Pension Risk Management with Funding and Buyout Options

Transcription

1 Pension Risk Management with Funding and Buyout Options Samuel H. Cox, Yijia Lin, Tianxiang Shi Department of Finance College of Business Administration University of Nebraska-Lincoln FIRM 2015, Beijing July 4-5, 2015 Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

2 Outline Introduction 1 Introduction Pension De-risking Securitization of Pension Risk with Options 2 Basic Framework 3 Example: Hypothetical Pension Options Sensitivity Analysis Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

3 Pension De-risking Introduction Pension De-risking Securitization of Pension Risk with Options In recent years, there has been a surge of interest from defined benefit pension plan sponsors in de-risking their plans. Strategies include, longevity hedges and pension buyouts (Lin et al., 2015). Driven by growing pension deficits unprecedented market swings to sustained declines in interest rates; Milliman 100 Pension Funding Index (PFI) decreased to 79.6%, down from 83.5% in December 2014 (Milliman, 2015) unanticipated improvements in mortality rates Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

4 Key Pension De-risking Tools Pension De-risking Securitization of Pension Risk with Options Buy-out: each individual pensioner is issued a policy so that their pension is provided directly by an insurance company; pension liabilities are completely removed from the pension firm s balance sheet. Longevity hedge: such as a longevity swap, allows a pension plan to transfer its high-end longevity risk to a third party, whilst retaining direct control of the assets. Buyouts are more effective in improving firm value under the enterprise risk management framework (Lin et al. 2015). Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

5 Challenge of the Buyout Market Pension De-risking Securitization of Pension Risk with Options Buyouts generally have had lower business volume than Longevity swaps in recent years, even though buyouts can create greater value: capital intensive and relatively expensive: In December 2014, the price of a buyout annuity transaction across US, UK, Ireland and Canada was 14% higher on average than the equivalent accounting liability (Mercer LLC, 2014). expensive for firms with underfunded plans: firms have to satisfy a minimum funded status by infusing cash to cover their funding deficits. Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

6 Objective and Methodology Pension De-risking Securitization of Pension Risk with Options Objective: to explore well-designed structures that make buyouts easier, especially for underfunded plans Methodology: propose two gap type options that provide buyout financing. pension funding option and pension buyout option create a transparent pension funding index based on market indices and publicly available mortality tables Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

7 Pension Funding Index Pension De-risking Securitization of Pension Risk with Options To increases market liquidity and reduce moral hazard and adverse selection. pension funding index at time t, PFI t, PFI t = PAI t PLI t PLI t : pension liability index based on N(0) retired life cohort aged x 0 at time 0, PLI t = N(t) Pa x0+t t = 1, 2,, where N(t) is the number of survivors at t, P is the annual payment, and a x0+t is the immediate life annuity factor. PAI t : pension asset index Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

8 Pension Asset Index Introduction Pension De-risking Securitization of Pension Risk with Options PAI t : determined by the value of a market portfolio composed of I indices at time t. I PAI t = A i,t 1 (1 + r i,t ), i = 1, 2,, I ; t = 1, 2, i=1 A i,t 1 : amount invested in index i at time t 1 r i,t : return of index i in period t PAI 0 : predetermined pension asset with investment weights w i Periodic portfolio adjustment (when t = 1, 2, ) PA t = I A i,t = PAI t + k UL t 1 {ULt>0} N(t)P i=1 UL t : funding deficit UL t = PLI t PAI t + N(t)P k: amortization factor Portfolio rebalance can be easily incorporated. Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

9 Pension Options for Fully Funded Plans Pension De-risking Securitization of Pension Risk with Options Pension funding option: to make up funding deficits and reduce external financing cost in order to satisfy a minimum funding requirement for a buyout transaction Pension funding option payoff: F w t = NA PLI 0 for t = 1, 2,..., n. NA: notional amount n: option period/term z: trigger funding index level K: strike funding level { PLI t (K PFI t ) if PFI t < z 0 if PFI t z, Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

10 Pension De-risking Securitization of Pension Risk with Options Pension Options for Fully Funded Plans Cont d Pension buyout option: not only fill the funding gap (up to K), but also cover the required buyout risk premium. Pension buyout option payoff: B w t = NA PLI 0 { PLI t (K PFI t + R t ) if PFI t < z 0 if PFI t z, for t = 1, 2,..., n. R t : estimated buyout risk premium at time t (Lin et al, 2015) Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

11 Pension De-risking Securitization of Pension Risk with Options Pension Options for Under-Funded Plans Pension funding option payoff: Ft u = NA 0 if PFI t K PLI PLI t (K PFI t ) if z < PFI t < K 0 0 if PFI t z Pension buyout option payoff: Bt u = NA PLI t R t if PFI t K PLI PLI t (K PFI t + R t ) if z < PFI t < K 0 0 if PFI t z Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

12 Pension Asset Index model Basic Framework Pension fund assets: S&P 500 index A 1,t, Merrill Lynch corporate bond index A 2,t and 3-month T-bill A 3,t. S&P 500 index A 1,t : Merton s jump-diffusion process (Merton, 1976) A 1,t = A 1,0 exp (α 1 1 N 1t 2 σ2 1 λ 1 k 1 )t + σ 1 W1t P + α 1 : instantaneous expected return σ 1 : instantaneous volatility W1t P : standard Brownian motion with mean 0 and variance t N 1t : Poisson process with arrival of λ 1 per unit of time Y 1j : standard normal with mean m 1 and s.d. s 1 k 1 : expected percentage change in the S&P 500 index Risk neutral Esscher measure is selected (Gerber and Shiu, 1994) Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24 j=1 Y 1j

13 Pension Asset Index model Cont d Basic Framework Processes of A i,t, i = 2, 3, as a geometric Brownian motion: da i,t A i,t = α i dt + σ i dw P it, i = 2, 3 α i : instantaneous expected return σ i : instantaneous volatility of asset i : standard Brownian motion with mean 0 and variance t W P it S&P 500 index A 1,t and Merrill Lynch corporate bond index A 2,t are correlated with Cov(W P 1t, W P 2t) = ρσ 1 σ 2 t, Monthly data from 1988 to 2010 are used to estimate parameters Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

14 Pension Valuation Rate Basic Framework Interest rate risk in pension risk management should be carefully considered. Incorporate the dynamics of pension valuation rate into pension options pricing. Cox-Ingersoll-Ross (CIR) model (Cox et al., 1985) dr p,t = ν (θ r p,t ) dt + σ p rp,t dw P p,t, ν: mean-reversion rate θ and σ p : long-term mean and instantaneous volatility W P p,t: standard Brownian motion Equally weighted average of US funding yield curve segment rates from August, 2008 to March, 2015 are used to estimate parameters. Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

15 Basic Framework Lee and Carter (1992) s Mortality Model One-year death rate q x,t for age x (x = 0, 1, 2, ) in year t (t = 1, 2,, K) ln q x,t = κ x + b x γ t + ɛ x,t, γ t = γ t 1 + g + e t, e t N(0, σ γ ) κ x and b x : age-specific parameters g: drift rate ɛ x,t and e t : normal errors with mean zero U.K. male population mortality tables from 1950 to 2003 are used Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

16 Basic Framework Option Pricing Formulas For Fully Funded Pensions Funding option for n-year term, in terms of the percentage of the nominal amount: PF w = 1 PLI 0 E Q [ e rτw N (τ w ) Pa x0 +τ w (K PFI τw ) +] N (t): survival evolution based on transformed mortality rates τ w = inf {t : PFI t < z, t {1, 2,..., n}} ( if the option is not triggered) Buyout option for n-year term: PB w = 1 PLI 0 E Q [ e rτw N (τ w ) Pa x0 +τ w ( (K PFIτw ) + + R τw )] = PF w + PR w Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

17 Basic Framework Option Pricing Formulas For Under-Funded Pensions Funding option for n-year term: PF u = 1 PLI 0 E Q [ e rτu N (τ u ) Pa x0 +τ u (K PFI τu ) +] where τ u = inf {t : PFI t > z, t {1, 2,..., n}} ( if the option is not triggered) Buyout option for n-year term: PB u = 1 PLI 0 E Q [ e rτu N (τ u ) Pa x0 +τ u ( (K PFIτu ) + + R τu )] = PF u + PR u Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

18 Risk Parameters and Assumptions Example: Hypothetical Pension Options Sensitivity Analysis Pension Funding Index The pension cohort has the same mortality experience as the U.K. male population At time 0, all plan participants reach the retirement age x 0 = 65 At time 0, 10,000 pensioners with annual survival benefit 60,000 per pensioner Pension asset weights (w 1, w 2, w 3 ) = (0.5, 0.45, 0.05), rebalanced annually amortization factor k = 1/5.95 Pension valuation rate: ν = , θ = 4.78%, σ p = 0.03, r p,0 = 4.8% Risk-free interest rate: r = 4% Market price of longevity risk (using Wang transform): λ EIB = , based on the European Investment Bank (EIB) bond issued in November 2004 Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

19 Example: Hypothetical Pension Options Sensitivity Analysis Pension Option Premiums for Fully Funded Plans Table 1: Life-Time Funding and Buyout Option Premiums for Fully Funded Plans Initial Funding Ratio Trigger Strike Funding Option Premium Buyout Add-on PFI 0 z K PF w PR w % 3.23% % 100% % % 1.65% % % Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

20 Example: Hypothetical Pension Options Sensitivity Analysis Pension Option Premiums for Under-Funded Plans Table 2: Life-Time Funding and Buyout Option Premiums for Under-Funded Plans Initial Funding Ratio Trigger Strike Funding Option Premium Buyout Add-on PFI 0 z K PF u PR u % 4.99% % 80% % % 3.79% % % 3.77% % 75% % % 2.87% % Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

21 Sensitivity Analysis Result Example: Hypothetical Pension Options Sensitivity Analysis Impact of Option Maturity: the longer the tenor of a pension option, the higher the price of the option. Impact of Risk-Free Rate For fully funded plans, PF w is negatively associated with r. For under-funded plans, positive relation found between PF u and r. Buyout add-on premiums decrease significantly when r increases for both cases. Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

22 Sensitivity Analysis Result Cont d Example: Hypothetical Pension Options Sensitivity Analysis Impact of Pension Valuation Rate Compare to a constant pension valuation scenario r p = 4.78%. both the funding and buyout options for fully funded plans require lower premiums at the constant pension valuation rate. Similar trend for PR u. For under-funded plans, most PF u s are actually higher at the constant pension valuation rate scenario. Impact of Longevity Risk For fully funded plans, PF w and PR w increase as λ increases. For under-funded plans, PF u decreases as λ increases. For under-funded plans, PR u increases as λ increases, even if the chance of triggering the option is lower. Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

23 Conclusion Introduction Example: Hypothetical Pension Options Sensitivity Analysis We provided new ways to reduce pension risk with buyouts. We introduced a transparent pension funding index based on market indices and publicly available mortality tables to increase market liquidity and reduce moral hazard and adverse selection problems. We illustrated how to price these new pension de-risking securities while recognizing investment risk, longevity risk, and interest rate risk. Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24

24 Example: Hypothetical Pension Options Sensitivity Analysis Thank You! Tianxiang Shi (University of Nebraska-Lincoln) FIRM /24