ΠΑΝΠΙΣΜΙΟΝΠΙΡΑΙΩ ΧΟΛΝΧΡΜΑΣΟΟΙΚΟΝΟΜΙΚΝ & ΣΑΣΙΣΙΚ ΣΜΗΜΑΝΣΑΣΙΣΙΚΗ & ΑΦΑΛΙΣΙΚΗΝΠΙΣΗΜΗ ΠΡΟΓΡΑΜΜΑΝΜΣΑΠΣΤΧΙΑΚΩΝΝΠΟΤΩΝΝ ΣΗΝΝΑΝΑΛΟΓΙΣΙΚΗΝΠΙΣΗΜΗ ΚΑΙ ΙΟΙΚΗΣΙΚΗΝΚΙΝΤΝΟΤ ΑΝΑΛΤΝΚΙΝΤΝΟΤ ΓΙΑ ΧΑΡΣΟΦΤΛΑΚΙΑΝΡΑΝΣΩΝΝΩ ουλέζαμνηηάλδομ δπζωηαδεάνλγαέα Πεδλαδάμ, ΙαθουάλδομΝ2ί16 1
UNIVERSITY OF PIRAEUS SCHOOL OF FINANCE & STATISTICS DEPARTMENT OF STATISTICS & INSURANCE SCIENCE M.SC. IN ACTUARIAL SCIENCE AND RISK MANAGEMENT RISK ANALYSIS FOR OF LIFE ANNUITIES STOCHASTIC PORTFOLIOS OF LIFE ANNUITIES RISKINESS ANALYSIS FOR A LARGE PORTFOLIO Sourilas Dimitrios Dissertation Thesis Piraeus, January 2016 2
Χ,,,.,,.. 3
.. Wiener, Ornstein-Uhlenbeck...,, Wiener Ornstein-Uhlenbeck. 4
Abstract In this paper we will present two stochastic approaches which are used for modeling interest randomness. In particular, we will be modeling the force of interest and the force of interest accumulation function. For the above purpose, we will use the stochastic Wiener process and the Ornstein-Uhlenbeck one. The implicit behavior of the force of interest will be investigated by studying the expected value of the force of interest accumulation function. Further, we will provide upper and lower bounds of the present value of a series of cash flows where the discount is within a specific stochastic discount process. Finally, we will present an application for a temporary life annuity which concerns an individual aged, showing the applicability of the above Wiener and Ornstein-Uhlenbeck stochastic process. 5
Ω,.. Σχαί.... 100%.,. Ά. Έ. Ό. 2008.., ( II)... Panjer & Bellhouse [18] 6
1980. Ά Beekman & Fuelling 1993 [6], Parker [19] 1994, DeSchepper & Goovaerts 1992 [11], o Denuit 1999 [10], Dufresne 1990 [13] Aitchison & Brown 1963 [4]. :...., Σκίνηη BrownΤ. Brown. Girsanov.. (Wiener, Ornstein-Uhlenbeck, White Noise)..,,. 7
ΠΡΙΧΟΜΝΑ ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..ΙΙΙΙΙ4 ΙΙΙΙΙΙΙΙΙΙΙ.ΙΙΙΙΙΙ..ΙΙΙΙΙ...6 1 Έ Χ 1.1 Π ΙΙΙΙΙΙΙ10 1.2 Έ ΙΙΙΙΙΙΙΙ.ΙΙΙΙ11 1.3 ΙΙΙΙΙΙΙΙΙΙΙΙ13 1.4 ΙΙΙΙΙΙΙΙΙ.ΙΙΙΙΙΙ..14 1.5 Έ ΧΙΙΙΙΙΙΙΙΙΙΙΙΙ.Ι.18 1.5.1 Χ ΩΙΙΙΙΙΙΙΙΙΙΙΙΙ18 1.5.2 ΙΙΙΙΙΙ..19 2 Π Έ 2.1 Π & Ι..21 2.2 Χ Ι.Ι23 2.2.1 ΙΙΙΙΙΙΙ...Ι24 2.2.2 ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ.24 2.3 ΙΙΙΙΙΙΙΙΙ25 2.4 BrownΙΙΙΙΙΙΙΙΙΙΙΙΙΙ.ΙΙΙ26 2.4.1 Brown W(t)ΙΙΙΙΙΙΙΙ29 2.5 BrownΙΙΙΙΙΙΙΙ..ΙΙΙ31 2.5.1 ΙΙΙΙΙΙΙΙΙΙΙΙΙ...ΙΙ33 2.6 Ornstein-UhlenbeckΙΙΙΙΙ..ΙΙ33 2.7 BrownΙΙΙΙΙΙΙΙΙΙΙ..34 2.8 GirsanovΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..ΙΙΙ.36 8
3 3.1 ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..39 3.2 ΙΙΙΙΙΙΙ.ΙΙΙΙΙΙΙ.ΙΙΙ41 3.3 Έ ΙΙΙΙΙΙΙΙΙΙΙ..ΙΙ42 3.3.1 WienerΙΙΙΙΙΙ.ΙΙΙ42 3.3.2 Ornstein-UhlenbeckΙΙΙ..42 3.4 Έ ΙΙΙΙ.43 3.4.1 White Noise ( )ΙΙΙ.Ι43 3.4.2 WienerΙΙΙΙΙΙΙΙΙΙΙΙΙ..Ι44 3.4.3 Ornstein-UhlenbeckΙΙΙΙΙΙ..ΙΙ45 3.5 ΙΙΙΙΙΙΙΙΙΙΙΙ..Ι45 3.6 ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ...Ι..47 3.7 Έ ΙΙΙΙ.Ι50 3.7.1 Έ Ornstein-UhlenbeckΙΙ.ΙΙΙΙΙΙΙΙΙ..ΙΙΙΙ50 3.7.2 Έ Ornstein-Uhlenbeck ΙΙ..ΙΙΙΙΙΙ...51 3.8 Ά Ό ΙΙΙΙ.ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..54 3.9 Χ ΙΙΙ...61 3.10 ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..Ι63 9
ΕΦΑΑΙΟ 1 ασικές αθηατικές Έοιες στη ρηατοοικοοική,..,. 1.1 Π...,. (Present Value (PV))... : = +.. 10
2000ά 3 3% : = =. +. 1830.28ά 3% 3 2000ά. 2000ά 3 1830.28. ω (Accumulated Value (AV)). (1.1). : = +.. n n. 1.2 Έ ( ) (Accumulative Value).,.. : = = ( )... 11
: =.... Έ : =. H, 200ά 200ά 500ά. Έ : 1.1 ( ) =. ( =. H : = = = () = [ln ] = = 12
= =.. 1.3, (fixed interest rate) (variable interest rate). ( ) (,,...),.,.,. (1.3) : = 13
1.4. (annuity).. : (immediate annuity). (due annuity). (unit annuity). (perpetuity).,, (.χ. ). : = = + = = = + +. (1.4),.. Έ. () n = + + + +. 14
=. (1.5). 1.1 n = + + + + + + +. =. (1.6). 1.2 = n = n.. n n. Έ. 15
n = + + + +. 1.3. 1.3 : n = + + + + + + + +. (1.8) 1.4.. Ό.. ( ). 16
1.5. = + + + + +. = + + + + + + 17
1.5 Έ.Χ.. 1.5.1 Χ Ω 1.2 Έ.. ( )... 1.1. = {,,,,,}. {}, {}, {}, {}, {}, {}.. 18
1.5.2 Έ. : 1.3 : = < = : i. ii. iii. lim = lim = : =.. : 1.4 Έ,.,,,, =,. 19
lim,, =, +.... 20
ΦΑΑ 2 Έ,. Brown. Brown., Girsanov. 2.1 Π & EUR/CHF 5 6. Χt. = 7. 5 6.. Έ. : 2.1 ( ) (..) (..)., =,, 21
[, ], [, [, ) < ( )....... 2.2., = ω, ω, Χ = (), (path) Χ. 2.2 : 22
5, [,]. 5. 2 =.,.,,..,.,,.9. (, ).,. 2.2 Χ,.... =, Υ,. :,. ( [24]). 2.1,,,,.,,.,. 23
2.2.1 =,, =,, =,,, =,,. =,,,,.,.. : = =, H :, =, = [ ],, : =, =, 2.2.2 Έ...... 24
2.3 Έ Χ =, R. : +h +h, h + h, + h < <,, Ι, 2.3 Έ 0 1. = =,,,, : = + + +,,,..,,,,,, : = + + + Ό. Beekman & Fuelling ( [6]), Parker ( [19]-[21]), Dufrense ( [13]) Goovaerts & De Schepper ( [11]). 3.8 Goovaerts. 25
2.4 Brown H Brown ( ). Brown. = {}. Έ Brown =. Wt : i. =. ii., > + 0 s. iii.. iv. H t. Ό w = 0 Brown Brown Wiener. 2.1. Ό = + (ii). 2.2 [ + ] =.,, + = + =. Χ [ + = [ + + ] 26
= [ + ] + [ ] = + = (ii) [ + ]. martingale, ( [16] ). 2.4 martingale : [ + ] =. 2.3 [ ] =. Χ martingale [ ] = [ + ] = = 2.4 W Brown [] = {, }, t, s. >. Brown = 2.1 [] = [ ] = & [] = [ ] = =. : [] = [ ] [] = = [ ] =. = [ + ] = [ ] + +[ ] = + [ ][] = + = {, } 27
= ( 3). Brown. h + h : + h = + h h. [, Τ] h =. : = [h ( h)] = h h = [ h ] = [h] = : = = [ h ] = = h = = [h] = : [ h ] = = = Brown 0 h. lim h,. : = Τ = Ά 0. : = 28
=. ω. h : =. 2.4.1 Brown W(t) Χ Brown. ω.. {} ( [16]). 2.5 Χ [, ]... {}, =,, : [ ] = [] =. Brown : [h h] = = = (h h) = = h h = < 29
2.5 Έ Brown : [h h] = = Έ : [h h] = = h h = = h h = = h =, [h h] = = = = h = h = h = = = = Brown [, ]. Brown. Brown. = 30
2.5 Brown Brown 0 1. -. : + h = h + + h h (binomial). h h. Χ [, ] h =. : = + h = + ( h ) = (..) h = (normal) 0. Ω : = + Brown. : = = + (2.2) Brown. = = + =. 31
H., +. Brown. 2.6 Έ Brown. = + +,. (2.1) = + +. 2.7 Έ Brown =. <.... Έ : [] = + = +. =. [] = =. =. Ά < = <. = <. =. 2.8. Brown : =. +. =. 32
=. =. 2.5.1 (2.2) : -.. Ornestein-Uhlenbeck 2.9.. Brown 0 =. =.. Έ ( ) =. =. =.. Ά :..... < <. = ( ) ( ) =.. =.. = == =. 2.6 Ornstein-Uhlenbeck.. Ό. Brown. (2.2). Έ (2.2) : = ( ) + (2.3) 33
( ),, Brown. (2.3), (2.3) ( ),., R + Ornestein-Uhlenbeck. (2.3) : =. Έ : : = + + = [ ] = : = = + : = + ( ) +. (2.3) Brown Brown. 2.7 Brown Brown Brown. Ό Brown :.. 34
. Brown. Ό Brown = +. = = : = + = + H ω Brown. 2.10 Brown: =. +. =. =.. =. =. =. 2.11 Brown 0.10. h = 22.926. Έ h h =0.12. =.. =. h = 35
2.8 Girsanov. : = +. martingale.. ( [24]). 2.6 Q ( ). ( [24]). 2.7 Έ = + Brown martingale. 36
= + Έ Girsanov Brown martingale. : = (2.4) : = +... 2.12 =. + =. + =.. =. =.. =. =. =.. 37
2.13 : =. +. 0.14. Έ =. =.. : =. +. =. 38
ΦΑΑ 3.., ( )... 3.1 - t 0 [ ] (3.1) [ ] (3.2) Έ,,,,,,. = + + + (3.1) : =,,, { [ < ], } = 39
=,,, {, } = = {,,, R + + + = t}, t R (3.1) (3.2),,,. [21].,,,. Έ G :,,, [,,, ],, R (3.3) : [ >, >,, > ],,,,, R (3.4) ( [10]),,,, (3.5),,,,,, R. (3.3) (3.4),,, =,,, = = =. POD (Positive Orthant Dependent)., =,,,., =,,,. 40
3.2 Έ. = (3.6) =. (Lognormal) [ ] [] - : [( ) ] = [ ] = { [ ] + [] (3.7) O (3.7) [4]. 0 1. = =,,,, : = + + + (3.8) =,,,,. (3.8). (3.8). Ω, (,,, )..,,,,,, : = + + + (3.9). = {, } {, },. 41
3.3 Gaussian 3.4 Gaussian. 3.3 Έ. : Wiener Ornstein-Uhlenbeck 3.3.1 Wiener Έ : = + (3.10), R Wiener. : : [ ] = (3.11) [, ] =, (3.12) (3.11) (3.12) [15]. 3.3.2 Ornstein-Uhlenbeck Έ : = + (3.13) R, Ornstein-Uhlenbeck =. = + (3.14) 42
Χ [5] (3.13) : [, ] = [] = (3.15) +, (3.16) = (3.17) : [, ] = +, (3.18) 3.4 Έ. : White Noise Wiener Ornstein-Uhlenbeck.. 3.4.1 White Noise ( ) R + : ~, (3.19)., R + White Noise.. White Noise Wiener 43
( [5] [10]). ( (3.6)) Wiener [] = (3.20) [, ] =, (3.21) ( (3.20) (3.21) [5]). 3.4.2 Wiener Έ Wiener. : = +, (3.22) Χ 3.3.1 : : [ ] = (3.23) [, ] =, (3.24) ( (3.6)) : [] = (3.25) [, ] = [, ] (3.26) : [, ] = (3.27) 44
3.4.3 Ornstein-Uhlenbeck Ό Ornstein-Uhlenbeck : = + > (3.28) = ( [5]). [, ] = [ ] = (3.29) +, (3.30) = ( (3.16)) ( ) Gaussian : [] = (3.31) [, ] =, + [ + + + ] (3.32) (3.31) (3.32) [19] 3.3 3.4 ([] = ).. 3.5 Έ - n. = =. 45
. ( [7]). n : [ ] = [ = ] = = [ ] (3.34) (3.2) [ ] = { [] + [] } (3.35) : [ ] = [ ] = = = = = [ ] (3.36) = [ ] (3.37) : Ό ~, (3.38) = [] [] [] (3.39) = [] + [] + [] + [, ] + [, ] + [, ]. (3.2) : [ ] = { + } (3.41) 46
3.6, ( [15] ). 1. ( =. =. ). Wiener White Noise... Ornstein-Uhlenbeck. ( [6])........ ( [6]).,.... 47
( ). Wiener. 2... 48
3.. 49
3.7 Έ. Ό,.. ( ). Έ s < t. 3.7.1 Έ Ornstein-Uhlenbeck. (3.12) [ =, = ] = [ + + =, = ] = + [ =, = ] (3.41) 50
Ό < (3.41) : [ =, = ] = + [ = ] (3.42) (3.42) [6]. [ =, = ] = +, < (3.43) 3.7.2 Έ Ornstein-Uhlenbeck (3.1) [ =, = ] = [ =, = ] = [ + =, = ] (3.44) =, (3.44) : [ =, = ] = + [ =, = ] = + [ =, = ] (3.45) u < s : [ =, = ] = + [ = ] (3.46) Ornstein-Uhlenbeck ( [5]) (3.46) : [ =, = ] = + + = + + (3.47) Wiener Ornstein- Uhlenbeck.. 51
. 4 [ =, = ]. [ =, = ]........ Χ, > =. =.. 3.1 =.. 52
3.2, > =. =.... 53
3.8 Ά Ό 3.1 =. Χ Wiener. Ό ( (3.9)) : = + 54
. Goovaerts et al (1999) = =. 0. : = exp{ } =. [0,1]. : : [ > ] = = (3.50) exp( ) = =. (3.1) (3.5) (3.9). (3.5) =, =. =.. =. (3.49). 2 4 POD.. [, ] < < < + ( [10] ). = + POD. 55
56
57
: = (3.52), - = δ, {, t } (3.53) Ornstein-Uhlenbeck {, t } Gaussian : = + (3.54), [, ], (3.55), =, + { + + ( ) ( + )} (3.56) : = =.,. ( Goovaerts et al (1999)) : = exp{, } =. [,]. 5 =., =., =. =.. 7 = 1 3. 6 8 POD. 58
59
60
3.9 Χ. [ < + ] = [ > ] =,,,, + + + +. : ; = [ ; ] ; = {, =, =,,, (3.8).,.. : ; = [ ] = + [ ] ; : [ ; ] = + [ ] = + [ ] [ ; ] ;. [ ; ] 45 (3.10) =. =.. (3.52) =., =,, =. =.. 61
: Ά [ ;. =. ] (3.10) =, = : Ά [ ;., =,, =. =. ] (3.52) =, = 62
Makeham ( [7]).. 3.10. Wiener Ornstein-Uhlenbeck. Χ,,.. White Noise,, Wiener.,, [ =, = ]. [ =, = ]. Έ.,., 63
. 64
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