Optimal investment strategies and intergenerational risk sharing for - - PowerPoint PPT Presentation

Outline

Optimal investment strategies and intergenerational risk sharing for target benefit pension plans

Barbara Sanders with Suxin Wang and Yi Lu

Department of Statistics and Actuarial Science Simon Fraser University

July 29, 2017

Outline

Outline

1

Introduction

2

TBP Model Model formulation Solution to the optimization problem

3

Illustrations

4

Conclusion

Barbara Sanders Simon Fraser University July 2017 2 / 26

Introduction TBP model Illustrations Conclusion

Target Benefit Plans

Key features: Predefined contribution level Sponsor liability limited to contributions Target benefit level Actual benefits vary Collective asset pool Members bear risk collectively

Barbara Sanders Simon Fraser University July 2017 3 / 26

Introduction TBP model Illustrations Conclusion

Target Benefit Plans

Key features: Predefined contribution level Sponsor liability limited to contributions Target benefit level Actual benefits vary Collective asset pool Members bear risk collectively

Barbara Sanders Simon Fraser University July 2017 3 / 26

Introduction TBP model Illustrations Conclusion

Target Benefit Plans

Practical objectives: Provide adequate benefits Maintain stability Respect intergenerational equity Key question: Given some starting asset value and contribution commitment, how should assets be invested and benefits be paid out to achieve these goals?

Barbara Sanders Simon Fraser University July 2017 4 / 26

Introduction TBP model Illustrations Conclusion

Target Benefit Plans

Practical objectives: Provide adequate benefits Maintain stability Respect intergenerational equity Key question: Given some starting asset value and contribution commitment, how should assets be invested and benefits be paid out to achieve these goals?

Barbara Sanders Simon Fraser University July 2017 4 / 26

Introduction TBP model Illustrations Conclusion

Stochastic optimization in pension literature

DB optimization: asset mix and contribution rate

Cairns (1996, 2000), Haberman and Sung (2004), Josa-Fombellida and Rincon-Zapatero (2001, 2004, 2008), Ngwira and Gerard (2007), etc.

DC optimization: asset mix and payout pattern

Gerrard et al. (2004), He and Liang (2013, 2015), etc.

Gollier (2008): asset mix, benefit payout, dividend policy Cui et al. (2011): asset mix, contribution rate, benefit payout

Barbara Sanders Simon Fraser University July 2017 5 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Dynamics of financial market

Risk-free asset S0(t) dS0(t) = r0S0(t)dt, t ≥ 0, where r0 represents the risk-free interest rate. Risky asset S1(t) dS1(t) = S1(t)[µdt + σdW(t)], t ≥ 0, where µ is the appreciation rate of the stock, σ is the volatility rate, and W(t) is a standard Brownian motion.

Barbara Sanders Simon Fraser University July 2017 6 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Plan membership

The fundamental elements of demographic model: n(t) : density of new entrants aged a at time t, s(x) : survival function with s(a) = 1 and a ≤ x ≤ ω. The density of those who attain age x at time t is n(t − (x − a))s(x), x > a.

Barbara Sanders Simon Fraser University July 2017 7 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Salary process

Dynamics of salary rate for a member who retires at time t: dL(t) = L(t)

• αdt + ηdW(t)
• ,

t ≥ 0, where α ∈ R+ and η ∈ R. W is a standard Brownian motion correlated with W, such that E[W(t)W(t)] = ρt. For a retiree age x at time t (x ≥ r), define assumed salary at retirement (x − r years ago) as

• L(x, t) = L(t)e−α(x−r),

t ≥ 0, x ≥ r.

Barbara Sanders Simon Fraser University July 2017 8 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

The time-age structure of the pension plan

Barbara Sanders Simon Fraser University July 2017 9 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Contribution process

Individual contribution rate for active member aged x at time t ≥ 0: C(x, t) = c0(x)eαt, a ≤ x < r. Aggregate contribution rate in respect of all active members at time t: C(t) = r

a

n(t − x + a)s(x)C(x, t)dx, t ≥ 0.

Barbara Sanders Simon Fraser University July 2017 10 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Benefit payment process

Individual pension payment rate at time t: for a new retiree aged r: B(r, t) = f(t)L(t) for an existing retiree aged x > r: B(x, t) = f(t) L(x, t)eζ(x−r) = f(t)L(t)e−(α−ζ)(x−r)

Barbara Sanders Simon Fraser University July 2017 11 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Benefit payment process

Individual pension payment rate at time t: for a new retiree aged r: B(r, t) = f(t)L(t) for an existing retiree aged x > r: B(x, t) = f(t) L(x, t)eζ(x−r) = f(t)L(t)e−(α−ζ)(x−r)

Barbara Sanders Simon Fraser University July 2017 11 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Benefit payment process

Aggregate pension benefit rate for all retirees at time t: B(t) = ω

r

n(t − x + a)s(x)B(x, t)dx = I(t)f(t)L(t), t ≥ 0. The updated aggregate target benefit is B∗eβt.

Barbara Sanders Simon Fraser University July 2017 12 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Fund dynamics

The pension fund dynamic can be described as

• dX(t) = π(t) dS1(t)

S1(t) + (X(t) − π(t)) dS0(t) S0(t) + (C(t) − B(t))dt,

X(0) = x0.

Barbara Sanders Simon Fraser University July 2017 13 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

The objective function

Let J(t, x, l) be the objective function at time t with the fund value and the salary level being x and l. It is defined as         

J(t, x, l) = Eπ,f T

t

• B(s) − B∗eβs2 − λ1
• B(s) − B∗eβs

e−r0sds +λ2

• X(T) − x0er0T2 e−r0T
• ,

J(T, x, l) = λ2

• X(T) − x0er0T2 e−r0T.

The value function is defined as φ(t, x, l) := min

(π,f)∈Π J(t, x, l),

t, x, l > 0.

See Ngwira and Gerrard (2007), He and Liang (2015). Barbara Sanders Simon Fraser University July 2017 14 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Using variational methods and Itˆ

• ’s formula, we get the

following HJB equation satisfied by the value function φ(t, x, l): min

π,f

• φt +
• r0x + (µ − r0)π + C1(t)eαt − fl · I(t)
• φx + αlφl

+ 1 2π2σ2φxx + 1 2η2l2φll + ρσηlπφxl + fl · I(t) − B∗eβt2 − λ1

• fl · I(t) − B∗eβt

e−r0t

• = 0.

Barbara Sanders Simon Fraser University July 2017 15 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

Solution to the optimization problem

Optimal strategies are π∗(t, x, l) = − δ 2σ [2x + Q(t)] , f ∗(t, x, l) = 1 l · I(t) λ1 2 + λ2 2 (2x + Q(t)) P(t) + B∗eβt

• ,

where δ = (µ − r0)/σ is the Sharp Ratio. The corresponding value function is given by φ(t, x, l) = λ2e−r0tP(t)[x2 + xQ(t)] + K(t).

Barbara Sanders Simon Fraser University July 2017 16 / 26

Introduction TBP model Illustrations Conclusion Model formulation Solution to the optimization problem

P(t) =   

1 λ2(T−t)+1,

r0 = δ2,

r0−δ2 λ2+(r0−δ2−λ)e−(r0−δ2)(T−t) ,

r0 = δ2, Q(t) =      2er0t T

t C1(s)e(α−r0)sds − B∗(T − t) − x0

• ,

β = r0, 2er0t T

t C1(s)e(α−r0)sds − B∗ (e(β−r0)T −e(β−r0)t) β−r0

− x0

• ,

β = r0, K(t) = λ2 T

t

e−r0t

• P(s)Q(s)
• C1(s)eαs − B∗eβs

− 1 4

• δ2 + λ2P(s)
• Q(s)
• − λ2

1

4

• ds.

Barbara Sanders Simon Fraser University July 2017 17 / 26

Introduction TBP model Illustrations Conclusion

Assumptions for numerical illustrations

a = 30, r = 65, ω = 100. Force of mortality follows Makeham’s Law. n(t) = 10 for all t ≥ 0, implying a stationary population. B∗ = 100, β = 0.025. Cost-of-living adjustment rate ζ = 0.02. r0 = 0.01, µ = 0.1, σ = 0.3, ⇒ δ = 0.3. α = 0.03, η = 0.01; initial salary rate L(0) = 1. Correlation coefficient ρ = 0.1; λ1 = 15, λ2 = 0.2. X(0) = 2500, c0 = 0.1.

See Dickson et al. (2013) Barbara Sanders Simon Fraser University July 2017 18 / 26

Introduction TBP model Illustrations Conclusion

Numerical analysis

Percentiles of π∗(t)/X ∗(t) and f ∗(t)

2 4 6 8 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 14 16 18 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Barbara Sanders Simon Fraser University July 2017 19 / 26

Introduction TBP model Illustrations Conclusion

Numerical analysis

Sample paths of f ∗(t) and B(t)

2 4 6 8 10 12 14 16 18 20 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 2 4 6 8 10 12 14 16 18 20 60 80 100 120 140 160 180 200 220

Barbara Sanders Simon Fraser University July 2017 20 / 26

Introduction TBP model Illustrations Conclusion

Numerical analysis

Effects of asset returns

1 500 1000 0.8 0.08 1500 0.07 2000 0.6 0.06 2500 0.05 3000 0.4 0.04 0.03 0.2 0.02 0.2 1 0.25 0.3 0.35 0.8 0.08 0.4 0.07 0.45 0.6 0.06 0.5 0.05 0.55 0.4 0.04 0.03 0.2 0.02

Barbara Sanders Simon Fraser University July 2017 21 / 26

Introduction TBP model Illustrations Conclusion

Numerical analysis

Effects of salary and target benefit growth rates

1000 0.05 1200 1400 1600 0.05 1800 0.04 0.045 2000 0.04 2200 0.035 2400 0.03 0.03 0.025 0.02 0.02 0.35 0.05 0.4 0.45 0.05 0.5 0.04 0.045 0.55 0.04 0.6 0.035 0.65 0.03 0.03 0.025 0.02 0.02

Barbara Sanders Simon Fraser University July 2017 22 / 26

Introduction TBP model Illustrations Conclusion

Numerical analysis

Medians of f ∗(t) for different values of λ1 and λ2

2 4 6 8 10 12 14 16 18 20 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 2 4 6 8 10 12 14 16 18 20 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5

Barbara Sanders Simon Fraser University July 2017 23 / 26

Introduction TBP model Illustrations Conclusion

Conclusion

We apply the Black-Scholes framework for plan assets, and consider a correlated salary process. We consider three key objectives of plan trustees (benefit adequacy, stability and intergenerational equity). We derive closed form expressions for optimal investments and payouts. The model is useful for identifying combinations of inputs that can meet stakeholders’ stated objectives.

Barbara Sanders Simon Fraser University July 2017 24 / 26

Introduction TBP model Illustrations Conclusion

References

Cui, J., De Jong, F ., and Ponds, E. (2011). Intergenerational risk sharing within funded pension schemes. Journal of Pension Economics and Finance, 10(01):1-29. Dickson, D. C., Hardy, M. R., and Waters, H. R. (2013). Actuarial mathematics for life contingent risks. Cambridge University Press. Gollier, C. (2008). Intergenerational risk-sharing and risk-taking of a pension fund. Journal of Public Economics, 92(5):1463-1485. He, L. and Liang, Z. (2015). Optimal assets allocation and benefit outgo policies of DC pension plan with compulsory conversion claims. Insurance: Mathematics and Economics, 61:227-234. Ngwira, B. and Gerrard, R. (2007). Stochastic pension fund control in the presence of Poisson jumps. Insurance: Mathematics and Economics, 40(2):283-292.

Barbara Sanders Simon Fraser University July 2017 25 / 26

Introduction TBP model Illustrations Conclusion

Questions?

Barbara Sanders Simon Fraser University July 2017 26 / 26