Policyholder Exercise Behavior for Variable Annuiies including Guaraneed Minimum Wihdrawal Benefis 1 2 Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, 11h Floor; Alana, GA 30303; USA Email: horsen@gsu.edu June 2011 1 We graefully acknowledge sponsorship by he Sociey of Acuaries. 2 Join work wih Dr. Daniel Bauer, Georgia Sae Universiy
Page 2 / 23 Overview 1 Inroducion 2 A Lifeime Uiliy Model for Variable Annuiies 3 Resuls 4 Conclusions and Fuure Research
Page 3 / 23 Inroducion 1 Inroducion Moivaion Risk-Neural Valuaion Approach 2 A Lifeime Uiliy Model for Variable Annuiies 3 Resuls 4 Conclusions and Fuure Research
Page 4 / 23 Inroducion Moivaion Variable Annuiies: Popular long-erm invesmen vehicles Tax-deferred growh Invesmen evolves according o underlying (risky) porfolio Uncerain payou Guaraneed Minimum (Deah / Income / Accumulaion / Wihdrawal) Benefis Insurers offer guaraneed paymens Policyholders can purchase securiy Similar o (combinaion of) financial opions
Page 5 / 23 Inroducion Moivaion Wihdrawal uncerainy Could miigae or inensify insurer s exposure o invesmen and/or moraliy risks Ineracions non-rivial Affecs pricing and risk managemen Insurers in rouble Disinermediaion in 1970s Equiable Life closed o new business in 2000 The Harford acceped $3.4B in TARP money in June 2009 afer losing $2.75B in 2008, hur by invesmen losses and he cos of VA guaranees
Page 6 / 23 Inroducion Risk-Neural Valuaion Approach Used in acuarial lieraure o price variey of opions: Milevsky and Posner (2001): GMDB Ulm (2006): Real opion o ransfer Zaglauer and Bauer (2008): Paricipaing life insurance conracs To analyze wihdrawal behavior for GMWBs: Milevsky and Salisbury (2006) Bauer, Kling and Russ (2008) Opimal sopping problem, akin o pricing American pu opion Exercise / Wihdraw if exercise value exceeds coninuaion value Wors-case scenario, calculae correc upper bound VA marke incomplee: canno sell or repurchase policy a is risk-neural value Wihdrawing means giving up possible guaranees and ax benefis
Page 7 / 23 A Lifeime Uiliy Model for Variable Annuiies 1 Inroducion 2 A Lifeime Uiliy Model for Variable Annuiies The Model Bellman Equaion Implemenaion in a Black-Scholes Framework Parameer Assumpions 3 Resuls 4 Conclusions and Fuure Research
Page 8 / 23 A Lifeime Uiliy Model for Variable Annuiies The Model Consider wihdrawal decisions in life-cycle model wih ouside invesmen PH maximizes expeced lifeime uiliy Consumpion and bequess Iniial wealh W 0 Annual (deerminisic) income I Invess P 0 in VA wih finie mauriy T, remainder in ouside accoun Includes GMWB, possibly oher guaranees Reurn-of-invesmen guaranees Oher ypes possible, a cos of larger sae space All guaranee accouns idenical o benefis base, G Annual guaranee fee φ as percenage of concurren accoun value
Page 9 / 23 A Lifeime Uiliy Model for Variable Annuiies The Model VAs grow ax-deferred Wihdrawals axed on las-in firs-ou basis Early wihdrawal ax (10%) if PH wihdraws prior o age 59.5 Resric all acions o policy anniversary daes Four sae variables VA accoun X Ouside accoun A Benefis base G Tax base H Three choice variables Wihdrawal amoun w Consumpion C Risk allocaion in ouside accoun ν
Page 10 / 23 A Lifeime Uiliy Model for Variable Annuiies Bellman Equaion V (y ) = max C,w,ν u C (C )+e β E [q x+ u B (b +1 S +1 ) + p x+ V +1 (y +1 S +1 )], (1) y (A, X, G, H ), ( ) + X + = X w, A + = A + I + w C fee I fee G axes, fee I = s max { w min(g W, G W ) }, fee G = s g (w fee I ) I {x+<59.5}, axes = τ min{w fee I fee G, (X H ) + }, ( G w ) + : w g ( { } ) W G +1 = min G w, G X + + X : w > g W ( ( ) ) + + H +1 = H w X H, [ ( ) ] + A +1 = A + ν S+1 κ ν S S+1 1, S [ ] X +1 = X + e φ ν X S+1, S b +1 = A +1 + max{x +1, GD +1 }, ν 0, i ν (i) = 1,,
Page 11 / 23 A Lifeime Uiliy Model for Variable Annuiies Implemenaion in a Black-Scholes Framework Solve by recursive dynamic programming: (I) Creae appropriae sae space grid (II) For = T : for all grid poins (A, X, G, H), compue V T (A, X, G, H). (III) For = T 1, T 2,..., 1: Given V +1, calculae V (A, X, G, H) recursively for each (A, X, G, H) on he grid using an approximaion of he inegral in (1) Discreize reurn space and evaluae via Green s funcion Gauss-Hermie quadraure (IV) For = 0: For he given saring values A 0 = W 0 P 0, X 0 = P 0, G 0 = G 1 = P 0 and H 0 = H 1 = P 0, compue V 0 (W 0 P 0, P 0, P 0, P 0 ) recursively from Equaion (1)
Page 12 / 23 A Lifeime Uiliy Model for Variable Annuiies Parameer Assumpions Policyholder is 55 years old, T = 15 years o mauriy P 0 = 100K ; W 0 = 2 P 0 = 200K ; I = 40K CRRA(γ = 3) uiliies; B = 1; β = r τ = 30%, κ = 15% Guaranee fee φ = 50 bps Surrender fee s = 5%, { g W 0 : 5 = 20, 000 : > 5 r = 4%, µ = 8%, σ = 15% Meron raio: µ r 0.08 0.04 = σ2 γ 0.15 2 3 0.5926 ν X = 100% equiy exposure in VA
Page 13 / 23 Resuls 1 Inroducion 2 A Lifeime Uiliy Model for Variable Annuiies 3 Resuls Wihdrawal Behavior Pricing and Sensiiviies 4 Conclusions and Fuure Research
Page 14 / 23 Resuls Wihdrawal Behavior Lile wihdrawal aciviy (approx. 12K per PH on average) No wihdrawals during accumulaion period No premaure wihdrawals in 67% of cases PH empies guaranee accoun in 6% of cases < 1% chance of excessive wihdrawal during conrac phase Two main reasons o wihdraw premaurely: VA accoun below ax base (approx. 7K on average) Nuanced paerns Ineracion of in-he-moneyness of guaranee, ax consideraions and excess wihdrawal charge VA accoun much greaer han ouside accoun (approx. 5K on average) To reduce overall risk exposure (Meron raio)
Page 15 / 23 Resuls Wihdrawal Behavior Fig. 1: =14, A 15 x 104 = 180K, Gṫ = 100K, H = 100K. w 14 max(guaranee,x ) 12 10 w 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X x 10 4
Page 16 / 23 Resuls Wihdrawal Behavior Fig. 3: =10, A 15 x 104 = 180K, Gṫ = 100K, H = 100K. w 14 max(guaranee,x ) 12 10 w 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 X x 10 4
Page 17 / 23 Resuls Wihdrawal Behavior Fig. 7: =10, A 8 x 104 = 20K, Gṫ = 100K, H = 100K. w C A + I = 60K 7 6 5 w,c 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 X x 10 5
Page 18 / 23 Resuls Pricing and Sensiiviies Guaranee fee of φ = 50 bps sufficien o cover expeced coss In-he-moneyness appears o be OK proxy for pricing Differen source o wihdrawals Eliminaing excess wihdrawal fee increases ne profis (win-win) Base Case w/d if X G s = 0% E Q [Fees] 6, 252 5, 925 5, 907 E Q [Excess-Fee] 19 0 0 E Q [Guaranee] 4, 558 4, 761 2, 136 %(Guaranee > 0) 24.34% 33.97% 31.94% E[agg. w/d] 12, 084 14, 180 16, 374 E[agg. w/d & 6] 0 0 4, 374 E[agg. w/d & X H ] 6, 953 14, 119 6, 047 E[agg. w/d & X > H ] 5, 030 0 5, 816
Page 19 / 23 Resuls Pricing and Sensiiviies Wihdrawal paerns highly sensiive o volailiy Considering axes imporan BC: σ = 15% σ = 20% σ = 25% No Taxes E Q [Fees] 6, 252 6, 047 5, 152 4, 734 E Q [Excess-Fee] 19 62 1, 006 64 E Q [Guaranee] 4, 558 8, 384 11, 533 4, 746 %(Guaranee > 0) 24.34% 36.48% 45.57% 28.16% E[agg. w/d] 12, 084 29, 914 85, 879 88, 791 E[agg. w/d & 6] 0 54 13, 356 82 E[agg. w/d & X H ] 6, 953 17, 615 27, 877 19, 500 E[agg. w/d & X > H ] 5, 030 10, 674 31, 221 69, 033
Page 20 / 23 Conclusions and Fuure Research 1 Inroducion 2 A Lifeime Uiliy Model for Variable Annuiies 3 Resuls 4 Conclusions and Fuure Research
Page 21 / 23 Conclusions and Fuure Research Develop lifeime-uiliy model o analyze wihdrawal behavior for VA wih guaranees Numerically solve policyholder s decision making problem in Black-Scholes framework Reurn-of-invesmen GMWB Infrequen wihdrawals PH wihdraws when VA accoun is below ax base Ineracion of in-he-moneyness of guaranee, ax consideraions and excess w/d fee PH wihdraws when VA accoun is large To lower overall risk exposure
Page 22 / 23 Conclusions and Fuure Research Exend policyholder environmen Unemploymen Risk Subjecive moraliies Wihdrawal paerns highly sensiive w.r.. volailiy σ Sochasic volailiy framework Alernaives o EUT Epsein-Zin preferences Correlaion Aversion
Page 23 / 23 THANK YOU!