Innovative retirement products An Chen , University of Ulm joint with: Peter Hieber (Technical University of Munich) Jakob Klein (Allianz Life) Manuel Rach (University of Ulm) Thorsten Sehner (University of Ulm) 2. ISM-UUlm Joint Workshop Risk and Statistics October 8-10, Ulm
Page 2 An Chen | Innovative retirement products | Content ◮ This talk “Innovative retirement products” is based on two working papers: ◮ An Chen, Peter Hieber and Jakob Klein (2019): “Tonuity: A Novel Individual-Oriented Retirement Plan”. Astin Bulletin , Forthcoming. ◮ An Chen, Manuel Rach and Thorsten Sehner (2019): “On the optimal combination of annuities and tontines”, Preprint
Page 3 An Chen | Innovative retirement products | We start with the first paper Chen & Hieber & Klein (2019)
Page 4 An Chen | Innovative retirement products | Motivation I Ageing society: How to ensure pension security ? ◮ Desirable products (from policyholders’ perspective) ◮ not too costly ◮ providing good protection against longevity risk ◮ secure cash flows in advanced ages
Page 5 An Chen | Innovative retirement products | Motivation II: retirement products ◮ Annuity ◮ longevity protection ( √ ) ◮ Solvency II : Annuity products get more expensive (more risk capital needed). ◮ Tontine ◮ Popular 17th century (FR, GB), today “Le Conservateur” , Sabin (2010), Milevsky and Salisbury (2015, 2016) ◮ not good longevity protection ◮ low risk capital required ( √ ) ⇒ Chen & Hieber & Klein (2019): Tontine/annuity = Tonuity Chen & Rach & Sehner (2019): Other combinations of tontines and annuities
Page 6 An Chen | Innovative retirement products | Annuity and Tontine: Payoff Single premium P 0 at time t = 0. Annuity : payoff c ( t ) ( t ≥ 0) until death (residual life time ζ > 0): b A ( t ) := 1 { ζ> t } c ( t ) . Tontine : homogeneous cohort of size n receives payoff nd ( t ) ( t ≥ 0). Each tontine holder receives: � nd ( t ) 1 { ζ> t } if N t > 0, N t b OT ( t ) := . 0, else where N t is the number of surviving policyholders at time t .
Page 6 An Chen | Innovative retirement products | Tontine: example 1st year 2nd year 3rd year d ( 1 ) = 800, N 1 = 8 d ( 2 ) = 800, N 2 = 7 d ( 3 ) = 720, N 3 = 7 nd ( 2 ) / N 2 ≈ 914 nd ( 3 ) / N 3 ≈ 823 nd ( 1 ) / N 1 = 800
Page 7 An Chen | Innovative retirement products | Actuarially fair pricing I: a simple mortality model ...is used by the insurer to price the retirement products (c.f. Lin and Cox (2005)): (1) Get survival probabilities t p x = P ( ζ > t ) , t ≥ 0 from past data (best-estimate survival probability) (2) Draw a mortality shock ǫ , true survival probabilities are ( t p x ) 1 − ǫ . ( systematic mortality risk ) ◮ ǫ is a r.v. with density f ǫ ( ϕ ) and support on ( −∞ , 1 ) (3) Conditional on ǫ = ϕ , the number of survivors is binomially distributed i.e. N ϕ ( t ) ∼ Bin ( n , t p 1 − ϕ ) , ( unsystematic mortality x risk )
Page 8 An Chen | Innovative retirement products | Actuarially fair pricing II ◮ Premium of the annuity: �� ∞ � P 0 = P A e − rt 1 { ζ> t } c ( t ) dt 0 = E 0 � ∞ � 1 t p 1 − ϕ e − rt c ( t ) = f ǫ ( ϕ ) d ϕ d t x −∞ 0 ◮ Premium of the tontine: � � n � � ∞ � 1 � P 0 = P OT e − rt 1 − t p 1 − ϕ = 1 − f ǫ ( ϕ ) d ϕ d ( t ) dt x 0 0 −∞
Page 9 An Chen | Innovative retirement products | Policyholder’s utility ◮ Policyholder follows constant relative risk aversion ( CRRA ) utility u ( x ) = x 1 − γ 1 − γ , with risk aversion γ ∈ [ 0 , ∞ ) \ { 1 } . ◮ Assumption: the policyholder without bequest motives would choose c ( t ) or d ( t ) to maximize � � ∞ � � � e − ρ t 1 { ζ> t } u E χ ( t ) d t , 0 with χ ( t ) = c ( t ) (annuity) or χ ( t ) = nd ( t ) / N ( t ) (tontine), subjective discount factor ρ , given an actuarially fair premium.
Page 10 An Chen | Innovative retirement products | Theorem (Optimal payout function: Annuity and Tontine) (a) For an annuity product, we obtain − 1 � ∞ γ ( r − ρ ) t · P 0 · γ − r ) t ¯ 1 e ( r − ρ c ∗ ( t ) = e , t p x d t 0 t p x := E [ t p x 1 − ǫ ] . (e.g. Yaari (1965)) where ¯ (b) For a tontine product, we obtain γ ( r − ρ ) t · P 0 1 d ∗ ( t ) = e κ n ,γ,ǫ ( t p x ) · γ , � λ ∗ � 1 � � 1 γ 1 − ( 1 − t p x 1 − ǫ ) n E with suitable κ n ,γ,ǫ and λ ∗ . (e.g. Chen, Hieber and Klein (2019))
Page 11 An Chen | Innovative retirement products | Sketch of a proof , (a), well-known (e.g. Yaari (1965)): ◮ Budget constraint: 1 ∞ � ∞ � � ! e − rt c ( t ) t p 1 − ϕ e − rt P 0 = f ǫ ( ϕ ) d ϕ d t = t p x m ǫ ( − log t p x ) c ( t ) dt . x 0 0 −∞ ◮ Write down the Lagrangian function for λ > 0: 1 1 ∞ ∞ � � � � e − rt c ( t ) e − ρ t t p 1 − ϕ t p 1 − ϕ � � � � L c , λ := f ǫ ( ϕ ) d ϕ · u c ( t ) d t + λ P 0 − f ǫ ( ϕ ) d ϕ d t x x 0 0 −∞ −∞ � λ · e ( ρ − r ) t � − 1 γ . ◮ First-order condition: c ∗ ( t ) = ◮ From budget constraint: � ∞ � � γ . λ ∗ = P − γ e ( r − ρ γ − r ) t t p x · m ǫ ( − log t p x ) d t 0 0
Page 12 An Chen | Innovative retirement products | Numerical example: parameter choices net premium pool size risk aversion P 0 = 10 000 n = 100 γ = 10 risk-free rate subjective discount rate cost of capital rate r = 4 % ρ = 4 % CoC = 6 % initial age Gompertz-law mortality shock ǫ ∼ N ( −∞ , 1 ) ( µ, σ 2 ) x = 65 m = 88 . 721, b = 10 µ = − 0 . 0035, σ = 0 . 0814 b � b � x − m t t p x = e e 1 − e
Page 13 An Chen | Innovative retirement products | Numerical example Optimal payouts c ∗ ( t ) and d ∗ ( t ) . Distribution n · d ∗ ( t ) / N ( t ) .
Page 14 An Chen | Innovative retirement products | Risk capital charge: Risk margin according to Solvency II product risk capital charge F 0 n = 10 101.32 n = 100 tontine 10.89 n = 1 000 1.33 annuity 483.51 Risk capital charges F 0 = CoC · � ∞ t = 0 e − r ( t + 1 ) · SCR ( t ) for different pool sizes n .
Page 15 An Chen | Innovative retirement products | Drawbacks Tontine/Annuity Both products have advantages / disadvantages, mainly: ◮ For an annuity, the insurance company takes the aggregate mortality risk . This increases the cost of risk capital provision (a tontine does not). ◮ A tontine leads to a volatile payoff at old ages (an annuity does not). Chen, Hieber and Klein (2019) suggest one way of combining both products ( Tontine/Annuity = Tonuity )?
Page 16 An Chen | Innovative retirement products | Tonuity: Payoff Idea : Switch between tontine and annuity payoff: nd [ τ ] ( t ) b [ τ ] ( t ) := 1 { 0 ≤ t < min { τ,ζ }} + 1 { τ ≤ t <ζ } c [ τ ] ( t ) , N ( t ) with switching time τ : ◮ A tonuity with switching time τ = 0 is an annuity ◮ A tonuity with switching time τ → ∞ is a tontine ◮ Volatile tontine payoff at old ages is replaced by a secure annuity payoff
Page 17 An Chen | Innovative retirement products | Conclusion of Chen, Hieber and Klein (2019) ◮ Tonuities combine beneficial features of annuities, tontines: ◮ Reduced solvency capital provision (tontine). ◮ Secure income at old ages (annuity). ◮ Each individual can choose an optimal tonuity product (with a corresponding switching time τ ), depending on longevity risk aversion , pool size , cost-of-capital rate .
Page 18 An Chen | Innovative retirement products | Moving to the second paper Chen & Rach & Sehner (2019) : In addition to tonuities, further innovative products are introduced/analyzed: ◮ Antine ◮ Portfolio of Annuities and Tontines
Page 19 An Chen | Innovative retirement products | Antine: Payoff Alternative Idea : Switch between annuity and tontine payoff: n b [ σ ] ( t ) = 1 { 0 ≤ t < min { σ, ζ }} c [ σ ] ( t ) + 1 { σ ≤ t < ζ } N ( t ) d [ σ ] ( t ) with switching time σ : ◮ An antine with switching time σ = 0 is a tontine ◮ An antine with switching time σ → ∞ is an annuity
Page 20 An Chen | Innovative retirement products | Portfolio ◮ The policyholder can now combine annuities and tontines by simultaneously investing in both products to a certain extent. ◮ The resulting payoff of this portfolio is given by b AT ( t ) = b A ( t ) + b OT ( t ) .
Page 21 An Chen | Innovative retirement products | Expected discounted lifetime utility ◮ A policyholder with an initial wealth v follows constant relative risk aversion ( CRRA ) utility u ( x ) = x 1 − γ 1 − γ with risk aversion γ ∈ [ 0 , ∞ ) \ { 1 } . ◮ Assumption: the policyholder without bequest motives would choose b ( t ) to maximize �� ∞ � � � e − ρ t u ( b ( t )) 1 { ζ ǫ > t } d t U { b ( t ) } t ≥ 0 := E , 0 under a budget constraint, where b ( t ) is the contract payoff from the various retirement products.
Page 22 An Chen | Innovative retirement products | Budget constraint: Expected value principle ◮ Premium of the annuity ( m ǫ ( s ) = E [ e s ǫ ] ) �� ∞ � � ∞ P A e − rt b A ( t ) dt e − rt 0 = E = t p x m ǫ ( − log t p x ) c ( t ) dt 0 0 � P A 0 = ( 1 + C A ) P A 0 ◮ Premium of the tontine: � � n � � ∞ � 1 � 1 − t p 1 − ϕ P OT e − rt = 1 − f ǫ ( ϕ ) d ϕ d ( t ) dt x 0 0 −∞ P OT � = ( 1 + C OT ) P OT 0 0 ◮ Note: C A > C OT ≥ 0
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