Longevity Risk and Hedge Effects in a Portfolio of Life Insurance Products with Investment Risk

Transcription

1 Longevity Rik and Hedge Effect in a Portfolio of Life Inurance Product with Invetment Rik Ralph Steven Anja De Waegenaere Bertrand Melenberg PRELIMINARY VERSION: Augut, Abtract Future payment of life inurance product depend on the uncertain evolution of urvival probabilitie. Thi uncertainty i referred to a longevity rik. Exiting literature how that the effect of longevity rik on ingle life annuitie can be ubtantial, and that there exit a natural hedge potential from combining ingle life annuitie with death benefit or urvivor wap. However, the effect of financial rik and portfolio compoition on thee hedge effect i typically ignored. The aim of thi paper i to quantify the hedge potential of combining different life inurance product and mortality linked aet when an inurer face both longevity and invetment rik. We how that the hedge potential of combining different mortality-linked product depend ignificantly on both the product mix and the aet mix. Firt, ignoring the preence of other liabilitie, uch a urvivor annuitie, can lead to ignificant overetimation of the hedge effect of death benefit or urvivor wap on longevity rik in ingle life annuitie. Second, invetment rik ignificantly affect the hedge potential in a portfolio of life inurance product. When invetment rik increae the hedge potential of urvivor annuitie increae and of death benefit decreae. Finally, we how that the hedge potential of urvivor wap i not only ignificantly affected by bai rik, but alo by invetment rik. Keyword: Life inurance, life annuitie, death benefit, urvivor wap, rik management, financial rik, longevity rik, inolvency rik, capital adequacy. Department of Econometric and OR, Tilburg Univerity, CentER for Economic Reearch and Netpar. Correponding author: Department of Econometric and OR Tilburg Univerity, CentER for Economic Reearch and Netpar, PO Box, LE Tilburg, The Netherland, Phone: +, Fax: +, A.M.B.DeWaegenaere@uvt.nl. Department of Econometric and OR, Department of Finance, Tilburg Univerity, CentER for Economic Reearch and Netpar.

2 Introduction Our goal in thi paper i to quantify longevity rik in portfolio of life inurance product, taking into account the potential effect of invetment rik on the impact of longevity rik. Specifically, our focu i on the potential interaction between liability product mix effect, and invetment mix effect. Exiting literature ugget that uncertainty regarding the future development of human life expectancy potentially impoe ignificant rik on penion fund and inurer ee, e.g., Olivieri and Pitacco, ; Brouhn, Denuit, and Vermunt, ; Coette et al., ; Dowd, Cairn, and Blake,. Exiting literature alo how that the natural hedge potential that arie from combining life annuitie and death benefit may be ubtantial ee, e.g., Cox and Lin, ; and Wang et al.,. Thee analye quantify longevity rik in annuity portfolio by determining it effect on the probability ditribution of the preent value of all future payment, for a given, determinitic, and contant term tructure of interet rate. A drawback of thi approach, however, i that it doe not allow to take into account the poible interaction between longevity rik and financial rik, i.e., it i a liability only approach. Hári et al. quantify longevity rik in portfolio of ingle life annuitie in the preence of financial rik by determining it effect on the volatility of the funding ratio at a future date. They define the funding ratio a the ratio of the value of the aet over the bet etimate value of the liabilitie. They find that financial rik can ignificantly affect the impact of longevity rik on funding ratio volatility. A drawback of a funding ratio approach i that it require pecifying the probability ditribution of the fair value of the liabilitie at a future date. In recent year there ha been coniderable interet in developing pricing model for longevity linked aet and liabilitie ee, e.g., Blake and Burrow, ; Dahl, ; Lin and Cox, ; and Denuit, Devolder, and Goderniaux,. Unfortunately, however, the lack of liquidity for trade in longevity linked aet and/or liabilitie make it very difficult to calibrate thee model. Our goal in thi paper i twofold. Firt, we quantify the impact of longevity rik, a well a interaction between financial rik and longevity rik, in a run-off approach. Specifically, we quantify rik by determining the minimal required buffer i.e., aet value in exce of the bet etimate value of the liabilitie, uch that the probability that the inurer or penion fund will be able to pay all future liabilitie i ufficiently high ee, e.g., Olivieri and Pitacco,. The ize of the buffer will be affected by longevity rik, which arie due to uncertain deviation in the future liability payment from their current bet etimate, and by financial rik, which arie due to uncertainty in future return on aet. We how that, even when financial rik and longevity rik

3 are aumed independent, financial rik ignificantly affect the effect of longevity rik on the minimal required buffer, i.e., interaction between financial and longevity rik hould not be ignored. Second, we quantify the effect of potential interaction between liability product mix effect and aet mix effect on the overall rikine of a portfolio of life inurance product. Exiting literature mainly focue on the effect of longevity rik on ingle life annuitie, for inured with given characteritic ee, e.g., Olivieri, ; Olivieri and Pitacco, ; Coette et al., ; and Hári et al.,. Life inurer and penion fund, however, may hold other longevity linked liabilitie, uch a, e.g., lat urvivor annuitie and death benefit inurance. Becaue the payment of thee different life inurance product typically have different enitivitie to change in mortality rate, inurer with a diverified portfolio of liabilitie may be le enitive to longevity rik. For example, Cox and Lin how empirically that a life inurer who ha % of it buine in annuitie and % in death benefit prize it annuitie on average % higher than an inurer who ha % of it buine in annuitie and % of it buine in death benefit. Thi indicate that inurer with death benefit liabilitie may have a competitive advantage. In addition, the impact of longevity rik on a portfolio of life inurance product may alo depend ignificantly on the characteritic of the inured population. For example, becaue longevity trend for male and female are not perfectly correlated, inurer with a more balanced gender mix may be le affected by longevity rik. The exiting literature on liability mix effect focue on the hedge potential of death benefit in portfolio of life annuitie, and ue a liability only approach to quantify the rik reduction due to liability mix ee, e.g., Wang et al.,,. We extend thi analyi by quantifying the impact of invetment rik on the natural hedge potential of combining life inurance product with different enitivitie to longevity rik, taking into account that other longevity linked liabilitie may affect the hedge potential of death benefit in portfolio of life annuitie. Analyzing the effect of product and aet mix on the overall rik i important for two reaon. Firt, taking into account interaction between financial and longevity rik may lead to more accurate olvency meaure. For example, we find that while both partner penion liabilitie and death benefit provide ome hedge effect for longevity rik in old-age penion liabilitie, the hedge potential of partner penion typically increae when invetment rik increae, but the oppoite Many defined benefit penion fund offer both old-age penion inurance and partner penion inurance. The latter conit of a urvivor annuity that yield periodic payment if the partner of the inured i alive and the inured ha paed away. The Retirement Equity Act of REA amended the Employee Retirement Income Security Act of ERISA to introduce mandatory poual right in penion plan.

4 hold for the hedge potential of death benefit. Second, inurer may be able to reduce their enitivity to longevity rik by reditributing their rik. Thi may allow them to reduce longevity rik without a counterparty which take over longevity rik. Our reult indicate that the extent to which inurer may benefit from uch mutual reinurance depend not only on their liability portfolio, but alo on their invetment trategie. Finally, the impact of longevity rik on the inurer olvency may be reduced by inveting in longevity-linked aet uch a, e.g., a longevity bond or a urvivor wap. Becaue the payment of uch intrument are baed on actual urvival of a reference population, they may be ued to partially hedge longevity rik. Exiting literature how that the hedge potential can be affected ignificantly by bai rik, i.e., reidual rik due to difference in characteritic of the inured population and the reference population, ee, e.g., Dowd, Cairn, and Blake. In thi paper we how that the hedge potential of uch longevity linked aet not only depend on bai rik, but alo on invetment rik. The paper i organized a follow. Section preent the model and give a formal definition of the rik meaure. Section how how invetment rik affect the impact of longevity rik in ingle life annuitie, urvivor annuitie, and death benefit, repectively. In Section we quantify the effect of the interaction between liability mix effect and aet mix effect. Section deal with the effect of liability and aet mix on the hedge potential of urvivor wap. Section conclude. The model Thi ection i organized a follow. In Subection. we define the life inurance liabilitie that we conider. In Subection. we define how we quantify rik in portfolio that are enitive to both longevity rik and financial rik. Subection. provide a brief dicuion of the model that are ued to forecat death rate, interet rate, and aet return. A complete decription of thee model can be found in Appendice A and B.. Liabilitie In thi ubection we define the life inurance liabilitie that we conider. In addition to traditional old-age penion, which take the form of a ingle life annuity, penion fund and inurer typically alo offer different other type of life inurance product, uch a partner penion and death benefit. A partner penion provide the partner of a deceaed participant with a life long annuity payment. The death benefit conit of

5 a ingle payment at the moment the inured die. Formally, we conider the following three type of liabilitie: i A ingle life annuity, which yield a nominal yearly old-age penion payment of, with a lat payment in the year the inured die; ii A urvivor annuity, coniting of a nominal yearly partner penion payment of every year that the poue outlive the inured; iii A death benefit, coniting of a nominal ingle death benefit payment of, in the year that the participant die. The liabilitie conit of a tream of payment in future period. Throughout the paper, we conider a fixed and given year t =, and quantify the rik in the payment in future year τ, for a portfolio with given characteritic. Becaue our focu i on the interaction between product mix and aet mix effect, we will conider portfolio coniting of everal product, with varying weight. Specifically, let P = {oa, pp, db} denote the et of life inurance product, and let I denote the et of inured. Becaue in any future period, the level of the payment depend on whether the inured i alive, and, in cae of partner penion, whether the partner i alive, relevant characteritic are thoe that affect the probability ditribution of their remaining lifetime. In addition to age, remaining lifetime alo ignificantly depend on gender and time. Therefore, we characterize a participant by a vector x, g, where x = x, g = g, if p {oa, db}, x = x, y, g = g, g, if p = pp, where x denote the age of the inured, g {m, f} denote the gender of the inured, and, in cae of partner penion inurance, y denote the age of the partner, and g {m, f} denote her/hi gender. Moreover, let p g x,t denote the probability that an x-year-old at date-t with gender g will urvive at leat another year; τ p x g = p g x, pg x+, pg x+τ,τ denote the date- probability that an x-year-old at date- with gender g will urvive at leat another τ year. The net cah outflow of life inurance product i affected by two type of longevity rik: non-ytematic longevity rik becaue, conditional on given urvival probabilitie, whether an individual urvive an additional year i a random variable;

6 ytematic longevity rik becaue the urvival probabilitie for future date τ, p g x,τ, are random variable. While non-ytematic longevity rik i diverifiable i.e., the rik become negligible when portfolio ize i large, thi i not the cae for ytematic longevity rik. Therefore, throughout the paper we aume that portfolio are large enough for non-ytematic longevity rik to be negligible, and focu on the impact of ytematic longevity rik. Then, for any given future year τ, the liability payment for an old-age penion inurance, a partner penion inurance, and a death benefit inurance repectively, are given by ee, alo, e.g., Gerber, : L p,τ x, g = τ p g x, = τ p g x = τ p g x τ p g x, for p = oa old-age penion, τp g y, for p = pp partner penion, for p = db death benefit. Then, the total payment of the life inurer in year τ i the um of the payment for all product and for all inured, i.e., δ i,p L p,τ x i, g i, L τ = i I p P where δ i,p denote the inured right of inured i for penion product p.. Quantifying longevity rik We quantify longevity rik by determining the minimal ize of the buffer, which i defined a the aet value in exce of the bet etimate value of the liabilitie, uch that the probability that the inurer or penion fund will not be able to pay all future liabilitie i ufficiently mall. Specifically, let u denote T for the maximum number of year the inurer ha to make payment, and let L for the random payment in period. Then, the current i.e., date bet etimate of the liabilitie equal expected preent value of future payment, which i given by: BEL = T = ] E [ L P, For example, in a portfolio with annuitie and death benefit, T = x max + x min, where x max i uch that the probability that an individual reache the age x max + i zero, and x min i the age of the younget inured.

7 where P denote the current market value of a zero-coupon bond with maturity, i.e., the preent value of paid out in period. We expre the initial aet value A a the bet etimate value of the liabilitie plu a buffer that i a percentage of the bet etimate value, i.e., A = + c BEL. The value of the inurer aet at a future date depend on realized liability payment, and realized aet return. Indeed, for any given, the value of the aet at date + i given by: A + = A L + R +, for all =,, T, where R + denote the return on invetment between time and +. Conequently, the terminal aet value, i.e., the remaining aet value at time T, i given by: T T T A T = A + R L + R τ. = = τ=+ We quantify rik in portfolio of life inurance product with a given invetment trategy by determining the minimal value of c uch that the probability that the inurer or penion fund will not be able to pay all future liabilitie i ufficiently mall, i.e., P A T < A = + c BEL ε. The minimal required buffer percentage c depend on the probability ditribution of the terminal value of the aet, A T, which, in turn, depend not only on on the initial aet value A, and the liability payment L, but alo on the invetment trategy. Recall that we expre the initial aet value A a the bet etimate value of the liabilitie, BEL, plu a buffer which i expreed a a percentage of the bet etimate value. We allow for the cae where the inurer ue a different invetment trategy on both part. The two portfolio will be referred to a the bet etimate portfolio and the buffer portfolio, repectively. In addition, we want to take into account that penion fund or inurer may wih to do ome form of duration matching. Therefore, we define the following trategie: bet etimate portfolio: for every duration =,, T, the bet etimate preent value correponding to duration, i.e., the amount

8 ] E [ L P, i reinveted in a portfolio that yield return rτ be, in period τ =,, ; in period, the value E [ L ] P + rτ be, i ued to pay the liabilitie τ= in period ; any hortage or exce i taken from, or reinveted in, the buffer portfolio; buffer portfolio: i reinveted in a portfolio that yield return rτ bu τ =,, T. in period Note that wherea the value of the buffer portfolio i affected by both longevity rik and invetment rik, the value of the bet etimate portfolio i only affected by invetment rik. For example, when the buffer portfolio i inveted in equity and the bet etimate portfolio in zero-coupon bond, a lower return on the aet, or a higher than expected realization of the liabilitie, lead to a maller proportion of aet inveted in equity. With the above decribed invetment trategy, we obtain the following reult. Propoition The required minimal buffer value i given by c = Q ǫ L BEL, where L = BEL + T = ] L E [ L P τ= τ= + rbu τ and Q ε L denote the ε quantile of L. + r be, τ, Proof. Given the above decribed invetment trategy, and given, it follow from that the terminal aet value atifie: A T = c BEL T = = [ + c BEL L] + r bu + T T = = ] E [ L P τ= + r be, τ L T τ= + r bu τ + r bu, with L a defined in. Therefore, the terminal aet value A T i nonnegative iff A = + c BEL L,

9 The reult now follow immediately from. The random variable L can be interpreted a follow. Conditional on any given future aet return rτ bu and rτ be,, and cah flow L, L repreent the value of the aet needed at date to pay all future liability payment. For the ake of intuition, conider for example the cae where all aet would yield a determinitic and contant annual return r. Then, L i given by: L = T = L + r, i.e., the dicounted preent value of all future liabilitie. Thu, the tandard approach in which longevity rik i quantified by determining it effect on the probability ditribution of the preent value of liabilitie can be een a a pecial cae of our model. Taking into account that aet return are uncertain, however, implie that L i not only affected by longevity rik, but alo by financial rik. Therefore, we decompoe L into four component, i.e., where L = BEL + L long + L inv + L interact, i BEL i the determinitic component: BEL repreent the value of the aet that would be needed at date to pay all future expected liability payment, when thee ] expected liabilitie are cah-flow matched, i.e., for all duration, the amount E [ L P i inveted in zero coupon bond with maturity. Thi implie that the aggregate return over period for thi part of the bet etimate portfolio equal + rτ be, =. P τ= ii L long i the pure longevity rik component: L long = T = ] L E [ L τ= + E [rbu τ ]. Thi component repreent the value of the aet that would be needed at date to pay all future unexpected liability payment i.e., payment in exce of the bet etimate value, in abence of financial rik, i.e., when the aet return would be determinitic and equal to the expected return on the buffer portfolio, i.e., E [ ] rτ bu.

10 ii L inv i the pure invetment rik component: L inv = T = ] ] E [ L E [ L P + rτ be, τ=. + rτ bu τ= Thi component repreent the value of the aet that i needed at date in exce of BEL to pay all future expected liability payment due to deviation of the return on the bet etimate portfolio from the cah-flow matching return, i.e., due to + rτ be,. P τ= iii L interact i the interaction invetment and longevity rik component: L interact = T ] L E[ L = τ= + rbu τ. + E [rbu τ ] τ= Thi component repreent the value of the aet that i needed at date to pay all future unexpected liability payment due to deviation of the return on the buffer portfolio from it expected return, i.e., due to rτ bu E [ ] rτ bu. It reflect the interaction between longevity rik and financial rik; it i non zero only if there i both longevity rik and financial rik.. Modeling mortality rate and aet return In thi ubection we briefly decribe the tochatic forecat model we ue to forecat the probability ditribution of the future urvival probabilitie p g x, for, future return on equity, and future term tructure of interet rate. For the probability ditribution of the future urvival probabilitie we include proce rik, parameter rik, and model rik. To incorporate model rik, we etimate three clae of urvival probability model, namely the Lee-Carter cla of model, the Cairn-Blake-Dowd cla of model, and the P-Spline model Currie, Durbin, and Eiler,, and generate cenario for future urvival rate from each cla of model. To etimate the parameter in each model, we ue age-, gender-, and time-pecific number of death and expoure to death for the Netherland, obtained from the Human Mortality Databae. For a detailed decription of the future urvival probabilitie model and etimation technique, and for parameter etimate, we refer to Appendix B. We generate cenario for future urvival probabilitie; cenario from Lee-Carter -type model with three different pecification, namely the Lee-Carter

11 model cenario, the Brouhn, Denuit, and Vermunt model cenario, and the Coete et al. model cenario; cenario from Cairn-Blake-Dowd model with four different pecification, allowing for a quadratic term in the age effect, and/or contant/diminihing age effect in the cohort effect each pecification cenario; and cenario from the P-Spline model with one pecification. To forecat the future probability ditribution of the aet return, we ue a Vaicek model to forecat the future probability ditribution of the term tructure of interet rate, combined with a Brownian motion with drift to model the tock price. We include proce rik and parameter rik. To incorporate parameter rik we etimate the parameter uing Generalized Method of Moment GMM. We allow for dependence between the term tructure of interet rate and the equity return in proce rik by allowing the reidual term in the equity return and the term tructure of interet rate to be dependent, and in parameter rik, by etimating the parameter for the future term tructure of interet rate and tock price imultaneouly. To etimate the parameter of the probability ditribution of future term tructure of interet rate and equity return, we ue the daily intantaneou hort rate, the daily interet rate on a year Dutch government bond, and the daily return on the Dutch tock index AEX, obtained from the Datatream. For a more detailed decription of the term tructure of interet rate and equity return model and etimation technique, and for parameter etimate, we refer to Appendix A. Effect of interet rate rik In thi ection we invetigate how invetment rik affect the impact of longevity rik in ingle life annuitie, urvivor annuitie, and death benefit, repectively. In the traditional liability only approach, longevity rik i quantified by determining it effect on the dicounted preent value of the liability payment, given a contant and determinitic interet rate, i.e., by invetigating the ditributional characteritic of L a given in. A hown in the previou ection, however, uncertain deviation from expected liabilitie imply that financial rik cannot be fully hedged, and may affect the impact of longevity rik in a nontrivial way. In thi ection we illutrate the impact of financial rik on the required ize of the buffer, by comparing the benchmark cae in to the cae where interet rate are uncertain. To quantify the effect of interet rate rik, we compare two invetment trategie. The firt invetment trategy i a naive one, where in every year, the remaining aet value after payment of the liabilitie i reinveted in one-year zero-coupon bond. Since

12 penion liabilitie often have a long duration, thi naive invetment trategy may have ubtantial invetment rik. We therefore alo conider an invetment trategy that eliminate interet rate rik in the liabilitie in the bet etimate cenario. We refer to thi invetment trategy a the expected liability hedge trategy. Thu, In the naive invetment trategy, both the bet etimate value BEL, and the buffer c BEL are reinveted in one-year zero-coupon bond, o that r be, τ = r bu τ = r τ, for all τ =,, T, and =,, T, where r τ denote the one-year interet rate in year τ. Therefore, the required buffer percentage c i given by, with L = T L. τ= + r τ = In the expected liability hedge trategy, the bet etimate value, BEL, i inveted in a portfolio of zero-coupon bond that cah flow matche the bet etimate value ] of the liabilitie in each future period, i.e., for every duration, the amount E [ L P i inveted in zero coupon bond with maturity. The buffer c BEL i inveted in one-year zero-coupon bond. Specifically, τ= + rτ be, =, P + rτ bu = + r τ, Then, the required buffer percentage c i given by, with L = BEL + T = ] L E [ L τ= + r τ. Under thi trategy, the bet etimate portfolio eliminate invetment rik in the bet etimate cenario for the liabilitie. Indeed, for each time to maturity, the face value of the zero-coupon bond equal the current bet etimate of the cah flow in year. Thu, all hedgeable rik i eliminated, and invetment rik only arie due to uncertain deviation in realized liability payment, which affect the value of the buffer portfolio. We ue the model decribed in the Appendix to imulate the probability ditribution of future invetment return and urvival probabilitie. We then ue thee imulated

13 ditribution to determine the minimum value of the buffer that i needed in order to reduce the probability of ruin to.% i.e., ε =., for the two invetment trategie, and ingle life annuitie, urvivor annuitie, and death benefit, repectively. We conider two type of inured, namely male and female aged. In cae of urvivor annuitie, the partner of a male inured i a female aged ; the partner of a female inured i a male aged. The reult are diplayed in Table. It i intuitively clear that the effect of longevity rik a well a of financial rik on the required buffer may depend ubtantially on the duration of the liabilitie. Therefore, the firt column in Table diplay the duration of the bet etimate of the liabilitie, which i given by: ] T = P E[ L Duration = ]. T = P E[ L Table : Capital requirement for life inurance product Product Duration c naive c elh,y c LO Male ingle life annuity..%.%.% Female ingle life annuity..%.%.% Male urvivor annuity..%.%.% Female urvivor annuity..%.%.% Male death benefit..%.%.% Female death benefit..%.%.% The table diplay the duration firt column, and the minimal required buffer for the naive invetment trategy only one-year zero-coupon bond, econd column, the expected liability hedge trategy a defined in third column, and for the benchmark liability-only cae lat column. We oberve that the minimal required buffer percentage depend heavily on the invetment trategy. Firt, compared to the naive invetment trategy with only one-year zero-coupon bond, the expected liability hedge trategy with a buffer portfolio inveted in one-year zero-coupon bond ignificantly reduce the reerve requirement for each life inurance product. Second, the liability only approach may lead to ignificant underetimation of the minimum required buffer. Indeed, even when all hedgeable financial rik i avoided i.e., under the expected liability hedge trategy, the required buffer i till ignificantly larger, except for death benefit, than uing the liability only approach. Hence, even for an inurer who ha thi conervative invetment trategy, there i till ignificant invetment rik. Thi occur becaue The age difference i baed on the average age difference in married couple ee, e.g., Brown and Poterba,.

14 longevity rik implie that the future cah flow cannot be fully cah-flow matched uing zero-coupon bond. Note that the high reerve requirement uing the invetment trategy with only one-year zero-coupon bond i partly due to a lower expected yearly return on bond with a hort duration intead of a long duration. Aet and liability mix effect In thi ection we quantify the effect of product mix ratio of inured right for the different life inurance product and gender mix ratio of male inured right over total inured right on the required olvency buffer, for different invetment trategie. To highlight the effect of the interaction between longevity rik and invetment rik, we conider the cae where the inurer eliminate all hedgeable invetment rik by inveting the bet etimate value in a portfolio of zero-coupon bond that matche the bet etimate value of the liabilitie in each future period. Financial rik then only arie due to longevity rik, becaue invetment rik in the aet needed to cover unexpected deviation from the bet etimate value cannot be hedged. Therefore, we conider the following invetment trategie: BEL i inveted in the expected liability hedge trategy, a defined in, o that L = BEL + T = ] L E [ L τ= + rbu τ. the buffer i inveted in one of the following portfolio: % one-year zero-coupon bond; % one-year zero-coupon bond, % equity; % one-year zero-coupon bond, % equity; % equity. We conider two type of inured, male and female inured aged. The partner of a male inured if preent i aged ; the partner of a female inured if preent i aged. Now let u denote: δ p,g = i I,g i =g δ i,p, for g {m, f}, and p P, γ = δoa,m δ oa,m+δ oa,f w g d g = δpp,g δ oa,g = δ db,g δ oa,g for g {m, f}, for g {m, f},

15 i.e., δ p,g denote the total inured right for product p for inured with gender g; γ i the fraction of male ingle life annuitie right relative to the total ingle life annuitie right, w g i the fraction of urvivor annuitie right relative to ingle life annuitie right for gender g, and d g i the fraction of death benefit relative to ingle life annuitie right for gender g. Then, it i verified eaily that the aggregate liabilitie in year, a defined in atify: L = γ [ Loa,, f + w f δ oa,m + δ L pp,,, f, m + d f L db,, f] oa,f + γ [ Loa,, m + w m L pp,,, m, f + d m L ] db,, m, where L oa,, L pp,, and L db, are a defined in. Thu the effect of product and gender mix i fully characterized by γ, w g, and d g. In Subection. we invetigate the hedge effect of product and gender mix in portfolio of ingle life and urvivor annuitie, without death benefit. In Subection. we include death benefit.. Gender and product mix in life inurance product In thi ubection we invetigate the effect of product and gender mix on longevity rik in portfolio of ingle life and urvivor annuitie without death benefit liabilitie, i.e., d m = d f =. We alo invetigate how thee product and gender mix effect are affected by the invetment trategy. In order to reduce the number of parameter, we conider the cae where product mix i.e., the ratio of urvivor annuity right over ingle annuity right i the ame for both gender, i.e., w = w m = w f. The left panel in Figure illutrate the effect of gender mix i.e., the ratio γ of male inured right over total inured right on the required buffer percentage c, in portfolio of ingle life and urvivor annuitie, for three different product mixe: w = top panel, w =. middle panel, and w =. bottom panel. The right panel in Figure illutrate the effect of product mix i.e., the ratio w of inured right for urvivor annuitie over inured right for ingle life annuitie on the required buffer percentage c, for three different gender mixe: % male inured right γ =, top panel, % female inured right γ =, middle panel, and % male inured right and % female inured right γ =., bottom panel. In each cae we conider four different invetment trategie for the buffer portfolio, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line. We oberve the following:

16 Figure : Reerve requirement in portfolio of ingle life and urvivor annuitie. c x % c x %. γ. w c x % c x %. γ. w c x % c x %. γ. w Left column of panel: the effect of gender mix. The panel diplay the required buffer percentage c a a function of γ, in portfolio of ingle life and urvivor annuitie where the ratio of urvivor annuity right over ingle life annuity right equal w = top panel, w =. middle panel, and w =. bottom panel. Right column of panel: the effect of product mix. The panel diplay required buffer percentage c a a function of w, in portfolio of ingle life and urvivor annuitie where the ratio of male inured right over total right equal γ = top row, γ = middle row, and γ =. bottom row. In each cae we conider four different invetment trategie for the buffer portfolio, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line.

17 For every liability mix, reerve requirement are ignificantly affected by unhedgeable invetment rik. An increae in equity lead to a higher expected return, but it alo yield a higher probability that the realized return i lower than expected. The firt effect dominate when the fraction of equity i lower than %. The latter effect dominate for more riky invetment trategie. Depending on the liability mix, the required buffer percentage when the buffer portfolio i fully inveted in equity i almot % higher than when only % of the buffer portfolio i inveted in equity. With regard to liability mix effect i.e., effect of gender and product mix, we oberve two effect: i For every aet mix, combining male and female liabilitie provide hedge effect, but thee effect are only ignificant when the fraction of urvivor annuity right i ufficiently low. ii The effect of including urvivor annuitie in a portfolio of ingle life annuitie depend on gender mix. Thi occur becaue there are two oppoite effect. On the one hand urvivor annuitie can reduce reerve requirement becaue they are negatively correlated with ingle life annuitie. On the other hand, they are more affected by the uncertainty in future urvival probabilitie becaue they have a longer duration. For a portfolio with half male and half female right, the latter effect dominate. With regard to the interaction between longevity rik and invetment rik, we oberve two effect: i a expected, the effect of invetment rik increae when the duration i higher, i.e,. the effect i larger for ingle life annuitie combined with urvivor annuitie, than for ingle life annuitie only, and larger for annuitie for female than for male; ii liability mix effect i.e., effect of gender and product mix are tronger when invetment rik i higher.. Natural hedge potential of death benefit In thi ubection we quantify the effect of death benefit on reerve requirement in portfolio with varying product mixe i.e., varying ratio of inured right for urvivor annuitie over inured right for ingle life annuitie, and invetigate how thee hedge

18 effect are affected by invetment rik. We focu on the cae where product mix i identical for both gender, i.e., w = w m = w f and d = d m = d f. A dicued earlier, the exiting literature typically quantifie longevity rik by invetigating it effect on the probability ditribution of the dicounted preent value of the liabilitie, for a contant and determinitic interet rate. The following propoition how that when financial rik i ignored, longevity rik in ingle life annuitie can be completely hedged by death benefit. Propoition Conider an immediate ingle life annuity for an x-year old with gender g, with an annual payment of, and a death benefit with a ingle payment of δ in the year of deceae for an x-year old with gender g. If R = r for all, and δ = +r, r the terminal aet value A T i unaffected by longevity rik. Proof. Let R τ = r for all τ, and let L τ = τ p g x + δ τ p g x τ p g x. Then, it follow from that: A T T = + c BEL T + r = T = + c BEL τ p g x + δ τ= T = + c BEL τ= L + r δ + δ +r + r τ = + c BEL δ = c BEL. τ p g x + r τ [ T The lat equality follow from BEL = E = Tp x g τ p g x τp g Tp g x x δ + r δ p g T x ] L +r = δ, δ = +r, r p g x =, and =, becaue by aumption, the probability that the inured reache age x + T i negligibly mall. Therefore, the terminal aet value i given by A T = c BEL +r T, which i independent of urvival rate. Propoition ugget that the hedge potential of including death benefit in portfolio of ingle life annuitie may be ignificantly overetimated if financial rik i ignored. Figure how that in order to properly quantify the hedge effect, it i important to take into account both product mix and aet mix effect. The figure diplay the effect of death benefit on the minimal required buffer percentage c for portfolio of life inurance product, for given invetment trategie. The left panel in Figure diplay the minimum required buffer a a function of d, the ratio of the inured right

19 for death benefit over ingle life annuitie, in portfolio with only ingle life annuitie. The right panel diplay the minimum required buffer a a function of d, for portfolio of ingle life annuitie and urvivor annuitie with w =.. In both cae, we conider three different gender mixe, % male inured right γ =, top panel, % female inured right γ =, middle panel, and % male inured right and % female inured right γ =., bottom panel, and four different invetment trategie for the buffer portfolio, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line. A expected, death benefit can ignificantly reduce the reerve requirement. However, the effect depend trongly on product mix and aet mix. The hedge effect i lower in portfolio with both ingle life and urvivor annuitie than in portfolio with only ingle life annuitie, ince urvivor annuitie partially hedge longevity rik in ingle life annuitie, and i lower when the invetment trategy i more riky. Hedge effect of mortality linked aet In thi ection we invetigate the effect of the mortality linked aet in the aet portfolio on the probability of ruin for a life inurer. More preciely, we invetigate the effect of longevity wap on the reerve requirement in a portfolio of life inurance product. Dowd et al. dicu the mechanim and ue of urvivor wap a an intrument for managing, hedging, and trading mortality-dependent rik intead of longevity bond. A urvivor wap can be defined a a wap involving at leat one future tochatic mortality-dependent payment. Given thi definition, the mot baic cae of a urvivor wap i an exchange of a ingle fixed payment for a ingle mortality-dependent payment. More preciely, let ref denote a reference population. Then, at time t =, party A agree with party B that A pay to B at time τ > the amount Kτ, ref known at time and B pay to A at the amount Sτ, ref, which depend on mortality rate and are thu currently tochatic. The payment made in thi agreement are that party B pay A if Kτ, ref < Sτ, ref the amount Sτ, ref Kτ, ref and party B pay A if Kτ, ref > Sτ, ref the amount Kτ, ref Sτ, ref. Hence, the payment of the urvival wap equal: SSτ, ref = Sτ, ref Kτ, ref, where Sτ, ref i the random mortality-dependent payment and Kτ, ref i the fixed payment. Typically, Kτ, ref and Sτ, ref are determined uch that there i no cah

20 Figure : Reerve requirement in portfolio of ingle life and urvivor annuitie and death benefit c x % c x % d d c x % c x % d d c x % c x % d d The panel diplay the required buffer percentage c a a function of d in portfolio of life inurance product, where the ratio of urvivor annuity right over ingle life annuity right equal w = left column, and w =. right column. In both cae, we conider three gender mixe, γ = top row, γ = middle row, and γ =. bottom row, and four aet mixe for the buffer portfolio, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line.

21 tranfer at the time of the iue. However, there i currently no publicly traded market in longevity-linked product and hence we do not oberve the market price of longevity rik. To avoid making aumption regarding the price of the wap, we et Kτ, ref equal to the current expected payment in period τ, and aume that there i a cah tranfer at the time of iue which equal the current over the counter price of the urvivor wap. The urvival wap we conider in thi paper i one where the payment i linear in the number of urvivor in the underlying population. Auming that non ytematic longevity rik i negligible, thi yield: Sτ, ref = N i ref Kτ, ref = N E [ i ref τp gi xi τp gi xi ], where xi and gi denote the age and the gender of individual i in the reference population. A problem with urvivor wap i to obtain a reference population. A natural reference group from the point of view of the inurer i the population of the inurer. However, the inurer may then have more information about the population than the eller of the urvivor wap. Since the inurer may have thi private information, buying a urvivor wap can be interpreted a a ignal that the reference group ha low mortality probabilitie, and hence the price of the urvivor wap would be high, ee Biffi and Blake. Another problem with the natural reference group from the point of view of the inurer i the tradeability of the urvivor wap, i.e., when every life inurer ha a different reference group, many different urvivor wap are needed. Thi would lead to much higher tranaction cot for the eller of the urvivor wap, ince he ha to put extra effort in etimating the ize of longevity rik in the urvivor wap. In order to eliminate the private information problem and to increae the tradeability, the whole population of a country i often choen a reference group, ince the information on thi reference group i the ame for the iuer and buyer of the wap. An example i the firt longevity bond iued by European Invetment Bank/Bank National de Pari In recent year there ha been coniderable interet in developing pricing model for longevity linked aet and liabilitie, ee, e.g., Blake and Burrow, Dahl, Lin and Cox, and Denuit, Devolder, and Goderniaux. Unfortunately, however, the lack of liquidity for trade in longevity linked aet and/or liabilitie make it very difficult to calibrate thee model. The longevity bond wa iued by the EIB and managed by BNP Pariba. The face value wa million, and wa primarily intended for purchae by U.K. penion fund. The urvivor wap involved yearly coupon payment that were tied to an initial annuity payment of million indexed to the urvivor rate of Englih and Welh male aged year in. The longevity bond wa withdrawn prior iue.

22 announced in November, which had a reference population the Englih and Welh male at age in. In Subection. we invetigate how the hedge effect of urvivor wap depend on liability mix and aet mix. To iolate thee effect, we aume no bai rik. In Subection. we invetigate how the hedge effect of urvivor wap i affected by bai rik. In order to focu on the effect of unhedgeable financial rik on the reduction in longevity rik from the mortality linked aet, we will conider the expected liability hedge trategy with buffer portfolio a defined in Section.. Vanilla urvivor wap and product mix In thi ubection we invetigate the effect of vanilla urvivor wap on buffer requirement for an inurer with a portfolio of life inurance product. The vanilla urvivor wap V SSref conit of a portfolio of urvivor wap with all time to maturity. We ue two different vanilla urvivor wap, namely one with reference group the whole male population aged i.e., ref = m and another with reference group the whole female population aged i.e., where ref = f. Let m be the number of vanilla urvivor wap with reference population male, and f with reference group female. Then, the minimal required initial value of the aet in order to limit the probability of ruin to ε i given by: A = BEL + c m, f BEL + V V SS m, f, where c m, f BEL denote the required buffer in exce of the bet etimate value and the price of the urvivor wap, and V V SS m, f i the date- price of the vanilla urvivor wap. Becaue the liability payment, net of payoff from longevity wap i given by: L = L m SS, m f SS, f, it follow from Propoition and that c m, f = Q εl m, f BEL, where T L m, f = BEL + = ] L E [ L m SS, m f SS, f. τ= + r τ bu

23 Becaue we do not make aumption regarding the price of the wap, we cannot determine the optimal fraction of urvivor wap, i.e., the one that minimize the required aet value A. However, for any given portfolio of urvivor wap m, f, we can determine the relative attractivene of the vanilla urvivor wap for different liability mixe and aet mixe. Moreover, for any given aet mix, we can determine the maximum price of the portfolio of urvivor wap under which a lower aet value i ufficient to cover all future liabilitie with probability at leat ε with urvivor wap than without urvivor wap. Thi maximum price i given by: V max V SS m, f = [c, c m, f ] BEL. We now invetigate the hedge potential urvivor wap in portfolio with varying product and aet mixe. We alo determine the maximum price under which inveting in urvivor wap lead to lower capital requirement in each cae. In order to reduce the number of parameter, let = m δ oa,m = f δ oa,f with δ oa, a defined in. Thi implie that when there i no bai rik, equal the fraction of ingle life annuity right for which longevity rik i fully hedged by vanilla urvivor wap, for both male and female. Figure and diplay the minimum required buffer, and the maximum price a defined in both a percentage of the bet etimate value of the liabilitie, repectively, a a function of. We conider two product mixe, only ingle life annuitie left panel, and ingle life and urvivor annuitie with w =. right panel, three gender mixe, γ = top row, γ = middle row, and γ =. bottom row, and four aet mixe, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line. From Figure we oberve that urvivor wap can ignificantly reduce reerve requirement in portfolio of life inurance product. However, the effect depend trongly on product mix and aet mix. Not urpriingly, the fact that there i no bai rik implie that for a portfolio of only ingle life annuitie, longevity rik can be fully eliminated by urvivor wap with =. For a portfolio with alo urvivor annuitie, however, the maximal rik reduction i attained at either < or >. Thi occur becaue urvivor annuitie to ome extent can provide a natural hedge for ingle life annuitie, but on the other hand are alo affected more trongly by longevity rik becaue they have longer duration. The firt effect dominate for a portfolio with only female inured, wherea the econd effect dominate for a portfolio coniting of half male and half female inured right.

24 Figure : Reerve requirement in portfolio of annuitie and vanilla urvivor wap without bai rik c x % c x %.... c x % c x %.... c x % c x %.... The figure diplay the minimal required buffer percentage, c, a a function of for portfolio with only ingle life annuitie w =, left panel, and for portfolio with ingle life and urvivor annuitie w =.; right panel. In both cae, we conider three gender mixe, γ = top row, γ = middle row, and γ =. bottom row, and four aet mixe for the buffer portfolio, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line.

25 Figure : Maximum price of vanilla urvivor wap without bai rik p x % p x %.... p x % p x %.... p x % p x %.... The figure diplay c, c,, i.e., the maximum price a a percentage of BEL, a a function of for portfolio with only ingle life annuitie w =, left panel, and for portfolio with ingle life and urvivor annuitie w =.; right panel. In both cae, we conider three gender mixe, γ = top row, γ = middle row, and γ =. bottom row, and four aet mixe for the buffer portfolio, % equity bold, % equity and % one-year zero-coupon bond dahed line, % equity and % one-year zero-coupon bond dotted line, and % one-year zero-coupon bond olid line.

26 . Vanilla urvivor wap with bai rik In the previou ubection we have oberved that vanilla urvivor wap can reduce the reerve requirement in a portfolio of life inurance product ubtantially. For a portfolio coniting of only ingle life annuitie it can even eliminate longevity rik. In thee calculation we have aumed that there i no bai rik, i.e., the mortality rate of the individual in the reference group for the vanilla urvivor wap are equal to the mortality rate of the inured in the portfolio of life inurance product. Dowd, Cairn, and Blake invetigate the hedge effect of a longevity bond where there exit bai rik, becaue the mortality rate of the inured differ from the reference group of the longevity bond. They ue a longevity bond which i baed on -year-old male to reduce longevity rik for an annuity for a -year-old male. Another type of bai rik in the hedge effect of thee mortality linked aet arie from the contruction of the product. A mortality linked aet with reference group the whole population in a country, for example the announced EIB longevity bond which would depend on urvivor probabilitie of Englih and Welh male aged year in, may alo lead to bai rik, ince it i known that urvival probabilitie for inured are generally higher than for the whole population ee, e.g., Denuit,. In thi paper we quantify the bai rik of a urvivor wap which i baed on the whole population intead of the population of the inurer. A pointed out in Brouhn et al. the mortality rate of individual with a life inurance product are generally lower than for individual without a life inurance product. We ue the Cox-type relational model that ha been uccefully applied in Brouhn et al. and Denuit to account for advere election. Specially, we let logµ h x,t = α h + β h logµ g x,t, where α h and β h i the peed of the future mortality improvement of the group h relative to the general population with gender g. We ue the etimated parameter value from Denuit which are given in Table for group inured and individual inured, for both male and female. Notice that β h <, which implie that the peed of the future mortality improvement in the inured population i mall er than the correponding peed for the general population. Thi occur becaue the advere election oberved in the Belgian individual life market i o trong that the future improvement for the inured population are weaker than for the general population.

27 m, group f, group m, individual f, individual α h β h.... Table : Parameter etimate of the Cox-type relational model. Source: Denuit. We aume that the reference population of the vanilla urvivor wap i the general population of male and the general population of female. We adjut the mortality rate of the inured in the portfolio uing equation. In Figure and we diplay the effect of the vanilla urvivor wap baed on a reference group of the general population for group inured and for individual inured, repectively. If the inured belong to group individual inured, we aume that the partner alo belong to group individual inured. A expected, the hedge potential of urvivor wap reduce ignificantly when there i bai rik. However, it i alo ignificantly affected by invetment rik. For example, in portfolio with urvivor annuitie for male inured, the hedge potential of urvivor wap reduce ignificantly when invetment rik become higher. The difference in urvivor probabilitie may be due to for intance ocial economic tatu, living condition, income, which are typically the ame for both poue.