Recent Developments in Longevity Risk Modelling with o ge ty s ode g t Application to Longevity Risk Management Management Sam Cox & Hal Pedersen (Presenter) (University of Manitoba) (University of Manitoba) Longevity and Pension Funds February 4, 2011 Axa, Paris
A Glance at the Data Before I started working on mortality B f I t t d ki t lit modelling for risk management I assumed that changes in mortality were pretty that changes in mortality were pretty simple. If mortality was improving then the actuary y p g y ought to simply use a bit more conservative table. Some of my colleagues told me during S f ll t ld d i informal discussions that they had their own way of handling improving mortality own way of handling improving mortality. Some of these colleagues even wanted to talk about it.
A Glance at the Data One person felt that taking the ratio of O f lt th t t ki th ti f q_x+1/q_x and then scaling this down by 95% or some other percentage and 95% or some other percentage and reconstructing the mortality rates after the scaling has served him well in the past. Actuaries need real solutions to real business problems and most actuaries are inclined to construct their own solution if inclined to construct their own solution if one they are satisfied with is not available. Evidently if one scales the mortality in this Evidently, if one scales the mortality in this fashion and starts at age 0 then one obtains a sequence of curves such as the f ll following. i
A Glance at the Data Mortality Improvements (Slope Scaled) [Initial Curve Illustrative Gompertz] Mortalit Impro ements (Slope Scaled) [Initial C r e Ill strati e Gompert ] 0.2000 0.1500 q_x new q_x_1 q_x new q_x_2 0.1000 new q_x_3 new q_x_4 0.0500 0.0000 0 0 20 0 40 0 60 60 80 80 100 00 120 0 140 0 Age x
A Glance at the Data m_x for US Male Mortality 0.80000 0.70000 0.60000 1950 0.50000 1960 0.40000 1970 1980 0.30000 1990 2000 0.20000 0.10000 0.00000 65 68 71 74 77 80 83 86 89 92 95 98 101 104 107
A Glance at the Data When I first looked at mortality data in detail I could not help but be struck by the rate at which it appears to be changing even over a hi h it t b h i period of one decade. US Male Population Mortality US Male Population Mortality q_x 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 65 0.02098 0.02023 0.01963 0.01932 0.01881 0.01852 0.01784 0.01759 0.01717 0.01703 70 0.03142 0.03032 0.03001 0.02925 0.02930 0.02838 0.02686 0.02691 0.02568 0.02495 75 75 0.04755 0.04755 0.04760 0.04760 0.04655 0.04655 0.04572 0.04572 0.04506 0.04506 0.04456 0.04456 0.04208 0.04208 0.04133 0.04133 0.04001 0.04001 0.03944 0.03944 80 0.07524 0.07170 0.07250 0.07007 0.07016 0.06885 0.06624 0.06552 0.06396 0.06213 As you know, the data is even more striking for younger ages. Even if there is year to year noise in the data the trend is strong.
A Glance at the Data If such trends are extrapolated for If h t d t l t d f extended periods of time, one will have what is surely unrealistic life tables what is surely unrealistic life tables. Of course, these trends change over time and the duration of particular trends seen p in the historical data also vary. The following French male mortality data ill illustrates aspects of the variations in t t t f th i ti i trends and their duration. Apparently different rates of improvement Apparently, different rates of improvement occur and for various durations.
0.00 0.10 0.20 0.30 0.40 0.50 0.60 1 1816 1 1824 French Male Mortality Data (1816 ‐ 2007) 1 1832 1 1840 1 1848 1 1856 1864 1 1 1872 1 1880 1888 1 1 1896 1904 1 1912 1 1 1920 A Glance at the Data 1928 1 1 1936 1 1944 1952 1 1960 1 1968 1 1976 1 1984 1 1 1992 2000 2 100 80 60 60 40 20 0 0
A Glance at the Data French Male Mortality Data (1816 ‐ 2007) 0.12 0.10 0.08 20 0.06 40 0.04 60 0.02 0.00 1816 1824 1832 1840 1848 1856 1864 1872 1880 1888 1896 1904 1912 1920 1928 1936 1944 1952 1960 1968 1976 1984 1992 2000
A Glance at the Data The changes in mortality rates seem to be Th h i t lit t t b volatile when measured annually. If we average the mortality rates over non If we average the mortality rates over non- overlapping trailing windows such as 5 years then the qualitative nature of the y q data might be easier to follow. If we look at ratio of this time series to a b base period, then one measure of i d th f mortality improvement is obtained. The following is US male mortality post The following is US male mortality post WWII.
A Glance at the Data US M l M US Male Mortality ‐ Reduction Factors on 5 ‐ year Trailing t lit R d ti F t 5 T ili Averages on 1947 Base Rates (1952, 1957, ..., 2007) 1.20 1.00 0 10 0.80 20 20 30 40 0.60 50 60 60 0.40 70 80 90 0.20 100 0.00 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007
Lee-Carter Model The Lee-Carter model remains the benchmark model for mortality modelling. The parameters a and b are age dependent parameters and k is a time dependent process the reflects the temporal variation in mortality. Estimated over an age range and a time range [t_0,t_N].
Lee-Carter Model In order to use the model for the risk- I d t th d l f th i k management simulation of mortality, one must adopt a stochastic process for k that must adopt a stochastic process for k that permits simulation going forward. The determination of such a stochastic process is an involved issue and one that is not fully settled in the literature, particularly in the case of the multi factor particularly in the case of the multi-factor version of the model. However for the moment let us suppose However, for the moment, let us suppose that such a stochastic process is adopted. We then simulate mortality as: y
Lee-Carter Model Traditionally the series k is treated as a Traditionally, the series k is treated as a random walk beyond the last data point.
Lee-Carter Model Traditionally the series k is treated as a Traditionally, the series k is treated as a random walk beyond the last data point.
Lee-Carter Model Lee ‐ Carter Mortality Projected Using Random Walk Drift Only (US Both Sexes Grouped) 0.12 0 12 0.1 0.08 2007 2017 0.06 2027 2027 2037 0.04 2047 2057 0.02 0 50 ‐ 54 55 ‐ 59 60 ‐ 64 65 ‐ 69 70 ‐ 74 75 ‐ 79 80 ‐ 84 85 ‐ 89
Lee-Carter Model The parameter estimates for this US Th t ti t f thi US grouped both gender data 1933-2007 are shown below shown below. a b 0 0.8 0.6 -5 0.4 0.2 -10 0 0 10 20 30 0 10 20 30 k 8 6 4 2 0 0 20 40 60 80
Lee-Carter Model A change of data to US male ages 65-85 A h f d t t US l 65 85 from 1950 has the parameters shown below. below b b a -2 0.35 0.3 -3 0.25 -4 0.2 -5 5 0 10 20 30 0 10 20 30 k 4 3 2 1 1 0 0 20 40 60
Lee-Carter Model The time series for k is not a relatively The time series for k is not a relatively straight line anymore but it can be approximated as such in the latter part of th the series. (Was also in other data set.) i (W l i th d t t ) Some authors have suggested using a multi factor version of Lee Carter multi-factor version of Lee-Carter. For most data sets, the first principal component captures the majority of the component captures the majority of the variation and the next few principal components capture virtually all of the variation. i ti However, for a multi-factor model one must be able to model the k’s to simulate must be able to model the k s to simulate for risk management.
Lee-Carter Model Three-factor model with initial fit to 2007 Three factor model with initial fit to 2007 (US both sexes grouped from 1950). a b 0 0.5 0 -5 -0.5 0 5 -10 -1 0 10 20 30 0 10 20 30 Explaining the Variance k 1.005 4 1 2 0.995 0 0.99 0.985 0 98 -2 0 20 40 60 0 10 20 30
Lee-Carter Model It appears that it might be reasonable to It appears that it might be reasonable to use random walks with drift to model all three factors. three factors. There are data sets for which some of the factors are non-linear and this is a potential roadblock to using a multi-factor f Lee-Carter. Instead of fitting a multi factor Lee Carter Instead of fitting a multi-factor Lee-Carter and then dealing with the dynamics of the k time series on an ad hoc basis, perhaps , p p we can force a linear structure on a the model?
Lee-Carter Model Borrowing an idea from the term structure Borrowing an idea from the term structure of interest rates, we look at the structure of the change in the logarithms of the of the change in the logarithms of the mortality rates. Fitting to the log differences is in contrast to the usual estimation of Lee-Carter which fits to the estimation of Lee Carter which fits to the logs of mortality rates. We assume that the change in the logs of We assume that the change in the logs of mortality rates can be captured with a linear drift. Performing principal components yield the following estimates for US both sexes grouped from 1950 grouped from 1950.
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