; FAQ 824: PV Factor - How to Calculate

FAQ 824: PV Factor - How to Calculate

Problem:

How are the tiered PV Factors calculated under Table Maintenance?


Solution:

The details for calculating a tiered PV Factor for a single life annuity based on a sample participant and sample plan are illustrated below. The illustration will be based on an understanding of commutation functions since they are included in the Commutation Functions report and will be used to check a participant's PVF. A brief explanation of commutation functions is provided here.

Overview of Commutation Factors
Dx represents the value of paying $1 reflecting interest and mortality at age x.
Nx represents the value of a series of payment of $1 reflecting interest and mortality starting at age x and ending at the last age of the given mortality table. This assumes a single payment at the integer age. The value N12x assumes monthly payments throughout the year.


Sample development of a PV Factor
Participant Data
Current Age: 54
Retirement Age (Benefit commencement age): 62
Plan Assumptions
Tiered Rates and Factors from related Commutation Reports for RP21C U Mortality:
Tiered Rate 1 Segment Rate: 3.92% for the first 5 years
Tiered Rate 2 Segment Rate: 4.46% for the following 15 years
Tiered Rate 3 Segment Rate: 4.88% for year 20 and thereafter.

The plan assumes no pre-retirement mortality.

You will use Table Maintenance and the Functions \ Commutation Functions to generate a new commutation report for each of the 3 tiered rates. For example, for the second tiered rate of 4.46%, you will print a Commutation Function report using the RP21C U mortality table, Interest Rate 1, 2, and 3 set to 4.46 and age range of 54 to 119. Repeat for the 3rd tiered segment rate. In this example, you do not need to generate a 1st Tiered Rate Commutation Functions report because there are no annuity payments within the first 5 years. For each Commutation Functions report, you will find the commutation factors at certain ages to build the PVF result.

From the Tiered Rate 2 (4.46%) Commutation Functions report:
Dx@ Age 62 using 4.46%: 66,703.52
N12x@ Age 62 using 4.46%: 944,795.58
N12x@ Age 74 (after 20 years, the 3rd segment rate will apply) using 4.46%: 348,193.60

From the Tiered Rate 3 (4.88%) Commutation Functions report:
Dx@ Age 62 using 4.88%: 52,221.21
N12x@ Age 74 using 4.88%: 252,390.89


Calculation Steps:
1. Calculate the value of monthly payments using interest and mortality (you will use the N12x and Dx values) from age 62 to 73 discounted to age 62 all using the Tier Rate 2 = (N12.62 - N12.74) / D62 = (944,795.58 - 348,193.60) / 66,703.52 = 8.94409
a. Note that by subtracting N12.74 from N12.62, this leaves you with only the series of payments for ages 62 to 73
b. You use the D62 factor since payments are discounted to the age at which the retirement (and the use of the mortality) begins.

2. Discount Step 1 at Tier Rate 2 interest from Retirement Age 62 to Current Age 54 = 8.94409 x (1.0446)^(54-62) = 6.30864
a. You discount at interest only because there is no pre-retirement mortality specified.

3. Calculate the value of monthly payments using interest and mortality from age 74 to perpetuity discounted to age 62 using the Tier Rate 3 = (N12.74) / D62 = (252,390.89) / 52,221.21 = 4.83311
a. Use the D62 factor since payments are discounted to the age at which the retirement (and the use of the mortality) begins.

4. Discount Step 3 at Tier Rate 3 interest from Retirement Age 62 to Current Age = 4.83311 x (1.0488)^(54-62) = 3.30130

5. The total PVF factor is (Step 2 + Step 4) x 12 = (6.30864 + 3.30130) x 12 = 115.3193

The present value factor for a single-life annuity of a participant aged 54 retiring at 62 using these tiered rates is 115.3193.

The 115.3193 is the same result you would get by using the Table Maintenance Calc PV Factors function given the same inputs for the participant and the plan assumptions.