SOCIETY OF ACTUARIES
EXAM MLC Models for Life Contingencies
ADDITIONAL MLC SAMPLE QUESTIONS AND SOLUTIONS
Copyright 2016 by the Society of Actuaries
319. Kevin is a participant in a defined benefit pension plan at DMN Pharmaceuticals. You are
given the following facts:
(i)
(ii)
(iii)
(iv)
Kevin was born December 31, 1980.
Kevin was hired on January 1, 2011 with an annual salary of 35,000.
Kevin’s salary has increased each year on January 1 by 3% in 2012 through 2015.
The annual accrued benefit as of any date under the pension plan is 2% of the average
annual salary over the three years prior to that date multiplied by the number of years
of service as of that date. The accrued benefit is payable annually on the first of the
month following the participant’s birthday, beginning on the first of the month
following the 65th birthday.
A valuation is performed as of December 31, 2015 using the Traditional Unit Credit cost method
and the following assumptions:
a)
b)
c)
d)
e)
In the future, Kevin’s salary will increase by 3% each year on January 1 (including on
January 1, 2016) as long as Kevin remains employed by DMN.
The retirement assumption is a single decrement of 100% at age 65.
All other decrements combined equal 5% at July 1 each year before age 65.
i = 0.04
a65  11.0
Calculate the actuarial liability for the retirement decrement under this valuation.
(A)
2770
(B)
2790
(C)
2810
(D)
2830
(E)
2850
Solution: (Key = B)
Following example 10.10 on page 360 of the textbook, the actuarial liability under the
Traditional Unit Credit cost method is the actuarial present value of the accrued benefit,
which is calculated using the service and final average salary as of the valuation date. Thus
the final average salary is the average of the salaries in the years 2013, 2014, and 2015,
which is 35,000 x (1.032 + 1.033 + 1.034)/3 = 38,257.
Valuation Date
12/31/2015
Average Salary
38,257
Years of Service
5
Accrued Benefit
3826
The actuarial liability is the actuarial present value (as of the valuation date) of the accrued
benefit and is given by
AL = 3826 x 11.0 x (0.95)30 / 1.0430 = 2785.
320. Kevin is a participant in a defined benefit pension plan at DMN Pharmaceuticals. You are
given the following facts:
(i)
(ii)
(iii)
(iv)
Kevin was born December 31, 1980.
Kevin was hired on January 1, 2011 with an annual salary of 35,000.
Kevin’s salary has increased each year on January 1 by 3% in 2012 through 2015.
The annual accrued benefit as of any date under the pension plan is 2% of pay in the
prior 12-month period multiplied by the number of years of service as of that date. The
accrued benefit is payable monthly on the first of each month, beginning on the first of
the month following the 65th birthday.
A valuation is performed as of December 31, 2015 using the Traditional Unit Credit cost method
and the following assumptions:
a)
b)
c)
d)
e)
f)
In the future, Kevin’s salary will increase by 3% each year on January 1 (including on
January 1, 2016) as long as Kevin remains employed by DMN.
The retirement assumption is a single decrement of 100% at age 65.
All other decrements combined equal 5% at July 1 each year before age 65.
i = 0.04
a65  11.0
Annuity values are calculated assuming deaths are uniformly distributed over each year
of age.
Calculate the normal contribution for the retirement decrement under this valuation.
(A)
560
(B)
590
(C)
620
(D)
650
(E)
680
Solution. (Key = D)
The normal contribution under the Traditional Unit Credit cost method is the actuarial
present value of the difference between the expected accrued benefit one year from the
valuation date and the accrued benefit at the valuation date.
As of Date
Pay in Prior Year
Years of Service
12/31/2015
12/31/2016
39,393
40,575
5
6
Accrued/
Expected Accrued
Benefit
3939
4869
Thus the expected accrued benefit on 12/31/2016 minus the accrued benefit on 12/31/2015 is
4869 – 3939 = 930.
(12)
Using UDD to approximate a65
at 4% interest we have
id
i  i(12)
(12)

1.000127
and


 0.464889
i(12)  d (12)
i(12)  d (12)
  (12)  a65   (12)  10.5365
 (12) 
(12)
a65
The normal contribution is the actuarial present value (as of the valuation date) of this
difference and is given by
NC = 930 x 10.5365 x (0.95)30 / 1.0430 = 648.
321. Kira is a participant in a defined benefit pension plan at DMN Pharmaceuticals. You are
given the following facts:
(i)
(ii)
(iii)
(iv)
Kira was born December 31, 1980.
Kira was hired on January 1, 2011 with an annual salary of 35,000.
Kira’s salary has increased each year on January 1 by 3% in 2012 through 2015.
The annual accrued benefit as of any date under the pension plan is 2% of the 3-year
final average salary as of that date multiplied by the number of years of service as of
that date. The accrued benefit is payable annually on the first of the month following
the participant’s birthday, beginning on the first of the month following the 65th
birthday.
A valuation is performed as of December 31, 2015 using the Projected Unit Credit cost method
and the following assumptions:
a)
b)
c)
d)
e)
In the future, Kira’s salary will increase by 3% each year on January 1 (including on
January 1, 2016) as long as Kira remains employed by DMN.
The retirement assumption is a single decrement of 100% at age 65.
All other decrements combined equal 5% at July 1 each year before age 65.
i = 0.04
a65  11.0
Calculate the actuarial liability for the retirement decrement under this valuation.
(A)
6660
(B)
6760
(C)
6860
(D)
6960
(E)
7060
Solution: (Key = B)
Under the Projected Unit Credit cost method, the actuarial liability is the actuarial present
value of the pay-projected benefit, which is calculated using the service as of the valuation
date. We have the following information.
Projected Final Average Salary at
65
Service at valuation date
Pay-Projected Benefit
35,000 x (1.0332 + 1.0333 +1.0334)/3 = 92,859
5
.02 x 5 x 92,859 = 9286
The actuarial liability is the actuarial present value (as of the valuation date) of the payprojected benefit and is given by
AL = 9286 x 11.0 x (0.95)30 / 1.0430 = 6760.
322. Kira is a participant in a defined benefit pension plan at DMN Pharmaceuticals. You are
given the following facts:
(i)
(ii)
(iii)
(iv)
Kira was born December 31, 1980.
Kira was hired on January 1, 2011 with an annual salary of 35,000.
Kira’s salary has increased each year on January 1 by 3% in 2012 through 2015.
The annual accrued benefit as of any date under the pension plan is 2% of the 3-year
final average salary as of that date multiplied by the number of years of service as of
that date. The accrued benefit is payable annually on the first of the month following
the participant’s birthday, beginning on the first of the month following the 65th
birthday.
A valuation is performed as of December 31, 2015 using the Projected Unit Credit cost method
and the following assumptions:
a)
b)
c)
d)
e)
In the future, Kira’s salary will increase by 3% each year on January 1 (including on
January 1, 2016) as long as Kira remains employed by DMN.
The retirement assumption is a single decrement of 100% at age 65.
All other decrements combined equal 5% at July 1 each year before age 65.
i = 0.04
a65  11.0
Calculate the normal contribution for the retirement decrement under this valuation.
(A)
1050
(B)
1150
(C)
1250
(D)
1350
(E)
1450
Solution: (Key = D)
Under the Projected Unit Credit cost method, the normal contribution is the actuarial present
value of the difference between the expected pay-projected benefit one year after the
valuation date and the pay-projected benefit at the valuation date. We have the following
information.
Projected Final Average Salary at
65
Pay-projected benefit at 12/31/2015
Pay-projected benefit at 12/31/2016
Difference
35,000 x (1.0332 + 1.0333 +1.0334)/3 = 92,859
.02 x 5 x 92,859 = 9,286
.02 x 6 x 92,859 = 11,143
11,143 – 9,286 = 1857
The normal contribution is the actuarial present value (as of the valuation date) of this
difference and is given by:
NC = 1857 x 11.0 x (0.95)30 / 1.0430 = 1352.
Sample WA Question 22
On December 31, 2015, a defined benefit plan member, who is exactly age 55, has 25 years of
service. Her salary in 2015 was 50,000.
The annual accrued benefit as of any date is 1.6% of the three-year final average salary as of that
date, multiplied by years of service as of that date. The pension is payable as a monthly single
life annuity, with the first payment due at retirement.
The valuation assumptions are as follows:







Exits from employment follow the Illustrative Service Table, except that all lives
surviving in employment to age 61 retire at that time.
All retirements occur on the employee’s birthday.
After retirement, mortality follows the Illustrative Life Table.
Annuities are valued using the 2-term Woolhouse formula.
Salaries are increased by 3% each year on January 1.
All contributions are paid on January 1 each year.
i = 0.06
(a) (i) Calculate the actuarial liability for the member at December 31, 2015 using the Projected
Unit Credit (PUC) cost method.
(ii) Calculate the normal contribution for 2016 using PUC.
(b) (i) Calculate the actuarial liability for the member at December 31, 2015 using the
Traditional Unit Credit (TUC) cost method.
(ii) Calculate the normal contribution for 2016 using TUC.
Solution
Based on the December 31, 2015 valuation, let ABx be the accrued benefit at age x, with pay and
service projected to age x, and let PBx be the benefit with pay projected to age x, but with service
as of the valuation date.
(a) (i) The actuarial liability as of December 31, 2015 is
V  PB a
P
0
(12)
60 60
( )
d60( r ) 5
(12) l61
v  PB61a61 ( ) v6
( )
l55
l55
We have
 (1.03)3  (1.03)4  (1.03)5 
PB60  (.016)(25)(50, 000) 
  22,517
3


 (1.03)4  (1.03)5  (1.03)6 
PB61  (.016)(25)(50, 000) 
  23,192
3


Using the 2-term Woolhouse approximation we have
11
11
(12)
(12)
a60
 a60 
 10.6871 and a61
 a61 
 10.4458
24
24
which gives
P
 150, 072
0V
(ii) The NC in this case is 0V P / 25 = 6002.
(b) (i) The TUC case is similar to PUC except that the accrued benefit is used in place of the
pay-projected benefit. Hence we replace the PB60 and PB61 in the above with AB55.
We have
 1  (1.03)1  (1.03)2 
AB55  (.016)(25)(50,000) 
  19, 423
3


P
and thus 0V  (19, 423)(6.5014)  126, 277 .
(ii) For the NC, we must calculate expected increase in the accrued benefit over 2016
 1.03  1  1.031 
AB56  AB55  (.016)(26)(50, 000) 
  19, 423
3


 20,806  19, 423  1383
NC = (1383)(6.5014) = 8991.