DB plan sponsors
An Chen University of Bonn
joint with Filip Uzelac (University of Bonn) Southwestern University of Finance and Economics
Chengdu, June 2012
Pension Benefit Guaranty Corporation
Pension Benefit Guaranty Corporation (PBGC):
B established under the Employee Retirement Income Security Act of 1974
B strengthening the retirement-income security of Americans B providing afederal insurance programto more than 44 million
American workers and retirees in more than 27,500defined benefit (DB) pension plans
♦ DB plan: An employer-sponsored retirement plan where employee benefits are prespecified and based on an employee’s salary and years of service .
♦ Investment risk and portfolio management are entirely under the control of the sponsoring company
♦ The sponsoring company is obliged to close insurance with PBGC
Beneficiary
(Employee)
-Sponsoring company (Employer)
DB pension plan
6
?
? 6
Pension fund
PBGC
PBGC insurancePension Benefit Guarantee Corporation
The recent development of the PBGC
B In the PBGC 2010 Annual Report, the PBGC reported a cumulative year-end deficit of $23 billion
B Some recent prominent cases for termination: Delphi, Lehman Brothers, Circuit City
A variety of drivers
B the recent sharp decline in the stock market beginning in 2000 B general decline in the interest rate
B ineffective premiums B · · · ·
Premium calculation
Prevailing premium calculation:
B set by Congress and paid by sponsoring company B In 2010, about 70% flat premiums and 30% variable rate
premiums (solely based on thepension funds’ underfunding)
⇒ Premiums not risk-based!
B Sponsoring companies have a claim on their pension fund’s surpluses (but limited liability)
⇒ Sponsoring companies have an incentive to invest their pension funds in riskier assets, when the pension fund is insured.
A risk-based premium calculation
President George W. Bush proposed risk-based premiums for the PBGC in his 2006 and 2007 government’s budget plan
In February 2011, President Obama backed a risk-based PBGC premium calculation endorsed by Bush in the 2012 government’s budget plan
Josh Gotbaum said this proposal would strengthen the defined benefit system because
B it would encourage companies to reduce risk in their plans
Existing literature: vanilla put option framework
Contingent claim approach: pension insurance (PBGC support) modelled as avanilla putoption
B The underlying asset of the put is the assets of the pension fund
B The exercise price of the put is the promised pension payment B Literature: Sharpe (1976), Bodie and Merton (1993), and
Marcus (1987), Chen, Ferris and Hsieh (1994), Lewis and Pennacchi (1994), Bodie (1996), Boyce and Ippolito (2002)
This paper
This paper contributes to literature on pricing PBGC insurance by incorporating
first, the support of the PBGC is a secondary guarantee provided to the pension funds
second, the pension fund can be prematurely terminated B premature termination is triggered by theunderfunding of the
sponsoring company(distress termination);
B premature termination means the pension fund is trusted by the PBGC
Focus of this paper:
Develop a theoreticalrisk-basedmodel for the PBGC insurance in acontingent claim framework
Empirically illustrate our theoretical pricing formula for the 100 biggest American DB sponsoring companies.
Main results
We explicitly model the pension insurance provided by the PBGC taking account of the above two realistic aspects
We are able to obtain an analytical valuation formula for the insurance premium
premium increasing in thecorrelationbetween the assets of the pension fund and of the sponsoring company
We empirically illustrate our theoretical pricing formula for the 100 biggest American DB sponsoring companies.
Our result clearly casts doubt on the current practice where about 70% of the PBGC premiums charged are flat.
We observe that the funding ratio, theleverage, and the asset volatilityof the sponsoring companies are three key risk factors in a risk-based premium calculation.
Agenda
Introduction (
√)
Model setup
Numerical results
Empirical exercise
Conclusion
Defined benefit plan
The pension plan is issued at time t0= 0 to a representative beneficiary
The benefits are paid out as alump-sum paymentat the beneficiary’s retirement date R: BR
B BR is the prespecified benefit related to an employee’s salary and years of service.
B We assume that the pension liability BR is deterministic.
Pension fund’s asset process
The pension fund invests in a constant equity-bond-mix
The pension fund trades in the risk free bond F and the risky asset A in a self-financing way starting with initial wealth X0
dXt =θ · Xt·dAt At
+ (1 − θ) · Xt· dFt Ft
=Xt(r + θ(µ − r )) + θ σ XtdWt1
dAt =µAtdt + σAtdWt1 (risky asset) dFt =rFtdt (risk-free asset)
B θ is thefraction of wealthinvested in the risky asset At and the remaining (1 − θ) fraction invested in the risk free bond F B W1is a standard Brownian motion under the market
probability measure P
Sponsoring company’s asset process
The sponsoring company’s assets also follow Black-Scholes dynamics with instantaneous rate of return µ
c> 0 and volatility σ
c> 0
dC
t=µ
cC
tdt + σ
cC
t(ρdW
t1+ p
1 − ρ
2dW
t2)
B W2is again a standard Brownian motion under the market measure P, independent of W1.
B The sponsoring corporation’s and the pension fund’s assets are correlated with acorrelation coefficientρ ∈ (−1, 1).
Premature termination
Assumption: the sponsoring company initiates default if it is unable to pay its outstanding corporate debt φ C
0e
gtplus an additional buffer ( − 1) φ C
0e
gt:
τ = inf{t|C
t≤ φ C
0e
gt}, > 1, φ < 1.
If τ ≤ R, the pension plan is closed at time τ .
If τ > R, the pension plan is naturally closed at the maturity date R.
Payoff structure of the sponsor support
The sponsoring company provides the
primarysupport to the underfunded pension fund.
When the sponsoring company defaults
if the pension fund is sufficiently funded, the sponsoring company does not have to balance any deficits of the pension fund;
if the pension fund is underfunded, then the sponsoring company will fully cover the deficit if the buffer is sufficient or it will only partly cover it if the deficits exceed the buffer.
Sponsor guarantees: Φ
c(τ ) (the payment sponsor makes at
τ ); Φ
c(R) (the payment sponsor makes at time R)
Sponsor support given premature termination Φ
c(τ )
Deficit of the pension fund at τ :
(BRe−(R−τ )− Xτ)1{Xτ<BRe−(R−τ )}
The size of the primal support Φc(τ ) is given by
Φc(τ ) =
BRe−r (R−τ )− Xτ, BRe−r (R−τ )− Xτ< ( − 1)φC0eg τ ( − 1)φC0eg τ, BRe−r (R−τ )− Xτ> ( − 1)φC0eg τ
The first term: the buffer is sufficient to cover all the deficits of the pension fund.
The second term: the buffer is insufficient to cover all the deficits of the pension fund.
Sponsor support given natural termination Φ
c(R)
Deficit of the pension fund at R: (BR− XR)1{XR<BR} The size of the primal support Φc(R) is given by Φc(R) =
BR− XR, CR > φC0egR+ (BR − XR)
CR− φC0egR φC0egR< CR < φC0egR+ (BR− XR)
For the extreme good performance of the sponsoring company, the sponsor covers all the deficits of the pension fund
For a moderate performance, the sponsor covers a part of the deficits of the pension fund.
The entire support provided by the plan sponsor:
Φc = Φc(τ )1{τ ≤R}+ Φc(R)1{τ >R}.
Payoff profile of the PBGC insurance
The amount of residual deficits the PBGC covers is capped.
Upon premature termination, the insurance of the PBGC is G (τ ) = min{ ¯G e−r (R−τ ), (BRe−r (R−τ )− Xτ) − Φc(τ )}
Upon natural termination, the insurance of the PBGC is G (R) = min{ ¯G , (BR− XR)1{XR<BR}− Φc(R)}
More compactly, the insurance of the PBGC G = G (τ )1{τ ≤R}+ G (R)1{τ >R}.
Risk-neutral pricing
The payoff profile for the PBGC insurance can be understood as exotic options ⇒ Risk-neutral pricing
B One canfirst adjust the probability measuresuch that it incorporates the effect of risk, then takes the expected discounted value under the adjusted measure.
B This new measure is the so-calledrisk-neutral probability measure P∗and the valuation method isrisk-neutral pricing.
Risk-based premium of the PBGC insurance
Therisk-based premiumpaid by the sponsoring company to the PBGC is the expected discounted insurance payoff under the risk neutral probability measure:
G0=E∗e−rRG (R)1{τ >R} + E∗e−r τG (τ )1{τ ≤R} where E∗ denotes the expected value under the risk-neutral measure P∗.
Analytical solutions are obtained.
Sketch of solution
Write down the assets processes under the risk-neutral measure Premium (premature termination)
B At time τ , it holds Cτ =C0exp
r −1 2σc2
τ + σcy
= φC0eg τ ⇒ y B Substitute this in Xτ (depending on ρy +p
1 − ρ2z where z ∼ N(0, 1) )
B Transform the condition Xτ > BRe−r (R−τ )− ( − 1)φC0eg τ to z > d1(τ )
B Use the density of the first-hitting τ Premium (natural termination)
B Use the property of iterated expectations
B Use the fact that ln X and ln C follow the cumulative bivariate normal distribution with correlation coefficient ρ.
Parameter choices
For this numerical illustration, we fix the relevant parameters as follows:
X
0=100, B
R= 240, σ = 0.20, C
0= 100, R = 15,
=1.05, r = 0.05, σ
c= 0.25, g = r , θ = 0.6,
G =120. ¯
PBGC premium
φ = 0.3 φ = 0.4 φ = 0.5 φ = 0.6 ρ = −0.5 1.179 1.480 1.881 2.351 ρ = −0.25 1.560 1.934 2.404 2.879
ρ = 0 1.932 2.376 2.908 3.410
ρ = 0.25 2.313 2.825 3.408 3.913
ρ = 0.5 2.727 3.308 3.901 4.427
A risk-based PBGC premium for varying
debt ratio φand
correlation coefficient ρ.Empirical exercise: firm-specific parameters
We acquired a data set of the 100 largest US Corporates and their defined benefit plans from P&I Investments
The initial wealth X0and C0are normalized to 100 Bˆ0i =X0(αi)−1,
⇒ ˆBRi = ˆB0ierR
αi: the funding ratio of pension fund i
...Firm-specific parameters
Equity holding of the pension fund:
θˆi=θiE+ θiPE+ θiHF + θiRE+1
2(θiA+ θiO)
θiE: the observed percentage pension fund i invested in equity θiPE: the share invested in private equity
θiHF: the share in hedge funds θiRE: the share in the real estate θiA the share in alternatives
θiO the share sponsor i invested in other assets.
Estimated debt ratio ˆφi= LiLiB
B+EMi
Sponsor’s asset volatility σc: historical data approach.
Other non-firm-specific parameters
The risk-free rate r will be approximated by the long-term Treasury Bill.
The growth rate of the sponsor liabilities g set equal to r
An estimate ˆσ of the volatility of the risky asset A: S&P-500 is the traded reference portfolio all pension funds invest
An estimate for the capped amount ¯G : fraction c of the maximum pension liabilities, i.e ˆG = max¯ ic ˆBRi..
The regulatory parameter and the correlation coefficient ρ are difficult to estimate and kept constant across sponsors and pension funds.
...Other non-firm-specific parameters
Non-firm-specific parameters:
r =g = 0.0413; ˆ σ = 0.2022; ρ = 0.5; R = 15;
=1.05; ˆ G = 0.4 ∗ 282.371 = 112.948. ¯
Estimation results for a representative subsample
Sponsor ˜ˆGi φˆi σˆci αi BˆRi θˆi
3M 2.329 0.186 0.174 0.940 197.660 0.678
Aetna 6.475 0.696 0.146 0.901 206.215 0.755
American Electric 7.840 0.680 0.159 0.803 231.382 0.693
Ashland 6.660 0.578 0.186 0.754 246.419 0.470
AT&T 4.060 0.475 0.229 0.883 210.419 0.527
Bank of America 0.175 0.938 0.086 1.123 165.450 0.632 Goodyear Tire & Rubber 13.824 0.836 0.139 0.658 282.371 0.667 Baxter International 3.578 0.266 0.213 0.784 236.990 0.630
Boeing 5.768 0.579 0.166 0.833 223.049 0.459
Caterpillar 4.856 0.472 0.140 0.826 224.939 0.736
Coca-Cola 3.450 0.215 0.178 0.754 246.419 0.697
Consolidated Edison 6.634 0.634 0.075 0.749 248.064 0.763 Dominion Resources 2.912 0.551 0.147 1.137 166.831 0.656
Dow Chemical 6.022 0.542 0.147 0.749 248.064 0.593
˜ˆG : risk-based PBGC premiumi ˜ˆG is expressed in % in Bi R. ˆφi: estimated debt ratio; ˆσc: estimated
sponsor vola; αi: funding ratio; ˆ
BRi: promised pension benefit; ˆθ: estimated equity holding
Estimation results for a representative subsample
Sponsor ˜ˆGi φˆi σˆci αi BˆiR θˆi
Eli Lilly 3.918 0.315 0.197 0.861 215.795 0.808
Exxon Mobil 2.754 0.288 0.144 0.822 226.034 0.609
FedEx 2.607 0.287 0.186 0.918 202.396 0.500
General Dynamics 5.442 0.418 0.180 0.677 274.446 0.722 Hewlett-Packard 3.619 0.468 0.227 0.865 214.800 0.392 Honey International 4.569 0.396 0.189 0.813 228.536 0.732
IBM 2.461 0.331 0.230 0.980 189.592 0.528
JP Morgan 0.365 0.922 0.078 1.301 142.813 0.814
United Technology 3.116 0.332 0.170 0.916 202.838 0.614 Walt-Disney 3.991 0.305 0.174 0.703 264.296 0.600
Wels-Fargo 5.729 0.875 0.060 0.932 199.356 0.641
Coca-Cola 3.450 0.215 0.178 0.754 246.419 0.697
˜ˆG : risk-based PBGC premiumi ˜ˆG is expressed in % in Bi R. ˆφi: estimated debt ratio; ˆσc: estimated
sponsor vola; αi: funding ratio; ˆ
BRi: promised pension benefit; ˆθ: estimated equity holding
Summary
We model the pension insurance provided by the PBGC taking account of distress termination
Assuming both the pension fund’s and the plan sponsor’s assets follow Black-Scholes dynamics and are correlated, we obtain a closed-form pricing formula for the risk-based premium.
Our empirical study clearly disputes the flat premium. The use of a variable rate premium is partly consistent with our results.
The funding ratio is the most significant driver of the risk-based premium. However, other financial risk factors like leverage, asset volatility of the sponsoring companies play a very important role in the risk-based premium.