1. Unfunded Actuarial Liabilities For a Defined Benefit Plan: UAL t = AL t – F t ”Funding Ratio” = F t / AL t We can calculate this Unfunded Actuarial Liability at any point in time It is often useful to project what we think the UAL will be in a year if all assumptions play out as expected UAL t+1 = AL t+1 – F t+1 Next year’s Asset Fund, F t+1, can be thought of as the following pieces: F t+1 = F t + I + C – P, where I = Interest earnings on the fund during the year C = Dollars contributed to the fund during the year P = Purchases of annuities for people leaving the plan during the year AL t+1 = AL t plus the pieces that bring it forward one year (NC, interest, decrement, etc)

2. Unfunded Actuarial Liabilities We can use the expectations about next year’s UAL to determine what gains or losses we might have on the defined benefit plan In general, UAL t+1 = UAL t (1 + i) - (Difference in interest actually earned LESS expected interest earned on Fund) - (Contributions actually made LESS Normal Costs expected in Actuarial Liability calculations) - (Reductions in liabilities due to actual terminations due to death or service LESS expected reductions for death or service) - (Reductions in liabilities due to actual retirements LESS expected retirements) So…. If all actuarial assumptions go as planned (interest, terminations, etc) then UAL t+1 = UAL t (1 + i)

3. Unfunded Actuarial Liabilities Typically, one or more of the assumptions will not play out as expected (or “can’t” due to fractional deaths, etc) So the plan will have an Actuarial Gain or Actuarial Loss. The Gain or Loss will comprised of the interest and liability components in the definition of UAL t+1 Actuarial Gain = (Difference in interest actually earned LESS expected interest earned on Fund) + (Reductions in liabilities due to actual terminations due to death or service LESS expected reductions for death or service) + (Reductions in liabilities due to actual retirements LESS expected retirements) The contribution piece is not part of this since it is primarily controlled by the plan sponsor.

4. Unfunded Actuarial Liabilities Alternatively, in terms of Unfunded Actuarial Liabilities, UAL t+1 = UAL t (1 + i) - (Contributions actually made LESS Normal Costs expected in Actuarial Liability calculations) - Actuarial Gain Actuarial Gain = UAL t (1 + i) - UAL t+1 - (C – NC t (1 + i) + I C ) Actuarial Gain = (UAL t + NC t ) (1 + i) – C – I C – UAL t+1 where I C is any additional interest generated from contributions made to the fund during the year

5. Unfunded Actuarial Liabilities Example – SOA Course 5 Examination, Problem No. 6 Benefit = 1.5% FS YOS for 0 < YOS < 10 +2.0% FS (YOS – 10) for 10 < YOS i = 6%; s = 4%; No pre-retirement decrements; r = 65; Retirement Annuity Factor = 12 Assets on 1/1/2011 = $300,000 Assets on 1/1/2012 = $320,000 Contributions on 12/31/2011 = $5,000 PUC Funding Method; Two Employees Employee A: entered plan on 1/1/2001 with e = 30; x = 40; S 40 = $30,000 Employee B: entered plan on 1/1/1981 with e = 30; x = 60; S 60 = $50,000 Actuarial Liability as of 1/1/12 = $350,000 Determine UAL as of 1/1/2011 & Actuarial Gain during 2011

6. Unfunded Actuarial Liabilities UAL as of 1/1/2011: Actuarial Liability for Employee A on 1/1/2011 =B 40 Retirement Annuity Discount from retirement back to age 40 =[1.5% (30,000)(1.04) 24 10] 12 (1.06) -25 =$32,351.30 Actuarial Liability for Employee B on 1/1/2011 =B 60 Retirement Annuity Discount from retirement back to age 60 =[[1.5% (50,000)(1.04) 4 10] + [2.0% (50,000)(1.04) 4 20]] 12 (1.06) -5 =$288,481.51 Total Plan Liability = $32,351.30 + $288,481.51 = $320,732.81 UAL as of 1/1/2011 = $320,732.81 - $300,000 = $20,732.81

7. Unfunded Actuarial Liabilities Actuarial Gain during 2011: Actuarial Gain = (UAL t + NC t ) (1 + i) – C – I C – UAL t+1 Normal Cost for Employee A during 2011 = 2.0% (30,000)(1.04) 24 1 12 (1.06) -25 =$4,300.17 Normal Cost for Employee B during 2011 =2.0% (50,000)(1.04) 4 1 12 (1.06) -5 =$10,490.24 Total Plan Normal Cost = $4,300.17 + $10,490.24 = $14,790.41

8. Unfunded Actuarial Liabilities Actuarial Gain during 2011: Actuarial Gain = (UAL t + NC t ) (1 + i) – C – I C – UAL t+1 Actuarial Gain = ($20,732.81 + $14,790.41)(1.06) – 5000 - 0 - ($350,000 - $320,000)a Actuarial Gain = $2,654.61

9. Actuarial Gains and Losses For a Defined Benefit Plan: Total Experience Gain = Investment Gain + Liability Gain = Earning more than assumed/expected on asset fund + Having better than expected experience on mortality, salary, withdrawal assumptions, etc

10. Actuarial Gains and Losses Investment Gain = Actual Asset Fund at End of Year – Expected Asset Fund at End of Year = Act F – Exp F Liability Gain = Expected Actuarial Liability at End of Year – Actual Actuarial Liability at End of Year = Exp AL – Act AL Total Experience Gain = (Act F – Exp F) + (Exp AL – Act AL)

11. Actuarial Gains and Losses Example: Annual Retirement Benefit = $600 per Year of Service Funding Method = TUC Actual fund performance = 10% versus 5% expected in Commutation Function Sheet 2 employees with e = 25 and x = 60; r = 65 1 dies during the year Asset Fund at BOY = $80,000; Normal Cost paid at BOY = $6,000 Calculate Total Experience Gain = (Act F – Exp F) + (Exp AL – Act AL)

12. Actuarial Gains and Losses Total Act AL under TUC at x = 60 is 2*(600)(35)(7.94)(.5393) = $179,845.76 Expected AL under TUC at x = 61 is $179,845.76 increased for interest, survival and next payment of Normal Cost = $208,863.33 Actual AL under TUC at x = 61 is 1*(600)(36)(7.94)(.6088) = $104,411.63 Liability Gain = $208,863.33 - $104,411.63 = $104,451.70

13. Actuarial Gains and Losses Act F is (80,000 + 6,000) * 1.10 = $94,600 Exp F is (80,000 + 6,000) * 1.05 = $90,300 Investment Gain = $94,600 - $90,300 = $4,300 Total Experience Gain = $4,300 + $104,451.70 = $108,751.70

14. Individual Level Premium We have been talking over the last classes about people entering a defined benefit pension plan that already exists However, sometimes employees are already in service when a plan is introduced Two ways to handle this: –Create a initial Supplemental Liability and calculate normal costs from the entry date (like under EANLD) –Start with no initial liability and fund at a more rapid pace from the inception date forward

15. Individual Level Premium The ILP cost method is commonly used for the second of the two methods Normal costs are derived from plan inception through retirement, and account for already accrued service PVFNC = PVFB NC (ä (T) a:r-a| ) = B r (D (T) r / D (T) a ) ä r (12) Rearrange and solve for NC NC [(N (T) a - N (T) r ) / D (T) a ] = B r (D (T) r / D (T) a ) ä r (12)

16. Individual Level Premium The ILP cost method is commonly used for the second of the two methods Normal costs are derived from plan inception through retirement, and account for already accrued service AL x = PVFB x - PVFNC x AL x = [B r (D (T) r / D (T) x ) ä r (12) ] - NC (ä (T) x:r-x| ) AL x = [B r (D (T) r / D (T) x ) ä r (12) ] - NC (N (T) x - N (T) r ) / D (T) x

17. Aggregate Cost Methods As we discussed early in this section of the class, funding can be done on an individual basis or on an aggregate basis All that we’ve discussed so far has been individual methods: TUC PUC EANLD EANLP ILP We’ll take a quick look at a couple aggregate methods

18. Individual Aggregate IA method (an oxymoron?) has components that are both individual and aggregate in nature Individual: Normal Cost for entire plan is still the sum of normal costs of each individual Aggregate: Normal Cost for each individual is affected by the amount of excess funding that exists in the plan as a whole

19. Individual Aggregate Back to our “Entrepreneurial Company” exercise: We all begin as new employees 10 years ago, now we are established enough to create a defined benefit plan New DB plan recognizes past service credits What are different ways to fund the plan?

20. Individual Aggregate Use a purely Individual Method, calculate current Actuarial Liability, and gather assets of that amount –Good funding, but expensive at first Use the Individual Level Premium method, and cram the funding in the shorter time to retirement –Normal costs can be very high Determine how much you can afford to post as plan assets now (call it “F”), and fund the difference going forward –Potentially a balance of current and future funding

21. Individual Aggregate Basics of IA method: F = amount of excess funding that exists in the plan F P = amount of the excess funding that is allocated to participant P This amount of excess funding helps to reduce the amount of liability that must be funded, so normal costs are reduced For each individual: PVFNC = PVFB - F P

22. Individual Aggregate Key question: How do you take F and divide it up in to pieces in order to figure out F P for each individual?? Common ways to do this: Divide the fund equally among all participants: F P = F / n Prorate F by the present value of the projected retirement benefit Prorate F by the present value of the accrued benefit Prorate F by the current AL for each participant Make up your own creative way….

23. Individual Aggregate Let’s work through an example with 2 participants F = $1,000 Retirement benefit = 1.65% of final salary times Years of Service r = 65, s =.03 Participant 1: e = 25, x = 35, S x = $80,000 Participant 2: e = 35, x = 45, S x = $120,000 How do we cost the plan?

24. Individual Aggregate Let’s work through an example with 2 participants Determing F P by using Present Value of Projected Retirement Benefit F = $1,000 Retirement benefit = 1.65% of final salary times Years of Service r = 65, s =.03 Participant 1: e = 25, x = 35, S x = $80,000 Participant 2: e = 35, x = 45, S x = $120,000 How do we cost the plan?

25. Individual Aggregate Let’s work through an example with 2 participants Determing F P by dividing evenly between participants F = $1,000 Retirement benefit = 1.65% of final salary times Years of Service r = 65, s =.03 Participant 1: e = 25, x = 35, S x = $80,000 Participant 2: e = 35, x = 45, S x = $120,000 How do we cost the plan?

26. Aggregate Cost Method Now let’s look at a cost method that truly is totally aggregate in nature The exercise, while incorporating individual information, is really geared towards figuring out the Normal Cost for the entire plan, rather than each person individually It incorporates the concept of the “average working-life annuity”

27. Aggregate Cost Method Average working-life annuity = ä (T) = Σ [( s N (T) x - s N (T) r ) / s D (T) x ] S x Σ S x = Present Value of all Future Salaries Paid Total Current Salaries

28. Aggregate Cost Method Using the average working-life annuity notation, we then have: TNC ä (T) = Σ PVFB – F Read this as “The total normal cost for the plan paid under the assumption of the average working-life annuity, equals the total present value of future benefits for the plan less any excess funding”

29. Aggregate Cost Method Of course, sometimes we want TNC to be a level percent of total salary so we use TNC = U Σ S x Special note: We can express level dollar as a subset of level percent of salary… What adjustments need to be made? What happens if we set s = 0, and S x = 1?

30. Other Funding Methods For Defined Benefit Plans implemented after employees already have accrued service: Individual Level Premium showed a way to do fast funding with no initial cash outlay Could also immediately fund an amount to pick up EANLD at the point in scale But other “in-between” alternatives exist

31. Frozen Initial Liability (EAN) FIL (EAN) method calculates for each individual what the current Actuarial Liability would be under EANLD But instead of picking up point in scale, we choose a time period over which to amortize this liability – called the “Frozen Initial Liability” (FIL) This amortization cost of FIL is called the “Supplement Cost” (SC) Meanwhile, we are using the EANLD Normal Cost to do the remaining funding Total Cost in a year = NC + SC

32. Frozen Initial Liability using TUC Method calculates for each individual what the current Actuarial Liability would be under TUC Again, we choose a time period over which to amortize this liability – called the “Frozen Initial Liability” (FIL) This amortization cost of FIL is called the “Supplement Cost” (SC) Meanwhile, we are using the TUC method to do the remaining funding Total Cost in a year = NC + SC