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