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A risk-based premium: What does it mean for DB plan sponsors

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(1)

DB plan sponsors

An Chen University of Bonn

joint with Filip Uzelac (University of Bonn) Southwestern University of Finance and Economics

Chengdu, June 2012

(2)

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

(3)

Beneficiary

(Employee)

 -

Sponsoring company (Employer)

DB pension plan

6

?

? 6

Pension fund

PBGC

PBGC insurance

(4)

Pension 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 · · · ·

(5)

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.

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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

(7)

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)

(8)

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.

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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.

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Agenda

Introduction (

)

Model setup

Numerical results

Empirical exercise

Conclusion

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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.

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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

(13)

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

c

C

t

dt + σ

c

C

t

(ρdW

t1

+ p

1 − ρ

2

dW

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).

(14)

Premature termination

Assumption: the sponsoring company initiates default if it is unable to pay its outstanding corporate debt φ C

0

e

gt

plus an additional buffer ( − 1) φ C

0

e

gt

:

τ = inf{t|C

t

≤  φ C

0

e

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.

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Payoff structure of the sponsor support

The sponsoring company provides the

primary

support 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)

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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.

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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}.

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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}.

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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 Pand the valuation method isrisk-neutral pricing.

(20)

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=Ee−rRG (R)1{τ >R} + Ee−r τG (τ )1{τ ≤R} where E denotes the expected value under the risk-neutral measure P.

Analytical solutions are obtained.

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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.

(22)

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. ¯

(23)

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 φ

and

correlation coefficient ρ.

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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 =X0i)−1,

⇒ ˆBRi = ˆB0ierR

αi: the funding ratio of pension fund i

(25)

...Firm-specific parameters

Equity holding of the pension fund:

θˆiiE+ 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.

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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.

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...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. ¯

(28)

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

(29)

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

(30)

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.

(31)

Thank you for your attention!

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