Download Valuation and Asset Liability Matching for Defined

Valuation and Asset Liability Matching
for Defined Benefit Pension Plans
Ryan Purcell and Jeremy Hilado
Faculty Advisor: Mike Ludkovski
Department of Statistics and Applied Probability
Introduction
Pension plan valuations are difficult due to uncertain obligations owed far into the future. We investigate the impact of various model assumptions
and methods to minimize fluctuations in net plan value.
Data
Expected Present Value of Obligations
We used a model data set provided by Towers
Watson which contains 100 employee records for
two years, 2011 and 2012, and includes:
Pension benefits are paid out as life annuities of amount A. A is a function of years of service, ending salary, and age. The Expected Present Value EPV of benefits for employee k at
retirement age x is given by the equation below;
0.3
i. Age of employee
ii. Years of service
EPVk = A
iii. Annual salary
100−x
X
vt
t−1
Y
j=0
t=1
iv. Status; active (working), vested (not working
but owed future benefits), or retired.
We make the following textbfassumptions in performing our valuation:
i. Employees cannot die before retirement
ii. Employees retire or are terminated before age
66
iii. Employee salaries increase by 4% each year
iv. The discount rate used to calculate the
present value of future benefits is 5%
Projected Obligation
Thousands Expected Benefit Payout Per Year 700 p x+ j ,
where v = 1/(1 + i) is the discount factor and p x
is the probability of surviving year x.
Probability of Death (1-px)
Mortality By Age
The vertical axis gives the
conditional probability of death at
each age 1-px.[2]
0.25
0.2
0.15
0.1
0.05
0
65
70
75
80
85
90
95
Age
Projected Benefit Obligation
• Projected Benefit Obligation PBO is an accounting standard used in calculating pension
plan obligations. Under the PBO method, when calculating EPV, age and service are
projected to the retirement date, but years of service are not.
• We tested how PBO was affected by changes in the salary increase assumption. The true
average salary increase for 2011 was 3.55%. Assuming yearly salary increases of 3.55%
results in a 2.7% decrease in total PBO.
• PBO is very sensitive to salary increase assumptions. A 1% change in the yearly salary
increase assumption results in a change in total PBO of over 5%.
650 • A roll-forward gives an estimate of the following year’s PBO assuming no change in
employee makeup. It also provides a way of checking our assumptions are proper.
600 550 500 • The total discrepancy between 2012 PBO and the roll-forward estimate was $157,000.
The change was due to new hires, change in employee status, and employees who became
deceased.
450 400 350 300 250 200 150 100 0 0 2013 2015 2017 2019 2021 2023 2025 2027 2029 2031 2033 2035 2037 2039 2041 2043 2045 2047 2049 2051 2053 2055 2057 2059 2061 2063 2065 2067 2069 2071 2073 2075 2077 2079 2081 2083 2085 2087 2089 2091 2093 50 Status
Active
Vested
Retired
Total
PBO and Roll-Forward Values in Thousands of Dollars
2011 PBO Roll-Forward 2012 2012 PBO Roll-Forward 2013
6,911
6,865
6,226
7,051
984
1,033
835
877
2,629
2,554
3,233
3,151
9,803
10,451
10,294
11,078
Asset Liability Management
• With cash flows projected 81 years into the future, the present value of the plan is greatly affected by fluctuations in the yield curve [1].
• To protect against interest rate changes, we matched the duration of bonds with the duration of future cash flows. We found a single
equivalent rate at the beginning of 2011 that produced the same present value of cash flows as the yield curve.
• The yield curve was highly volatile in 2011. This caused the single equivalent rate to fluctuate between 4.3% and 5.6%. Based on this range,
we felt a 5% discount rate in the valuation was reasonable.
• Due to changes in the yield curve, present value of liabilities increased by 18% over the year. Shown as the purple line below.
Percent Change of Net Liabili3es by Month 25.00% 20.00% 15.00% 10.00% 30yr Bond 10yr and 30yr • The duration of liabilities ranged from 14 to 16 years over 2011.
Hedging the risk using bonds with duration of 14.5 years would
shrink the change in value of assets and liabilities to 1.7% at the
end of the year (green line).
14.5yr Liabili;es 5.00% • Using a portfolio of 30 and 10 year bonds with a combined duration of 14.5 years would match the liabilities more closely and
shrink the net loss to 1.29% after one year (red line).
1/
1/
11
2/
1/
11
3/
1/
11
4/
1/
11
5/
1/
11
6/
1/
11
7/
1/
11
8/
1/
11
9/
1/
11
10
/1
/1
1 11
/1
/1
1 12
/1
/1
1 1/
1/
12
2/
1/
12
3/
1/
12
4/
1/
12
5/
1/
12
0.00% -­‐5.00% • Due to the decrease in the interest rate, using only 30-year bonds
would result in a profit of 15%.(blue line).
-­‐10.00% -­‐15.00% -­‐20.00% • We conclude that the 30 and 10 year bond portfolio would provide an effective hedge to interest rate risk given its low volatility
despite large changes in the yield curve over year.
References
[1] Society of Actuaries. Pension Discount Curve and Liability Index, Society of Actuaries, 01 Mar. 2012. Web. Apr. 2012. <http://www.soa.org/Files/Xls/
pen-discount-curve-0312.xls>.
[2] Society of Actuaries. RP-2000 Mortality Rates Table, Society of Actuaries, 01 Jan. 2000. Web. Feb. 2012. <http://www.soa.org/files/pdf/rp00_mortalitytables.pdf>.
• Thank you to Towers Watson and Mike Ludkovski for their time and guidance on this project.