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Computing joint life refund annuities

QUESTION: How are joint life cash refund annuities computed?

ANSWER: ProAdmin determines the total present value of a 100% joint life refund annuity which is 100% joint life annuity plus an additional albeit decreasing lump sum amount. It then blends this value with a refund annuity - where the balance paid as a lump sum upon the last death of the member and beneficiary - to obtain the k% Joint life refund annuity.

Example: suppose for member age 58 and beneficiary 55, 100% monthly joint and survivor annuity payments are $1,000 ($12,000/year) and the initial guaranteed amount is $100,000.

If the participant and beneficiary both die 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 lump sum payment is viewed as decreasing life insurance payable upon the second death (of participant and beneficiary).

For this, ProAdmin first determines how many payments would completely pay off the guaranteed amount using 100% joint and survivor annuity. In this case: 100,000/1000 = 100 monthly payments or 8 years 4 months of annual payments.

Since this 100% Joint & Survivor annuity pays $1,000 per month when either, or both, the member or the beneficiary is alive, 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 year  		   Average cumulative   
  payments made  
 1  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 payment  		        Check  
  Year     Sum of average
payments made  		   Starting with 100 payments
or 8.333333 years of payments  		   Lump Sum     Sum of average
payments made +
lump sum  
  1     6,500     8.333333 - (13/24) = 7.791667     7.91667 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 = 6.791667     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     1.91667 - 1 = 0.591667     0.791667 x 12000 = 9,500     100,000  
  9     102,500          0       

General assumptions are (1) deaths occur at mid-year and (2) payments are made through the average middle of the year.  

We now need the lump value of the guaranteed amount.  For each year, the present value factor of the remaining guaranteed amount is based on an interest factor and the death of the last one, the member or the beneficiary, to survive. The present value is the sum of two amounts: one based on the beneficiary dying before the member and the other is based on the member dying before the beneficiary.  

In the case where the member dies last, for each year, we use probabilities of the beneficiary dying at any time before mid-year, the member’s survival to the beginning of the year and the member’s death during the year. We can develop the present value factors for years 1 through 8 using the following probabilities:

The steps are:

  1.      Develop factor nry based on the beneficiary's age.
  2.      Develop the present value factor in the case where the member dies last.
  3.      Sum the present values from step 2 for years 1 through 8.
  4.      Develop factor nrx based on the member's age.
  5.      Develop the present value factor in the case where the beneficiary dies last.
  6.      Sum the present values from step 5 for years 1 through 8.

For this example we are using UP84 mortality at 5% interest. 

  Step 1 - Develop annual factors nry based on the beneficiary's age.
  Year (n)     Y     qy     py     npy     1-(qx/2)     1-[npy * {1-(qy/2)}]     nr55  
1 55 0.009033 0.990967 1 0.995484 1 - (1.000000 * 0.995484) 0.004517
2 56 0.009875 0.990125 0.990967 0.995063 1 - (0.990967 * 0.995063) 0.013926
3 57 0.010814 0.989186 0.981181 0.994593 1 - (0.981181 * 0.994593) 0.024124
4 58 0.011863 0.988137 0.970571 0.994069 1 - (0.970571 * 0.994069) 0.035186
5 59 0.12952 0.987048 0.959057 0.935240 1 - (0.959057 * 0.993524) 0.047154
6 60 0.014162 0.985838 0.946635 0.992919 1 - (0.946635 * 0.992919) 0.060068
7 61 0.015509 0.984491 0.933229 0.992246 1 - (0.933229 * 0.992246) 0.074008
8 62 0.017010 0.982990 0.918755 0.991495 1 - (0.918755 * 0.991495) 0.089059

 

​

 Step 2 - Develop annual present value factors in the case where the member (x) dies last.
Year (n) x y # of annual payments in lump sum. Mortality factor Interest Calculation:
Number of payments * (n-1)px * qx * nry * interest 		Yearly Present Value
1 58 55 7.791667 px * qx * ry v(13/24) 7.791667 (1.000000) (0.011863) (0.004517) (0.973918) 0.000407
2 59 56 6.791667 1px * qx+1 * 1ry v(1+(13/24)) 6.791667 (0.988137) (0.012952) (0.013926) (0.927541) 0.001123
3 60 57 5.791667 2px * qx+2 * 2ry v(2+(13/24)) 5.791667 (0.975339) (0.014162) (0.024124) (0.883372) 0.001705
4 61 58 4.791667 3px * qx+3 * 3ry v(3+(13/24)) 4.791667 (0.961526) (0.015509) (0.035186) (0.841307) 0.002115
5 62 59 3.791667 4px * qx+4 * 4ry v(4+(13/24)) 3.791667 (0.946614) (0.017010) (0.047154) (0.801245) 0.002307
6 63 60 2.791667 5px * qx+5 * 5ry v(5+(13/24)) 2.791667 (0.930512) (0.018685) (0.060068) (0.763090) 0.002225
7 64 61 1.791667 6px * qx+6 * 6ry v(6+(13/24)) 1.791667 (0.913125) (0.020517) (0.074008) (0.726753) 0.001805
8 65 62 0.791667 7px * qx+7 *7ry v(7+(13/24)) 0.791667 (0.894391) (0.022562) (0.089059) (0.692145) 0.000985
 Step 3 - Sum the yearly present values for years 1 through 8. Sum = 0.012671

 

 Step 4 - Develop annual factors nrx based on the member's age.  
Year (n) x qx px npx 1 - (qx/2) 1-[npx * {1 - (qx/2)}] nr58
1 58 0.011863 0.988137 1.000000 0.994069 1 - (1.000000 * 0.994069) 0.005931
2 59 0.012952 0.987048 0.988137 0.993524 1 - (0.988137 * 0.993524) 0.018262
3 60 0.014162 0.985838 0.975339 0.992919 1 - (0.975339 * 0.992919) 0.031568
4 61 0.015509 0.984491 0.961526 0.992246 1 - (0.961526 * 0.992246) 0.045930
5 62 0.017010 0.982990 0.946614 0.991495 1 - (0.946614 * 0.991495) 0.061437
6 63 0.018685 0.981315 0.930512 0.990658 1 - (0.930512 * 0.990658) 0.078182
7 64 0.020517 0.979483 0.913125 0.989742 1 - (0.913125 * 0.989742) 0.096242
8 65 0.022562 0.977438 0.894391 0.988719 1 - (0.894391 * 0.988719) 0.115699

 

 Step 5 - Develop the present value factor in the case where the beneficiary (y) dies last.
Year (n) x y # of annual payments in lump sum. Mortality factor Interest Calculation:
Number of payments * (n-1)py * qy * nrx * interest 		Yearly Present Value
1 58 55 7.791667 py * qy * rx v(13/24) 7.791667 (1.000000) (0.009033) (0.005931) (0.973918) 0.000407
2 59 56 6.791667 1py * qy+1 * 1rx v(1+(13/24)) 6.791667 (0.990967) (0.009875) (0.018262) (0.927541) 0.001126
3 60 57 5.791667 2py * qy+2 * 2rx v(2+(13/24)) 5.791667 (0.981181) (0.010814) (0.031568) (0.883372) 0.001714
4 61 58 4.791667 3py * qy+3 * 3rx v(3+(13/24)) 4.791667 (0.970571) (0.011863) (0.045930) (0.841307) 0.002132
5 62 59 3.791667 4py * qy+4 * 4rx v(4+(13/24)) 3.791667 (0.959057) (0.012952) (0.061437) (0.801245) 0.002319
6 63 60 2.791667 5py * qy+5 * 5rx v(5+(13/24)) 2.791667 (0.946635) (0.014162) (0.078182) (0.763090) 0.002233
7 64 61 1.791667 6py * qy+6 * 6rx v(6+(13/24)) 1.791667 (0.933229) (0.015509) (0.096242) (0.726753) 0.001814
8 65 62 0.791667 7py * qy+7 *7rx v(7+(13/24)) 0.791667 (0.918755) (0.017010) (0.115699) (0.692145) 0.000991
 Step 6 - Sum the yearly present values for years 1 through 8. Sum = 0.012734

The sum of the present values (0.012671 + 1.012735 = 0.025406) constitutes the total present value factor for the additional amount needed due to the guaranteed amount. The total present value is 100% joint life annuity factor of 14.705790 plus the additional present value factor: 14.705790 + 0.025406 = 14.731195. The additional amount needed to cover this life insurance is 0.025406 x 12000 = $ 304.86.

When computing other survivor percentages (k) for the joint life refund, ProAdmin uses the k% joint life annuity factor plus an additional factor. The additional factor is an interpolation between two present value factors:

  1. factor for the remaining guaranteed amount based on the member’s single life refund annuity, using interpolation factor (1-k)
  2. factor for the remaining guaranteed amount based on the 100% Joint life refund annuity, using interpolation factor k.

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.