Risk Sources in a Life Annuity Portfolio: Decomposition and Measurement Tools

Transcription

1 University of Nebraska - Linoln DigitalCommons@University of Nebraska - Linoln Journal of Atuarial Pratie Finane Department 2000 Risk Soures in a Life Annuity Portfolio: Deomposition and Measurement Tools Mariarosaria Coppola University of Naples Federio II, oppola@dms.unina.it Emilia Di Lorenzo University of Naples Federio II, diloremi@dmsna.dms.unina.it Marilena Sibillo University of Sassari., sibillo@ssmain.uniss.it Follow this and additional works at: Part of the Aounting Commons, Business Administration, Management, and Operations Commons, Corporate Finane Commons, Finane and Finanial Management Commons, Insurane Commons, and the Management Sienes and Quantitative Methods Commons Coppola, Mariarosaria; Lorenzo, Emilia Di; and Sibillo, Marilena, "Risk Soures in a Life Annuity Portfolio: Deomposition and Measurement Tools" (2000). Journal of Atuarial Pratie This Artile is brought to you for free and open aess by the Finane Department at DigitalCommons@University of Nebraska - Linoln. It has been aepted for inlusion in Journal of Atuarial Pratie by an authorized administrator of DigitalCommons@University of Nebraska - Linoln.

2 Journal of Atuarial Pratie Vol. 8, 2000 Risk Soures in a Life Annuity Portfolio: Deomposition and Measurement Tools Mariarosaria Coppola, * Emilia Oi Lorenzo, t and Marilena Sibillo* Abstrat The paper onsiders a model for a homogeneous portfolio of whole life annuities immediate. The aim is to study two risk fators: the investment risk and the insurane risk. A stohasti model of the rate of return is used to study these risk fators. Measures of the insurane risk and the investment risk for the entire portfolio are suggested. The problem of the longevity risk is presented, and its onsequenes with different projetions of the mortality tables are analyzed. The model is applied to some onrete ases, and several illustrations show the importane of the two omponents of the riskiness in terms of the number of policies in the portfolio. Understanding these risks will allow insurane ompanies to ontrol, to some extent, the overall risk of their annuity portfolios. Key words and phrases: Ornstein-Uhlenbek proess, investment risk, insurane risk, longevity risk, moments of insurane funtions *Mariarosaria Coppola, Ph.D., is a researher in the area of stohasti analysis of finanial and insurane risk at the University of Naples "Federio II". Dr. Coppola's address is: Dipartimento di Matematia e Statistia, Faolta' di Eonomia, Universita' degli Studi di Napoli "Federio II", via Cintia, Complesso Monte S.Angelo Napoli, ITALY. Internet address: oppola@dms.unina.it temilia Di Lorenzo, Ph.D., is an assoiate professor of financial mathematis at the University of Naples "Federio II". She has been grant-holder at the Italian National Researh Counil and researher at the University of Salerno. Dr. Di Lorenzo's address is: Dipartimento di Matematia e Statistia, Faolta' di Eonomia Universita' degli Studi di Napoli "Federio II", via Cintia, Complesso Monte S.Angelo Napoli, ITALY. Internet address: diloremi@dmsna.dms.unina.it *Marilena Sibillo, Ph.D., is an assoiate professor of finanial mathematis at the University of Sassari. Dr. Sibillo's address is: Istituto Eonomio e Aziendale, Faolta' di Eonomia, Universita' degli Studi di Sassari, Via Rolando Sassari, ITALY. Internet address: sibillo@ssmain.uniss.it Work performed as part of the researh projet "Interazione fra rishio finanziario e rishio assiurativo: modelli e strumenti di ontrollo" of the Programma di Riera di Interesse Nazionale 1998 "Modelli per la gestione di rishi finanziari, assiurativi e operativi." 43

3 44 Journal of Atuarial Pratie, Vol. 8, Introdution Most of the problems faed by an insurer managing a portfolio of life insurane poliies are based on the investment risk (due to interest rates) and insurane risk (due to mortality) and on their interations. Beause of the nature of these risks, most of the researh has been done on the present value of a single poliy within a framework whereby both interest rates and mortality are random. Reently the fous has shifted to similar problems onerning an entire portfolio of poliies. Among the ontributions in this area are Norberg (1993), Parker (1993), (1994a), (1994b), (1996), and (1997), and Frees (1998). Norberg (1993) gave the first two moments of the present value of stohasti payment streams and applied them to a portfolio of temporary insurane ontrats. Parker (1993) studied moments of the present value of future ash flows modeling the fore of interest by (i) a white noise, (ii) a Wiener proess, and (iii) an Ornstein-Uhlenbek proess. Parker found moments of the present value of a portfolio of benefits relating to life policies (1994a) and endowment insurane policies (1994b) by modeling the fore of interest using a Vasiek model; see Vasiek (1977). Parker (1996) proposed two methods to obtain the limiting distribution of the present value of a portfolio of benefits. Parker (1997) provided an interesting paper on the interation between investment and insurane risks for a portfolio of life insurane policies with random urtate future lifetimes. Using the Vasiek model for the rate of return Parker onsidered the variane as a measure of the riskiness of a portfolio and divided it into insurane and investment risks. Frees (1998) showed the utility of the oeffiient of determination for quantifying the relative importane of eah soure of unertainty where there are more than two soures of risks. The aim of the paper is to study the risk of an annuity portfolio by dividing this risk into two omponents: an investment risk and an insurane risk. We offer some ways of ontrolling these by means of the variability measures of the expeted value of the life annuities portfolio with respet to eah of these two omponents. In dealing with a portfolio of life insurane poliies, it is well-known that the effet of aidental deviations of mortality an be redued by using pooling tehniques. But as pointed out in Maroo and Pitao (1998) and Olivieri (1998), however, in the ase of a portfolio of life annuities, a phenomenon not ontrollable by pooling tehniques is the longevity risk, whih is the systemati deviations of the atual number of deaths from the expeted number of deaths due to the improvements in future mortality. The longevity risk produes atuarial losses

4 Coppa/a, Oi Lorenzo, and Sibilla: Risk Soures 45 in the ase of a life annuity portfolio, while in the ase of life insurane ontrats it produes atuarial gains. For these reasons it seems partiularly useful to inlude suitable projetions of mortality improvements in the ase of a life annuity portfolio. In Setion 2 we propose the random variables in a portfolio of homogeneous whole life annuities immediate and we obtain the first two moments of the present value of the portfolio and of the average ost per poliy. Setion 3 presents a desription of the stohasti proess used to model the instantaneous rate of return, while in Setion 4 we onsider the two soures of risk and their measures for the entire portfolio; the longevity risk is introdued also. In Setion 5, the model is applied and several illustrations onerning the importane of the two omponents of the riskiness, as they relate to the number of policies in portfolio, are presented. 2 Portfolio of Life Annuities Let us onsider a portfolio of homogeneous whole life annuityimmediate policies. These policies are assumed to have been issued to lives eah age x and pay an annual benefit of one unit payable at the end of eah year to eah of the survivors. For i = 1, 2,...,, let Ti be the random variable representing the urtate-future-lifetime of the ith life insured and let Zi be the random variable representing the present value of the lifetime annuity benefits for the ith annuitant if Ti = 0; if Ti = 1, 2,..., (1) where: y(t) = f~ Dsds, t > 0, with Ds being the random instantaneous rate of return at time s that is used for disounting the payments. Moreover we suppose (see, for example, Bowers et al., 1987, Chapters 3 and 8, and Parker 1994a) that the following assumptions hold: (i) For i = 1, 2,...,, the TiS are independent and identially distributed;

5 46 Journal of Atuarial Pratie, Vol. 8, 2000 (ii) Given knowledge of y(h) for h = 1,2,..., the ZiS are independent and identially distributed for i = I, 2,..., ; and (iii) For i = I, 2,...,, the TiS and 8s are mutually independent. The random Zi variables are independent only when onditioning on the knowledge of the sequene of y (h) s for h = I, 2,... In general they are not independent, as the same rates of return are used for disounting the payments. For our valuations it is neessary to ompute the first and the seond moments of Zi that are: 00 E[Zd = E[E[Zi ltd] = L hpxe[e-y(h)] h=l 00 E[zl] = L hpxe [e- 2y (h)] h=l 00 h-l + 2 L hpx L E[e-y(r)e-y(h)]. h=2 r=l The proof of equation (3) is easily derived as follows: Proof: (2) (3) E[zl] = E[E[Zll {y(h)}h=l]] 00 h = L E[( L e- y (k»)2h h=l k=l 00 h [1 qx = L E[( L e- y (k»)2](hpx - h+1px) h=l k=l 00 { h+l h } = E[e- 2Y (l)]px + h~l E[(k~l e- y (k»)2] - E[(k~l e- y (k»)2] h+1px 00 h-l = E[e-2Y (l)px + L hpx ( L 2e- y (r) e-y(h) + e- 2y (h»)] h=2 r=l and equation (3) holds. o Let Z () denote the total present value for the entire portfolio of armuities, i.e.,

6 Coppola, Oi Lorenzo, and Sibillo: Risk Soures 47 Z() = L Zi. i=l (4) The first two moments of Z() are: 00 E[Z()] = L hpxe[e-y(hl] h=l E[Z(C)2] = E[L zl + L ZiZj] i=l = L E[ZI] + L E[ZiZj]. i=l i.j=j i"j (5) (6) Next we need an expression for E[ZiZj]. But, by virtue of assumptions (i), (ii), and (iii) (Parker 1994a), E[ZiZj] = E[E[ZiZj I {y(h)}h=l]] = E[E[Zi I {y(h)}h=l]e[zj I {y(h)}h=l]] = E[E[ZI I {y(h)}h=de [Z2 I {y(h)}h=l]] = E[ZlZ2] TJ Tz = E[ L e-y(hl L e-y(kl] h=l TJ k=l Tz = E[E[ L e-y(hl L e-y(kl I {y(r) };'=l]] h=l k=l = E[ L hpxe-y(hl L kpxe-y(kl] h=l k=l = L L hpx kpxe[e-y(hl-y(kl]. h=lk=l Therefore equation (6) an be written as: (7)

7 48 Journal of Atuarial Pratie, Vol. 8, L L L hpxkpxe[e-y(h)-y(k)] i,j~l ifj = E[Zl] h=l k=l ( - 1) L L hpxkpxe[e-y(h)-y(k)]. (8) h=lk=l Finally, from equations (5) and (8), we an obtain the variane of Z(). For our analysis it will be useful to onsider the average ost per poliy, Z () /, of the portfolio under onsideration. 3 Stohasti Rate of Retu rn One of the problems faing insurane ompanies is the finanial risk arising from flutuations of their rate of return. To investigate this problem we follow Di Lorenzo, Sibillo, and Tessitore (1997) and model the instantaneous global rate of return (y(t)) as a sum of two omponents: a deterministi omponent (0 (t)) and a stohasti omponent (X(t)) that desribes the deviations of the instantaneous global rate of return from its expeted value, o(t). This means that Y(t) an be written as: Y(t) = o(t) + X(t). (9) We suppose that o(t) is determined by foreasts based on the existing investments. In addition, {X(t),O ::0; t < +oo} is an Ornstein Uhlenbek proess, with parameters {3 > 0 and u > 0 and initial value X(O) = o. X(t) is haraterized by the following stohasti differential equation: dx(t) = -{3X(t)dt + udw(t) (10) where W(t) is a standard Wiener (Brownian motion) proess. It follows from equation (9) that the stohasti present value at time o of a payment of one monetary unit at time t is given by:

8 Coppa/a, Oi Lorenzo, and Sibilla: Risk Soures 49 e-y(t) = e- fi Y(s)ds = e- fi(8(s)+x(s))ds where = v (t)f(t) (11) v (t) = e- fi 8(s)ds (12) and F(t) = e-fjx(s)ds. (13) Clearly v (t) is the deterministi disounting fator and F (t) is the stohasti disounting fator. F(t) is log normally distributed with parameters -E[f6 X(s)ds], and Var[f6 X(s)ds] and its rth moment about the origin is given by the formula E[(F(t))r] = exp{-re[i: X(s)ds] + ~r2var[i: X(s)ds]}. (14) Using the fat that E[X(t)] = 0 and letting: CP(t) = Var[ I: X(s)ds] (15) we obtain (Crow and Shimizu 1988): E[F(t)] = e~<p(t) (16) and Var[F(t)] = e<p(t) [e<p(t) -1]. (17) Finally, aording to Di Lorenzo, Sib ill 0, and Tessitore (1997), the autoovariane funtion an be written as follows: where: Cov[F(h),F(k)] = e~(<p(h)+<p(k))[e<l>(h,k) -1] (18) <fj(h, k) = Cov[ Ia h X(s)ds, I: X(s)ds].

9 50 Journal of Atuarial Pratie, Vol. 8, Measures of Soures of Unertainty As Frees (1998) points out, it is important to identify the fators affeting the total risk. To this end, we will onsider mortality and stohasti interest as risk fators and make atuarial valuations using an instantaneous total rate of return (interest inome plus apital gains and losses) represented by the stohasti proess defined in equations (9) and (10). Moreover, we will take into aount the mortality omponent, both relating to the riskiness aused by random mortality deviations, and to the riskiness aused by improvements in mortality trend. After identifying the risk fators, we must study ways to manage them. The risk ontrol tools are different depending on the risk omponents onsidered. For example, The risk due to random deviations of the numbers of deaths from their expeted values an be ontrolled by means of pooling tehniques and reinsurane; The investment risk an be ontrolled by various well-known finanial risk management tehniques suh as immunization tehniques and hedging strategies (Frees 1998); and The longevity risk (due to an improved mortality trend) an be ontrolled by using projeted mortality tables that are onstruted on the basis of foreasts of the future mortality trend (Maroo and Pitao 1998 and Olivieri 1998). In light of the above onsiderations, it is important to quantify the ontribution of eah risk fator to the total riskiness of the portfolio. It is for this purpose that we want to study the mortality and investment omponents of the life annuity portfolio onsidered in Setion Insurane and Investment Risk Measures For valuation purposes, it seems reasonable to adopt a simple measure of the two risk omponents affeting the portfolio. We adopt a well-known formula for the deomposition of the variane and apply it to the variane of the present value of the annuity portfolio. First we observe that Var[Z()], the variane of the present value of the portfolio onsidered in our study, an be deomposed in two ways as follows (Parker 1997): Var[Z()] = E[Var[Z() I {Td~=l]] + Var[E[Z() I {Td~=l]] (19)

10 Coppola, Di Lorenzo, and Sibillo: Risk Soures 51 and Var[Z ()] = E[Var[Z () I {y (k)} k= 1]] + Var[E[Z() I {y(k)}k=l]] (20) In equation (19), Var[E[Z() I {Tdf=l]] provides a measure of the variability of Z() aused by ash flows onneted to random events (mortality, survival), after averaging out the effet of the stohasti disounting fators. Thus, we have the following definition: Definition 1. The insurane risk measure is Var[E[Z() I {Tdf=rJ]. Analogously, E[Var[Z() I {Tdf=l]] is an average over ash flows onneted to random events of the variability in Z () due to the stohasti rate of return, and it an be onsidered as an investment risk measure. In equation (20), however, Var[E[Z() I {y(k)}]] is a measure of the variability of Z() due to the effet of the stohasti disounting fators as the effet of random events onneted with mortality and survival have been averaged out, so it is a measure of the investment risk. Thus, we have the following definition: Definition 2. TheinvestmentriskmeasureisVar[E[Z() I {y(k)}k=rj]. We hoose equation (20) for our valuations, beause, as Parker (1997) explains, it allows us to learly relate the risk omponents to the number of poliies. We get: Var[E[Z() I {y(k)}k=l]] = Var[E[I Zi I {y(k)}k=rj] i=l also given by: 00 = Var[ I hpxe-y(hl] h=l = 2 I I hpxkpxcov[e-y(hl, e-y(kl] (21) h=lk=l Var[E[Z() I {y(k)}k=l]] = 2 I I hpxkpxe[e-y(hl-y(kl] h=lk=l 00 - ( I hpxe [e- y (hl])2 h=l

11 52 Journal of Atuarial Pratie, Vol. 8, 2000 and E[Var[Z() I {y(k)}k=dj = E[Var[I Zi I {y(k)}k=l]] i=l = E[Var[Zi I {y(k)}k=l]] = E[E[zll {y(k)}k=l]] - E[(E[Zi I {y(k)}k=1])2]. (22) With regard to the average ost per poliy, Z () /, we get: Z() Var[E[-- I {y(k)}k=l]] = I I hpxkpxcov(e-y(h), e-y(k») (23) h=l k=l and E[Var[Z~C) I {y(k)}k=l]] = ~(E[E[ZII {y(k)}k=dj - E[[E[Zi I {y(k)}k=1]]2]). (24) 4.2 The Longevity Risk Together with the risk due to aidental deviations of death frequenies from their expeted values, the improvements of mortality trends at adult ages have onsequenes on all life insurane ontrats. As life annuities are ontrats pertaining to survival benefits, the alulation of present values should be based on mortality tables with built-in mortality projetions, beause unexpeted improvements in future mortality at the older ages ould result in an underestimation of future osts and result in atuarial losses. Definition 3. The longevity risk is the systemati deviation of the atual number of deaths from their expeted values aross the older ages. By analyzing mortality trend in terms of survival funtions, two aspets known as retangularization and expansion emerge. Retangularization refers to the higher onentration of deaths around the mode of the urve of deaths, lowering the risk for the insurer. Expansion refers to the random advanement of the mode of urve of deaths toward the ultimate life time (Olivieri and Pitao 1999) and hene a higher risk for the insurer. Longevity risk is the result of retangularization and expansion ating jointly (Maroo and Pitao 1998). It an be mitigated by using projeted mortality tables;, that is, tables onstruted on the basis of a foreast of the future mortality trend (Pitao 1998).

12 Coppola, Di Lorenzo, and Sibillo: Risk Soures 53 5 Numerial Illustrations Let us onsider a portfolio of whole life annuities immediate as desribed in Setion 2. We will quantify the insurane and investment risks on the basis of equations (21) to (24) and four different mortality tables. Following Olivieri (1998), we assume that the basi distribution of future lifetimes an be represented by a Weibull distribution, i.e., the survival funtion from age 0 to age x, s(x), is given by: s(x) = e-(x/ix}j', x > 0, where ()( > 0 and y > 0 are onstant parameters. The projeted survival funtion from age 0 to age x is also assumed to follow a Weibull distribution. The basi mortality table and the three projeted tables with inreasing survival probabilities are based on the parameters ()( and y suggested by Olivieri (1998). These parameter values are given below. Parameter Values Survival Tables ()( Basi 82.7 Pessimisti Projetion 83.5 Realisti Projetion 85.2 Optimisti Projetion 87.0 y The parameters f3 and (J of the fore of interest proess (equation (9)) used in our alulations are determined in a manner similar to Di Lorenzo, Sibillo, and Tessitore (1997). As the Ornstein-Uhlenbek proess, X(t), (equation (9)) represents the deviations of the fore of interest from its expeted values, we use the differenes between the atual observed rates and the orresponding foreasted rates. Then by means of the ovariane equivalene priniple (pandit and Wu 1983 and Parker 1994), we an estimate f3 and (J from these differenes. Using data from Italian short-term (three months) bonds, regularly reported in Statistial Bulletin, we obtain 6 = 0.09, f3 = 0.11, and (J = Tables 1 and 2 show the mean, variane, investment risk omponent, and insurane risk omponent of the present value of a portfolio of annuities issued at age 65. Table 1 is based on = 15, while Table 2 is based on = 1000.

13 S4 journal of Atuarial Pratie, Vol. 8, 2000 Tables 3 and 4 show the mean, variane, investment risk omponent, and insurane risk omponent of the present value of the average ost per poliy of a portfolio of annuities issued at age 6S. Table 3 is based on = IS, while Table 4 is based on = Tables Sand 6 show the mean, variane, investment risk omponent, and insurane risk omponent of the present value of a portfolio of annuities issued at age 4S. Table 5 is based on = 15, while Table 6 is based on = Tables 7 and 8 show the mean, variane, investment risk omponent, and insurane risk omponent of the present value of the average ost per poliy of a portfolio of annuities issued at age 4S. Table 7 is based on = IS, while Table 8 is based on = 1000.

14 Coppola, Di Lorenzo, and Sibillo: Risk Soures 55 Table 1 Present Value of Annuity Portfolio at Age 65 with = 15 Projetions Basi Pessimisti Realisti Optimisti E[Z()] l Var[Z()] Var[E[Z () I {y} ]] E[Var[Z () I {y} ]] Table 2 Present Value of Annuity Portfolio at Age 65 with = 1000 Projetions Basi Pessimisti Realisti Optimisti E[Z()] Var[Z() ] Var[E[Z () I {y} ]] E[Var[Z() I {y}]] From Tables 1 and 2 we observe that the mean value of Z() inreases with the projetion; the global variane, for = 15, dereases, exept for the optimisti projetion, while it always inreases for = Analyzing the two risk omponents we note that for both values of the finanial risk inreases with the projetion, while the insurane risk dereases. Tables 3 and 4 show a similar behavior to Tables 1 and 2, respetively. The numerial results for the global variane are onfirmed if we study it as funtion of the :

15 56 Journal of Atuarial Pratie, Vol. 8, 2000 Table 3 Present Value of Average Cost per Poliy at Age 65 with = 15 Projetions Basi Pessimisti Realisti Optimisti E[ Z()] Var[ z()] Var[E[ Z~C) I {y}]] E[Var[ Z~C) I {y}]] Table 4 Present Value of Average Cost per Poliy at Age 65 with = 1000 Projetions Basi Pessimisti Realisti Optimisti E[ Z()] Var[ Z()] Var[E[z~C) I{y}]] E[Var[z~C)I{Y}]] Z()] _ V [ ( -1) ar pess C C = Var[ Z() heal = ( - 1) = Z()] _ ( -1) V ar [-- opt = So the variane related to the pessimisti projetion is greater than the variane related to the realisti projetion for < 16; moreover, the variane related to the realisti projetion is greater than the variane related to the optimisti projetion for < 14.

16 Coppa/a, Oi Lorenzo, and Sibilla: Risk Soures 57 Table 5 Present Value of Annuity Portfolio at Age 45 with = 15 Projetions Basi Pessimisti Realisti Optimisti E[Z()] Var[Z()] Var[E[Z() I {y}]] E[Var[Z() I {y}]] Table 6 Present Value of Annuity Portfolio at Age 45 with = 1000 Projetions Basi Pessimisti Realisti Optimisti E[Z()] Var[Z()] Var[E[Z() I {y}]] E[Var[Z() I {y}]] For all values of, the finanial risk inreases and the insurane risk dereases when the projetion inreases. We observe that the dereasing behavior of the insurane risk is stronger when the number of poliies is small. From a mathematial point of view, we an justify this behavior by means of equation (24) in whih the dependene of E[Var[ Z~C) I {y(k)}]] on is evident. For every fixed survival table, the global variane of Z~C) dereases as inreases. In partiular, the finanial risk takes the same value (from equation (23) we see that Var[E[ Z~C) I {y (k)} ]] does not depend on ), while the insurane risk dereases to zero as tends to infinity (see equation (24)). We an repeat analogous onsiderations about Tables 5, 6, 7, and 8. Observe that for x = 45 the global variane always inreases; in fat we have:

17 58 Journal of Atuarial Pratie, Vol. 8, 2000 Table 7 Present Value of Average Cost per Poliy at Age 65 with = 15 Projetions Basi Pessimisti Realisti Optimisti E[ Z~) ] Var[ Z() ] Var[E[ Z~) I {y}]] l E[Var[ Z~) I {y}]] Table 8 Present Value of Average Cost per Poliy at Age 65 with = 1000 Projetions Basi Pessimisti Realisti Optimisti E[ Z() ] Var[ Z(C)] l Var[E[z~C) I{y}]] l E[Var[ Z~C) I {y}]] Var[Z(C) ]pess = ( -1) C C = C Var[Z() ]real = ( -1) = Var[Z() ]opt = ( -1) = The variane related to the pessimisti projetion is greater than the variane related to the realisti projetion for < 10; moreover, the variane related to the realisti projetion is greater than the variane related to the optimisti projetion for < 8.

18 Coppola, Oi Lorenzo, and Sibilla: Risk Soures 59 6 Summary and Conluding Remarks We have analyzed and quantified two risk soures for a portfolio of life annuities: the investment risk and the insurane risk. This analysis was done in a framework in whih both mortality and rates of returns are random. The global rate of return is modeled as the sum of two omponents: a deterministi one, whih onsiders the existing investments of the ompany, and a stohasti one, representing the deviations of the real rate of return from its antiipated values. The stohasti omponent is an Ornstein-Uhlenbek proess with a mean reversion level of zero. We also onsider the longevity risk, the risk due to the improvements in mortality trend. The effets of the mortality improvements are investigated using different projeted mortality tables. On the basis of the numerial examples presented, we may onlude that the insurane risk dereases when the projetion inreases. On the other hand, the finanial risk inreases when the projetion inreases, beause the ompany ould be exposed for a longer period to a risk of systemati nature. Moreover, the mean value of the present value of the ash flows onneted to the portfolio inreases when the projetion inreases, beause the insurer ould bear bigger osts. In onlusion, the numerial results presented in Setion 6 show how the use of projeted mortality tables allows the insurer to front the risk of greater osts and how the exposure to the finanial risk and to the insurane risk varies, depending on the longevity of the lives insured. One area for future researh is the development of the model presented in the paper, fousing on the effet of the randomness of the projetions in the valuations onerning the onsidered portfolio. Suh researh an lead to the determination of the systemati risk omponent due to the type of randomness depited by the survival funtions used for onstruting mortality tables. Referenes. Arnold, L. Stohasti Differential Equations: Theory and Appliations. New York, N.Y.: John Wiley & Sons, Beekman, J,A. and Fuelling,.P. "Interest and Mortality Randomness in Some Annuities." Insurane: Mathematis and Eonomis 9 (1990):

19 60 Journal of Atuarial Pratie, Vol. 8, 2000 Beekman, J.A. and Fuelling, CP. "Extra Randomness in Certain Annuity Models." Insurane: Mathematis and Eonomis 10 (1991): Bowers, N.L., Gerber, H.U., Hikman, ].C, Jones, D.A., and Nesbitt, C]. Atuarial Mathematis. Shaumburg, Ill.: Soiety of Atuaries, Crow, E.L. and Shimizu, K. (Editors) Lognormal Distribution, Theory and Appliations. New York, N.Y.: Marel Dekker, In., Di Lorenzo, E., Sibillo, M., and Tessitore, G. "A Stohasti Model for Finanial Evaluations. Appliations to Atuarial Contrats." Proeedings of the Seond International Conferene on "Applied Stohasti Models and Data AnalYSis." Held in Capri, Italy. (1997): Frees, E.W. "Stohasti Life Contingenies with Solveny Considerations." Transations of the Soiety of Atuaries 42 (1990): Frees, E.W. "Relative Importane of Risk Soures in Insurane Systems." North Amerian Atuarial Journal (1998): Gard, T.C Introdution to Stohasti Differential Equations. Pure and Applied Mathematis. New York, N.Y.: Marel Dekker, In., Maroo, P. and Pitao, E. (1998) "Longevity Risk and Life Annuity Reinsurane." Transations of the 26th International Congress of Atuaries. Held in Birmingham, England. 6 (1998): Norberg, R. "A Solveny Study in Life Insurane." In Proeedings of the 3rd AFIR International Colloquium (1993): Norberg, R. "Stohasti Calulus in Atuarial Siene." Surveys in Applied and Industrial Mathematis, Olivieri, A. "Per una Quantifiazione del Longevity Risk." Ani del Convegno AMASES. Held in Genova, Italy. (1998): Olivieri, A. and Pitao, E. Funding Sikness Benefits for the Elderly. Dipartimento di Matematia Appliata alle Sienze E., Stat. e Att. "B. De Finetti," Universita di Trieste, Trieste, Italy, Pandit, S.M. and Wu, S.M. Time Series and System Analysis with Appliations. New York, N.Y.: John Wiley and Sons, Parker, G. "Distribution of the Present Value of Future Cash Flows." In Proeedings of the 3rd AFIR International Colloquium (1993): Parker, G. "Moments of the Present Value of a Portfolio of Poliies." Sandinavian Atuarial Journal (1994a): Parker, G. "Stohasti Analysis of a Portfolio of Endowment Insurane Poliies." Sandinavian Atuarial Journal (1994b):

20 Coppola, Oi Lorenzo, and Sibillo: Risk Soures 61 Parker, G. "A Portfolio of Endowment Poliies and its Limiting Distribution." Astin Bulletin 26, no. 1, (1996): Parker G. "Stohasti Analysis of the Interation Between Investment and Insurane Risks." North Amerian Atuarial Journal (1997): Pitao, E. Tavole di mortalita proiettate e loro impieghi in ambito attuariale. Appunti introduttivi. Quaderni del Dipartimento di Matematia Appliata aile Sienze Eonomihe, Statistihe e Attuariali "E. de Finetti" n. 1/1997, Universita di Trieste, Trieste, Italy, Vasiek, O. "An Equilibrium Charaterization of the Term Struture." Journal of Finanial Eonomis 5, no. 2, (1977):

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