ΠΑΝ ΠΙΣ ΜΙΟΝΠ ΙΡΑΙΩ ΧΟΛ ΝΧΡ ΜΑΣΟΟΙΚΟΝΟΜΙΚ Ν & ΣΑΣΙΣΙΚ ΣΜΗΜΑΝΣΑΣΙΣΙΚΗ & ΑΦΑΛΙΣΙΚΗΝ ΠΙΣΗΜΗ ΠΡΟΓΡΑΜΜΑΝΜ ΣΑΠΣΤΧΙΑΚΩΝΝΠΟΤ ΩΝΝ ΣΗΝΝΑΝΑΛΟΓΙΣΙΚΗΝ ΠΙΣΗΜΗ ΚΑΙ

Transcript

1 ΠΑΝΠΙΣΜΙΟΝΠΙΡΑΙΩ ΧΟΛΝΧΡΜΑΣΟΟΙΚΟΝΟΜΙΚΝ & ΣΑΣΙΣΙΚ ΣΜΗΜΑΝΣΑΣΙΣΙΚΗ & ΑΦΑΛΙΣΙΚΗΝΠΙΣΗΜΗ ΠΡΟΓΡΑΜΜΑΝΜΣΑΠΣΤΧΙΑΚΩΝΝΠΟΤΩΝΝ ΣΗΝΝΑΝΑΛΟΓΙΣΙΚΗΝΠΙΣΗΜΗ ΚΑΙ ΙΟΙΚΗΣΙΚΗΝΚΙΝΤΝΟΤ ΑΝΑΛΤΝΚΙΝΤΝΟΤ ΓΙΑ ΧΑΡΣΟΦΤΛΑΚΙΑΝΡΑΝΣΩΝΝΩ ουλέζαμνηηάλδομ δπζωηαδεάνλγαέα Πεδλαδάμ, ΙαθουάλδομΝ2ί16 1

2 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

3 Χ,,,.,,.. 3

4 .. Wiener, Ornstein-Uhlenbeck...,, Wiener Ornstein-Uhlenbeck. 4

5 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

6 Ω,.. Σχαί %.,. Ά. Έ. Ό , ( II)... Panjer & Bellhouse [18] 6

7 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

8 ΠΡΙΧΟΜΝΑ ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..ΙΙΙΙΙ4 ΙΙΙΙΙΙΙΙΙΙΙ.ΙΙΙΙΙΙ..ΙΙΙΙΙ Έ Χ 1.1 Π ΙΙΙΙΙΙΙ Έ ΙΙΙΙΙΙΙΙ.ΙΙΙΙ ΙΙΙΙΙΙΙΙΙΙΙΙ ΙΙΙΙΙΙΙΙΙ.ΙΙΙΙΙΙ Έ ΧΙΙΙΙΙΙΙΙΙΙΙΙΙ.Ι Χ ΩΙΙΙΙΙΙΙΙΙΙΙΙΙ ΙΙΙΙΙΙ Π Έ 2.1 Π & Ι Χ Ι.Ι ΙΙΙΙΙΙΙ...Ι ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ ΙΙΙΙΙΙΙΙΙ BrownΙΙΙΙΙΙΙΙΙΙΙΙΙΙ.ΙΙΙ Brown W(t)ΙΙΙΙΙΙΙΙ BrownΙΙΙΙΙΙΙΙ..ΙΙΙ ΙΙΙΙΙΙΙΙΙΙΙΙΙ...ΙΙ Ornstein-UhlenbeckΙΙΙΙΙ..ΙΙ BrownΙΙΙΙΙΙΙΙΙΙΙ GirsanovΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..ΙΙΙ.36 8

9 3 3.1 ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ ΙΙΙΙΙΙΙ.ΙΙΙΙΙΙΙ.ΙΙΙ Έ ΙΙΙΙΙΙΙΙΙΙΙ..ΙΙ WienerΙΙΙΙΙΙ.ΙΙΙ Ornstein-UhlenbeckΙΙΙ Έ ΙΙΙΙ White Noise ( )ΙΙΙ.Ι WienerΙΙΙΙΙΙΙΙΙΙΙΙΙ..Ι Ornstein-UhlenbeckΙΙΙΙΙΙ..ΙΙ ΙΙΙΙΙΙΙΙΙΙΙΙ..Ι ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ...Ι Έ ΙΙΙΙ.Ι Έ Ornstein-UhlenbeckΙΙ.ΙΙΙΙΙΙΙΙΙ..ΙΙΙΙ Έ Ornstein-Uhlenbeck ΙΙ..ΙΙΙΙΙΙ Ά Ό ΙΙΙΙ.ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ Χ ΙΙΙ ΙΙΙΙΙΙΙΙΙΙΙΙΙΙΙ..Ι63 9

10 ΕΦΑΑΙΟ 1 ασικές αθηατικές Έοιες στη ρηατοοικοοική,..,. 1.1 Π...,. (Present Value (PV))... : =

11 2000ά 3 3% : = = ά 3% ά. 2000ά ω (Accumulated Value (AV)). (1.1). : = +.. n n. 1.2 Έ ( ) (Accumulative Value).,.. : = = ( )... 11

12 : =.... Έ : =. H, 200ά 200ά 500ά. Έ : 1.1 ( ) =. ( =. H : = = = () = [ln ] = = 12

13 = =.. 1.3, (fixed interest rate) (variable interest rate). ( ) (,,...),.,.,. (1.3) : = 13

14 1.4. (annuity).. : (immediate annuity). (due annuity). (unit annuity). (perpetuity).,, (.χ. ). : = = + = = = + +. (1.4),.. Έ. () n =

15 =. (1.5). 1.1 n = =. (1.6). 1.2 = n = n.. n n. Έ. 15

16 n = : n = (1.8) Ό.. ( ). 16

17 1.5. = =

18 1.5 Έ.Χ Χ Ω 1.2 Έ.. ( ) = {,,,,,}. {}, {}, {}, {}, {}, {}.. 18

19 1.5.2 Έ. : 1.3 : = < = : i. ii. iii. lim = lim = : =.. : 1.4 Έ,.,,,, =,. 19

20 lim,, =,

21 ΦΑΑ 2 Έ,. Brown. Brown., Girsanov. 2.1 Π & EUR/CHF 5 6. Χt. = Έ. : 2.1 ( ) (..) (..)., =,, 21

22 [, ], [, [, ) < ( ) , = ω, ω, Χ = (), (path) Χ. 2.2 : 22

23 5, [,] =.,.,,..,.,,.9. (, ).,. 2.2 Χ,.... =, Υ,. :,. ( [24]). 2.1,,,,.,,.,. 23

24 2.2.1 =,, =,, =,,, =,,. =,,,,.,.. : = =, H :, =, = [ ],, : =, =, Έ

25 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

26 2.4 Brown H Brown ( ). Brown. = {}. Έ Brown =. Wt : i. =. ii., > + 0 s. iii.. iv. H t. Ό w = 0 Brown Brown Wiener Ό = + (ii). 2.2 [ + ] =.,, + = + =. Χ [ + = [ + + ] 26

27 = [ + ] + [ ] = + = (ii) [ + ]. martingale, ( [16] ). 2.4 martingale : [ + ] =. 2.3 [ ] =. Χ martingale [ ] = [ + ] = = 2.4 W Brown [] = {, }, t, s. >. Brown = 2.1 [] = [ ] = & [] = [ ] = =. : [] = [ ] [] = = [ ] =. = [ + ] = [ ] + +[ ] = + [ ][] = + = {, } 27

28 = ( 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

29 =. ω. h : = Brown W(t) Χ Brown. ω.. {} ( [16]). 2.5 Χ [, ]... {}, =,, : [ ] = [] =. Brown : [h h] = = = (h h) = = h h = < 29

30 2.5 Έ Brown : [h h] = = Έ : [h h] = = h h = = h h = = h =, [h h] = = = = h = h = h = = = = Brown [, ]. Brown. Brown. = 30

31 2.5 Brown Brown : + h = h + + h h (binomial). h h. Χ [, ] h =. : = + h = + ( h ) = (..) h = (normal) 0. Ω : = + Brown. : = = + (2.2) Brown. = = + =. 31

32 H., +. Brown. 2.6 Έ Brown. = + +,. (2.1) = Έ Brown =. <.... Έ : [] = + = +. =. [] = =. =. Ά < = <. = <. = Brown : =. +. =. 32

33 =. = (2.2) : -.. Ornestein-Uhlenbeck Brown 0 =. =.. Έ ( ) =. =. =.. Ά :..... < <. = ( ) ( ) =.. =.. = == =. 2.6 Ornstein-Uhlenbeck.. Ό. Brown. (2.2). Έ (2.2) : = ( ) + (2.3) 33

34 ( ),, Brown. (2.3), (2.3) ( ),., R + Ornestein-Uhlenbeck. (2.3) : =. Έ : : = + + = [ ] = : = = + : = + ( ) +. (2.3) Brown Brown. 2.7 Brown Brown Brown. Ό Brown :.. 34

35 . Brown. Ό Brown = +. = = : = + = + H ω Brown Brown: =. +. =. =.. =. =. = Brown h = Έ h h =0.12. =.. =. h = 35

36 2.8 Girsanov. : = +. martingale.. ( [24]). 2.6 Q ( ). ( [24]). 2.7 Έ = + Brown martingale. 36

37 = + Έ Girsanov Brown martingale. : = (2.4) : = =. + =. + =.. =. =.. =. =. =.. 37

38 2.13 : = Έ =. =.. : =. +. =. 38

39 ΦΑΑ 3.., ( ) t 0 [ ] (3.1) [ ] (3.2) Έ,,,,,,. = (3.1) : =,,, { [ < ], } = 39

40 =,,, {, } = = {,,, 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

41 3.2 Έ. = (3.6) =. (Lognormal) [ ] [] - : [( ) ] = [ ] = { [ ] + [] (3.7) O (3.7) [4] = =,,,, : = (3.8) =,,,,. (3.8). (3.8). Ω, (,,, )..,,,,,, : = (3.9). = {, } {, },. 41

42 3.3 Gaussian 3.4 Gaussian. 3.3 Έ. : Wiener Ornstein-Uhlenbeck Wiener Έ : = + (3.10), R Wiener. : : [ ] = (3.11) [, ] =, (3.12) (3.11) (3.12) [15] Ornstein-Uhlenbeck Έ : = + (3.13) R, Ornstein-Uhlenbeck =. = + (3.14) 42

43 Χ [5] (3.13) : [, ] = [] = (3.15) +, (3.16) = (3.17) : [, ] = +, (3.18) 3.4 Έ. : White Noise Wiener Ornstein-Uhlenbeck White Noise ( ) R + : ~, (3.19)., R + White Noise.. White Noise Wiener 43

44 ( [5] [10]). ( (3.6)) Wiener [] = (3.20) [, ] =, (3.21) ( (3.20) (3.21) [5]) Wiener Έ Wiener. : = +, (3.22) Χ : : [ ] = (3.23) [, ] =, (3.24) ( (3.6)) : [] = (3.25) [, ] = [, ] (3.26) : [, ] = (3.27) 44

45 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] ([] = ) Έ - n. = =. 45

46 . ( [7]). n : [ ] = [ = ] = = [ ] (3.34) (3.2) [ ] = { [] + [] } (3.35) : [ ] = [ ] = = = = = [ ] (3.36) = [ ] (3.37) : Ό ~, (3.38) = [] [] [] (3.39) = [] + [] + [] + [, ] + [, ] + [, ]. (3.2) : [ ] = { + } (3.41) 46

47 3.6, ( [15] ). 1. ( =. =. ). Wiener White Noise... Ornstein-Uhlenbeck. ( [6]) ( [6]).,

48 ( ). Wiener

49 3.. 49

50 3.7 Έ. Ό,.. ( ). Έ s < t Έ Ornstein-Uhlenbeck. (3.12) [ =, = ] = [ + + =, = ] = + [ =, = ] (3.41) 50

51 Ό < (3.41) : [ =, = ] = + [ = ] (3.42) (3.42) [6]. [ =, = ] = +, < (3.43) 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

52 . 4 [ =, = ]. [ =, = ] Χ, > =. = =.. 52

53 3.2, > =. =

54 3.8 Ά Ό 3.1 =. Χ Wiener. Ό ( (3.9)) : = + 54

55 . 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 56

57 57

58 : = (3.52), - = δ, {, t } (3.53) Ornstein-Uhlenbeck {, t } Gaussian : = + (3.54), [, ], (3.55), =, + { + + ( ) ( + )} (3.56) : = =.,. ( Goovaerts et al (1999)) : = exp{, } =. [,]. 5 =., =., =. =.. 7 = POD. 58

59 59

60 60

61 3.9 Χ. [ < + ] = [ > ] =,,,, : ; = [ ; ] ; = {, =, =,,, (3.8).,.. : ; = [ ] = + [ ] ; : [ ; ] = + [ ] = + [ ] [ ; ] ;. [ ; ] 45 (3.10) =. =.. (3.52) =., =,, =. =.. 61

62 : Ά [ ;. =. ] (3.10) =, = : Ά [ ;., =,, =. =. ] (3.52) =, = 62

63 Makeham ( [7]) Wiener Ornstein-Uhlenbeck. Χ,,.. White Noise,, Wiener.,, [ =, = ]. [ =, = ]. Έ.,., 63

64 . 64

65 ΙΙΟΡΑΙΑ [1] A. N. Χ, 1. (2003) [2] Χ... (2006) [3] Χ... (2006) [4] AITCHISON, J. and BROWN, J.A.C. the Lognormal Distribution, 176 pp., Cambridge University Press. (1963) [5] ARNOLD, L. Stochastic Differential Equations : Theory and Applications, 228 pp., John Wiley & Sons, New York (1974) [6] BEEKMAN, J.A. and FUELLING, C.P. Interest and Mortality Randomness in Some Annuities Insurance: Mathematics and Economics 9, (1990) [7] BOWERS, N.L., GERBER, H.U., HICKMAN, J.C., JONES, O.A. and C.J. NESBITT. Actuarila Mathematics. Society of actuaries, Itasca, Illinois (1996) [8] BUHLMANN, H. Stochastic Discounting Insurance: Mathematics and Economics 1 I, I (1992) [9] CANADIAN INSTITUTE OF ACTUARIES Rapport sur les Statistiques Economiques Canadiennes, (1993) [10] DENUIT, M., GENEST, C. and E. MARCEAU. Stochastic bounds on sums of dependent risks. Insurance: Mathematics & Economics 25, (1999) [11] DE SCHEPPER, A. and M.J GOOVAERTS. Some further results on annuities certain with random interest. Insurance: Mathematics & Economics 11, (1992) 65

66 [12] DE SCHEPPER, A. TEUNEN, M. and M.J. GOOVAERTS. An analytical inversion of a Laplace transform related to annuity certain. Insurance: Mathematics & Economics 14, (1994) [13] DUFRESNE, D. The distribution of a perpetuity, with applications to risk theory and pension funding. Scandinavian Actuarial Journal (1990) [14] FREES, E.W. Stochastic life contingencies with solvency considerations. Transactions of the Society of Actuaries XLII, (1990) [15] KARLIN, S. and TAYLOR, H. M. A Second Course in Stochastic Processes, 542 pp., Academic Press, San Diego. (1981) [16] MARCEL B. FINAN A Discussion of Financial Economics in Actuarial Models. Arkansan Tech University (2014) [17] MOOD, A.M., GRAYI3II.L, F.A. and BOES, D.C. Introduction to the Theory of Statistics, 567 pp., McGraw-Hill. (1974) [18] PANJER, H.H and BELLHOUSE, D.R Stochastic Modeling if Interest Rates with Applications to Life Contingencies. Journal of Risk and Insurance 47, (1980) [19] PARKER, G. Limiting distribution of the present value of a portfolio. ASTIN Bulletin 94-1, (1994) [20] PARKER, G. Two Stochastic approaches for discounting actuarial functions. ASTIN Bulletin 24, (1994) [21] PARKER, G. Stochastic analysis of a portfolio of endowment policies. Scandinavian Actuarial Journal, (1994) [22] STEPHEN G. KELLISON The Theory of Interest. University of Central Florida (2009) [23] SZEKLI, R. Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics 97. Springer Verlag Berlin. (1995) [24] HOMAS MIKOSCH Elementary Stochastic Calculus with Finance in View. World Scientific Publishing Company (2009) 66