The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract


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1 The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous law of mortality was itroduced by Gompertz i 1825 i [G]. I 1860 Makeham itroduced i [M] a modificatio to obtai aother law of mortality. Both these laws assume that the populatio uder cosideratio is stable. The two laws differ by a costat term i the force of mortality. The updated approach to the study of populatio presume that mortality chages over time. The differece stems from observig that the expectacy of life chages over the years. There were made various attempts to itroduce dyamic life tables, ad the LeeCarter model has this advatage. The LeeCarter model i [LC] describes better the mortality uder chages over time as was show recetly by ([MR]). We ited to study the differece arisig from a fixed chage i the force of mortality ad the Gompertz ad Makeham cases may serve to demostrate such a chage. The differece i the expectacy of life i both cases affects directly the correspodig life tables, ad cosequetly the auities (for life, term auitie ad deferred ), as well as the assuraces (whole life, term ad edowmet), ad the premiums ad reserves i the various cases. The chages i a stable life table model may serve to evaluate the chages that arise i terms of a sesitive aalysis of various assurace plas. Cosequetly there results a tool to cope with evaluatig the effect of chage i the force of mortality. Keywords : force of mortality ; Expectacy of life ; Auities, a,, Assuraces, A, A ; Premiums ad Reserves i various cases. 1 1 of 6 Electroic copy available at:
2 1. Auities We let G deote the "Gompertz case", M the "Makeham case" ad GM the "Gompertz ad Makeham cases". We follow the defiitios ad otatios of [Z1], [Z2], ad [Z3]. I G the force of mortality is give by, [, ad i M the force of mortality is give by, [, for some fixed. We may have 0, 0 0, i the last case both models coicide. Note that we may replace by where ad the results that will follow will be symmetric to the results that we obtai whe we replace for (e.g. for ad for ad so o ad for. Recall that the expectacy of a live aged i these models is ad respectively. Let be the force of iterest, ad let (µ,δ) be the PV of the cotiuous auity of 1 p.a. for a live aged ad we have (e.g.[z1]) (µ,δ+ε) ε, δ ad i particular we have the equalities (µ,δ+ ), δ, ad,,. As is well kow, 0 ad sice (e.g.[ Z]) we may deduce :. :, 0, Recall the approximatio (e.g. [Z1] ) : µ, δ ε µ, δ µ, δ where deotes the PV of the cotuuous auity of 1 p.a. for a live aged icreasig by 1 aually. The followig approximatio results :. : The approximatio, µ, 0 holds. The results for geeralizes to, evaluatig i the same table but i differet itersets :. :, δ ) (,δ+ ad a similar geeralizatio hold for the approximatio i the same table ad iterset :. :, δ (,δ), δ The result follows from, δ (,δ+ ) ad the approximatio (,δ+ ) (,δ), δ. It seems that oe ca derive appropriate approximatios for temporary auities a ad deferred auities, as well as for due auities ad i arrear auities. 2 of 6 Electroic copy available at:
3 2. Assuraces Let (µ,δ) be the PV of the cotiuous whole life issurace of 1 for a live aged. The result of comparrig, evaluatig i the same table but i differet itersets yields :. : (µε,δ) µ, δ ε ε (µ,δε). From the well kow equality (µ,δ)δ (µ,δ)=1 there follows the equality (µ,δε)= 1 δ ε (µ,δε). Recall that : (µε,δ)= 1 δ (µ ε, δ) ad we may derive the equality : (µε,δ) µ, δ ε ε (µ,δε) as claimed. Note that sice, δ, 1 we may coclude i a similar way that,,, that is or there follows the equality : A, A, a,. Recall from [Z1] thatwe have the approximatio µ, δ ε µ, δ µ, δ where µ, δ deotes the PV of the cotiuous whole life assurace of 1 for a live aged icreasig aually by 1. ad a similar geeralizatio hold for the approximatio i the same table ad iterset :. : µ, δ ε µ, δ µ, δ ε µ, δ ε. From the approximatio µ, δ ε µ, δ µ, δ (e.g. [Z1]) ad from propositio 3 the result follows as stated. I particular we may derive :. (,δ),δ)δ, δ. The equality,δ)δ (,δ)=1 implies,δ)1 δ (,δ) ad the equality (,δ)δ (,δ)=1 implies (,δ)=1 δ (,δ). Therefore (,δ)=1 δ (,δ ad by [Z1] we may derive the approximatio, δ 1 δ (,δ)δ, δ, ad sice,δ)1 δ (,δ) the approximatio (,δ),δ)δ, δ follows as stated. It seems that oe ca derive appropriate approximatios i a similar way for term assurace 1 A, edowmet assurace various assurace plas paid at the ed of the year of death.. A,ad deferred assurace, as well as for the 3 of 6
4 3. Premiums The premium for the whole life isurace may be derived from. This implies, 1, ad i particular, 1,. Furthermore we have the equality, 1,. Recall the approximatio :,,,, δ. Thus we may state the followig resultig equatio ad approximatio :. :,,..,,, δ, The result follows from the equality,, ad the approximatio,,, δ,. Note that startig with the equalities :, 1, Ad, 1,, it results that,, Ad i particular we may derive the equality : P, P,. It seems that oe ca derive appropriate approximatios i a similar way for premiums for the various assurace plas ad pesio schemes (cosidered as deferred auities).. 4. Reserves The reserve, i time for the whole life isurace for a live aged may be derived from Ad we obtai :, ad i a similar way for M µ, δ (µ,δ ad (µ,δ)= 1δ (µ,δ), 1,,, 1,, ad for G we have, 1,,. Recall that,, ad,, 4 of 6
5 Ad we may derive :. : Ad there results the approximatio :,,.,,,,, δ,, δ, I a similar way if we start with the equalities :, 1,, ad, 1,,, we have, V, holds., ad i particular the equality V, It seems that oe ca derive appropriate approximatios i a similar way for reserves i the various assurace plas ad pesio schemes.. 5. Approximatig higher derivatios Recall [Z2] for relatios of higher derivatives of with ad. A life table satisfies the weak decreasig assumptio e.g. whe we presume ati selectio, i which case the premium decreases whe the force of iterest icreases. I this case the iequality holds. Therefore we have ad we ca derive the iequality 1, that is 1 ad cosequetly :.. I particular followig [Z2] start with the equality ad use iductio, to obtai the equalities A x a x for all 0 where = that is :. 0, 0. The proof of this propositio is a direct cosequece of the observatio : of 6
6 Refereces [FL] Föllmer, H., Leukert, P., Quatile hedgig. Fiace Stoch. 3, [G ] Gompertz, B. " O the Nature of the Fuctio Expressive of the Law of Huma Mortality, ad o a New Mode of Determiig the Value of Life Cotigecies." Phil. Tras. of the Royal Soc. of Lodo, Vol. 115 (1825), pp [M] [LC] Makeham, W. M. "O the Law of Mortality ad the Costructio of Auity Tables." J. Ist. Act. ad Assur. Mag. 8, , Lee, R.D., Carter, L.R., Modellig ad forecastig U.S. mortality. J. Amer. Statist. Assoc. 87 (14), [MR] A.Melikov, Y.Romaiuk "Evaluatig the performace of Gompertz, Makeham ad Lee Carter mortality models for risk maagemet with uitliked cotracts" IME Volume 39(3)2006, Pp [Z1] Zaks A., Auities uder chages i Life Tables ad chages i the Iterest, Pravartak J. Is. & Risk Ma.2009(8) [Z2] [Z3] Auities uder chages i Life Tables ad chages i the Iterest II, Pravartak J. Is. & Risk Ma.2009(9) Auities uder chages i Life Tables ad chages i the Iterest III, or Premium Icreases whe Iterest Drcreases, Pravartak J. Is. & Risk Ma.2009(10) of 6
Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
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