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Example: Annually using beginning of plan year factors

QUESTION: How are late retirements computed when calculations are performed annually using factors from the beginning of the plan year (BOPY)?

ANSWER:. Calculations performed Annually using factors from the Beginning of plan year use the actuarial basis as of the calculation date and compute two annuities:  one is an immediate annuity for the member's age at the beginning of the plan year, and the other is a deferred annuity from the member's age at the beginning of the plan year to the member's age at the calculation date.  The ratio of the immediate over the deferred annuity provides the incremental increase factor for the value of the benefit at the end of the previous plan year.

An example calculation for a member who wishes to retire on 11/1/2022 is presented below.  The basis for late retirement is PPA assumptions which use the required 417(e) mortality table and quarterly interest rates.  We shall examine each line of the calculation to understand how the calculation progresses. 

Benefit Formula Components
Type: Late Retirement - Calculations performed annually at BOPY
    Member Age at Accrued PPA Mort Table Immediate Deferred Annuity Incremental Late Benefit
Line Calc Date Calc Date BOPY Benefit PPA Interest Rates Annuity at BOPY from BOPY to Calc Date Retirement Factor Component
1  04/01/2017   65y   65y   1000   2017 1.79 - 3.80 - 4.71%   13.521544   13.521544   1   1000.00 
2  12/31/2017   65y 9m   65y   1000   2017 1.96 - 3.77 - 4.62%   13.552576   12.812149   1.057791   1057.79 
3  12/31/2018   66y 9m   65y 9m  1000  2018 3.08 - 4.19 - 4.58%  13.358633 12.374212 1.079409 1141.79
4  07/12/2019   67y 4m   66y 9m  1000  2019 3.01 - 4.11 - 4.41%  13.082554 12.509463 1.045813 1194.10
5  12/31/2019   67y 9m   66y 9m  1000  2019 2.72 - 3.76 - 4.33%  13.415572 12.431887 1.079126 1232.13
7  12/31/2020  68y 9m 67y 9m 1000  2020 1.08 - 2.78 - 3.47%  14.282191 13.291837 1.074508 1323.94
8  12/31/2021  69y 9m 68y 9m 1000  2021 0.61 - 2.84 - 3.54%  13.815013 12.823026 1.077360 1426.36
9  10/29/2022  70y 7m 69y 7m 1000  2022 3.23 - 4.59 - 4.69%  11.614066 10.797159 1.075659 1534.28
10  11/01/2022  70y 9m 69y 9m 1000  2022 3.23 - 4.59 - 4.69%  11.614066 10.633777 1.092186 1557.85

The immediate and deferred for one year annuities in the chart below will be used in each line of the chart above. It is assumed that interpolating integral annuity factors for non-integral ages are understood.  We shall focus on determining the deferred annuities.

Administration factors
 Age  417(e) table; PPA interst rates Immediate annuity; Immediate annuity for age + 1   One-year deferred annuity   One-year deferred annuity for age + 1;
65 2017  1.79%   3.80%   4.71%  13.521544 13.174347 12.533564 12.186950
65 2017 1.96% 3.77% 4.62% 13.552576 13.201367 12.565340 12.214713
65 2018 3.08% 4.19% 4.58% 13.604497 13.274279 12.621572 12.291759
66 2019 3.01% 4.11% 4.41% 13.341842 12.996125 12.359098 12.013784
66 2019 2.72% 3.76% 4.33% 13.683568 13.326240 12.699580 12.342656
67 2020 1.08% 2.78% 3.47% 14.603709 14.175019 13.613022 13.184776
68 2021 0.61% 2.84% 3.54% 14.139042 13.707004 13.146689 12.715139
69 2022 3.23% 4.59% 4.69% 11.876422 11.526614 10.895740 10.546456

The next charts examine each line of the calculation.  Note that, for purposes of the calculation, beginning of the plan year is the later of the member's NRD and the actual beginning of the plan year.  Line 1 illustrates this rule.

Line 1: The member has reached Normal Retirement age of 65.
 Calc date   Age at
 Calc Date 		  Plan Year 
 Begin Date 		  Age at 
 Plan Year 
 Begin Date 		  Immediate Annuity 
 for Age at Plan  
> Year Begin Date 		  Period 
 Between Ages 		  Deferred Annuity for 
 age at Plan Year 
 Begin Date deferred 
 for the period 		  Incremental Late 
 Retirement Factor 		  Benefit at Calc Date 
 04/01/2017  65y  04/01/2017  65y 13.521544 0 months 13.521544 1 1,000.00
Plan year begin date is the later of the actual plan year begin date of 01/01/2017 and NRD of 04/01/2017.


Line 2:
 Calc date   Age at
 Calc Date 		  Plan Year 
 Begin Date 		  Age at 
 Plan Year 
 Begin Date 		  Immediate Annuity 
 for Age at Plan  
> Year Begin Date 		  Period 
 Between Ages 		  Deferred Annuity for 
 age at Plan Year 
 Begin Date deferred 
 for the period 		  Incremental Late 
 Retirement Factor 		  Benefit at Calc Date 
The annuities changed from the values in line 1 since the basis changed.
12/31/2017 65y 9m 04/01/2017 65y 13.552476 9 months 12.812149 1.057791 1.057.79
The deferred annuity is derived by interpolating values between two annuities: the immediate annuity and the 1 year deferred annuity Immediate Annuity:
65y = 13.552476		 Deferred 1y annuity:
1| 65y = 12.565340		 Deferred 9 month annuity from age 65 0.75| 65 =
[(3/12) * 13.552476] +
[(9/12) * 12.565340]
= 12.812149


Line 3: 
 Calc date   Age at
 Calc Date 		  Plan Year 
 Begin Date 		  Age at 
 Plan Year 
 Begin Date 		  Immediate Annuity 
 for Age at Plan  
> Year Begin Date 		  Period 
 Between Ages 		  Deferred Annuity for 
 age at Plan Year 
 Begin Date deferred 
 for the period 		  Incremental Late 
 Retirement Factor 		  Benefit at Calc Date 
12/31/2017 Benefit at previous date 1,057.79
12/31/2018 66y 9m 01/01/2018 65y 9m 13.356833 1 year 12.374212 1.079409 1,141.79
The deferred annuity is derived by interpolating values between two deferred annuities: 1 year deferred annuity at age 65 and 1 year  deferred annuity at age 66. Deferred 1y annuity
age 65:
1|65y = 12.621572		 Deferred 1y annuity
age 66:
1|66 = 12.291759		 Deferred 1y annuity
from age 65y 9m:
1|65y 9m =
[(3/12) * 12.621572] +
[(9/12) * 12.291759]
= 12.374212


Line 4: 
 Calc date   Age at
 Calc Date 		  Plan Year 
 Begin Date 		  Age at 
 Plan Year 
 Begin Date 		  Immediate Annuity 
 for Age at Plan  
> Year Begin Date 		  Period 
 Between Ages 		  Deferred Annuity for 
 age at Plan Year 
 Begin Date deferred 
 for the period 		  Incremental Late 
 Retirement Factor 		  Benefit at Calc Date 
12/31/2018 Benefit at previous date 1,141.79
07/12/2019 67y 4m 01/01/2019 66y 9m 13.082554 7 months 12.509463 1.045813 1,194.10
The value for the deferred annuity is derived by first interpolating values between two deferred annuities: 1 year deferred annuity at age 66 and 1 year deferred annuity at age 67 to provide a 1 year deferred annuity for age 66y 9m; Deferred 1y annuity
age 66:
1|66y = 12.359098		 Deferred 1y annuity
age 67:
1|67 = 12.013784		 1year deferred annuity
from age 66y 9m:
1|65y 9m
=[(3/12) * 12.359098] +
[(9/12) * 12.013784]
= 12.100113
Then interpolating between the immediate annuity and new deferred annuity for a deferral of 7 months. Deferred 1y annuity
age 66:
1|66y = 12.359098		 Deferred 1y annuity
age 66y 9m:
1|66y9m = 12.100113		 7 month deferred annuity
from age 66y 9m:
(7/12)|65y 9m =
[(5/12) + 13.359098] +
[(7/12) * 12.100113]
= 12.509463


Lines 5, 6 and 7: Because the deferral period is one year in all cases, the lines will be combined in one chart
 Calc date   Age at
 Calc Date 		  Plan Year 
 Begin Date 		  Age at 
 Plan Year 
 Begin Date 		  Immediate Annuity 
 for Age at Plan  
> Year Begin Date 		  Period 
 Between Ages 		  Deferred Annuity for 
 age at Plan Year 
 Begin Date deferred 
 for the period 		  Incremental Late 
 Retirement Factor 		  Benefit at Calc Date 
12/31/2019 Benefit at previous date 1,194.10
5-12/31/2019 67y 9m 01/01/2019 66y 9m 13.415572 1 year 12.431887 1.079126 1,232.13
The deferred annuity is derived by interpolating values between two deferred annuities: 1 year deferred annuity at age 66 and 1 year deferred annuity at age 67. Deferred 1y annuity
age 66:
1|66 = 12.699580		 Deferred 1y annuity
age 67:
1|67 = 12.342656		 Deferred 1y annuity
from age 65y 9m:
1|65y 9m =
[(3/12) * 12.699580] +
[(9/12) * 12.342656]
= 12.431887
6-12/31/2020 68y 9m 01/01/2020 67y 9m 14.282191 1 year 13.291837 1.074508 1,323.94
The deferred annuity is derived by interpolating values between two deferred annuities: 1 year deferred annuity at age 67 and 1 year deferred annuity at age 68. Deferred 1y annuity
age 66:
1|67 = 13.613022		 Deferred 1y annuity
age 67:
1|68 = 13.184776		 Deferred 1y annuity
from age 65y 9m:
1|65y 9m =
[(3/12) * 13.613022] +
[(9/12) * 13.184776]
= 13.291837
7-12/31/2021 69y 9m 01/01/2021 68y 9m 13.815013 1 year 12.823026 1.077360 1,426.36
The deferred annuity is derived by interpolating values between two deferred annuities: 1 year deferred annuity at age 68 and 1 year deferred annuity at age 69. Deferred 1y annuity
age 66:
1|68 = 13.146689		 Deferred 1y annuity
age 67:
1|69 = 12.715139		 Deferred 1y annuity
from age 65y 9m:
1|65y 9m =
[(3/12) * 13.146689] +
[(9/12) * 12.715139]
= 12.823027


Line 8: 
 Calc date   Age at
 Calc Date 		  Plan Year 
 Begin Date 		  Age at 
 Plan Year 
 Begin Date 		  Immediate Annuity 
 for Age at Plan  
> Year Begin Date 		  Period 
 Between Ages 		  Deferred Annuity for 
 age at Plan Year 
 Begin Date deferred 
 for the period 		  Incremental Late 
 Retirement Factor 		  Benefit at Calc Date 
12/31/2021 Benefit at previous date 1,
10/29/2022 70y 7m 01/01/2019 69y 9m 11.614066 10 months 10.797159 1.075659 1,534.28
The deferred annuity is derived by first interpolating values between two deferred annuities: 1 year deferred annuity at age 66 and 1 year deferred annuity at age 67 to provide a 1 year deferred annuity for age 66y 9m; Deferred 1y annuity
age 66:
1|66y = 12.359098		 Deferred 1y annuity
age 67:
1|67 = 12.013784		 1 year deferred annuity
from age 66y 9m:
1|65y 9m
=[(3/12) * 10.895740] +
[(9/12) * 10.546456]
= 10.633777
Then interpolating between the immediate annuity and new deferred annuity for a deferral of 7 months. Deferred 1y annuity
age 66:
1|66y = 12.359098		 Deferred 1y annuity
age 66y 9m:
1|66y9m = 10.633777		  10 month deferred annuity
from age 66y 9m:
(10/12)|65y 9m =
[(2/12) * 11.614066] +
[(10/12) * 10.633777]
= 10.797159