(MS, ) Problem 1

Transcription

1 MS, 7.6.4) AKTUAREKSAMEN KONTROL I FINANSIERING OG LIVSFORSIKRING ved Københavns Universie Sommer 24 Skriflig prøve den 4. juni 24 kl All wrien aids are allowed. The wo problems of oally 3 quesions comprise 6 pages. Problem In his problem we shall sudy porfolio opimizaion under a predefined paymen sream give bealingssrøm). Throughou, n > is a fied ime horizon. During he lecures we have sudied a porfolio opimizaion problem where an invesor opimizes final wealh subjec o a predefined paymen rae y. This problem was formalized by ma E π γ X n)γ, where he wealh dynamics are given by dx ) = r + π α r)) X ) d yd + πσx ) dw ), X ) =, where we have used he noaion from he lecures, i.e. π represening he proporion of wealh invesed in a Black-Scholes marke sock inde, ec. The sign on y defines y as a predefined consumpion sream for y posiive) give forbrugsbealingssrøm). The siuaion of a negaive y hen describes he siuaion of a predefined income sream give indkomsbealingssrøm). We shall here insead sudy he case where he predefined paymen sream consiss of a lump sum paymen sumbealing) a ime n only. Thus, he problem can be formalized by ma E π γ X n)γ, )

2 where dx ) = r + π α r)) X ) d + πσx ) dw ) ydε n ), X ) =. Here ε n ) indicaes { n}, i.e. ε n ) = n, ) ). where Argue ha he opimizaion problem ) is equivalen o he problem ma E X n) y)γ, 2) π γ dx ) = r + π α r)) X ) d + πσx ) dw ), X ) =. 2 Define he opimal value funcion V, ) relevan for he opimizaion problem 2), wrie down he Bellman equaion for V, ) including he side consrain V n, ) = γ y)γ, imposed by 2)). Show ha he opimal invesmen proporion π in erms of he derivaives of) V, ) is given by he usual proporion π = α r σ 2 3 Guessing a soluion on he form V, ) V, ) ). V, ) = γ f ) γ g )) γ will lead o ordinary differenial equaions for f and g wih side condiions according o he side condiions for V, ). Wha are he erminal condiions slubeingelser) for f n) and g n)? The ordinary differenial equaion for g ) becomes g ) = rg ). Wrie down an eplici soluion g ) and give an inerpreaion. 4 Deermine he opimal amoun invesed in socks and eplain how he invesmen porfolio can be said o consis of a par hedging y and a residual par invesed according o a fied proporion α r in he risky asse. γ σ 2 2

3 I is possible o generalize he resuls from above o a siuaion where he predefined paymen y is a funcion h of he sock price, i.e. y = h S n)). In his case he value funcion is defined by V, s, ) = ma E π γ and he Bellman equaion becomes V, s, ) = min π V n, s, ) = γ h s))γ. X n) h S n)))γ S ) = s, X ) =, V, s, ) r + π α r)) 2 V, s, ) 2 π 2 σ 2 V s, s, ) αs 2 V ss, s, ) σ 2 s 2 V s, s, ) sπσ 2, 5 Deermine he opimal porfolio π in erms of he derivaives of) V, s, ). Show ha if V can be wrien on he form wih funcions f and g differen from he ones inroduced in 3) V, s, ) = γ f ) γ g, s)) γ, 3) hen π = α r g, s)) + g γ σ 2 s, s) s. 6 The guess 3) will lead o an ordinary differenial equaion for f and a parial differenial equaion for g. Wha are he erminal condiions for f n) and g n, s)? The parial differenial equaion for g, s) becomes g, s) = rg, s) rsg s, s) 2 σ2 s 2 g ss, s). Wrie down a sochasic represenaion formula for g, s) and give an inerpreaion recall maerial in financial mahemaics). Wha can you now say abou he opimal porfolio found in 5? Problem 2 In his problem we sudy opimal dividend payous for a emporary life annuiy ophørende livrene). The benefis are paid for by a single premium indskud) a ime and run by he rae of from ime unil ime n or deah whichever occurs firs. Deah a ime occurs by he inensiy µ ). The free 3

4 reserve de frie reserver/bonuspoeniale) is invesed in a riskfree asse only wih consan rae of ineres r. The dividends are paid ou a ime by he rae δ ) if he annuian modageren af livrenen) is alive. The insurance company wishes o pay ou he free reserves so as o maimize epeced power uiliy of he epeced dividend payous. We formalize he problem described above by he dynamics of he free reserves dx ) = rx ) d I ) δ ) d, X ) =, and he opimizaion problem ma E δ γ I ) δ )γ d. Here I ) indicaes he annuian being alive a ime. Since all dividend paymens sop upon deah he only value funcion of ineres is he value funcion for he annuian being alive. This value funcion reads V, ) = ma E δ γ I s) δγ ds X ) =, I ) =. Argue, e.g. by reference o On Meron s Problem for Life Insurers ha he Bellman equaion for he problem above is given by V, ) = inf V, ) r δ) δ γ δγ + µ ) V, ). 4) 2 Solve he opimizaion problem given above, i.e. specify he value funcion V, ), he opimal conrol δ, and he ime dependen funcion f appearing in he value funcion. You can eiher solve direcly on he basis of 4) or refer o formulas in On Meron s Problem for Life Insurers or boh o double-check. The opimally conrolled free reserve follows he dynamics dx ) = rx ) d I ) δ ) d = r I ) ) X ) d, f ) X ) =, wih soluion R X ) = e r fs))ds Is). 4

5 3 Inerpre he value E e r I ) δ ) d. Argue for he following line of equaliies E e r I ) δ ) d = E e r I ) = E R f ) e r fs))ds d e R µs)ds dñ ), where Ñ ) couns he number of deahs equals or ), occurring wih inensiy, for an individual. f) 4 Show ha if he annuian survives wih probabiliy, i.e. µ ) = for n, hen he epeced presen value of fuure dividends equals. Hin: noe ha for all µ, for n, such ha Ñ n) =.almos surely.). f) Show ha oherwise he epeced presen value of fuure dividends is less han. Now we eend he model in which he life annuiy is consruced such ha a disabiliy sae invalideilsand) is inroduced. The disabiliy inensiy is denoed by σ whereas he moraliy inensiy from he sae disabled is denoed by ν. The moraliy inensiy from he sae acive is he µ given above. We assume ha ν > µ. The recovery inensiy rehabilieringsinensie), i.e. he inensiy for going from disabled o acive, is. The life annuiy is sill paid ou as long as he insured is alive independenly of his sae of healh. 5 Define he value funcions V a, ) and V i, ) corresponding o he wo saes acive and disabled. Wrie down he sysem of Bellman equaions for V a, ) and V i, ). The soluions can be wrien as V a, ) = γ f a ) γ γ and V i, ) = γ f i ) γ γ, respecively. Wrie down he opimal conrols δ a, ) and δ i, ) in erms of he funcions f a and f i. 6 Assume ha γ = corresponding o logarihmic uiliy). Wrie down he ordinary differenial equaions for f a and f i and show ha f a ) f i ). Hin: Form he derivaive f a ) f i )). Conclude ha δa,) δi,) and inerpre his observaion. Again we can calculae he epeced presen value of fuure dividends. We need he dynamics of he opimally conrolled free reserves 5

6 dx ) = r I ) ) X ) d, f Z) ) X ) =. We hen define he saewise reserves U a, ) = E e rs ) I s) δ Z) s, X s)) ds X ) =, Z ) = a, U i, ) = E e rs ) I s) δ Z) s, X s)) ds X ) =, Z ) = i. I is possible o derive a sysem of parial differenial equaions for U a and U i do no do his!), U a, ) = ru a, ) δ a, ) + µ ) U a, ) σ ) U i, ) U a, ) ) U a, ) r ), f a ) U a n, ) =, U i, ) = ru i, ) δ i, ) + ν ) U i, ) U i, ) r ), f i ) U i n, ) =. 7 Show ha U a, ) = g a ), U i, ) = g i ), where g i ) = g a ) = e R R s ντ)dτ s f i s) e e R s µτ)+στ)+ f a τ))dτ f i τ) dτ ds, ) f a s) + σ s) gi s) ds. 6

7 Konrol i finansiering og livsforsikring, WRITTEN EXAMINATION SUMMER 24, SKETCH OF SOLUTIONS. where Le X ) = X ) + yε n ). Then ) is equivalen o ) γ ma E X n) y, γ d X ) = r + π α r)) X ) d + πσ X ) dw ). Renaming X by X reads 2). 2 The value funcion The Bellman equaion V, ) = min π V, ) = ma E π V n, ) = γ y)γ. γ X n) y)γ X ) =. V, ) r + π α r)) 2 V, ) 2 π 2 σ 2, The opimizing porfolio d V, ) r + π α r)) 2 dπ V, ) 2 π 2 σ 2 3 = V, ) α r) V, ) 2 π σ 2 π=π = π = α r V ). 5) σ 2 V f n) =, g n) = y, g ) = e rn ) y. g ) is he presen value of he lump sum consumpion y. 4 Plugging he formulas V, ) = f ) γ + g )) γ, V, ) = γ ) f ) γ + g )) γ 2, 7

8 ino 5) gives he amoun π = α r g )). γ σ 2 Thus he risky invesmen is he usual proporion α r invesed afer he γ σ 2 financial value g is subraced from he wealh. Since y is hedged by riskless invesmen of he amoun g, he oal invesmen sraegy consiss of hedging y and invesing residually risky by γ 5 Opimizing porfolio d dπ α r σ 2. V, s, ) r + π α r)) 2 V, s, ) 2 π 2 σ 2 V s, s, ) αs 2 V ss, s, ) σ 2 s 2 V s, s, ) sπσ 2 = V, s, ) α r) V, s, ) 2 π σ 2 V s, s, ) sσ 2 = π = α r σ 2 V, s, ) V, s, ) V s, s, ) s V, s, ). π=π V, s, ) = γ f ) γ g, s)) γ, V, s, ) = f ) γ g, s)) γ, V, s, ) = γ) f ) γ g, s)) γ 2, V s, s, ) = γ) f ) γ g, s)) γ 2 g s, s). π = α r V, s, ) σ 2 V, s, ) V s, s, ) s V, s, ) = γ α r σ 2 g, s)) + g s, s) s. 6 The sochasic represenaion formula reads where under Q g, s) = E Q e rn ) h S n)) S ) = s, ds ) = rs ) d + σs ) dw Q ). The benefi h S n)) is hedged by invesing he amoun sg s in socks. Thus he opimal amoun can be inerpreed as a combinaion of hedging h S n)) and invesing he residual amoun in socks by he proporion α r. γ σ 2 The Bellman equaion is a special case of a wo sae version of 5) in On Meron s Problem for Life Insurers where π =, a ) =, a ) =, V =. 8

9 2 δ V, ) r δ) γ δγ µ ) V, ) Now guess on Insering in he Bellman equaion V, ) = γ f ) γ γ, V, ) = γ γ f ) γ γ f ), V, ) = f ) γ γ, V, ) γ = f ). γ γ f ) γ γ f ) = f ) γ γ f ) = γ ) γ r f n) =. r = r + µ )) f ), This is a Thiele equaion wih soluion f ) = δ=δ = δ = V, ) γ. ) ) γ µ ) f ) γ f ) γ f ) γ γ f ) µ ) γ f ) e R s r +µ ds. 3 The value??) is he financial value epeced presen value) of fuure dividend payous. If I ) =, hen I s) = for s. Then E e R n r I ) δ ) d = E e R r I ) R f ) e Is) r fs) d = E e R r I ) R f ) e r fs) d R = E I ) f ) e fs) d = = E e R µ R f ) e fs) d e R µ dñ ), where Ñ has he inensiy, since R e fs) f) f) is recognized as he moraliy densiy. 9

10 4 The proporion of paid back if µ = : E dñ ) = E Ñ n) =. The general µ ) proporion of paid back: E e R n µ dñ ) E dñ ) = E Ñ n) =. This is rue even hough Ñ has differen inensiies in he wo siuaions. 5 V a, ) = inf V a, ) r δ) δ γ δγ + µ ) V a, ) σ ) V i, ) V a, ) ), V i, ) = inf V i, ) r δ) δ γ δγ + ν ) V i, ). 6 δ a, ) = δ i, ) = f a ), f i ). f a ) = + µf a ) σ f i ) f a ) ), f a n) =, f i ) = + νf i ), f i n) =. f a f i = µ f i f a) σ f i f a) ν µ) f i. f a ) f i ) = e R s µ+σ ν µ) f i ds for ν µ. By he conrols above we have ha δa,). The proporion of wealh paid ou is larger for he disabled han for he acive. This is due o he increased moraliy since he disabled should ge dividends paid ou faser in order o ge any dividends a all relaive o he acive). 7 U i, ) = g i ), δi,) g i = f i + ν ) gi ) + g i ) f i, g i n) =, g i ) = e R R s ν s f i s) e f i ds.

11 Since here is no recovery his corresponds o he siuaion sudied in 5 wih parameers replaced by disabiliy parameers, however. U a, ) = g a ), g a = f + µ ) a ga ) σ ) g i ) g a ) ) + g a ) f, a g a n) =, g a ) = ) e R s µ+σ+ f a f a s) + σ s) gi s) ds.