On th dcomposition of lif xpctancy and limits to lif Ls Mayhw and David Smith Faculty of Actuarial Scinc Cass Businss School Longvity 10 Sptmbr 2014 Santiago Chil
Som ky dmographic issus What do trnds in lif xpctancy in diffrnt countris tll us about limits to lif xpctancy? Ar thr altrnativs to th us of mortality as a basis for forcasting lif xpctancy? Why dos lif xpctancy in som countris go down as wll as up? Why is th US falling bhind othr major countris?
Comparativ trnds in lif xpctancy in thr countris 52 lif xpctancy at ag 30 50 48 46 44 42 40 Japan E&W Poland 38 36 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 yar Mal trnds in lif xpctancy at ag 30 in Japan, England and Wals and Poland using quadratic functions (Sourc HMD)
Is thr vidnc for convrgnc in England and Wals data? ag to which givn % surviv 100 90 80 70 60 50 40 30 20 10 0 ag by which givn prcntag ar dad 100 90 80 70 60 50 40 95th Prcntil 90th Prcntil 80th Prcntil 70th Prcntil 60th Prcntil 50th Prcntil 40th Prcntil 30th Prcntil 20th Prcntil 10th Prcntil lif xpctancy at 30 30 30 35 40 45 50 55 60 65 70 lif xpctancy at ag 30 In th convrgnt cas lif xpctancy procds to a maximum. If w tak th priod 1840 to 1939 thr is clar vidnc for lif xpctancy to b on a convrgnt track although it has a long way to run. In both mal and fmal cass this is 67.5 at ag 30.
Is thr vidnc for convrgnc in England and Wals data? ag to which givn % surviv 100 90 80 70 60 50 40 30 20 10 0 Aftr 1939 thr was a clar chang of trnd lif xpctancy at 30 In th convrgnt cas lif xpctancy procds to a maximum. If w tak th priod 1840 to 1939 thr is clar vidnc for lif xpctancy to b on a convrgnt track although it has a long way to run. In both mal and fmal cass this is 67.5 at ag 30.
Forcasting lif xpctancy basd on pr-war convrgnt trnd lif xpctancy at ag 30 (mal) 70 65 60 55 50 45 40 35 mal actual 1941 to 1939 forcast 30 1825 1850 1875 1900 1925 1950 1975 2000 2025 2050 2075 2100 yar Th trnd prdictd from 1939 suggsts that a maximum would b approach from about 2050 whn progrss would lvl out lif xpctancy at ag 30 (mal) 70 65 60 55 50 45 40 35 mal actual 1941 to 1939 mal actual 1950 to 2009 forcast 30 1825 1850 1875 1900 1925 1950 1975 2000 2025 2050 2075 2100 yar In fact, th actual trnd is svrly rtardd from about 1950 to 1975 bfor rsuming its stp climb. Howvr, convrgnt bhaviour is rplacd by paralll bhaviour. Th sam occurs in fmals but th rtardation ffct is lss. So hardr to stimat uppr limit.
Altrnativ mthod basd on Motivation partial lif xpctancy Crat known limits by dividing lif cours into ag intrvals Idntity trnds within ach basd on known uppr limits Aggrgat ag intrvals to obtain lif xpctancy btwn any dsird ag intrval (.g. 30-100) Isolat and focus attntion on ag intrvals whr trnd ar hardr to stablish (.g. >100)
Partial lif xpctancy lx 1 lx 2 population lx 3 lx 4 x 1 x 2 3 ag x x x 4 5
Partial Lif Expctancy Dfin x as th futur xpctd lif of somon n x n: x m currntly agd x n btwn th ags of x n and x m. Hnc And 1 1 1 x 1 x1 : x 3 lx l 2 x l 1 x l 2 x l 3 x l 2 x3 lx 2 2 1 x3 1 1 l l l lx y x 2 1 2 1 y x x 1 3 1 1 1 x 1 x3: x 5 lx l 4 x l 3 x l 4 x l 5 x l 4 x5 lx 2 2 x5 1 1 l l l lx y x 2 1 4 y x x 3 5 So : : x x x x x x x 1 1 1 3 1 3 5
Aggrgation of partial lif xpctancy l l... 40 100 30 30 30:40 40 40:50 100 100:110 l30 l30 W can add partial lif xpctancis togthr to obtain ovrall lif xpctancy if w wish. Sinc w know that lif xpctancy cannot xcd tn yars in a tn yar intrval w can assss which ag groups hav rachd thir limit and which hav not. In principl this mans w can pin-point whr th gratst uncrtainty lis and thus focus our attntion on th ag rangs whr futur incrass in lif xpctancy will com from
Mal partial lif xpctancy by lading 30(30:40) 40(40:50) 50(50:60) 60(60:70) 70(70:80) 80(80:90) 90(90:100) 100(100: 110) country 1950-2009 1950 1960 1970 1980 1990 2000 2009 Swdn E&W E&W E&W Japan Swdn Swdn Swdn Swdn E&W E&W Japan Swdn Swdn Swdn Swdn Swdn Japan Japan Swdn Swdn Swdn Swdn Swdn Japan Japan Swdn Swdn Swdn Swdn Swdn Japan Japan Japan Swdn Swdn Swdn Swdn Japan Japan Japan Japan US US Swdn US Japan Japan Japan US US US US US Japan Japan (LC) 30 Swdn Swdn Swdn Japan Japan Japan Japan Tabl showing th lading country by tn yar ag group in ach dcad from 1950 onwards in E&W, Franc, Japan, Swdn and th US
Mal partial lif xpctancy by lagging country 1950-2009 30(30:40) 40(40:50) 50(50:60) 60(60:70) 70(70:80) 80(80:90) 90(90:100) 100(100: 110) 1950 1960 1970 1980 1990 2000 2009 Japan Japan US US US US US Japan US US Franc Franc Franc US Japan US US Franc Franc Franc US Japan US US E&W US US US Japan Japan E&W E&W E&W E&W US Japan Japan Japan E&W E&W E&W Swdn Japan Japan Japan Japan E&W Swdn Swdn Japan Japan Japan Japan Franc Swdn Swdn (LC) 30 Japan Japan US Franc US US US Tabl showing th lagging country by tn yar ag group in ach dcad from 1950 onwards in E&W, Franc, Japan, Swdn and th US
30(30:40) 40(40:50) 50(50:60) Fmal partial lif xpctancy by 60(60:70) 70(70:80) 80(80:90) 90(90:100) 100(100: 110) lading country 1950-2009 1950 1960 1970 1980 1990 2000 2009 Swdn Swdn Swdn Japan Japan Swdn Swdn Swdn Swdn Swdn Japan Japan Swdn Swdn Swdn Swdn Swdn Japan Japan Japan Japan Swdn Swdn Swdn Japan Japan Japan Japan Swdn Swdn Swdn Japan Japan Japan Japan US US Swdn Franc Japan Japan Japan US US US US US Japan Japan US US US US US Japan Japan (LC) 30 Swdn Swdn Swdn Franc Japan Japan Japan Tabl showing th lading country by tn yar ag group in ach dcad from 1950 onwards in E&W, Franc, Japan, Swdn and th US
30(30:40) 40(40:50) 50(50:60) 60(60:70) 70(70:80) 80(80:90) 90(90:100) 100(100: 110) Fmal partial lif xpctancy by lagging country 1950-2009 1950 1960 1970 1980 1990 2000 2009 Japan Japan US US US US US Japan Japan US US US US US Japan Japan US US US US US Japan Japan US US US US US Japan Japan Japan E&W E&W US US Japan Japan Japan E&W E&W US US Japan Japan Japan Japan E&W E&W Swdn Japan Japan Japan Japan Swdn Swdn Swdn 30 Japan Japan Japan E&W E&W US US Tabl showing th lagging country by tn yar ag group in ach dcad from 1950 onwards in E&W, Franc, Japan, Swdn and th US
Empirical rlationship btwn probability of survival and 10-yar lif xpctancy Probability of survival in th nxt tn yars vrsus lif xpctancy in th nxt tn yars: Ky A from ag 30 to 60 a gratr than 85% of surviving a dcad; B from ag 70 to 80 ~ 70% chanc; C from ag 80 to 90 ~ 35% chanc; D from ag 90 to 100 ~ 5% chanc
Partial lif xpctancis for Japans fmals From ag 30 1950 1960 1970 1980 1990 2000 2009 2030 2050 30(30:40) 30(40:50) 30(50:60) 30(60:70) 30(70:80) 30(80:90) 30(90:100) 9.8 9.9 9.9 10.0 10.0 10.0 10.0 10.0 10.0 9.2 9.6 9.8 9.9 9.9 9.9 9.9 10.0 10.0 8.4 9.1 9.4 9.6 9.7 9.7 9.7 9.9 9.9 6.9 7.9 8.4 9.0 9.2 9.3 9.4 9.7 9.9 4.4 5.4 6.2 7.3 8.0 8.4 8.7 9.4 9.7 1.4 1.9 2.5 3.7 4.9 5.9 6.5 8.4 9.2 0.1 0.2 0.3 0.5 1.0 1.9 2.5 5.7 8.1 30(100.100 ) 0.0 0.0 0.0 0.0 0.0 0.1 0.2 1.3 4.9 30 40.2 44.1 46.5 49.9 52.7 55.2 57.0 64.3 71.8 Tabl shows 10-yar partial lif xpctancis for Japans fmals from1950 with forcasts for 2030 and 2050. Bottom row shows full lif xpctancy at ag 30
Fitting trnd lins to partial lif Th trnd lins for Japans fmals ar basd on th following quations: fi ( t) A yi fi ( t) 1 xpctancis f ( t) i a i b t i Th quation fits most of th sris considrd vry accuratly although this is not th only variant
Yars Contribution from ach dcad of lif to Japans fmal lif xpctancy at 30 80 70 60 50 40 30 20 10 0 1950 1960 1970 1980 1990 2000 2009 2030 2050 Calndar Yar 100-110 90-100 80-90 70-80 60-70 50-60 40-50 30-40
How trnd lins compar lif xpctancy at 30 (fmals) 100 90 80 70 60 50 B D A C 40 30 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 A comparison of trnds in Japans fmal lif xpctancy at ag 30 basd on thr forcasting approachs: th quadratic, straight lin and limiting modls. yar Ky: A quadratic trnd; B basd on limiting modl with uppr limit of D; C basd on a straight lin trnd.
Conclusions Lif xpctancy is a crucial masur of human dvlopmnt but whn making forcasts trnd lins ar unboundd and can bcom unralistic Partial lif xpctancis offr a way around this, at last in part Rtrospctiv analysis of trnds in partial lif xpctancy show important stratgic shifts with th US falling to th bottom and Japan rising to th top