; FAQ 906: PPA/ARP Shortfall Payment Calculation using Yield Curve Rates

FAQ 906: PPA/ARP Shortfall Payment Calculation using Yield Curve Rates

Problem:

How do you calculate the outstanding balance for the prior year shortfall amortization installments and the shortfall amortization installment for the current year when using a yield curve?


Solution:

If your results use the 15-Year ARP21 shortfall amortization method, review the ARP21 Example.

If your results use the 7-year PPA shortfall amortization method, review the PPA Example.

The 15-Year amortization method became effective for all plan years beginning in 2022, therefore, the PPA Example reflects calculations for a 2021 valuation.

ARP21 EXAMPLE
Amortization Period: 15 years
Assumptions: 1/1/2022 valuation
Yield curve: 202112 (spot rates as of 12/2021)
Spot rates:
Year 1.0: 0.60%
Year 2.0: 1.06%
Year 3.0: 1.32%
Year 4.0: 1.50%
Year 5.0: 1.68%
Year 6.0: 1.87%
Year 7.0: 2.08%
Year 8.0: 2.28%
Year 9.0: 2.45%
Year 10.0: 2.61%
Year 11.0: 2.74%
Year 12.0: 2.84%
Year 13.0: 2.92%
Year 14.0: 2.97%

2021 shortfall amortization installment (determined using 15-year amortization period) = 50,000
2022 funding shortfall = 400,000
15-year installment factor calculation reflecting payments on 1/1 using the yield curve:
Installment 1 = 1.000000
Installment 2 = (1/1.0006)^1 = 0.994036
Installment 3 = (1/1.0106)^2 = 0.979132
Installment 4 = (1/1.0132)^3 = 0.961423
Installment 5 = (1/1.0150)^4 = 0.942184
Installment 6 = (1/1.0168)^5 = 0.920073
Installment 7 = (1/1.0187)^6 = 0.894792
Installment 8 = (1/1.0208)^7 = 0.865796
Installment 9 = (1/1.0228)^8 = 0.834976
Installment 10 = (1/1.0245)^9 = 0.804252
Installment 11 = (1/1.0261)^10 = 0.772864
Installment 12 = (1/1.0274)^11 = 0.742788
Installment 13 = (1/1.0284)^12 = 0.714587
Installment 14 = (1/1.0292)^13 = 0.687864
Installment 15 = (1/1.0297)^14 = 0.663820

Sum of Installments 1 through 15 = 12.778588
Sum of Installments 1 through 14 = 12.778588 - 0.663820 = 12.114769

Present value of the 14 remaining 2021 shortfall payments: 50,000 x 12.114769 = 605,738

2022 shortfall balance = 400,000 - 605,738 = -205,738
2022 shortfall payment = -205,738 / 12.778588 = -16,100


PPA EXAMPLE

Amortization Period: 7 years
Assumptions: 1/1/2021 valuation
Yield curve: 202012 (spot rates as of 12/2020)
Spot rates:
Year 1.0: 0.26%
Year 2.0: 0.42%
Year 3.0: 0.53%
Year 4.0: 0.66%
Year 5.0: 0.85%
Year 6.0: 1.08%

2020 shortfall amortization installment = 50,000
2021 funding shortfall = 400,000
7-year installment factor calculation reflecting payments on 1/1 using the yield curve:
Installment 1 = 1.00000
Installment 2 = (1/1.0026)^1 = 0.997407
Installment 3 = (1/1.0042)^2 = 0.991653
Installment 4 = (1/1.0053)^3 = 0.984267
Installment 5 = (1/1.0066)^4 = 0.974030
Installment 6 = (1/1.0085)^5 = 0.958563
Installment 7 = (1/1.0108)^6 = 0.937581

Sum of Installments 1 through 7 = 6.843500
Sum of Installments 1 through 6 = 6.843500 - 0.937581 = 5.905919

Present value of the 6 remaining 2020 shortfall payments: 50,000 x 5.905919 = 295,296

2021 shortfall balance = 400,000 - 295,296 = 104,704
2021 shortfall payment = 104,704 / 6.843500 = 15,300