Follow up on the TUC Method• In life insurance reserving and net

premium discussions, you may have discussed the fact that next year’s reserve value is a function of the current reserve value and the next net premium

• Similar logic exists for defined benefit pension funding

Follow up on the TUC Method• Successive Actuarial Liabilities in the

TUC method and functions of the previous AL and next NC

• ALx+1 = [ALx • (1+i) / (px)] + NCx+1

• Look familiar to a “reserve roll-forward”?• See the updated commutation function

sheet where the AL is calculated two different ways

Pension Benefits that use a Salary Scale in their formula• We talked last time about the TUC method,

where the actuarial liability puts blinders to the future

• Current actuarial liability under TUC only considers benefits accrued to date

• But often times individual cost methods will either…– (a) need to use salary as a way to define the pension benefits OR– (b) need to use salary to project forward pension benefits OR– (c) both

Ways to approach definitions of salary scales• Perhaps a specific increase factor for each

attained age x …. Almost like a mortality table has specific rates for each different attained age

• Create a general increase factor regardless of age … next year’s salary = last year’s salary • (1 + s)

• Question… which of the two is more realistic?– Remember, salary increases are a combination of promotional

increases (getting assigned larger job responsibilities) and general inflationary increases

Ways to approach definitions of salary scales• A general way to define future salary

via the table approach:• Sr-1 = Sx • (1 + s)r-1-x

• This is read as “expected final salary at age r-1 equals current salary at age x increased at rate s for the years between x and r-1”

Ways to approach definitions of salary scales• Or, in cases where you do not want to

use a general increase factor, you can use ratios

• Sr-1 = Sx • sr-1 / sx

• This is read as “expected final salary at age r-1 equals current salary at age x increased at the ratio between r-1 and x in the defined salary table”

Expected final salary• Why is expected final salary Sr-1 and not Sr?

• You retire when you reach age r, based upon your final salary at age r-1

• For example, you retire at age 65, but don’t earn a year’s salary while your 65; your final salary for pension calculations is at age 64

Salary under three common examples

• (1) The Final Salary plan• Assume benefit is 3.9% of final salary times

years of service• Then Bx = .039 • Sx • (1 + s)r-1-x • (x – e) for a

funding method that used projected salary• Remember, e was the age at which the

person entered the plan, so (x-e) represents service time

Salary under three common examples

• (2) The Final Average Salary plan• Assume benefit is 4.1% of final three-year average

salary times years of service• Then Bx = .041 • FAS • (x – e) for a funding method

that used projected salary• Where FAS = 1/3 [(Sx • (1 + s)r-1-x) + (Sx • (1 + s)r-2-x) +

(Sx • (1 + s)r-3-x)]• Notice you can factor some things out from inside

the brackets since they contain a lot of similar terms

Salary under three common examples

• (3) The Career Average Salary plan• Assume benefit is 5.5% of career average

salary times years of service• Then Bx = .055 • CAS • (x – e) for a funding

method that used projected salary• Where CAS = (1/(r-e)) • [(Sx • (1 + s)r-1-x) + (Sx •

(1 + s)r-2-x) + (Sx • (1 + s)r-3-x) + … + Se]

The Projected Unit Credit (PUC) Method

• The PUC method leverages heavily off TUC but adds the use of salary scales

• Salaries are projected forward to retirement age in order to incorporate them into the liability and normal cost calculations

The Projected Unit Credit (PUC) Method

• Actuarial Liability for each participant:–ALx = Bx • (D(T)

r / D(T)x) • är

(12)

–where Bx = annual pension benefit that has accrued to age x, but with a salary component projected forward to r

The Projected Unit Credit (PUC) Method

• Example: Consider a defined benefit pension plan who has an annual benefit equal to 2.5% of 2-Year Final Average Salary, times years of service

• Bx = .025 • Sx • [(1/2) • (1 / sx ) • (sr-1 + sr-2)] • (x-e)

• ALx = Bx • (D(T)r / D(T)

x) • är(12)

The Projected Unit Credit (PUC) Method

• Normal Cost for each participant:– NCx = bx • (D(T)

r / D(T)x) • är

(12)

– where bx = annual pension benefit that has been earned in last year, but with a salary component projected forward to r

The Projected Unit Credit (PUC) Method

• Let’s do some sample calculations

Level Cost Individual Cost Methods

• Again, we mentioned earlier that increasing cost methods, like TUC and PUC are just one way to do plan costing

• As our commutation function sheet example showed, the cost in years near retirement can easily be large multiples of what costs were at entry into the plan

Level Cost Individual Cost Methods• So, how to create a way to levelize the

cost – make it the same cost over time – even if it means doing some form of pre-funding of future benefits

• Introducing… Level Cost Methods• Doesn’t this sound just like annually

renewable term insurance versus whole life? It is the same idea.

Entry Age Normal Level Dollar• Under EANLD, the normal cost is defined so

that the actuarial present value of all future normal costs is equal to the actuarial present value of all future benefits

• PVFNC = PVFB

• Again, just like life reserving where in a level net premium method, PVFP = PVFB

Entry Age Normal Level Dollar

• PVFNC = PVFB• NC (ä(T)

e:r-e|) = Br • (D(T)r / D(T)

e) • är(12)

• Rearrange and solve for NC

Entry Age Normal Level Dollar

• This is a little bit different jump to make because you’ll notice that nothing in NC (ä(T)

e:r-e|) = Br • (D(T)r / D(T)

e) • är(12)

depends on the valuation age x• And it’s not supposed to – the methods just takes

all that needs to be funded and levelly spreads it over the earnings years between e and r

• Just like a life insurance net premium doesn’t care when you are 3 or 30 years into a whole life policy

Entry Age Normal Level Dollar

• Defining the Actuarial Liability at valuation age x is again similar to a reserve calculation

• ALx = PVFBx - PVFNCx

• ALx = [Br • (D(T)r / D(T)

x) • är(12)] - NC (ä(T)

x:r-x|)

• You can also do a retrospective accumulation of Normal Costs to get the Actuarial Liability