conference_MAF_22042014

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

A multivariate approach to project the long runrelationship of mortality indices for Canadian

provinces

A. Ntamjokouen S.Haberman G. Consigli

Vietri sul Mare, 24th Aprile 2014

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Agenda

Literature review on multi and single population;Analysis of Lee Carter parameters ;Order of integration for each of the 9 mortality indicesusing the ADF, PP, KPSS tests;Optimal value of lag of VAR;the Johansen cointegration test Analysis;The estimation of VECM and the VAR models and theforecasting of derived model.Pricing of annuities by cohorts for both males and females

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Motivations

One population model? Literature focuses on the modeling of 1population mortality rates

Lee Carter Model(1992);Lee Miller(2001);Booth Maindonal Smith Variant(2002);Hyndman and Ullah(2005);De Jong and Tickle(2006);Renshaw Haberman(2006) with cohort effect;Currie(2004) with P-Splines, and Currie(2006) with Ageperiod Cohort;Cairns-Blake-Dowd(2009).

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Literature

Modeling mortality rates to improve the Lee Carter modelLazar and Denuit(2009): common trends between 5 agegroups mortality;Darkiewicz(2004): Lee Carter validity as a cointegrationapproach;Njenga and sherris(2009): cointegration among HeligmanPollard;D’Amato(2013): Multipopulation longevity risk amongcountries;

Mortality indicesSalhi(2010) and Zhou et al(2012) on the basis risk.Sharon S. Yang et al. (2009) pricing of longevity bondsderivatives among 4 countries;

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Motivations

Why multiprovinces longevity risk?

Pricing of life insurance annuities accross countries orregions within a country;Engineering of longevity bonds derivatives;Hedging variations of life expectancy pattern.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Data

We colllect data from Canadian Human Mortality Databasefrom 9 provinces: Prince Edward Island(PEI), NovaScotia(NS), News Brunswick(NB), Quebec(Q), Ontario(Q),Manitoba(M), Saskatchewan(S), Alberta(A), BritishColumbia(BC) from 1921 to 2009;It is managed by the Department of Demography of theUniversité de Montreal in collaboration with the Max PlankInstitute for Demographic Research and the Department ofdemography of the University of California atBerkeley(CHMD).

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Data

We retrieve the mortality indices produced by the LeeCarter model for the 9 mortality provinces;The determination of order of integration for each of the 9mortality indices using the Augmented Dickey Fuller,Philips-Perron as well as KPSS Test;The computation of the optimal value of lag of the vector ofautoregressive model;the Johansen cointegration test which test thecointegration rank and specify which variable will enter inthe cointegrated equations and in the Vector of Errorcorrection model;The estimation of VECM and the VAR models and theforecasting of derived model.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Lee Carter Model for each of 9 provinces

We retrieved the singular mortality indice from the 9 provincesthrough Lee Carter model.The Lee carter Model is described as followed:

ln(m1(t, 1)) = a1,x + b1k1,t + e1,t (1)

where:ax describes the shape of age profile of mortality;bx coefficient describes the variation of death rates to variation in thelevel of mortality;kt is the mortality index;ex,t is the error term with ex,t ∼ N(0, σ2

u) is white noise which is theage feature mortality not captured by the model.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Males Mortality indices for each province in Canada

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Females Mortality indices for each province in Canada

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

VAR and VECM models

Since the two graphs show up common trends a priori, the testof integration(after ADF, PP and KPSS tests of integrationconfirm) is 1 for all the 9 provinces analyzed in this framework.

Sex AIC HQ SC FPEMales 6 1 1 1

Females 6 1 1 1

Table 1: The diagnostics tests of residuals under VAR model

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

VAR and VECM models

The VAR model is derived as described below: The vector ofautoregression for p lags is written in Lutkepohl(2005) as:

kt = A0 +A1kt−1 +A2kt−2 + ......Apkt−p + et (2)

where kt = (k1t, k2t, ....., kKt) for k = 1, .....,K time series,(A0.....Ai) are the coefficients and et is white noise.

According to Pfaff(2008), the VAR (p) can be converted into VECM asfollows:

∆kt = Γ1∆kt−1 + Γ2∆kt−2 + ...+ Γp−1∆kt−p+1 +A0 + et (3)

where Γi = −(I −A1 − .....−Ai), i = 1, ..., (p− 1)Π = −(I −Ai,−...−Ap) is a N-dimensional time series, A0 isthe intercept term, et is white noise.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

VAR and VECM models

If r = K the number, the number of cointegrated variables rwhich are stationary equals the rank(K) of π then the model willbe estimated by using the standard statistical model.

If r = 0 this means that there is no cointegrated relationshipsrelationships between the variables. The variables arestationary if we take the take the differences of variables above.

If 0 < r < K there exists 2 matrices α and β such that Γ = αβThere will be r cointegrating relationship or n− r commontrends. Variables into the VECM are all stationary.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Diagnostics of residuals for Males in Alberta

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Evidence of the cointegrated equations for Canadianprovincial Mortality level with critical values at 5%,10% and 1%

r test value 5% 10% 1%r <= 8 3.34 9.24 7.52 12.97r <= 7 11.38 19.96 17.85 24.6r <= 6 25.50 34.91 32 41.07r <= 5 46.40 53.12 49.65 60.16r <= 4 84.23 76.07 71.86 84.45r <= 3 127.73 102.14 97.18 111.01r <= 2 175.99 131.7 126.58 143.09r <= 1 229.25 165.58 159.48 117.2r = 0 300.68 202.92 196.37 215.74

Table 2: Evidence of the cointegrated equations for Canadianprovincial Mortality level with critical values at 5%, 10% and 1%

.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Summary of the cointegration Johansen Test

We run the cointegrated equation with various tests Trace andEigen for both females and males. - Analysis reveal commontrends with Trace and Eigen Values tests

Sex group cointegrated equation common factorsIndices Trace | Eigen Trace | Eigen

Females 5 | 5 4| 4Males 3 | 4 6|5

Table 3: Summary of the cointegration Johansen Test

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Backtesting of the two models VAR and VECM

Out-of-samples VAR(M) VAR(F) VECM(M) VECM(F)Portmanteau test 0.81 0.68 0.97 0.75JB Multivariate 0.18 0.31 0.04 0.16

Skewness 0.88 0.17 0.17 0.062Kurtosis 0.02 0.56 0.0507 0.59

Table 4: Diagnostics of residuals for VAR and VECM models in bothgenders cases

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Backtesting of the two models VAR and VECM

Sex group Females MalesOut-of-samples VAR | VECM VAR | VECMh=2005-2009 5.63% | 5.13% 6.85%| 5.73%h=2002-2009 6.66% | 6.52% 9.47%|10.96%h=2000-2009 12.89%|7.43% 8.42%|22.91%h=1995-2009 16.38%|9.79% 10.66%|2.45%h=1990-2009 19.36%|15.14% 29.67%|24.51%h=1984-2009 21.77%|16.80% 39.80%|30.01%

Table 5: The average MAPE for models VAR and VECM for the 9provinces

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

We observe from table 1 that VECM is more precise thanthe VAR model for females. The backtesting of the twomodels for females presents good performance accuracyoverall. The cointegrated models work better for this sexgroup;It is uncertain for 3 periods as to males for lengths periodincluding 2005-2009, 2002-2009, 2000-2009 which modelis better;Furthermore beyond the 10 years time horizons, errors aretoo large. Almost 30% are unexplained for males. This isdue to the fact that models VAR and VECM do not copevolatility of future mortality indices.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Volatility of the two models VAR and VECM

Out-of-samples Sex Historic VAR VECMh=1995-2009 Males 166.31 37.23 48.10

Females 98.16 91.19 78.51h=1990-2009 Males 172.9 52.17 59.75

Females 107.77 114.88 107.72h=1984-2009 Males 213.93 67.46 69.44

Females 124.45 139.94 136.18

Table 6: Comparison of volatility of historical mortality without-of-sample forecasts produced by models VAR and VECM with insample

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Projecting Males mortality indices for all otherprovinces with VAR models

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Projecting Females mortality indices for all otherprovinces with VAR models

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Forecasting Canadian Males Mortality indices from theVector of Error Correction model

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Forecasting Canadian females Mortality indices fromthe Vector of Error Correction model

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Hypothesis on the pricing methodology

The retirement age will be set to 65 age old regardless thecohort;Payments are made monthly and will be equal to 12;The actuarial present value of a yearly annuity of 1;The interest rate for the evaluation is 4% and the inflationrate is about 2%.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Pricing annuities of females cohorts 1960,1970,1980,1990 and 2000

Females (ARIMA) VAR VECMCohorts life time | APV life time | APV life time | APV

1960 80.89 | 7.74 81.13| 7.80 82.32| 8.521970 82.59 | 8.05 82.91| 8.12 84.59| 9.031980 84.13 | 8.33 84.39| 8.42 86.62| 9.491990 85.25 | 8.60 85.55| 8.70 88.05| 9.892000 86.18| 8.85 86.38| 8.96 89.19| 10.22

Table 7: Pricing annuities of females cohorts 1960, 1970,1980,1990and 2000

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Pricing annuities of Males cohorts 1960,1970,1980,1990 and 2000

Females ARIMA VAR VECMCohorts Life time | APV Life time | APV Life time | APV

1960 75.64 | 6.56 76.68 | 7.20 77.02| 7.421970 77.87 |7.02 79.45| 7.97 79.79| 8.151980 80.49| 7.44 82.87| 8.7 83.36| 8.811990 82.82 | 7.82 85.91| 9.34 86.22| 9.382000 84.51 | 8.16 88.1| 9.89 88.30| 9.87

Table 8: Pricing annuities of males cohorts 1960, 1970,1980,1990and 2000

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

Conclusion

Mortality indices from each province in Canada showdecrements and continuing declining with common trends;The two models show better fit for females gender butuncertainty in the case of males particularly beyond 10years period. Volatility is taken into account only partially;We project mortality indices in 50 yearsperiod for both twogenders and for the two models. VAR projections havenarrow shape like in D′Amato(2013) and VECM like inZhou(2013) and Njenga(2011);We performed the pricing of annuities by group of cohorts1960, 1970, 1980, 1990, 2000 and VECM presents betterperformance in terms of pricing as well as life expectancyover ARIMA and VAR models.

Agenda Review of Literature Database Methodology procedure Lee Carter Model theory Lee Carter Model theory Lee Carter Model theory Optimal lag for VAR model Therory of VAR and VECM models Therory of VAR and VECM models Diagnostics of residuals of VAR models Cointegration rank from Canadian provincial mortality Summary of Cointegration rank and Common factors Backtesting VAR and VECM out of sample Backtesting VAR and VECM out of sample Historical volatility of out-of-sample beyond 25 years Projection of VAR model for each Males Projection VAR models for each province Projection of VECM model for singular Male group Projection of VECM model for singular female group Pricing of annuities Pricing of annuities

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