Re-thinking the Life Tables for Assured Lives in Kenya

Transcription

1 Re-thinking the Life Tables for Assured Lives in Kenya Carolyn Njenga, PhD (UNSW, Australia) Strathmore University, Nairobi, Kenya A Technical Paper presentation at the TASK Convention, October 21 st and 22 nd 2015, Nairobi, Kenya Njenga (Strathmore University) Kenyan Mortality October / 34

2 Outline 1 Introduction 2 Literature Review 3 Methodology 4 The Data 5 Comparison of Male and Female Assured Lives Ordinary Life Assureds 6 Results for the Males 7 Results for the Females 8 A hybrid of the two? 9 Conclusions and Recommendations Njenga (Strathmore University) Kenyan Mortality October / 34

3 Introduction Changing the regulatory environment: Mortality assumptions for Assured Lives are stated in the Insurance Act. Previously, assured lives in Kenya were assumed to follow mortality expectations quantified in the United Kingdom (particularly A ). Then used KE tables (overestimated mortality and were not popular). Kenya needs updated mortality tables based on current experience. Data quality is poor therefore it is a tedious challenge to develop Kenyan Life Tables for Assured Lives. Njenga (Strathmore University) Kenyan Mortality October / 34

4 Introduction Aim: To improve the accuracy and fit of the Kenyan Life Table. In the tables that are being introduced in the KE Life tables - the cubic spline method was used as the primary method fitting the data from ages 16 to 55. A good result... and we can keep improving it. Inspired by the Continuous Mortality Investigations carried out in countries such as the UK. Njenga (Strathmore University) Kenyan Mortality October / 34

5 Literature Review Extract from the Core Technical Courses Graduation: the process of using statistical techniques to improve estimates provided by crude mortality rates. Purpose of graduation: produce a smooth set of mortality rates remove random sampling errors (as far as possible) improve the reliability of the mortality estimates. Njenga (Strathmore University) Kenyan Mortality October / 34

6 Literature Review In this study, the sample size is small. Initial exposed to risk for ages years for the period of study (from ) is: Females: 482,153 (809 deaths) Males: 714,607 (1,748 deaths). The crude estimates are each estimated independently from one another = estimates suffer independent sampling errors. Graduation uses additional information provided by the numbers of deaths at adjacent ages to improve the reliability of the point estimates. Njenga (Strathmore University) Kenyan Mortality October / 34

7 Literature Review How graduation is done... There are main three techniques used for graduation: 1 graduation by reference to a standard table (if the class of lives involved in the graduation is sufficiently similar to the class of lives whose experience formed the basis of a particular standard table) 2 graduation using a parametric formula (often used for reasonably large experiences) 3 graphical graduation (quick and dirty approach) - requires expert judgment. Njenga (Strathmore University) Kenyan Mortality October / 34

8 Methodology Notation: q s x: Death rates from the standard tables ˆq x : Estimated crude death rates at exact age x q x : Graduated death rates d x : number of deaths in [x, x + 1) E x : number of exposed to risk in [x, x + 1) We VERY briefly discuss the steps that are followed in performing: 1 Graduation by Reference to a Standard Table 2 Graduation using a parametric formula Njenga (Strathmore University) Kenyan Mortality October / 34

9 Methodology Graduation by Reference to a Standard Table Procedure. 1 Select a suitable existing standard mortality table 2 Seek a reasonably simple function f () that will link the estimates to the standard table q x = f (q s x) OR µ x = f (µ s x) 3 Determine the best-fitting parameter values using either maximum likelihood or (weighted) least squares 4 Calculate the graduated values 5 Test for goodness-of-fit, smoothness etc Njenga (Strathmore University) Kenyan Mortality October / 34

10 Methodology Graduation using a Parametric Formula Procedure. 1 Select a graduation formula or parametric family of curves. 2 Determine parameter values 3 Calculate graduated rates 4 Test for goodness-of-fit, smoothness etc Njenga (Strathmore University) Kenyan Mortality October / 34

11 The Data Description of the Data on Kenyan Assured Lives Data is available from with a gap in 2004 and It is categorized into: Ordinary Life Assureds Group Life Assured Annuitants The data is obtained from 21 life insurance companies. More than half the data was not usable. Vital information was missing. Limited data... with peculiar characteristics. Njenga (Strathmore University) Kenyan Mortality October / 34

12 Comparison of Male and Female Assured Lives Male and female assured lives experiences are compared in a quest to complete the first step in graduation with reference to a standard mortality table. Understanding the mortality features helps in selection of the standard table to use. Njenga (Strathmore University) Kenyan Mortality October / 34

13 Comparison of Male and Female Assured Lives Ordinary Life Assureds Generally... Exposed to risk: Males > Females Figure: Exposed-to-Risk Njenga (Strathmore University) Kenyan Mortality October / 34

14 Comparison of Male and Female Assured Lives Ordinary Life Assureds And... Excessive Male Mortality q x,t : Males > Females Proportion with q-males greater than q-females ALL Njenga (Strathmore University) Kenyan Mortality October / 34

15 Comparison of Male and Female Assured Lives Ordinary Life Assureds Lemons and Lemonade Figure: Piggyback the not-so-good mortality data on an existing Life Table from a population with similar characteristics e.g. South African Life Tables Njenga (Strathmore University) Kenyan Mortality October / 34

16 Comparison of Male and Female Assured Lives Ordinary Life Assureds Why South African Life Tables? From The Report of the South African Continuous Statistical Investigations Committee Assured Lives Mortality Investigation : 1 Techniques used to calculate the variables e.g. Exposed to risk are the same (based on number of policies rather than number of lives). 2 Scope was similar (covers only Whole Life and Endowment Assurance business) 3 Males were more than females 4 Excessive male mortality was experienced 5 Both countries are subject to similar prevalence of HIV/AIDS, lifestyles (including diet), security challenges, etc. 6 South Africa has a much higher life insurance penetration rate = a larger sample size. Njenga (Strathmore University) Kenyan Mortality October / 34

17 Comparison of Male and Female Assured Lives Ordinary Life Assureds The Current KE model: Based on parametric graduation Model proposed in original study was a weighted version of: CS : q x = b 0 + b 1 x + b 2 x 2 + b 3 x 3 with pivot ages at suitable ages (27, 36, 45, 54). Njenga (Strathmore University) Kenyan Mortality October / 34

18 Comparison of Male and Female Assured Lives Ordinary Life Assureds Alternative Model: based on graduation with Reference to a Standard Table Two models were proposed for the graduation and two standard tables (one select, S, and one ultimate, U) were used. N.B. The Kenyan data used does not include information from medical underwriting (the medical test could include an HIV test) and are therefore assumed to be ultimate. The models: and M1 : q x = a + b q s x (1) M2 : q x = (a + b x)q s x (2) Hence we will consider four models M1U, M1S, M2U and M2S. Njenga (Strathmore University) Kenyan Mortality October / 34

19 Comparison of Male and Female Assured Lives Ordinary Life Assureds Estimation of Parameters in equations (1) and (2) i.e. a and b using WLS and MLE Which estimation method is preferable? 1 WLS: 2 MLE: x w x (ˆq x q x ) 2 for w x = E x Deaths = D x Binom(E x, q x ) where q x = d x E x WLS is simpler but MLE can be used to quantify risk via confidence bounds. Njenga (Strathmore University) Kenyan Mortality October / 34

20 Results for the Males The Five models compared with WLS Estimates Njenga (Strathmore University) Kenyan Mortality October / 34

21 Results for the Males Comparing with Maximum Likelihood Estimates Njenga (Strathmore University) Kenyan Mortality October / 34

22 Results for the Males Tests on Male Graduation MODEL: M1S M1U M2S M2U CS BEST WEIGHTED LEAST SQUARES S-coef 8.36E E E E M1U χ ,224 M2S Sign Test MAXIMUM LIKELIHOOD S-coef 1.38E E E E M1U χ ,224 M2S Sign Test Likelihood 11, , , , M1U S-coef - smoothness measured by the sums of the squares of the differences (usually third-order) in graduated values (Bayo & Faber, 1983). Lower is GOOD! χ 2 - Goodness-of-fit. Lower is GOOD! p value - H 0 : median difference is zero I.E. NO BIAS IN GRADUATED VALUES. Use α = ( = there is Bias.) Njenga (Strathmore University) Kenyan Mortality October / 34

23 Results for the Females The Five models compared with WLS Estimates Njenga (Strathmore University) Kenyan Mortality October / 34

24 Results for the Females Comparing with Maximum Likelihood Estimates Njenga (Strathmore University) Kenyan Mortality October / 34

25 Results for the Females Tests on Female Graduation MODEL: M1S M1U M2S M2U WEIGHTED LEAST SQUARES S-coef 1.06E E E E-09 χ Sign Test MAXIMUM LIKELIHOOD S-coef 3.17E E E E-09 χ Sign Test Likelihood 13, , , ,755.6 S-coef - smoothness measured by the sums of the squares of the differences (usually third-order) in graduated values (Bayo & Faber, 1983). Lower is GOOD! χ 2 - Goodness-of-fit. Lower is GOOD! p value - H 0 : median difference is zero I.E. NO BIAS IN GRADUATED VALUES. Use α = ( = there is Bias.) Njenga (Strathmore University) Kenyan Mortality October / 34

26 A hybrid of the two? None of these is perfect. CS overestimates mortality in the tail (older ages) and M1 U does not capture the mortality hump for the younger ages. Solution: Use Graphical Graduation to create a hybrid model of the two. Njenga (Strathmore University) Kenyan Mortality October / 34

27 A hybrid of the two? Njenga (Strathmore University) Kenyan Mortality October / 34

28 A hybrid of the two? Hybrid model for males χ 2 value of and S = 8.942E 09 (better fit but less smooth) Njenga (Strathmore University) Kenyan Mortality October / 34

29 A hybrid of the two? Njenga (Strathmore University) Kenyan Mortality October / 34

30 A hybrid of the two? Njenga (Strathmore University) Kenyan Mortality October / 34

31 Life expectancy Alt Life expectancy CS A hybrid of the two? > 1 implies Expect to collect premiums for longer using Alt. LT (Cheaper Premium) Njenga (Strathmore University) Kenyan Mortality October / 34

32 Conclusions Conclusions and Recommendations M1 U is better than the CS (Cubic Spline) model for the older ages CS is better than M1 U for the younger ages CS values are much higher than the observed values = if the CS values are used to price the cost of life insurance will be inflated A hybrid of the two models gives a better fit Use Kenyan national mortality tables as the reference table then adjust for bias Njenga (Strathmore University) Kenyan Mortality October / 34

33 Conclusions and Recommendations Recommendations Better regulation (including risk-based reserving) and more affordable life insurance can result from being versatile in developing the Kenyan life table. Ideally: Use the CS Model for Reserving (It is conservative) Use the Alternative models for Pricing (perhaps subject to a company s own credibility factor) Njenga (Strathmore University) Kenyan Mortality October / 34

34 Conclusions and Recommendations Questions and Comments. Thank you for your time. Dr. Carolyn Njenga Njenga (Strathmore University) Kenyan Mortality October / 34