Dampening annual declines of disbursements with a reset provision provides a policy tool for converting highly unpredictable and possibly sharp declines in the short run into more predictable but much larger declines in the long run. Dampening reduces the size of declines at the front of the disbursement period, but the need for possible resets makes large declines at the back of the period more likely. The effect on large declines at the back of the period can be evaluated by simulating the future probability distributions of the disbursements with and without dampening. To illustrate the effects of dampening suppose that any overlimit disbursements are reduced by a quarter of any gap that exists between a disbursement and its limit, instead of by the full amount. Suppose also that the disbursement is reset to 90% of the limit if it exceeds 120% of the limit to avoid possibly running out of funds. The modifications of the model to simulate the disbursements under these conditions are described in a note.
Under these conditions, Chart 21 shows the effects of dampening on the disbursements at years 12, 24, and 36 when starting at a disbursement of 3.5%. The cumulative probability distributions of the disbursements without any dampening are shown by the solid curves and with the dampening by the dotted curves. For any of the years, the solid and dotted curves begin to move to the left at the same probability on the vertical axis because the dampening does not affect the chances that there will be a decline. For lower probabilities, the dotted curves are at first below the solid curves and then crossover and are above the solid curves. The lower dotted curves before the crossover show that the dampening is making these smaller declines more likely. The probability on the vertical axis at which a dotted curve crosses to the left of a solid curve shows the chances that dampening will cause a larger decline.
At year 12, the crossover probability is far below the probability of a decline showing that there is very little chance that dampening will cause any increase in declines at the front of the period. At years 24 and 36, however, the crossover probability has moved up closer to the probability of a decline. At the back of the period, dampening is more likely than not to increase the size of any declines that occur. The size of these increases on average is from about 15% to 20%. For a 3.5% initial disbursement, however, the chances of these increases in the declines is less than .05. Given the decreasing chances that the beneficiary will survive until later in the period, the risk that dampening will cause larger declines at the back of the period may be easy to accept to minimize any problems at the front of the period.
Suppose now that the initial disbursement is increased from 3.5% to 4.0% or 4.5%. The results in these cases are shown in Charts 22 and 23. Dampening has now moved the crossover probability up closer to the probability of a decline. Thus, dampening is now more likely to increase the size of any declines that may occur. Even at year 12, dampening has a .10 chance of increasing the size of declines when starting at 4.5%. Larger declines have become a concern at the front of the period as well as the back. At the back of the period the increases in the declines caused by dampening continue on average to be about 15% or 20%. Now, however, the chances of these increases occurring are significantly higher. By the end of the period when the initial disbursement is 3.5% the chance of an increase is only .05. When the initial disbursement increases to 4.0% that chance rises to .13, and to .25 for starting at 4.5%. For higher initial disbursements the risk of larger declines caused by dampening has become more difficult to accept to reduce earlier problems.
Note
As a variation on the basic model 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 and is called the Annual Correction Adjustment, or ACA. For b=1 this variation is the same as the basic model, and for b=0 the same as for an infinite discount rate, 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 example of dampening with a reset used in the illustration: b=.25, c=.2, and d=.1.
C(t) = F(t) + G’(t) + G”(t) + G”’(t) t = 2, 3,…, T
F(t) = If( C(t-1) <= C*(t), C(t-1), 0 ) t = 2, 3, …, T
G’(t) = If( And( C(t-1) > C*(t), c = 0 ), H(t), 0 ) 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. (The simulations in Pye (2017) were made with the relations as stated above, and they were so described in the text. Unfortunately, however, the last line of these relations in Pye(2017) is stated incorrectly.)
Dampen
Dampening Annual Declines
Dampening annual declines of disbursements with a reset provision provides a policy tool for converting highly unpredictable and possibly sharp declines in the short run into more predictable but much larger declines in the long run. Dampening reduces the size of declines at the front of the disbursement period, but the need for possible resets makes large declines at the back of the period more likely. The effect on large declines at the back of the period can be evaluated by simulating the future probability distributions of the disbursements with and without dampening. To illustrate the effects of dampening suppose that any overlimit disbursements are reduced by a quarter of any gap that exists between a disbursement and its limit, instead of by the full amount. Suppose also that the disbursement is reset to 90% of the limit if it exceeds 120% of the limit to avoid possibly running out of funds. The modifications of the model to simulate the disbursements under these conditions are described in a note.
Under these conditions, Chart 21 shows the effects of dampening on the disbursements at years 12, 24, and 36 when starting at a disbursement of 3.5%. The cumulative probability distributions of the disbursements without any dampening are shown by the solid curves and with the dampening by the dotted curves. For any of the years, the solid and dotted curves begin to move to the left at the same probability on the vertical axis because the dampening does not affect the chances that there will be a decline. For lower probabilities, the dotted curves are at first below the solid curves and then crossover and are above the solid curves. The lower dotted curves before the crossover show that the dampening is making these smaller declines more likely. The probability on the vertical axis at which a dotted curve crosses to the left of a solid curve shows the chances that dampening will cause a larger decline.
At year 12, the crossover probability is far below the probability of a decline showing that there is very little chance that dampening will cause any increase in declines at the front of the period. At years 24 and 36, however, the crossover probability has moved up closer to the probability of a decline. At the back of the period, dampening is more likely than not to increase the size of any declines that occur. The size of these increases on average is from about 15% to 20%. For a 3.5% initial disbursement, however, the chances of these increases in the declines is less than .05. Given the decreasing chances that the beneficiary will survive until later in the period, the risk that dampening will cause larger declines at the back of the period may be easy to accept to minimize any problems at the front of the period.
Suppose now that the initial disbursement is increased from 3.5% to 4.0% or 4.5%. The results in these cases are shown in Charts 22 and 23. Dampening has now moved the crossover probability up closer to the probability of a decline. Thus, dampening is now more likely to increase the size of any declines that may occur. Even at year 12, dampening has a .10 chance of increasing the size of declines when starting at 4.5%. Larger declines have become a concern at the front of the period as well as the back. At the back of the period the increases in the declines caused by dampening continue on average to be about 15% or 20%. Now, however, the chances of these increases occurring are significantly higher. By the end of the period when the initial disbursement is 3.5% the chance of an increase is only .05. When the initial disbursement increases to 4.0% that chance rises to .13, and to .25 for starting at 4.5%. For higher initial disbursements the risk of larger declines caused by dampening has become more difficult to accept to reduce earlier problems.
Note
As a variation on the basic model 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 and is called the Annual Correction Adjustment, or ACA. For b=1 this variation is the same as the basic model, and for b=0 the same as for an infinite discount rate, 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 example of dampening with a reset used in the illustration: b=.25, c=.2, and d=.1.
C(t) = F(t) + G’(t) + G”(t) + G”’(t) t = 2, 3,…, T
F(t) = If( C(t-1) <= C*(t), C(t-1), 0 ) 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. (The simulations in Pye (2017) were made with the relations as stated above, and they were so described in the text. Unfortunately, however, the last line of these relations in Pye(2017) is stated incorrectly.)
Posted November 2018 Revised July 2022