FAQ 824: PV Factor - How to Calculate

Problem: How are the tiered PPA PV Factors using segment rates 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 factors 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 ============================================ - When a Dx is divided by a Dy, it represents the present value of paying $1 reflecting interest and mortality at age y but paid at age x. - When an Nx is divided by a Dy, it represents the present value of a series of payments of $1 reflecting interest and mortality starting at age x and ending at the last age of the given mortality table but discounted to age y. This assumes a single payment at each integer age. The value N12x assumes monthly payments throughout the year using an 11/24ths annuity approximation. ============================================ Sample Development of a PV Factor - Data Needed ============================================ --------------------------------------------------- Participant Data --------------------------------------------------- Current Age: 54 Retirement Age (Benefit commencement age): 62 --------------------------------------------------- Plan Assumptions --------------------------------------------------- Tiered Rates and Factors from related Commutation Reports for RP26C U Mortality: Tiered Rate 1 Segment Rate: 4.75% for the first 5 years Tiered Rate 2 Segment Rate: 5.25% for the following 15 years Tiered Rate 3 Segment Rate: 5.74% for year 20 and thereafter. The plan assumes no pre-retirement mortality. --------------------------------------------------- Print Commutation Functions Reports --------------------------------------------------- Use Table Maintenance and the Functions \ Commutation Functions to generate a commutation report. - Input RP26C U as the Primary Mortality table. - For each of the 3 tiered rates, enter 4.75, 5.25, and 5.74 into the respective Interest Rate 1, 2, and 3 fields to generate three commutation reports, one for each rate. NOTE: 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. - Input Prim. Starting Age of 62 and Prim. Ending Age of 74. Press the Print button to generate the Commutation Functions reports. You will use the following commutation factors at certain ages to build the PVF result in the Calculation Steps below: From the Tiered Rate 2 (5.25%) Commutation Functions report: Dx@ Age 62 using 5.25%: 42,303.17 N12x@ Age 62 using 5.25%: 555,603.19 N12x@ Age 74 (after 20 years, the 3rd segment rate will apply) using 5.25%: 190,738.69 From the Tiered Rate 3 (5.74%) Commutation Functions report: Dx@ Age 62 using 5.74%: 31,865.67 N12x@ Age 74 using 5.74%: 131,620.17 ================================================= Sample Development of a PV Factor - Calculation Steps ================================================= The three tiered rates apply as follows from the participant's current age: - Tier Rate 1 (4.75%): Ages 54-58 (years 1 to 5) - Tier Rate 2 (5.25%): Ages 59-73 (years 6 to 20) - Tier Rate 3 (5.74%): Ages 74+ (year 21 onward) Since retirement begins at age 62 (within the Tier Rate 2 period), the calculation steps separate annuity payments into two segments: ages 62-73 (using Tier Rate 2 factors) and ages 74+ (using Tier Rate 3 factors) Step 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 = (555,603.19 - 190,738.69) / 42,303.17 = 8.62499 a. Note that by subtracting N12.74 from N12.62, this isolates the series of payments for ages 62 to 73 (the 12 years within the Tier Rate 2 period) b. You use the D62 factor since payments are discounted to the age at which the retirement (and the use of the mortality) begins. Step 2. Discount Step 1 at Tier Rate 2 interest from Retirement Age 62 to Current Age 54 = 8.62499 x (1.0525)^(54-62) = 5.72772 a. You discount at interest only because there is no pre-retirement mortality specified. Step 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 = (131,620.17) / 31,865.67 = 4.13047 a. Use the D62 factor since payments are discounted to the age at which the retirement (and the use of the mortality) begins. Step 4. Discount Step 3 at Tier Rate 3 interest from Retirement Age 62 to Current Age = 4.13047 x (1.0574)^(54-62) = 2.64293 Step 5. The total PVF factor is (Step 2 + Step 4) x 12 = (5.72772 + 2.64293) x 12 = 100.44776 The present value factor for a single-life annuity of a participant aged 54 retiring at 62 using these tiered rates is 100.4478. =========================================================== Comparison to Table Maintenance Function - Calc PV Factors Result =========================================================== Using the following inputs in the Calc PV Factors function, the resulting PVF is 100.4478: Primary Post-Retirement Mortality: RP26C U Rates are PPA Tiered: Yes Interest Rate 1: 4.75 (with Years: 5) Interest Rate 2: 5.25 (with Years: 15) Interest Rate 3: 5.74 Current Age: 54 Retirement Age: 62 The 100.4478 result matches the result you get using the manual calculation steps above.