Indifference fee rate 1 for variable annuities Ricardo ROMO ROMERO Etienne CHEVALIER and Thomas LIM Université d Évry Val d Essonne, Laboratoire de Mathématiques et Modélisation d Evry Second Young researchers meeting on BSDEs, Numerics and Finance. 07-09 July 2014, Bordeaux, France. 1 This research benefitted from the support of the Chaire Marchés en Mutation, Fédération Bancaire Française. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 1 / 31
Outline 1 Variable Annuities 2 Model 3 Indifference fees 4 Numerical Results Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 2 / 31
Outline Variable Annuities 1 Variable Annuities 2 Model 3 Indifference fees 4 Numerical Results Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 3 / 31
Variable Annuities What is a Variable Annuity? Variable annuity is a contract between a policyholder and an insurance company. The policyholder gives an initial amount of money to the insurer. It is invested in a reference portfolio until a preset date, until the policyholder withdraws from the contract or dies. At the end of the contract, the insurance pays an amount of money depending on the performance of the reference portfolio. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 4 / 31
Risks Variable Annuities Actuarial risks: mortality, longevity,.. Financial risks: volatility, interest rate,.. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 5 / 31
Variable Annuities Literature Bauer (2008) presents a general framework to define Variable Annuities (VA). Boyle and Schwartz (1977), extend the Black-Scholes framework to insurance issues. Milvesky and Posner (2001) apply risk neutral option pricing theory to value Guaranteed Minimum Death Benefits (GMDB) in VAs. Dai et al. (2008) HJB equation is derived for a singular control problem related to VA. Belanger et al. (2009) describes the GMDB pricing problem as an impulse control problem. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 6 / 31
Outline Model 1 Variable Annuities 2 Model 3 Indifference fees 4 Numerical Results Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 7 / 31
Model Model (main points) No restrictive assumptions on the reference portfolio and the interest rate dynamics (Markovianity of processes is not assumed): Incomplete market, not a unique risk-neutral measure. We introduce a methodology with BSDEs with a jump. Indifference pricing with continuous fees. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 8 / 31
Model Financial Market and Wealth Process Let (Ω, F, P) be a complete probability space, with F the Brownian filtration. Financial market: ds 0 t = r t S 0 t dt, t [0, T ], S 0 0 = 1, ds t = S t (µ t dt + σ t db t ), t [0, T ], S 0 = s > 0 where µ, σ and r are F-adapted bounded processes and σ is lower bounded by a positive constant. Discounted wealth process: X x,π t = x + t with strategy π and initial capital x. 0 π s (µ s r s )ds + t 0 π s σ s db s, Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 9 / 31
Model Exit time of a Variable Annuity Policy Let τ be the exit time which is the minimum time between: The time of death of the insured. The time of total withdrawal. The random time τ is not assumed to be an F-stopping time. We consider G := (G t ) t 0 with G t := F t σ(1 τ u, u [0, t]) for all t 0. Hypothesis Immersion of F in G: every F-martingale is a G-martingale. The process N. := 1 τ. admits an F-compensator. τ 0 λ t dt. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 10 / 31
Dynamics Model Discounted Account Value A p : da p t = A p t [ (µt r t ξ t p)dt + σ t db t ], t [0, T ], with initial value A 0, fee-rate p and withdrawal (ξ t ) 0 t T. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 11 / 31
Pay-off Model The discounted pay-off including the withdrawals at time T τ to the insured is: F (p) := F L (T, A p )1 {T <τ} + F D (τ, A p )1 {τ T } + T τ 0 ξ s A p s ds. Notice that F (p) is G T τ -measurable. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 12 / 31
Model Guarantees without withdrawals The common guarantees are: Constant guarantee: Gt Q (p) = A 0. Roll-up guarantee: Let be η > 0, then Gt Q (p) = A 0 (1 + η) t. Ratchet guarantee: Gt Q (p) = max(a p t 1, A p t 2,..., A p t ). For t an anniversary date. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 13 / 31
Ricardo ROMO (UniversitéFigure d Evry) : Guarantees Indifferenceand fee rate Account Value. Bordeaux, France 2014 14 / 31 Usual Guarantees Model 2.2 2 Account Process Ratchet Guarantee Roll up Guarantee 1.8 1.6 p 1.4 1.2 1 0.8 0 2 4 6 8 10 12 14 16 18 20 Anniversary Dates
Outline Indifference fees 1 Variable Annuities 2 Model 3 Indifference fees 4 Numerical Results Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 15 / 31
Indifference fees Finding the Indifference Fees The objective is to find a fee p such that sup π A F [0,T ] E [ U ( X x,π T )] = sup E [ U ( A 0 + X x,π T F (p ) )], π A G [0,T ] where A F [0, T ] (resp. A G [0, T ]) is the set of admissible strategies between the interval of time [0, T ] in F (resp. in G). Utility function where γ > 0. U(y) = e γy, y R, Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 16 / 31
Indifference fees The classical problem: V 0 = sup π AF [0,T ] {E [U (X π T )]} Hu, Imkeller and Muller (2004), Rouge and El Karoui (2000) Theorem The optimal value is V 0 = exp(γy 0 ), using the optimal strategy π t := µ t r t γσ 2 t + z t σ t, where y 0 and z are given by the BSDE ( dy t = ν2 t 2γ z tν t ) dt z t db t, y T = 0. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 17 / 31
Indifference fees Utility Maximization with VA (Step 1) V G (p) := sup π AG [0,T ] E [ U ( X A 0,π T F (p) )] Proposition The value function is V G (p) = where F (p) := F (p) + 1 γ log { sup E [ exp ( γ ( X A 0,π T τ F (p) ))], π A G [0,T τ] ess inf π A G [T τ,t ] ) ]} (X E [e γ A 0,π T X A 0,π T τ GT τ. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 18 / 31
Indifference fees Utility Maximization with VA (Step 2) Finding F (p) Proposition There exists a process Y (τ) such that ess inf E [ exp ( (γx A 0,π π A G T X A 0,π T τ )) ] (τ) G T τ = exp(γy T τ ), [T τ,t ] where (Y (τ), Z (τ) ) is solution of the BSDE { ] dy (τ) t = + ν tz (τ) t dt + Z (τ) t db t, [ ν 2 t γ Y (τ) T = 0. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 19 / 31
Indifference fees Utility Maximization with VA (Step 3) V G (p) := sup π AG [0,T ] E [ U ( X A 0,π T F (p) )] Theorem The value function is given by ( V G (p) = exp γ ( A 0 Y 0 (p) )), where (Y (p), Z(p), U(p)) is a solution of Y t (p) = F (p) + T τ t τ T τ Z s (p)db s t τ ( e γus(p) 1 λ s γ T τ t τ ) ν2 s 2γ ν sz s (p) ds U s (p)dh s, t [0, T ]. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 20 / 31
Indifference fees The Optimal Strategy The Strategy πt := ν t γσ t ν t γσ t + Zt(p) σ t, t [0, T τ), + Z (τ) t σ t, t [T τ, T ]. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 21 / 31
Methodology Recapitulation Indifference fees Indifference Fees { [ ( )]} sup π A F [0,T ] {E [U (XT π )]} = sup π A G [0,T ] E U X π,a 0 T F (p). Utility Maximization: Classical Utility Maximization Problem. V0 = exp(γy 0 ). Not the Classical ( Problem. V G (p) = exp γ ( A 0 Y 0 (p) )). Existence of the Indifference Fees. Y 0 (p ) A 0 = y 0. Simulations. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 22 / 31
Indifference fees Existence of the Indifference Fees Consider ψ(p) := Y 0 (p) y 0 A 0, p R. Proposition The function ψ is continuous and non-increasing on R. (i) For any p R, we have ψ(p) > 0 i.e., for any fee p, we have V G (p) < V F. (ii) For any p R, we have ψ(p) < 0 i.e., for any fee p, we have V G (p) > V F. (iii) There exist p 1 and p 2 such that ψ(p 1 )ψ(p 2 ) < 0. Then, there exists an indifference fee p. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 23 / 31
Outline Numerical Results 1 Variable Annuities 2 Model 3 Indifference fees 4 Numerical Results Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 24 / 31
Numerical results. Numerical Results We assume that r and µ are Markov chains taking values in the states spaces S r = {0, 0.01,..., 0.25} and S µ = {0, 0.01, 0.02,..., 0.3}. We give the following numerical values to parameters: γ = 1.3, λ = 0.05, ξ = 0, A 0 = 1, and, for the financial market parameters: r 0 = 0.02, µ 0 = 0.15. Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 25 / 31
Market Risk Numerical Results 0.12 0.1 0.08 p 0.06 0.04 0.02 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 σ Figure : Ratchet option (T = 20). Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 26 / 31
Actuarial Risk Numerical Results 0.4 0.35 0.3 0.25 p 0.2 0.15 0.1 0.05 0 6 8 10 12 14 16 18 20 22 24 26 28 Anniversary Dates Figure : Ratchet option (σ = 0.3). Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 27 / 31
Numerical Results Roll up Guarantee Risk 0.25 0.2 p 0.15 0.1 0.05 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 η Figure : Roll up option (T = 20, σ = 0.3). Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 28 / 31
The End Thank you! Ricardo ROMO ROMERO rickyromo@gmail.com Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 29 / 31
The End Questions? Ricardo ROMO ROMERO rickyromo@gmail.com Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 30 / 31
The End Their respective transitional matrix are: 1 if i = j, 2 1 if i = 1 and j = 2, 2 qi,j r 1 if i = 27 and j = 26, = 2 1 and if i = j + 1 and i 26, 4 1 if i = j 1 and i 2, 4 0 else, 1 if i = j, 2 1 if i = 1 and j = 2, 2 q µ 1 if i = 32 and j = 31, i,j = 2 1 if i = j + 1 and i 31, 4 1 if i = j 1 and i 2, 4 0 else, Ricardo ROMO (Université d Evry) Indifference fee rate Bordeaux, France 2014 31 / 31