ANSWER: ProAdmin determines the total present value of a life annuity plus an additional amount (lump sum) or additional amounts (installments).
Example: suppose monthly life annuity payments are $1,000 (12,000/year) and the initial guaranteed amount is $100,000.
Lump Sum (also known as Modified Cash Refund)
If a participant dies after receiving some monthly payments, but not enough to cover the guaranteed amount, then a lump sum payment is made to provide the difference (between the guaranteed amount and the payments already made). This concept is viewed as decreasing life insurance which becomes zero once the guaranteed amount has been paid in full.
For this, ProAdmin first determines how many payments would completely pay off the guaranteed amount. In this case: 100,000/1000 = 100 monthly payments or 8 years 4 months of annual payments.
If the annuity is paid monthly at the beginning of the month, then ProAdmin uses the factor 13/24 to determine the average number of payments made during the first year. In the second year the average total number of payments is: 1 + (13/24). "On average" the cumulative payments each year are:
Year Monthly averaging factor
for the first yearAverage cumulative
payments made1 13/24 6,500 2 18,500 3 30,500 4 42,500 5 54,500 6 66,500 7 78,500 8 90,500 9 102,500
Next, ProAdmin computes the number of payments remaining to be made, which when summed-up with the payments made, provide the guaranteed lump sum.
Number of annual payments that
comprise the lump sum paymentCheck Year Sum of average
payments madestarting with 100 payments or
8.333333 years of paymentsLump sum sum of average
payments made +
lump sum1 6,500 8.333333 - (13/24) = 7.791667 7.791667 x 12000 = 93,500 100,000 2 18,500 7.791667 - 1 = 6.791667 6.791667 x 12000 = 81,500 100,000 3 30,500 6.791667 - 1 = 5.791667 5.791667 x 12000 = 69,500 100,000 4 42,500 5.791667 - 1 = 4.791667 4.791667 x 12000 = 57,500 100,000 5 54,500 4.791667 - 1 = 3.791667 3.791667 x 12000 = 45,500 100,000 6 66,500 3.791667 - 1 = 2.791667 2.791667 x 12000 = 33,500 100,000 7 78,500 2.791667 - 1 = 1.791667 1.791667 x 12000 = 21,500 100,000 8 90,500 1.791667 - 1 = 0.791667 0.791667 x 12000 = 9,500 100,000 9 102,500
0
We can see that after 8 years and 4 months of monthly payments, the guaranteed amount has been remunerated.
Number of annual
payments for
Development of Present
Value factorsYear lump sum Mortality Interest Yearly Present Value
1 7.791667 qx v(13/24))
7.916667(qx)(v(13/24)) 2 6.791667 px * qx+1 v(1+(13/24)))
6.791667(px * qx+1)(v(1+(13/24))) 3 5.791667 2px * qx+2 v(2+(13/24)))
5.791667(2px * qx+2)(v(2+(13/24))) 4 4.791667 3px * qx+3 v(3+(13/24)))
4.791667(3px * qx+2)(v(3+(13/24))) 5 3.791667 4px * qx+4
v(4+(13/24)))
3.791667(4px * qx+2)(v(4+(13/24))) 6 2.791667 5px * qx+5
v(5+(13/24)))
2.791667(5px * qx+2)(v(5+(13/24))) 7 1.791667 6px * qx+6
v(6+(13/24)))
1.791667(6px * qx+2)(v(6+(13/24))) 8 0.791667 7px * qx+7
v(7+(13/24)))
0.791667(7px * qx+2)(v(6+(13/24))) 9
The sum of the present values (last column) constitutes the total present value factor for the additional amount needed due to the guaranteed amount.
Using the UP 84 mortality table and 5% interest and participant age 58, the values are:
Year Present Value factor With values
Yearly Present Value 1 7.916667(qx)(v(13/24)) 7.791667(.011863)(.973918) 0.090022 2 6.791667(px * qx+1)(v(1+(13/24))) 6.791667(.012798)(.927541)
0.080624
3 5.791667(2px * qx+2)(v(2+(13/24))) 5.791667(.013813)(.883372)
0.070669
4 4.791667(3px * qx+3)(v(3+(13/24))) 4.791667(.014912)(.841307)
0.060115
5 3.791667(4px * qx+4)(v(4+(13/24))) 3.791667(.016102)(.801245)
0.048918
6 2.791667(5px * qx+5)(v(5+(13/24))) 2.791667(.017387)(.763090)
0.037039
7 1.791667(6px * qx+6)(v(6+(13/24))) 1.791667(.018735)(.726753)
0.024394
8 0.791667(7px * q x+7 )( v(7+(13/24)) ) 0.791667(.020179)(.692145)
0.011057
9
Sum = 0.422838
The additional amount needed to cover this life insurance is 0.422838 x 12000 = $ 5,074.06.
The Detailed Results of the refund annuity with lump sum as benefit for any remaining balance, show the following items:
Item Heading Description 1 Commencement Date Annuity commencement date 2 Eligible ? Yes if the member is eligible for payment on that date; No otherwise. 3 Actual Member Age ProAdmin's internal age computation 4 Form Member Age Age per the applicable Age Definition. Factors are based on this age. 5 Interest Rate Applicable Interest rate: either the scalar rate, the first segment rate if using spot rates, or the immediate rate if using PBGC-style rates. 6 Guaranteed Amount Starting guaranteed amount 7 Annuity Form Value Single life annuity at the Form Member Age (item 4) 8 Life Ins. Form Value Value of the decreasing life insurance at the Form Member Age (item 4) 9 Total Ann+Ins Form Value (a) Item 7 + item 8 10 Normal Form (b)
Normal Form factor 11 Conversion Factor (b) / (a); when the refund annuity is the normal form, this factor is 1. 12 Normal Form Benefit The annual benefit payable under the Benefit Definition normal form of payment 13 Member Benefit Item 11 * Item 12 14 Relative value information, if any, follows
Installments
When the remaining balance to be paid in installments with the guaranteed period rounded up to one month (or year), the value of the total benefit is computed as a life annuity plus an annuity certain.One point to note: The value of the refund annuity with installments might be slightly larger than the value of the refund annuity with lump sum. This is because the determination of the period for the installments (=Initial guaranteed amount/payment amount) is rounded up to the next period and this slight increase produces a slightly higher present value.
The Detailed Results of the refund annuity with installments for any remaining balance, show the following items:
Item Heading Description 1 Commencement Date Annuity commencement date 2 Eligible ? Yes if the member is eligible for payment on that date; No otherwise. 3 Actual Member Age ProAdmin's internal age computation 4 Form Member Age Age per the applicable Age Definition. Factors are based on this age. 5 Interest Rate Applicable Interest rate: either the scalar rate, the first segment rate if using spot rates, or the immediate rate if using PBGC-style rates. 6 Guaranteed Amount (c) Starting guaranteed amount 7 Guaranteed Period (c) / (d) Certain period rounded up to the month (or year)
8 Annuity Form Value Member Age
Life annuity with period certain for Form Member Age (item 4) last birthday 9 Form Value Member Age +1 Life annuity with period certain for Form Member Age (item 4) next birthday 10 Form Value (a) Linear interpolation between item 8 and item 9 11 Normal Form (b) Normal form factor 12 Conversion Factor (b)/(a); when the refund annuity is the normal form, this factor is i. 13 Normal Form Benefit (d) The annual benefit payable under the Benefit Definition normal form of payment 14 Member Benefit Item 12 * item 13 15 Relative value information, if any, follows
The selection of installments as the payment form for the balance of the guaranteed amount is simply a continuation of the same annual amount until the guaranteed amount is exhausted. The number of payments needed to pay off the guaranteed amount is the guaranteed amount / payment amount (rounded up to the next month or year).
General assumptions are (1) deaths occur at mid-year and (2) payments are made through the average middle of the year.