Optimal Surrender Policy for Variable Annuity Guarantees

Transcription

1 Optimal Surrender Policy for Variable Annuity Guarantees Anne MacKay University of Waterloo January 31, 2013 Joint work with Dr. Carole Bernard, University of Waterloo Max Muehlbeyer, Ulm University Research funded by the Hickman Scholarship of the Society of Actuaries and NSERC Anne MacKay University of Waterloo Optimal Surrender for VA 1/22

2 Outline 1 Introduction 2 Optimal surrender boundary: GMAB case 3 Optimal exercise boundary: Path-dependent case 4 Conclusion Anne MacKay University of Waterloo Optimal Surrender for VA 2/22

3 Variable annuities and financial options ˆ Variable annuities: similar to mutual funds with additional guarantees ˆ VA guarantees paid for via a fixed fee rate throughout the life of the contract (not paid upfront) ˆ Decreases return on fund ˆ Impacts value of option ˆ VA contract can be surrendered (Knoller, Kraut, and Schoenmaekers (2011)) ˆ Financial needs ˆ Higher costs of opportunity ˆ Moneyness of the option Anne MacKay University of Waterloo Optimal Surrender for VA 3/22

4 Continuous fee and the surrender option ˆ Simple payoff at maturity: max(g, F T ) = F T + (G F T ) + ˆ Option paid by continuous fee set as percentage of fund: ˆ Fee is low when option value is high ˆ Incentive to surrender when fund value is high ˆ Surrender region: surrender benefit higher than payoff expected if contract is kept Anne MacKay University of Waterloo Optimal Surrender for VA 4/22

5 Setting Index value S t : ds t S t = r dt + σdw t Account value F t based on index: df t F t = (r c)dt + σdw t F t F s LN (log(f t ) + (r c σ2 2 )(t s), σ2 (t s)) Anne MacKay University of Waterloo Optimal Surrender for VA 5/22

6 Contract ˆ Accumulation benefit: max(f T, G), G = F 0 e gt, g < r ˆ Surrender benefit: e κ(t t) F t Anne MacKay University of Waterloo Optimal Surrender for VA 6/22

7 Integral representation for surrender option ˆ Similarities between surrender option and American option ˆ Can use techniques developed for American options (Kim (1990), Kim and Yu (1996), Carr, Jarrow, and Myneni (1992), Wu and Fu (2003)) ˆ Integral can be obtained in different ways: ˆ Finite number of surrender times (as in Kim (1990)) ˆ No-arbitrage arguments (as in Kim and Yu (1996)) Anne MacKay University of Waterloo Optimal Surrender for VA 7/22

8 Trading strategy for surrender option ˆ Confirm that optimal strategy is a threshold strategy ˆ Hold the VA below the surrender boundary B t ˆ When F t crosses B t from below, sell the VA and invest the proceeds ˆ When F t crosses B t from above, use the investment to buy the VA ˆ Payoff of portfolio is payoff of VA Anne MacKay University of Waterloo Optimal Surrender for VA 8/22

9 Gain from surrender Proposition The benefit associated with the surrender option between [t, t + dt] for an infinitesimal time step dt is given by e κ(t t) (c κ)f t dt Anne MacKay University of Waterloo Optimal Surrender for VA 9/22

10 Proof of Proposition ˆ Suppose surrender at t. ˆ Policyholder receives e κ(t t) F t = e κ(t t) e ct S t = e κt e (c κ)t S t ˆ To buy the VA at time t + dt, policyholder only needs e κ(t (t+dt)) F t+dt = e κt e (c κ)(t+dt) S t+dt ˆ Investment made at t becomes e κt e (c κ)(t+dt) S t+dt + e κt e (c κ)t S t e rdt (1 e (c κ)dt ) = e κ(t (t+dt)) F t+dt + e κ(t t) (c κ)f t dt + o(dt) Anne MacKay University of Waterloo Optimal Surrender for VA 10/22

11 Price of VA contract Price of VA with surrender V (t, F t ) = E[e rt max(g, F T ) F t ] + }{{} Maturity benefit T e κt (c κ)f t e ct e (c κ)u Φ ( d 1 (F t, B u, u, t) ) du t }{{} Surrender option ˆ Maturity benefit: Similar to vanilla option under Black-Scholes ˆ Surrender option: T t e r(u t) e κ(t u) (c κ)xf Fu (x F t )dxdu. B u Anne MacKay University of Waterloo Optimal Surrender for VA 11/22

12 Optimal exercise boundary condition ˆ At maturity B T = G T and along the surrender boundary, V (F t, t) = e κ(t t) F t = B t. ˆ Work backwards to solve for B t B t =v(f t, t) + e(f t, t) =e c(t t) B t e κ(t t) Φ(d 1 (B t e κ(t t), G T, T, t)) + e r(t t) G T Φ(d 2 (B t e κ(t t), G T, T, t)) T + (c κ)b t e (c κ)t e (c κ)u Φ(d 1 (B t e κ(t t), B u, u, t))du t Anne MacKay University of Waterloo Optimal Surrender for VA 12/22

13 Numerical example Assumptions: ˆ T = 15 ˆ G = F 0 = 100 ˆ κ = 0, unless otherwise indicated ˆ c = 0.91% (fair fee for maturity benefit) ˆ r = 0.03 ˆ σ = 0.2, unless otherwise indicated Anne MacKay University of Waterloo Optimal Surrender for VA 13/22

14 Optimal exercise boundary, sensitivity analysis: sigma 200 sigma=0.15 sigma=0.20 sigma=0.25 sigma= Underlying fund price F(t) Time t (Years)

15 Optimal exercise boundary, sensitivity analysis: kappa 200 k=0 k=c/4 k=c/3 k=c/2 180 Underlying fund price F(t) Time t (Years)

16 Geometric average ˆ Consider the payoff max(g T, Y T ), where Y T is the geometric average defined as Y t = exp 1 t t 0 lnf s ds ˆ The conditional distribution of Y u (Y t, F t ) for u > t is again log-normal with mean and variance given by M g t V g t = t u lny t + u t u = σ2 (u t)3 3u2 lnf t + r c σ 2 2 (u t) 2 2u Anne MacKay University of Waterloo Optimal Surrender for VA 16/22

17 Pricing formula Theorem Let V g (Y t, F t, t) denote the price at time t of the VA with guarantee G T and a surrender benefit equal to e κ(t t) Y t. Then V g (Y t, F t, t) can be decomposed into a European part v g (Y t, F t, t) and an early exercise premium e g (Y t, F t, t) where V g (Y t, F t, t) = v g (Y t, F t, t) + e g (Y t, F t, t), v g (Y t, F t, t) = e r(t t) e Mg t + V g t 2 Φ e g (Y t, F t, t) = e κt e rt T t ( ln(gt )+M g t +V g t V g t e u(κ r) e Vu,t 2 Y t u t F u t 2u t ) ) + e r(t t) G T Φ( ln(gt ) M g t V g, t E [k(u, F u, t)] du Anne MacKay University of Waterloo Optimal Surrender for VA 17/22

18 Particularities of the path-dependent case ˆ Optimal surrender behaviour depends on account value F t and geometric average Y t. Optimal surrender surface ˆ To solve for optimal surrender surface, need to consider many values F t at each time t. ˆ To simplify calculations, assume that B t (F t ) has the form B t (F t ) = max(g T e r(t t), a t + b t F t ) Anne MacKay University of Waterloo Optimal Surrender for VA 18/22

19 Numerical example Additional assumptions: ˆ T = 10 ˆ Payoff: max(y T, G T ) ˆ G T = F 0 e 0.025T ˆ c = Anne MacKay University of Waterloo Optimal Surrender for VA 19/22

20 Optimal Boundary F t time

21 Conclusion ˆ Integral representation for the surrender option ˆ Can retrieve optimal surrender boundary ˆ Can be used for path independent and path dependent payoffs Future work: ˆ Use for other types of fee structures ˆ Consider flexible premium (as in Chi and Lin (2013)) Anne MacKay University of Waterloo Optimal Surrender for VA 21/22

22 References Carr, P., R. Jarrow, and R. Myneni (1992): Alternative characterizations of American put options, Mathematical Finance, 2(2), Chi, Y., and S. X. Lin (2013): Are Flexible Premium Variable Annuities Underpriced?, working paper available at SSRN. Kim, I. J. (1990): The Analytic Valuation of American Options, The Review of Financial Studies, 3(4), Kim, I. J., and G. G. Yu (1996): An alternative approach to the valuation of American options and applications, Review of Derivatives Research, 1(1), Knoller, C., G. Kraut, and P. Schoenmaekers (2011): On the Propensity to Surrender a Variable Annuity Contract, Discussion paper, Working Paper, Ludwig Maximilian Universitaet Munich. Wu, R., and M. C. Fu (2003): Optimal exercise policies and simulation-based valuation for American-Asian options, Operations Research, 51(1), Anne MacKay University of Waterloo Optimal Surrender for VA 22/22

23 Thank you for your attention! Anne MacKay University of Waterloo Optimal Surrender for VA 23/22