Optimal Investment and Consumption Decision of Family with Life Insurance

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1 Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, Speaker 2 Deparmen of Mahemaical Sciences, KAIST 3 School of Compuaional Sciences, KIAS 4 Deparmen of Mahemaical Sciences, KAIST Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

2 Table of Conens 1 Inroducion 2 The Economy 3 The Opimizaion Problem 4 Properies of Opimal Soluions and Numerical Examples 5 Concluding Remarks Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

3 Inroducion Inroducion Two economic agens in family: Parens and Children. Parens lifeime is uncerain. We assume ha parens represen children s faher or moher wih labor income, and ha children have no labor income. While alive, parens receive deerminisic labor income unil T > 0. If parens die before T, he children have no income unil T and hey choose he opimal consumpion and porfolio wih remaining wealh combining he insurance benefi. Consider uiliy funcions of parens and children separaely. Maximize he weighed average of uiliy of parens and uiliy of children. Using he maringale mehod, analyic soluions for he value funcion and he opimal policies are derived. We analyze how he changes of he weigh of parens uiliy funcion and oher facors, such as family s curren wealh level and he fair discouned value of fuure labor income, affec he opimal policies and also illusrae some numerical examples. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

4 Inroducion Lieraure review(seleced) Yaari (1965) Richard (1975) Pliska and Ye (2007): sudied opimal life insurance and consumpion for a income earner whose lifeime is random and unbounded. Ye (2007): considerd opimal life insurance, consumpion and porfolio choice problem under uncerain lifeime using maringale mehod as we used o solve our problem. Bayrakar and Young (2008): solved he problem of maximizing uiliy of consumpion wih a consrain on he probabiliy of lifeime ruin, which can be inerpreed as a risk measure on he whole pah of he wealh process. Huang e al. (2008): invesigaed opimal life insurance, consumpion and porfolio choice problem under uncerain lifeime wih sochasic income process. They focussed on he effec of correlaion beween he dynamics of financial capial and human capial. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

5 The Economy Financial marke composed by wo asses Financial marke I is assumed ha here are one risk-free asse and one risky asse. Risk-free asse: Risky asse: ds 0 = rs 0 d (1) ds 1 = µs 1 d + σs 1 dw (2) W : a sandard Brownian moion on a complee probabiliy space (Ω, F, P) {F } T =0 is he P-augmenaion of he naural filraion generaed by W. r, µ, σ : consans Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

6 Definiions The Economy Conrol variables π(): amoun invesed in he risky asse S 1 a ime c p (): consumpion rae of parens a ime c c (): consumpion rae of children a ime I(): life insurance premium rae a ime Noaions w : deerminisic labor income of parens θ µ r σ : marke-price-of-risk: ζ e 0 (λy+s+r)ds : discoun process Z e θw 1 2 θ2 : exponenial maringale process H ζ Z : pricing kernel(sae-price-densiy) process Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

7 Uncerain life ime The Economy Law of moraliy λ y+ Le λ y+ be an insananeous force of moraliy curve (hazard rae), where y is he age of he breadwinner a iniial ime of he model. Then he condiional probabiliy of survival, from age y o y +, under he law of moraliy defined by λ y+ can be compued by p y e 0 (λy+s)ds. (3) Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

8 The Economy Family s wealh dynamics Life insurance benefi Family s insurance premium rae a ime is I() Receive lump sum paymen I(τ) λ y+τ a he parens deah ime τ. X : family s wealh a ime unil τ m min[τ, T ] Define M() X + I() λ y+ : oal legacy when he parens die a ime wih wealh X Family s wealh dynamics The family s wealh dynamics X saisfies he following SDE: dx = [rx + (µ r)π() c p () c c () I() + w ]d + σπ()dw (4) = [(r + λ y+ )X + (µ r)π() c p () c c () λ y+ M() + w ]d + σπ()dw, for 0 < τ m. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

9 The Economy Family s wealh dynamics Equivalen maringale measure For a given T, we define he equivalen maringale measure P(A) E[Z T 1 A ], for A F T. By Girsanov s heorem, W W + θ, 0 T, is a sandard Brownian moion under he new measure P. Family s wealh dynamics under P The wealh process (4) before τ m can be rewrien as dx = [rx c p () c c () I() + w ]d + σπ()d W (5) = [(r + λ y+ )X c p () c c () λ y+ M() + w ]d + σπ()d W, for 0 < τ m. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

10 Budge consrain The Economy Budge consrain We have he following budge consrain E [ T T ] T H s c p (s)ds + H s c c (s)ds + λ y+s H s M(s)ds + H T X T H (X + b ), for 0 < τ m, (6) where b T w s ζ s ζ ds a. a b is he fair discouned value of he parens fuure labor income from o τ m. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

11 The Opimizaion Problem Opimizaion problem of family Expeced uiliy a ime Family s expeced uiliy funcion U(, X ; c p, c c, π, I) wih an iniial endowmen X a ime, < τ m : U(, X ; c p, c c, π, I) [ τm ] T = E α 1 e δ(s ) u p (c p (s))ds + α 2 e δ(s ) u c (c c (s))ds. (7) u p (c): uiliy funcion of parens u c (c): uiliy funcion of children δ > 0: consan subjecive discoun rae α 1 0: consan weighs of uiliy funcion of parens α 2 0: consan weighs of uiliy funcion of children α 1 and α 2 saisfies α 1 + α 2 = 1. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

12 The Opimizaion Problem Opimizaion problem of family Power uiliy funcions wih lower bound of consumpion rae Uiliy funcion of parens u p (c): Uiliy funcion of children u c (c): u p (c) (c R p) 1 γp 1 (8) u c (c) (c R c) 1 γc 1 γ c. (9) > 0( 1): parens coefficien of relaive risk aversion γ c > 0(γ c 1): children s coefficien of relaive risk aversion R p 0: lower bound of parens consumpion rae R c 0: lower bound of children s consumpion rae Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

13 Meron s consan The Opimizaion Problem Assumpion 1 We define he Meron s consan K i, i = p, c, and assume ha i is always posiive, ha is, K i r + δ r + γ i 1 θ 2 > 0, i = p, c. γ i 2γ 2 i Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

14 The Opimizaion Problem Seps o find he value funcion Sep1 We firs solve he opimizaion problem afer τ m, τ m, Sep2 and hen solve he problem before τ m, < τ m, using maringale mehods. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

15 The Opimizaion Problem Seps o find he value funcion Sep1 We firs solve he opimizaion problem afer τ m, τ m, Sep2 and hen solve he problem before τ m, < τ m, using maringale mehods. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

16 The Opimizaion Problem Sep1: Opimizaion problem afer τ m Opimizaion problem for τ m T If parens die before T, ha is, τ < T, hen w = 0 for τ T. If τ < T, I() = 0 for τ T. Therefore, children have only wo conrol variables: heir consumpion c c (), and invesmen π(). Family s expeced uiliy funcion U c (, X ; c c, π) wih an iniial endowmen X a ime, τ m T : [ ] T U c (, X ; c c, π) = E e δ(s ) u c (c c (s))ds. (10) For τ m T, le A c (, X ) be he admissible class of he pair (c c, π) a ime for which he family s expeced uiliy funcion (10) is well-defined. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

17 The Opimizaion Problem Opimizaion problem for τ m T Lemma 1 For τ m T, he value funcion is V c (, X ) sup U c (, X ; c c, π) = e δ Φ(, X ), (c c,π) A c(,x ) where and { g()γc δ Φ(, X ) e X R c (1 e r(t ))} 1 γ c 1 γ c r g() 1 e Kc(T ) K c. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

18 The Opimizaion Problem Opimizaion problem for τ m T Lemma 1(Coninued) And he opimal policies are given by c c () = 1 g() { X R ( c 1 e r(t ))} + R c r and π () = θ σγ c { X R ( c 1 e r(t ))}. r Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

19 The Opimizaion Problem Opimizaion problem for < τ m Value funcion for < τ m The value funcion of he family a ime < τ m, V (, X ), is defined as follows: V (, X ) [ τm sup E e δ(s ) {α 1 u p (c p (s)) + α 2 u c (c c (s))} ds (c p,c c,π,i) A(,X ) +α 2 1 {τ<t } e δ Φ(τ, M(τ)) ] subjec o he budge consrain (6), where A(, X ) is he admissible class of he quadruple (c p, c c, π, I) a ime < τ m for which he family s expeced uiliy funcion (7) is well-defined. (11) Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

20 The Opimizaion Problem Value funcion for < τ m Theorem 1 The value funcion V (, X ) is given by { T V (, X ) = Y ν X + b R p e s (λ y+u+r)du ds R ( c r(t 1 e ))} r + α 1 γp 1 where Y ν γ ( p 1 X = α Y ν ) γp 1 γp T e s saisfies he following equaion: a 1 γp 1 + R p T ) 1 T (Y ν γp e s e s (λ y+u+r)du ds + R c r (λ y+u+k p)du ds + α 1 γc (λ y+u+k p)du ds + α 1 γc a Given and X, Y ν is uniquely deermined by he equaion (12). 2 2 γ ( c 1 γ c Y ν ) 1 (Y ν γc g() ) γc 1 γc g(), ( 1 e r(t )) b. (12) Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

21 The Opimizaion Problem Theorem 1(Coninued) For 0 < τ m, he opimal policies are given by and c p() = α 1 γp 1 ) 1 (Y ν γp { π () = θ T X + b R p σγ c I () λ y+ + θ σ γ c γ c = M () X = b R p T α e s 1 γp 1 + R p, c c () = α 1 γc 2 ) 1 (Y ν γp ) 1 (Y ν γc e s (λ y+u+r)du ds R c r T (λ y+u+r)du ds α 1 γp 1 + R c, ( r(t 1 e ))} e s (λ y+u+k p)du ds, (13) ) 1 T (Y ν γp e s (λ y+u+k p)du ds. (14) Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

22 Properies of Opimal Soluions and Numerical Examples Properies of Opimal Policies Proposiion 1 If α 1 (0, 1], he opimal life insurance premium rae I () decreases as he wealh level X increases. If α 1 = 0, he opimal life insurance premium rae I () is no affeced by X. Proposiion 2 If α 1 [0, 1), he opimal life insurance premium rae I () increases as he fair discouned value of fuure labor income b increases. If α 1 = 1, he opimal life insurance premium rae I () is no affeced by b. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

23 Properies of Opimal Soluions and Numerical Examples Properies of Opimal Policies Consumpion rae of Parens 7 6 Fix b=100 and vary X from 0 o 100 Fix X=10 and vary b from 90 o X +b Consumpion rae of Children X +b Invesmen X +b Figure 1: Relaions beween he opimal policies and X + b.( = 0, R p = 2, R c = 1, δ = 0.03, r = 0.04, µ = 0.06, σ = 0.3, A L = 0.005, B L = , T = 30) Remarks abou Figure 1 Solid lines represen he relaions beween he opimal policies and X, whereas doed lines represen he relaions beween he opimal policies and b. We can observe ha he opimal policies excep he opimal life insurance premium rae I () are deermined by X + b. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

24 Properies of Opimal Soluions and Numerical Examples Properies of Opimal Policies Life insurance premium rae Life insurance premium rae X +b α 1 =1 α 1 = X +b α 1 =0.5 Fix b=100 and varies X from 0 o 100 by 10 Fix X=10 and varies b from 90 o 190 by 10 Life insurance premium rae X +b Figure 2: Relaions beween he opimal policies and X + b.( = 0, R p = 2, R c = 1, δ = 0.03, r = 0.04, µ = 0.06, σ = 0.3, A L = 0.005, B L = , T = 30) Remarks abou Figure 2 Unlike oher opimal policies, I () is no deermined by X + b. I can be seen ha he curren wealh level X has a negaive effec on he opimal life insurance premium rae I (). On he oher hand, b has a posiive effec on I (). Therefore, he opimal life insurance premium rae I () is deermined no by X + b, bu by boh X and b. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

25 Properies of Opimal Soluions and Numerical Examples Properies of Opimal Policies Proposiion 3 The opimal life insurance premium rae I () decreases as he weigh of he uiliy funcion of parens α 1 increases from 0 o 1. Proposiion 4 The opimal invesmen π() decreases as α 1 increases from 0 o 1 if > γ c. The opimal invesmen π() increases as α 1 increases from 0 o 1 if < γ c. If = γ c, π() is no affeced by α 1. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

26 Properies of Opimal Soluions and Numerical Examples Properies of Opimal Policies Life insurance premium α 1 : weigh of uiliy funcion of parens 5 Consumpion rae of Parens α 1 : weigh of uiliy funcion of parens Consumpion rae of Children α 1 : weigh of uiliy funcion of parens =2, γ c =2 =2, γ c =3 =3, γ c =2 =1/2, γ c =1/2 =2, γ c =2 =2, γ c =3 =3, γ c =2 =1/2, γ c =1/2 =2, γ c =2 =2, γ c =3 =3, γ c =2 =1/2, γ c =1/2 Figure 3: Relaions beween α 1 and he opimal policies. ( = 0, X 0 = 10, R p = 2, R c = 1, δ = 0.03, r = 0.04, µ = 0.06, σ = 0.3, C w = 5.0, k w = 0.03, A L = 0.005, B L = , T = 30) Remarks abou Figure 3 Figure 3 illusrae he relaions beween α 1 and he opimal policies for differen risk aversion. Since α 1 is he weigh of he uiliy funcion of parens, i is obvious ha he opimal consumpion of parens c p() increases and he opimal consumpion of children c c () decreases as α 1 increases. The opimal life insurance premium rae I () decreases as α 1 increases. This is because beques moive is weak when α 1 is large. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

27 Properies of Opimal Soluions and Numerical Examples Properies of Opimal Policies Invesmen Invesmen Opimal Invesmen α 1 : weigh of uiliy funcion of parens Opimal Invesmen α 1 : weigh of uiliy funcion of parens =2, γ c =2 =1/2, γ c =1/2 =2, γ c =3 =3, γ c =2 Figure 4: Relaions beween α 1 and he opimal policies. ( = 0, X 0 = 10, R p = 2, R c = 1, δ = 0.03, r = 0.04, µ = 0.06, σ = 0.3, C w = 5.0, k w = 0.03, A L = 0.005, B L = , T = 30) Remarks abou Figure 4 Unlike oher opimal policies, he effec of α 1 on he opimal invesmen π () depends on risk aversion. If > γ c, π () decreases as α 1 increases, and π () increases as α 1 increases when < γ c. In oher words, if he risk aversion coefficiens of parens and children are differen, π () increases when he relaive imporance of less risk averse family member s uiliy increases. If = γ c, π () is no affeced by α 1. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

28 Concluding Remarks Concluding Remarks We invesigae an opimal porfolio, consumpion and life insurance premium choice problem of family. Analyic soluions for he value funcion and he opimal policies are derived by he maringale mehod. We analyze he properies of he opimal policies, where he emphasis is placed on he role of α 1 which is he weigh of parens uiliy funcion. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

29 Concluding Remarks Concluding Remarks We invesigae an opimal porfolio, consumpion and life insurance premium choice problem of family. Analyic soluions for he value funcion and he opimal policies are derived by he maringale mehod. We analyze he properies of he opimal policies, where he emphasis is placed on he role of α 1 which is he weigh of parens uiliy funcion. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

30 Concluding Remarks Concluding Remarks We invesigae an opimal porfolio, consumpion and life insurance premium choice problem of family. Analyic soluions for he value funcion and he opimal policies are derived by he maringale mehod. We analyze he properies of he opimal policies, where he emphasis is placed on he role of α 1 which is he weigh of parens uiliy funcion. Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

31 Concluding Remarks Thank you! Minsuk Kwak (KAIST) BFS2010, Torono June 25, / 28

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