Longevity Risk: Methods, Models, and Management

Mathematical Finance Seminar @ U of Southern California Los Angeles, CA – September 12, 2016

Daniel Bauer Georgia State University

This presentation is primarily based on a collaborations with Enrico Bifﬁs, Fausto Gozzi, and Nan Zhu. Financial support from the Society of Actuaries under a CAE Grant is gratefully acknowledged. Page 2 Longevity Risk | September 12, 2016 | Daniel Bauer

Introduction What is "mortality/longevity risk"? Is it a big deal? For whom?

Mortality Surface Models Motivation Factor Analysis of Mortality Forecasts Mortality Term Structure Models Self-consistency Condition Applications

Conclusion Page 3 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction

Introduction What is "mortality/longevity risk"? Is it a big deal? For whom?

Mortality Surface Models Motivation Factor Analysis of Mortality Forecasts Mortality Term Structure Models Self-consistency Condition Applications

Conclusion I Are we all getting older? If so, is there a limit? Is there risk/uncertainty?

I What does this mean for individuals, corporations, and economies?

Page 4 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction Some folklore

I Jeanne Louise Calment

I longest conﬁrmed human lifespan on record

I living to the age of 122 years, 164 days

I died in 1997 (she was born before the telephone was invented!)

I Signed a contingency contract (similar to a reverse mortgage) at age 90 that paid 2,500 francs ($500) a month for her apartment at death. Altogether, more than 900,000 francs ($180,000)

I Smoked till 119, loved chocolate ^¨ Page 4 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction Some folklore

I Jeanne Louise Calment

I longest conﬁrmed human lifespan on record

I living to the age of 122 years, 164 days

I died in 1997 (she was born before the telephone was invented!)

I Signed a contingency contract (similar to a reverse mortgage) at age 90 that paid 2,500 francs ($500) a month for her apartment at death. Altogether, more than 900,000 francs ($180,000)

I Smoked till 119, loved chocolate ^¨

I Are we all getting older? If so, is there a limit? Is there risk/uncertainty?

I What does this mean for individuals, corporations, and economies? Page 5 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction Vaupel et al. – We are getting older, and the trend has been fairly stable...

More information: Oeppen & Vaupel (2002) "Broken Limits to Life Expectancy." Science, 296: 1029-1031. Page 6 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction Olshansky et al. – ...but there may be a limit (?)

How did we get here? Why did life expectancy rise by 30 years in the last century? U.S. FEMALES Maximum Lifespan Potential = 120

Maximum Observed Age at Death = 105, 113 *

Period Life Expectancy at Birth = 49, 80 *

Modal Age at Death = 73, 88 *

2000

1900

More information: Olshansky, Carnes, & Désquelles (2001) "Demography: Prospects for Human Longevity." Science, 291: 1491-1492. Graph from presentation by J. Olshansky: http://www.aaas.org/spp/rd/Olshansky.pdf I Mitigation/Management (?)

I Transaction design / transfer risk to individuals

I Shift from DB to DC pensions, increase in retirement age, self-annuitization schemes, tontines, etc. ? But individuals not very well equipped to take on risk post retirement...1

I Securitization / mortality derivatives

I Various transactions to transfer mortality risk to investors, at a premium (Buy-Outs, Buy-Ins, Longevity/Survivor Swaps, Longevity Bonds, etc.) ? Longevity market has not lived up to expectations (reminiscent of mortgage-backed securities?) although picked up a little in recent past...

I Are these risks appropriately understood? Do existing models capture all the risks in an appropriate way?

Page 7 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction So there is risk/uncertainty... But how grave is it?

I E.g., how will the life expectancy change over time? → Individual: level of retirement savings → Insurance company: (risk of) liability of deferred annuity → Economy: Social security, retirement age

1Turner (2006) "Pensions, risks, and capital markets." JRI 73: 559-574. I Are these risks appropriately understood? Do existing models capture all the risks in an appropriate way?

Page 7 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction So there is risk/uncertainty... But how grave is it?

I Mitigation/Management (?)

I Transaction design / transfer risk to individuals

I Shift from DB to DC pensions, increase in retirement age, self-annuitization schemes, tontines, etc. ? But individuals not very well equipped to take on risk post retirement...1

I Securitization / mortality derivatives

1Turner (2006) "Pensions, risks, and capital markets." JRI 73: 559-574. Page 7 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction So there is risk/uncertainty... But how grave is it?

I Mitigation/Management (?)

I Transaction design / transfer risk to individuals

I Shift from DB to DC pensions, increase in retirement age, self-annuitization schemes, tontines, etc. ? But individuals not very well equipped to take on risk post retirement...1

I Securitization / mortality derivatives

I Are these risks appropriately understood? Do existing models capture all the risks in an appropriate way?

1Turner (2006) "Pensions, risks, and capital markets." JRI 73: 559-574. I (We need) Data:

I Popular: Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de.

I Other sources: CDC – NVSS or Wonder Berkeley Mortality database

Page 8 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction (Insurance / Financial) Mathematics

2 I (We need a) Model: Lee-Carter model (by far the most popular model, applied by policy makers, governmental agencies, and the broader scholarly literatures)

log{mx,t } = αx + βx κt + εx,t ; κt = θ + κt−1 + et

(mx,t is the central death rate for age group x in year t)

I Estimation typically in two step procedure using a singular value decomposition on {mx,t }x,t to identify κt .

2Lee & Carter (1992) "Modeling and forecasting US mortality." JASA, 87: 659-671. Page 8 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction (Insurance / Financial) Mathematics

2 I (We need a) Model: Lee-Carter model (by far the most popular model, applied by policy makers, governmental agencies, and the broader scholarly literatures)

log{mx,t } = αx + βx κt + εx,t ; κt = θ + κt−1 + et

(mx,t is the central death rate for age group x in year t)

I Estimation typically in two step procedure using a singular value decomposition on {mx,t }x,t to identify κt .

I (We need) Data:

I Popular: Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de.

I Other sources: CDC – NVSS or Wonder Berkeley Mortality database

2Lee & Carter (1992) "Modeling and forecasting US mortality." JASA, 87: 659-671. Page 9 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction Lee-Carter In Matlab:

l_mx_female = log(mx_female); alpha = mean(l_mx_female,2); for t=1:T l_mx_female(:,t) = l_mx_female(:,t) - alpha * ones(X,1); end; Sigma = cov(transpose(l_mx_female)); [eigvec,eigval] = eig(Sigma); beta_first = eigvec(:,X); kappa = transpose(beta_first) * l_mx_female; deltakappa = zeros(T-1,1); for t=1:T-1 deltakappa(t) = kappa(t+1) - kappa(t); end; theta_kap = mean(deltakappa); sigma_kap = std(deltakappa);

−1 0.18 2

1.5 0.16 −2 1

0.14 0.5 −3

0 0.12

−4 −0.5

0.1 −1

−5 0.08 −1.5

−2 −6 0.06 −2.5

−7 0.04 −3 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60

β x αx → Gompertz (1825): mx = α e βx κt Page 10 Longevity Risk | September 12, 2016 | Daniel Bauer Introduction Mortality probability and cohort life expectancy for 50-year old US female

-3 x 10 38 5.5 37 5 36

4.5 35

4 34

33 3.5 32 3 31

2.5 life expectancy at age 50 in years 30 one-year mortality rate at age 50 in years

2 29 1960 1970 1980 1990 2000 2010 2020 2030 1960 1970 1980 1990 2000 2010 2020 2030 year year

I One year mortality probability (q50) seems relatively straightforward

I Cohort life expectancy (Forward looking! Not period life expectancy as for Vaupel line!) increasing, although good bit of variation. The 95% conﬁdence region (red dashed) seems too narrow... Page 11 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

Introduction What is "mortality/longevity risk"? Is it a big deal? For whom?

Mortality Surface Models Motivation Factor Analysis of Mortality Forecasts Mortality Term Structure Models Self-consistency Condition Applications

Conclusion 2. Currently expected to live 25.1 more years. How uncertain is this number? What are the chances that it changes to 23.7 or 24.5 next year? ⇒ Risk in mortality projections

I While the two questions are related, and the distinction does not matter in theory, it is relevant for the statistical approach

I For personal ﬁnancial planning/household ﬁnance, insurers’ liability risk evaluation, and government/public economics, question 2 may be more suitable

Page 12 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Consider a 60 year old individual who faces aggregate mortality risk: 1. Currently expected to die with probability of 0.7% next year (mortality rate). How uncertain is this appraisal? What are the chances that the rate is 0.65% or 0.75%? ⇒ Risk in mortality rates I While the two questions are related, and the distinction does not matter in theory, it is relevant for the statistical approach

I For personal ﬁnancial planning/household ﬁnance, insurers’ liability risk evaluation, and government/public economics, question 2 may be more suitable

Page 12 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

2. Currently expected to live 25.1 more years. How uncertain is this number? What are the chances that it changes to 23.7 or 24.5 next year? ⇒ Risk in mortality projections Page 12 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

2. Currently expected to live 25.1 more years. How uncertain is this number? What are the chances that it changes to 23.7 or 24.5 next year? ⇒ Risk in mortality projections

I While the two questions are related, and the distinction does not matter in theory, it is relevant for the statistical approach

I For personal financial planning/household finance, insurers’ liability risk evaluation, and government/public economics, question 2 may be more suitable ⇒ We want to find & estimate tractable, coherent mortality models that adequately reflect risk in mortality projections

I Analogy in the motivation: Term structure models

Page 13 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

!! ! !!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!! "!!"!!"!

! ! "#$%! !!!!! &! ! ! '! ! ! ! !!!!!(! ! ! ! ! ! !!!!!()'!

I Current stochastic mortality models focus on stochastically forecasting mortality rates (question 1) → red in graphic

I This paper considers the risk in mortality projections (question 2) → blue in graphic I Analogy in the motivation: Term structure models

Page 13 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

!! ! !!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!! "!!"!!"!

! ! "#$%! !!!!! &! ! ! '! ! ! ! !!!!!(! ! ! ! ! ! !!!!!()'!

I Current stochastic mortality models focus on stochastically forecasting mortality rates (question 1) → red in graphic

I This paper considers the risk in mortality projections (question 2) → blue in graphic

⇒ We want to ﬁnd & estimate tractable, coherent mortality models that adequately reﬂect risk in mortality projections Page 13 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

!! ! !!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!! "!!"!!"!

! ! "#$%! !!!!! &! ! ! '! ! ! ! !!!!!(! ! ! ! ! ! !!!!!()'!

I Current stochastic mortality models focus on stochastically forecasting mortality rates (question 1) → red in graphic

I This paper considers the risk in mortality projections (question 2) → blue in graphic

⇒ We want to ﬁnd & estimate tractable, coherent mortality models that adequately reﬂect risk in mortality projections

I Analogy in the motivation: Term structure models

I Forecast yield curve p(T + 1, τ) based on p(t, τ) (q. 2!) Q R t+τ I p(t, τ) = Et [exp(− t rs ds)], so (in theory) modeling rt sufﬁces (q. 1!) I Necessity to consider cross-sectional data (persistence vs. transience, etc.) I However, there are key differences (age, P vs. Q, etc.) Page 14 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models Preview of Results: Forecasts of Life Expectancy

life expectancy at age 50 in years 30

29 1960 1970 1980 1990 2000 2010 2020 2030 year I Assume: Time-homogeneous, Gaussian model

dµt = (A µt + Λ) dt + Σ dWt

I SDE on some Hilbert space H, A = (∂τ − ∂x )

I Markov, time-homogeneous (?) [different / much milder than Markov assumption on hazard rate!]

I Gaussian [W ﬁn-dim. Brownian motion] → Not necessarily positive (???) → Use for now [tractability!] but also discuss non-negative models I Deﬁne: τ+l px (tj+1) τ+l+∆px−∆(tj ) Fl (tj , (τ, x)) = − log , tj+1 − tj = ∆ τ px (tj+1) τ+∆px−∆(tj ) ¯ Fl (tj ,(τ,x)) I Proposition: Vectors Fl (tj ) = ω(τ, x) × √ iid Gaussian. t −t j+1 j (τ,x)∈C˜

Page 15 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Given: (Forward) survival probabilities {τ px (t)|(τ, x) ∈ C}

I Easier to model/work with: (Forward) force of mortality / hazard rate ∂ µ (τ, x) = − log{ p (t)} t ∂τ τ x I Deﬁne: τ+l px (tj+1) τ+l+∆px−∆(tj ) Fl (tj , (τ, x)) = − log , tj+1 − tj = ∆ τ px (tj+1) τ+∆px−∆(tj ) ¯ Fl (tj ,(τ,x)) I Proposition: Vectors Fl (tj ) = ω(τ, x) × √ iid Gaussian. t −t j+1 j (τ,x)∈C˜

Page 15 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Given: (Forward) survival probabilities {τ px (t)|(τ, x) ∈ C}

I Easier to model/work with: (Forward) force of mortality / hazard rate ∂ µ (τ, x) = − log{ p (t)} t ∂τ τ x

I Assume: Time-homogeneous, Gaussian model

dµt = (A µt + Λ) dt + Σ dWt

I SDE on some Hilbert space H, A = (∂τ − ∂x )

I Markov, time-homogeneous (?) [different / much milder than Markov assumption on hazard rate!]

I Gaussian [W ﬁn-dim. Brownian motion] → Not necessarily positive (???) → Use for now [tractability!] but also discuss non-negative models Page 15 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Given: (Forward) survival probabilities {τ px (t)|(τ, x) ∈ C}

I Easier to model/work with: (Forward) force of mortality / hazard rate ∂ µ (τ, x) = − log{ p (t)} t ∂τ τ x

I Assume: Time-homogeneous, Gaussian model

dµt = (A µt + Λ) dt + Σ dWt

I SDE on some Hilbert space H, A = (∂τ − ∂x )

I Markov, time-homogeneous (?) [different / much milder than Markov assumption on hazard rate!]

I Regions: England/Wales (ENW), France (FRA), Japan (JPN), United States (USA), and West Germany (FRG)

I Genders: male and female

I Years: 1956-2006 / as much as we can

I Methods: 1. Lee-Carter

I Weighted-least-squares algorithm I Ages: 0-95 2. CBD-Perks

I Basic model w/o cohort effect I Ages: 25-95 3. P-splines

I Fixing the degree of freedom (df ) at 20 I Ages: 25-95

Page 16 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

"True" mortality forecasts from market place or insurance prices → not abundant/noisy... [key results same for "ofﬁcial" UK life tables] Page 16 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

"True" mortality forecasts from market place or insurance prices → not abundant/noisy... [key results same for "ofﬁcial" UK life tables] Raw data: (deterministic) mortality forecasts generated from rolling windows of past mortality experience [from Human Mortality Databases, www.mortality.org]

I Regions: England/Wales (ENW), France (FRA), Japan (JPN), United States (USA), and West Germany (FRG)

I Genders: male and female

I Years: 1956-2006 / as much as we can

I Methods: 1. Lee-Carter

I Weighted-least-squares algorithm I Ages: 0-95 2. CBD-Perks

I Basic model w/o cohort effect I Ages: 25-95 3. P-splines

I Fixing the degree of freedom (df ) at 20 I Ages: 25-95 Page 17 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models Surfaces of Eigenvectors: PC1

female, US, Lee-Carter male, France, Lee-Carter female, Ger, Lee-Carter

female, US, splines male, France, splines female, Ger, splines

→ Explains 90 − 97% of the variation! Page 18 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

female, US, Lee-Carter Page 19 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I First two factors explain vast majority of variation in most cases (mostly > 90% for ﬁrst factor, ﬁrst two factors >> 95%)

I Very similar shapes exhibited across different countries/genders/forecasting methods (at least for the ﬁrst factor) ⇒ PC1

I Systematic, increasing in age/term

I Forward forces of mortality for high ages in the far future more volatile than in the near future → slope factor

⇒ PC2

I Additional effect for higher ages

I Inverse relationship between high ages in the near and in the far future → twist factor

¯ 0 ¯ I Simple factor models by regressing Fl (tj ) on Yi = bi Fl (tj ) [akin to Diebold & Li (2006, JEconometrics), Duffee (2011), etc.] I Maths:

I Implies martingale property that yields drift condition, i.e. Λ = f (Σ)

I Prop.: µt allows for Gaussian ﬁnite-dimensional realization iff

Σ(τ, x) = C(x + τ) × exp{Mτ} × N

× × [M ∈ Rm m, N ∈ Rm d , C ∈ C([0, ∞), Rm) – semi-parametric form] R t ⇒ µt (τ, x) = µ0(τ + t, x − t) + 0 Λ(τ + t − s, x − t + s) ds Z t +C(x + τ) exp {M τ} exp {M (t − s)} N dWs 0 | {z } =Zt (state process)

Page 20 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Interpretation as forecasts:

I Expected values for future survival probs. comprised in the current surface ⇒ Expectation from (stochastic) forecast should coincide with "burnt in" values

I Cross-sectional restriction akin to no-arbitrage condition for interest rate term structure models Page 20 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Interpretation as forecasts:

I Expected values for future survival probs. comprised in the current surface ⇒ Expectation from (stochastic) forecast should coincide with "burnt in" values

I Cross-sectional restriction akin to no-arbitrage condition for interest rate term structure models

I Maths:

I Implies martingale property that yields drift condition, i.e. Λ = f (Σ)

I Prop.: µt allows for Gaussian ﬁnite-dimensional realization iff

Σ(τ, x) = C(x + τ) × exp{Mτ} × N

× × [M ∈ Rm m, N ∈ Rm d , C ∈ C([0, ∞), Rm) – semi-parametric form] R t ⇒ µt (τ, x) = µ0(τ + t, x − t) + 0 Λ(τ + t − s, x − t + s) ds Z t +C(x + τ) exp {M τ} exp {M (t − s)} N dWs 0 | {z } =Zt (state process) I Estimation:

I Use Maximum Likelihood Estimation that may or may not enforce self-consistency condition

I In addition, allow for non-systematic deviations ¯ obs ¯ mod I F (tj , tj+1) = F (tj , tj+1) + j , j ∼ N(0, α · diag{Σ}), α small

Page 21 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Identiﬁcation:

¯ I Relationship to Fl / PCA: Z τ+1 Z ∆ d Mv M(∆−s) Fl (τ, x) = E[Fl (τ, x)] + C(x + v) e dv × e N dWs τ 0 | {z } | {z } =O(τ,x) =Z∆

I Prop.: We can treat factors separately

I Find model by regressing on principal components

I Obtain M and N from regression on factor w/o functional assumptions on C(x) [identiﬁcation issues in higher dimensions – rely on examples from interest rate modeling to ﬁnd convenient shapes, in particular Björk and Gombani (1999, FinStoch)]

I Potentially use functional assumptions for capturing C(x) [parametric] Page 21 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models

I Identiﬁcation:

¯ I Relationship to Fl / PCA: Z τ+1 Z ∆ d Mv M(∆−s) Fl (τ, x) = E[Fl (τ, x)] + C(x + v) e dv × e N dWs τ 0 | {z } | {z } =O(τ,x) =Z∆

I Prop.: We can treat factors separately

I Find model by regressing on principal components

I Potentially use functional assumptions for capturing C(x) [parametric]

I Estimation:

I Use Maximum Likelihood Estimation that may or may not enforce self-consistency condition

I In addition, allow for non-systematic deviations ¯ obs ¯ mod I F (tj , tj+1) = F (tj , tj+1) + j , j ∼ N(0, α · diag{Σ}), α small Page 22 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models Associated Hazard Rate Based on our speciﬁc one-factor assumption, the aged-dependent hazard rate / (spot) force of mortality is Z t (2) µt (x) = µ0(t, x − t) + Λ(t − s, x − t + s) ds +C(x) × Zt 0 | {z } may use "baseline" if no µ0 e.g. (2) = (A + B exp{c x}) × (ξ2 + Zt ) where ( ) ! Z 1 −2b −b2 1 − ab dZ = d t = Z dt + dW t (2) 1 0 t a t Zt Example path:

8 140 Z2 W 7 Z1 120

6 100 5

80 4

3 60

2 40

1 20 0

0 ï1

ï2 ï20 0 4 8 12 16 20 0 4 8 12 16 20 t t (Non-negative model: OU → Feller square root) Page 23 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models Conﬁdence Intervals for Future LE: USA Female (After 1 yr)

44.4 16.2 44.2 16.1 44 16 43.8 15.9 43.6 15.8 43.4

15.7 43.2

43 15.6

42.8 15.5

42.6 15.4

42.4 15.3 Lee-Carter Non-parametric Factor Lee-Carter Non-parametric Factor

age 40 age 70

17 47

16.8 46

16.6 45 16.4 44 16.2

43 16

42 15.8

41 15.6

40 15.4

15.2 39

15 Factor Mortality Surface Non-negative Factor Mortality Surface Non-negative

age 40 age 70 Page 24 Longevity Risk | September 12, 2016 | Daniel Bauer Mortality Surface Models Cohort Effects

I Key difference to yield curve models: new generations / cohort effects (Renshaw & Haberman, 2006, IME; Cairns et al., 2009, NAAJ)

log{mx,t } = αx + βx κt + γt−x + εx,t

I Deterministic models: µt (τ, x) = µ0(τ + t, x − t), x ≥ t > 0, τ ≥ 0, µt (τ, x) = ψt−x (τ + x), 0 ≤ x < t

I Differential notation on H:

0 (0) µt = A µt − AB ψt 2 µ0 ∈ H, ψ ∈ Lγ (R+)

I Stochastic extension yield SPDEs with boundary noise

I Questions:

I Well-posedness (spaces?)

I Finite-dimensional realizations Page 25 Longevity Risk | September 12, 2016 | Daniel Bauer Conclusion

Introduction What is "mortality/longevity risk"? Is it a big deal? For whom?

Mortality Surface Models Motivation Factor Analysis of Mortality Forecasts Mortality Term Structure Models Self-consistency Condition Applications

Conclusion Page 26 Longevity Risk | September 12, 2016 | Daniel Bauer Conclusion Conclusion

I Having appropriate estimates for the risk in mortality projections is important

I Common approach may not be suitable to appraise risk of changes in life expectancies

I Research Directions:

I Continuous models with cohort effects

I Multiple populations, trade off risk?

I Application in household ﬁnance: Annuitization decision in portfolio context/inﬂuence of systematic mortality risk

I Asset pricing model with aggregate demographic uncertainty