chance that a reset will be necessary. Moreover, cutting the need for any budget reduction by a half significantly reduces the risk posed by unpredictable budget reductions. A major retrenchment may still be desirable to reduce the chances of low disbursements in the future, but the need to reduce annual budget risk is significantly reduced.
To illustrate the issues in dampening, a 4.0% initial disbursement will be assumed, as well as the Mid Returns. Instead of reducing overlimits to the limit, reductions of only a quarter or a half will be considered. To avoid the possibility of running out of funds, the annual budget will be reduced to 90% of the limit whenever the disbursement becomes more than 20% above the limit. The modifications of the model to incorporate dampening and the resets are discussed in a note.
The effects of the dampening and resets on the cumulative probability distributions of the disbursements at years 12 and 24 are shown in Charts 7.1 and 7.2 when the gap is reduced by a quarter. The dashed curves show the effect of the dampening. The charts show the dampening reduces the chances of small declines at either year as the dashed curves are somewhat below the solid curves for disbursements somewhat less than 4.0%. When the disbursements are somewhat further less than 4.0%, however, the dashed curves are above the solid curves indicating that the resets are increasing the chances of large declines. Moreover, the chances of larger declines by year 24 in Chart 7.2 appear sufficient to make the dampening undesirable.
When any overlimit gap is reduced by a half instead of a quarter, however, Charts 7.3 and 7.4 show that the resets no longer appear to be a serious problem. The dashed curves are below the solid curves for small declines, but no longer appear sufficiently above the solid curves for larger declines, to pose a problem. Nevertheless, cutting large budget reductions by a half does provide significant help in reducing the threat of annual budget declines. It appears that if a disbursement reaches its limit, and a major retrenchment is not undertaken, that there is good reason not to reduce any overlimit to the limit, but instead by only a half. Annual budget risk will then be a less serious problem.
Note
To add dampening to the basic model with a reset provision, six columns are added to the spreadsheet containing the relations given below. The entries in the rows of four of these columns, F(t), G’(t), G’’(t), and G’’’(t) are added together to get the value of C(t) each year. In particular, suppose that when C(t-1)>C*(t) that C(t)=C(t-1)−b*[C(t-1)−C*(t)] if C(t-1)−b*[C(t-1)−C*(t)]≤V(t). If C(t-1)−b*[C(t-1)− C*(t)]>V(t) then C(t)=V(t). In this relation b is a given constant that satisfies: 0≤b≤1. For b=1 there is no change from the basic model, and for b=0 there is no change from the basic model with an infinite RDR, for which C*(t)=V(t). In these relations c and d are given constants that provide the conditions for a reset, and satisfy c ≥ 0 and 0 ≤ d ≤ 1. For the examples of dampening used in the illustrations: b=.25 or .50, c=.2, and d=.1.
C(t)=F(t)+G’(t)+G”(t)+G”’(t) t = 2, 3, …, T
G’(t)=If(And(C(t-1)>C*(t),c=0),H(t),0) t = 2, 3, …, T
H(t)=If(C(t-1)–b*[C(t-1)–C*(t)]≤V(t),C(t-1)–b*[C(t-1)−C*(t)],V(t) t = 2, 3, …, T
G”(t)=If(And(C(t-1)>C*(t),C(t-1)≤[1+c]*C*(t)),H(t),0) t = 2, 3, …, T
G”’(t)= If(And(C(t-1)>[1+c]*C*(t),c>0),I(t),0) t = 2, 3, …, T
I(t)=If([1 –d]*C*(t)≤V(t),[1 – d]*C*(t),V(t)) t = 2, 3, …, T
In these expressions And( x, y ) is an Excel function that is true if both x and y are true, and false otherwise.
Posted November 2018 Revised July 2022 October 2024
Dampen
Dampening Annual Declines
When a disbursement reaches its limit, Retrench shows that there is a risk of large annual declines in the annual budget. One way of dealing with this risk is to undertake a major retrenchment, and reestablish a cushion to absorb worse than expected investment returns. A major retrenchment not only deals with the risk of annual budget declines, but also reduces the chances of lower disbursements later in the disbursement period. If a major retrenchment is not undertaken, however, there is another strategy to reduce the threat of unpredictable annual budget declines. Instead of reducing overlimit disbursements to the limit they can be reduced only part way.
Dampening the reduction of overlimit disbursements to the limit provides a way to reduce the threat of large annual reductions in the budget without undertaking a major retrenchment. Instead of reducing overlimits to the limit they can be reduced only a quarter or half the way. The problem with dampening, however, is that it runs the risk of running out of funds. The solution to that problem is to undertake a major retrenchment if the disbursement becomes too far above its limit. In this case, a major retrenchment is not voluntary, but necessary to avoid the possibility of running out of funds. This major retrenchment, however, causes the risk of much lower disbursements later in the disbursement period.
The likelihood that a major retrenchment will be needed when dampening, however, depends on the size of the reduction in the gap that is made with the dampening. If the reduction in the gap is sufficiently large it is possible that any need for a reset may be sufficiently small so as not to pose a risk of later declines. The possibility that a reset may cause the risk of declines later in the disbursement period can be tested with simulations. It turns out that reducing overlimits by only a quarter of the gap does cause a risk of declines later in the period. When the gap is reduced by a half when there is an overlimit, however, there is little chance that a reset will be necessary. Moreover, cutting the need for any budget reduction by a half significantly reduces the risk posed by unpredictable budget reductions. A major retrenchment may still be desirable to reduce the chances of low disbursements in the future, but the need to reduce annual budget risk is significantly reduced.
To illustrate the issues in dampening, a 4.0% initial disbursement will be assumed, as well as the Mid Returns. Instead of reducing overlimits to the limit, reductions of only a quarter or a half will be considered. To avoid the possibility of running out of funds, the annual budget will be reduced to 90% of the limit whenever the disbursement becomes more than 20% above the limit. The modifications of the model to incorporate dampening and the resets are discussed in a note.
The effects of the dampening and resets on the cumulative probability distributions of the disbursements at years 12 and 24 are shown in Charts 7.1 and 7.2 when the gap is reduced by a quarter. The dashed curves show the effect of the dampening. The charts show the dampening reduces the chances of small declines at either year as the dashed curves are somewhat below the solid curves for disbursements somewhat less than 4.0%. When the disbursements are somewhat further less than 4.0%, however, the dashed curves are above the solid curves indicating that the resets are increasing the chances of large declines. Moreover, the chances of larger declines by year 24 in Chart 7.2 appear sufficient to make the dampening undesirable.
When any overlimit gap is reduced by a half instead of a quarter, however, Charts 7.3 and 7.4 show that the resets no longer appear to be a serious problem. The dashed curves are below the solid curves for small declines, but no longer appear sufficiently above the solid curves for larger declines, to pose a problem. Nevertheless, cutting large budget reductions by a half does provide significant help in reducing the threat of annual budget declines. It appears that if a disbursement reaches its limit, and a major retrenchment is not undertaken, that there is good reason not to reduce any overlimit to the limit, but instead by only a half. Annual budget risk will then be a less serious problem.
Note
To add dampening to the basic model with a reset provision, six columns are added to the spreadsheet containing the relations given below. The entries in the rows of four of these columns, F(t), G’(t), G’’(t), and G’’’(t) are added together to get the value of C(t) each year. In particular, suppose that when C(t-1)>C*(t) that C(t)=C(t-1)−b*[C(t-1)−C*(t)] if C(t-1)−b*[C(t-1)−C*(t)]≤V(t). If C(t-1)−b*[C(t-1)− C*(t)]>V(t) then C(t)=V(t). In this relation b is a given constant that satisfies: 0≤b≤1. For b=1 there is no change from the basic model, and for b=0 there is no change from the basic model with an infinite RDR, for which C*(t)=V(t). In these relations c and d are given constants that provide the conditions for a reset, and satisfy c ≥ 0 and 0 ≤ d ≤ 1. For the examples of dampening used in the illustrations: b=.25 or .50, c=.2, and d=.1.
C(t)=F(t)+G’(t)+G”(t)+G”’(t) t = 2, 3, …, T
G’(t)=If(And(C(t-1)>C*(t),c=0),H(t),0) t = 2, 3, …, T
H(t)=If(C(t-1)–b*[C(t-1)–C*(t)]≤V(t),C(t-1)–b*[C(t-1)−C*(t)],V(t) t = 2, 3, …, T
G”(t)=If(And(C(t-1)>C*(t),C(t-1)≤[1+c]*C*(t)),H(t),0) t = 2, 3, …, T
G”’(t)= If(And(C(t-1)>[1+c]*C*(t),c>0),I(t),0) t = 2, 3, …, T
I(t)=If([1 –d]*C*(t)≤V(t),[1 – d]*C*(t),V(t)) t = 2, 3, …, T
In these expressions And( x, y ) is an Excel function that is true if both x and y are true, and false otherwise.
Posted November 2018 Revised July 2022 October 2024