A comparative study of two-population models for the assessment of basis risk in longevity hedges

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1 A comparative study of two-population models for the assessment of basis risk in longevity hedges Steven Haberman, Vladimir Kaishev, Pietro Millossovich, Andres Villegas Faculty of Actuarial Science and Insurance Cass Business School, City University London Longevity 11 Lyon 7-9 September 2015

2 2 Longevity Swaps longevity swaps: longevity risk transfer solution for pension schemes/annuity providers flexible alternative to buy-ins/buy-outs 2014 longevity swaps volume: 25.4bn more than the double the volume written in (Hymans Robertson LLP (2015)) two types of swaps indemnity based (bespoke): floating leg matches pension payments index based: floating leg depends on mortality index (eg LifeMetrics, Deutsche Börse,... )

3 2 Longevity Swaps longevity swaps: longevity risk transfer solution for pension schemes/annuity providers flexible alternative to buy-ins/buy-outs 2014 longevity swaps volume: 25.4bn more than the double the volume written in (Hymans Robertson LLP (2015)) two types of swaps indemnity based (bespoke): floating leg matches pension payments index based: floating leg depends on mortality index (eg LifeMetrics, Deutsche Börse,... )

4 2 Longevity Swaps longevity swaps: longevity risk transfer solution for pension schemes/annuity providers flexible alternative to buy-ins/buy-outs 2014 longevity swaps volume: 25.4bn more than the double the volume written in (Hymans Robertson LLP (2015)) two types of swaps indemnity based (bespoke): floating leg matches pension payments index based: floating leg depends on mortality index (eg LifeMetrics, Deutsche Börse,... )

5 3 Index Based Longevity Swap Fixed Premiums (hedge cost) Pension/Annuity Fund Annuity Payments Indexed Payments based on E&W mortality Investors/reinsurers

6 4 Indemnity based vs Index based Swap index based swap standardized more efficient, cheaper solution mismatch between pension payments and floating (index) cash flows basis risk almost all swap transactions so far were bespoke key issue: understanding basis risk components demographic risk: difference between the two populations book and index sampling risk: volatility in mortality experience structural risk: due to limited range of hedging instrument

7 4 Indemnity based vs Index based Swap index based swap standardized more efficient, cheaper solution mismatch between pension payments and floating (index) cash flows basis risk almost all swap transactions so far were bespoke key issue: understanding basis risk components demographic risk: difference between the two populations book and index sampling risk: volatility in mortality experience structural risk: due to limited range of hedging instrument

8 5 Where Does This Research Come From? research project sponsored by Life and Longevity Market Association and the Institute of Actuaries aim: develop a practical tool for analysing basis risk joint research group from Cass Business School and Hymans Robertson Phase I ( ) report available at actuaries.org.uk/sites/all/files/documents/ pdf/ifoa-llma-longevity-basis-risk-report.pdf

9 5 Where Does This Research Come From? research project sponsored by Life and Longevity Market Association and the Institute of Actuaries aim: develop a practical tool for analysing basis risk joint research group from Cass Business School and Hymans Robertson Phase I ( ) report available at actuaries.org.uk/sites/all/files/documents/ pdf/ifoa-llma-longevity-basis-risk-report.pdf

aim: develop a practical tool for analysing basis risk joint research group from Cass Business

10 5 Where Does This Research Come From? research project sponsored by Life and Longevity Market Association and the Institute of Actuaries aim: develop a practical tool for analysing basis risk joint research group from Cass Business School and Hymans Robertson Phase I ( ) report available at actuaries.org.uk/sites/all/files/documents/ pdf/ifoa-llma-longevity-basis-risk-report.pdf

11 Multi Population Mortality Models Basis risk assessment requires a two-population mortality model huge selection to choose from Narrowing down the long list of possible models? Define criteria for good practical model Review existing models vs criteria Define what modelling approach is appropriate in which cases

12 7 Size of pension scheme / annuity book Key Questions How long does the experience need to be for direct modelling? Alternative approach Direct modelling of book What models would be appropriate? How small can the book be for direct modelling? Sampling risk is the main determinant of basis risk Number of years of available data

13 Main Findings Size of pension scheme / annuity book 8 years Direct modelling of book Alternative approach CAE+Cohorts or M7-M5 25K lives Sampling risk is the main determinant of basis risk Number of years of available data 8

14 9 Universe of Multipopulation Models

15 10 Criteria to shortlist models (CMI wp 25, Cairns et al. (2008, 09), Haberman and Renshaw (2011)) Practical Easy to implement Transparent Parsimonious Compatible with available data Disentangle level and improvement differences Central estimates (deterministic) Consistent with historic data Consistent with expected mortality characteristics (e.g. compensation law) Goodness of fit of rates and rate differences Able to incorporate cohort effect

16 Criteria to shortlist models (CMI wp 25, Cairns et al. (2008, 09), Haberman and Renshaw (2011)) Practical Easy to implement Transparent Parsimonious Compatible with available data Disentangle level and improvement differences Central estimates (deterministic) Consistent with historic data Consistent with expected mortality characteristics (e.g. compensation law) Goodness of fit of rates and rate differences Able to incorporate cohort effect

17 11... Criteria Correlations Non-perfect correlations between year on year changes in mortality at different ages Non-perfect correlations between mortality rates in the two populations Reasonableness Forecast level of uncertainty in rates and rate differences Flexibility Handle different book sizes, lengths of back history, portfolio heterogeneity Include user view on key parameters Generate sample paths Incorporate parameter uncertainty in simulations

18 ... Criteria Correlations Non-perfect correlations between year on year changes in mortality at different ages Non-perfect correlations between mortality rates in the two populations Reasonableness Forecast level of uncertainty in rates and rate differences Flexibility Handle different book sizes, lengths of back history, portfolio heterogeneity Include user view on key parameters Generate sample paths Incorporate parameter uncertainty in simulations

19 ... Criteria Correlations Non-perfect correlations between year on year changes in mortality at different ages Non-perfect correlations between mortality rates in the two populations Reasonableness Forecast level of uncertainty in rates and rate differences Flexibility Handle different book sizes, lengths of back history, portfolio heterogeneity Include user view on key parameters Generate sample paths Incorporate parameter uncertainty in simulations

20 12 Shortlisting an Appropriate Model(s) Some criteria allowed us to narrow down the list of models based on their properties: for instance Non-perfect correlations between mortality rates in the two populations all models with a single period term (eg common factor) Compatible with available data co-integrated models Easy to implement/transparent Bayesian models, P-splines To further refine the remaining models need a common mathematical formulation and data sets to ease comparison

21 12 Shortlisting an Appropriate Model(s) Some criteria allowed us to narrow down the list of models based on their properties: for instance Non-perfect correlations between mortality rates in the two populations all models with a single period term (eg common factor) Compatible with available data co-integrated models Easy to implement/transparent Bayesian models, P-splines To further refine the remaining models need a common mathematical formulation and data sets to ease comparison

22 12 Shortlisting an Appropriate Model(s) Some criteria allowed us to narrow down the list of models based on their properties: for instance Non-perfect correlations between mortality rates in the two populations all models with a single period term (eg common factor) Compatible with available data co-integrated models Easy to implement/transparent Bayesian models, P-splines To further refine the remaining models need a common mathematical formulation and data sets to ease comparison

23 12 Shortlisting an Appropriate Model(s) Some criteria allowed us to narrow down the list of models based on their properties: for instance Non-perfect correlations between mortality rates in the two populations all models with a single period term (eg common factor) Compatible with available data co-integrated models Easy to implement/transparent Bayesian models, P-splines To further refine the remaining models need a common mathematical formulation and data sets to ease comparison

24 13 General Mathematical Formulation two populations: reference (R) and book (B) D i xt, i =D, R: number of deaths at age x, calendar year t, population i; similarly for the matching exposures E i xt reference population Dxt R Bin(Ext, R qx,t) R N logit qx,t R =αx R + βx j,r k j,r t + γt x R i=1 k R t =d + k R t 1 + ξ R t, ξ R WN(0, Σ R ) γ R c =φ 0 + φ 1 γ R c 1 + ɛ R c, ɛ R WN(0, σ 2 R)

25 General Mathematical Formulation two populations: reference (R) and book (B) D i xt, i =D, R: number of deaths at age x, calendar year t, population i; similarly for the matching exposures E i xt reference population Dxt R Bin(Ext, R qx,t) R N logit qx,t R =αx R + βx j,r k j,r t + γt x R i=1 k R t =d + k R t 1 + ξ R t, ξ R WN(0, Σ R ) γ R c =φ 0 + φ 1 γ R c 1 + ɛ R c, ɛ R WN(0, σ 2 R)

26 14... General Mathematical Formulation Book Dxt B Bin(Ext, B qx,t) B M logit qx,t B logit qx,t R =αx B + βx j,b k j,b t + γt x B i=1 k B t =Ψ 0 + Ψ 1 k B t 1 + ξ B t, ξ B WN(0, Σ B ) γ B c =ψ 0 + ψ 1 γ B c 1 + ɛ B c, ɛ B WN(0, σ 2 B) model the spread between mortality rates in the two populations level differences: (α B x ) improvement differences: (k B t ) VAR(1) implies not long run divergence cohort differences: (γ B c ) AR(1) implies not long run divergence

27 14... General Mathematical Formulation Book Dxt B Bin(Ext, B qx,t) B M logit qx,t B logit qx,t R =αx B + βx j,b k j,b t + γt x B i=1 k B t =Ψ 0 + Ψ 1 k B t 1 + ξ B t, ξ B WN(0, Σ B ) γ B c =ψ 0 + ψ 1 γ B c 1 + ɛ B c, ɛ B WN(0, σ 2 B) model the spread between mortality rates in the two populations level differences: (α B x ) improvement differences: (k B t ) VAR(1) implies not long run divergence cohort differences: (γ B c ) AR(1) implies not long run divergence

28 15 Reference population specification

29 ... Reference Population Specification England & Wales (EW), ages 60-89, years select Lee Carter+cohorts and M7 consistently with existing literature (Cairns et al. (2009), Haberman and Renshaw (2011)) Lee-Carter+cohorts logit q R x,t = α R x + β R x k R t + γ R t x M7 logit qx,t R = k 1,R t +(x x)k 2,R t +((x x) 2 σx)k 2 3,R t +γ R t x

30 17 Book Population Specification generate synthetic datasets based on Club Vita schemes and IMD (postcode based) national mortality data allows for changing some characteristics of the books backtesting sample books base case: period age range scheme size lives per year (large) four different distributions according to IMD typical split fit models by ML

31 17 Book Population Specification generate synthetic datasets based on Club Vita schemes and IMD (postcode based) national mortality data allows for changing some characteristics of the books backtesting sample books base case: period age range scheme size lives per year (large) four different distributions according to IMD typical split fit models by ML

32 17 Book Population Specification generate synthetic datasets based on Club Vita schemes and IMD (postcode based) national mortality data allows for changing some characteristics of the books backtesting sample books base case: period age range scheme size lives per year (large) four different distributions according to IMD typical split fit models by ML

33 18... Book Population Specification 1.2 Mortality ratio to England and Wales by age Typical Lives Mortality ratio to England and Wales by year Typical Amounts 1.1 Book / England and Wales Extreme Wealthy Extreme Deprived Book / England and Wales age year

34 19 Book Population Specification

35 20 Fitting (Rates and) Rates Differences

36 21... Fitting (Rates and) Rates Differences Some models are too simple to fit observed differences

37 22 Main Message avoid models with non parametric book specific age response term β B x not enough book data to estimate β B x may produce over-smoothed aggregate demographics metrics model behaves as if it implied perfect correlation example: RelLC+Cohort β x B vs. x age

38 23 Goodness of Fit vs Parsimony CAE+Cohort and M7-M5 have best compromise models with book specific cohort have the worst trade-off enough to capture level and and slope differences

39 24 CAE+cohorts & M7-M5 CAE+cohorts: logit q R x,t =α R x + β R x k R t + γ R t x logit q B x,t logit q R x,t =α B x + β R x k B t M7-M5: logit qx,t R =k 1,R t + (x x)k 2,R t + ((x x) 2 σ x))k 2 3,R t + γt x R logit qx,t B logit qx,t R =k 1,B t + (x x)k 2,B t

40 25 Reasonable Forecast Levels of Uncertainty (Cairns et al. (2011)) sources of uncertainty: process risk (PR) future trajectories of k B and γ B parameter uncertainty (PU) estimation of parameters of the model, computed via bootstrapping (Koissi et al. (2006), Renshaw and Haberman (2008)) sampling risk (SR) volatility of actual mortality experience (D B xt) focus on key metrics period truncated life expectancy value based hedge cohort truncated life expectancy cash-flow based hedge analyse robustness wrt book size and book length

41 Reasonable Forecast Levels of Uncertainty (Cairns et al. (2011)) sources of uncertainty: process risk (PR) future trajectories of k B and γ B parameter uncertainty (PU) estimation of parameters of the model, computed via bootstrapping (Koissi et al. (2006), Renshaw and Haberman (2008)) sampling risk (SR) volatility of actual mortality experience (D B xt) focus on key metrics period truncated life expectancy value based hedge cohort truncated life expectancy cash-flow based hedge analyse robustness wrt book size and book length

42 Reasonable Forecast Levels of Uncertainty (Cairns et al. (2011)) sources of uncertainty: process risk (PR) future trajectories of k B and γ B parameter uncertainty (PU) estimation of parameters of the model, computed via bootstrapping (Koissi et al. (2006), Renshaw and Haberman (2008)) sampling risk (SR) volatility of actual mortality experience (D B xt) focus on key metrics period truncated life expectancy value based hedge cohort truncated life expectancy cash-flow based hedge analyse robustness wrt book size and book length

43 26 Robustness wrt Book Size Book Book Reference model CAE + Cohorts M7 M5 variance risks PR PR+PU size 30 year curtailed period life expectancy at age 60 in 2020 Book Book Reference model CAE + Cohorts M7 M5 risks variance PR PR+PU size 25 year curtailed cohort life expectancy at age 65 in 2011

44 Robustness wrt Book History mean error in life expectancy Book Typical Lives Typical Amounts Extreme Wealthy Extreme Deprived CF + Cohorts CAE + Cohorts history length Gravity M7 M5 Forecast mean error in the forecast of 30 year period curtailed life expectancy at age 60 mean error in life expectancy diff Book Reference Typical Lives Typical Amounts Extreme Wealthy Extreme Deprived CF + Cohorts CAE + Cohorts history length Gravity M7 M5 27 Forecast mean error in the difference of 30 year period curtailed life expectancy at age 60 between the book and the reference

45 ... Robustness wrt Book History mean abs. error in life expectancy Book Typical Lives Typical Amounts Extreme Wealthy Extreme Deprived CF + Cohorts CAE + Cohorts history length Gravity M7 M5 Forecast mean absolute error in the forecast of 30 year period curtailed life expectancy at age mean abs. error in life expectancy diff Book Reference Typical Lives Typical Amounts Extreme Wealthy Extreme Deprived CF + Cohorts CAE + Cohorts history length Forecast mean absolute error in the difference of 30 year period curtailed life expectancy at age 60 between the book and the reference Gravity M7 M5

46 29 Conclusion robustness wrt book size for book sizes < lives, PR is unrealistically high distorting the basis risk assessment for book sizes < lives, PU is significant distorting the basis risk assessment robustness wrt book length History length shorter than 8 years have poor forecasting performance CAE+Cohort has the best out-of-sample performance both in terms of bias and accuracy for history lengths longer than 8 years no model outperform the others in terms of differences

47 29 Conclusion robustness wrt book size for book sizes < lives, PR is unrealistically high distorting the basis risk assessment for book sizes < lives, PU is significant distorting the basis risk assessment robustness wrt book length History length shorter than 8 years have poor forecasting performance CAE+Cohort has the best out-of-sample performance both in terms of bias and accuracy for history lengths longer than 8 years no model outperform the others in terms of differences

48 0 Hedge Effectiveness (Cairns et al. (2014)) measured by R 2 = 1 VAR(L h H) VAR(L) = ρ 2 where L liability, either book period (value hedge) or cohort (cash-flow hedge) life expectancy H hedging instrument corresponding quantity in the reference population h optimal hedge ratio ρ correlation coefficient between L and H focus on analysis wrt book size and length

49 31... Hedge Effectiveness correlation^ PR PR+PU PR+PU+SR CF + Cohorts CAE + Cohorts Gravity M7 M book size Value hedge example: 30 year curtailed period life expectancy at age 60 in 2020 correlation^ PR PR+PU PR+PU+SR CF + Cohorts CAE + Cohorts Gravity M7 M book size Cash flow hedge example: 25 year curtailed cohort life expectancy at age 65 in 2011

50 32... Hedge Effectiveness 1.00 PR PR+PU PR+PU+SR correlation^ CF + Cohorts CAE + Cohorts Gravity M7 M history length Value hedge example: 30 year curtailed period life expectancy at age 60 in PR PR+PU PR+PU+SR correlation^ CF + Cohorts CAE + Cohorts Gravity M7 M history length Cash flow hedge example: 25 year curtailed cohort life expectancy at age 65 in 2011

51 Main Conclusions Our analysis suggest that the most appropriate models are the M7-M5 (more flexible) and the Common Age Effect plus Cohorts For the estimation exercise to be reliable lives 8 years of available history are the minimum requirement

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