watsonwyatt.com Actuaries Club of the Southwest Generalized Linear Modeling for Life Insurers Jean-Felix Huet, FSA November 2, 29 Agenda Current method disadvantages GLM background and advantages Study case analysis Applications 1
Current Approach of Mortality Analysis (One-Way Analysis) Focus on limited risk factors that impact mortality Age, sex, smoker status, may extend to other factors Company experience is sub-divided into categories to examine the relationship of actual to expected mortality experience (A/E ratio) Limitations Mortality is simultaneously impacted by all risk factors The subdivision process is limited by the credibility of the experience developed for each sub category. Does not quantify the impact of each risk factor on the mortality result 2 Application of Predictive Modeling In Life Insurance Offer an alternative to analyze mortality experience compared dto Traditional l One-Way analysis One way analysis looks at a single risk factor at a time Predictive Modeling Approach will allow for interactions between all risk factors when analyzing the true impact of the factor under investigation 3
Benefits of Predictive Modeling In Life Insurance Better understanding of the factors influencing the mortality rates, lapse rates or option selection Better understanding of the interaction between factors Better understanding of the profitability of different lines of business Overall competitive advantages 4 Predictive Modeling Statistical model that relates an event (death) with a number of risk factors (age, sex, YOB, amount, marital status, etc.) Age Sex Y.o.B. Married Amount etc. Model Expected mortality 5
Generalized Linear Models (GLMs) Special type of predictive modelling A method that can model a number as a function of some factors For instance, a GLM can model Motor claim amounts as a function of driver age, car type, no claims discount, etc Motor claim frequency (as a function of similar factors) Historically associated with non-life personal lines pricing (where there was a pressing need for multivariate analysis) 6 Generalized Linear Models -1 E[Y] = µ = g ( X.β + ξ ) Var[Y] = φ.v(µ) / ω 7
Generalized Linear Models -1 E[Y] = µ = g ( X.β ) Observed thing (data) Some function (user defined) Some matrix based on data (user defined) as per linear models Parameters to be estimated (the answer!) 8 Further Theory and Background CAS 24 Discussion Paper Programme On CAS Exam 9 syllabus Copies available at www.watsonwyatt.com/glm 9
Generalized Linear Models What the mathematics means in practice: Probability of death in year = Base level for observed population Factor 1 (based on age/sex) Factor 2 (based on duration) Factor 3 (based on amount) Each factor is a series of multiplicative coefficients All factors considered simultaneously, allowing for correlations in the data automatically Allow for nature of the random process involved Provide information about certainty of result Robust and transparent Factor level Multiplier F.87 M 1. 1 Factors Often Found to Influence Mortality Age, sex, duration Amount of benefit Geographical Lifestyle cluster Product type (eg flat or escalating annuity) Medical information Calendar year of exposure Calendar year of birth 11
Data Considerations Bad news: we need around 1,+ events for a GLM analysis to work Good news: because we can have calendar year as a factor in the analysis to pick up trends, we can use more years than normally considered safe So GLMs viable with eg 2, annuities 5 years observation Can obtain more value from our data 12 GLM Steps Pre-modeling analysis Model iteration Model refinement Interpretation of the results 13
How to Read the Graphs All graphs show relative Qx of different categories of one factor against a base level identified by % label. Qx for other levels are x% higher or lower than the base level. Colors -.24 Green: Expected GLM results -.3 Orange: One-way relatives are -.36 the relative death rates for the factor before considering other factors simultaneously. Blue: 95% confidence interval. Tight confidence interval indicates statistical significance. Exposure The amount of exposure for a category is indicated by the bar on the x-axis..6 -.6 -.12 -.18 Annual Income Effect % -6% -14% -17% -29% <= 3K <= 5K <= 75K <= 1K > 1K Income relativities confidence interval estimate Oneway Approx 95% Unsmoothed Smoothed estimate 16 14 12 1 8 6 4 2 14 Example 1: Effect of Annuity Amount Income Effect.6 % 16 -.6-6% 14 12 -.12 -.18-15% -18% 1 8 6 -.24 4 -.3 -.36 36 <= 3K <= 5K <= 75K <= 1K > 1K Income -29% 2 Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate Results show evidence of reduced mortality with increased benefits 15
Example 2: Impact of age/sex Run 1 Model 2 - GLM - Significant 3 1319% 133% 865% 2.4 756% 672% 591% 521% 411% 461% 1.8 332% 369% 294% 255% 218% 186% 1.2 138% 16% 12% 119% 7% 85% 43% 56%.6 2% 31% % 9% -14% -8% -25% -2% -35% -35% -33% -3% -4% -36% -45% -51% -.6-57% -64% -68% 1133% 9 89% 8 593% 444% 495% 398% 7 351% 36% 265% 228% 6 195% 164% 133% 16% 5 69% 85% 44% 56% 3% 4 17% 5% -12% -5% -23% -17% 3-36% -3% -46% -41% 2-54% -5% -6% -57% -57% -58% -6% -6% -57% 1 55 58 61 64 67 7 73 76 79 82 85 88 91 > 96 M 57 6 63 66 69 72 75 78 81 84 87 9 > 93 <= 96-1.2 Twoway mapping of Mage and Msex Oneway relativities Restricted factor A mortality table is fitted using experience data and the variation of mortality by age is fixed in subsequent analysis of other risk factors 16 Example 3: Calendar Year Trend Run 1 Model 2 - GLM - Significant.1 7.8 6.6.4.2 5% 4% 4% 2% 1% % 5 4 3 2 -.2 1 -.4 4 22 23 24 25 26 27 Calendar year Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate Mortality improvements 1% per annum over previous six years 17
Example 4: Effect of Joint Life Status Joint Survivor Status.8 25.6.4 3% 2.2 -.2 % 15 1 -.4-4% 5 -.6 6 -.8 Single Life Joint Life Primary Joint Life Surviving Spouse Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate Evidence of broken heart syndrome which may influence pricing 18 Example 5: The Selection Effect Run 1 Model 2 - GLM - Significant.1 -.1 % 3 25 -.2 -.3 -.4 -.5-3% 2 15 1 -.6 -.7 5 -.8 8 <=5 5+ Duration Approx 95% confidence interval Smoothed estimate Selection effect is not conclusive 19
Example 6: Geographic Region Effect Geographic Region.4.3 18 16.2 17% 16% 15% 14.1 % 7% 7% 7% % % -2% 8% 2% 12 1 8 -.1-14% 6 4 -.2 2 -.3 1 2 3 4 5 6 7 8 9 1 11 12 13 Region Oneway relativities Approx 95% confidence interval Unsmoothed estimate Some regions were found to be statistically significant ( 4, 7, 1 and 13 ). However, we excluded this factor mainly because of the wide confidence interval for the other regions. 2 How to Derive Mortality Assumptions Mortality Table Calendar year Income Joint Status Mortality Assumption @ based on 27 and income < 35K 27 level, income > 1K Married with Joint Life Status Age Female Male Factor level Loading Factor level Loading Factor level Loading Female 55.795.955 22 5.% 35K.% Joint Life Alive -4.% 55.542 56.892.177 23 4.% 5K -6.% Surviving Spouse 3.% 56.68 57.978.121 24 4.% 75K -15.% Single.% 57.667 58.125.137 25 2.% 1K -18.% 58.699 59.13.1373 26 1.% >1K -29.% 59.683 6.913.1387 27.% 6.622 61.836.1394 61.57 62.83.1438 62.565 63.878.1518 63.599 64.956.1617 64.652 65.14.1721 65.79 66.1129.1835 66.769 67.123.197 67.838 68.135.2138 68.92 69.1483.2338 69.111 7.1613.2562 7.199 71.1726.282 71.1177 72.1842.359 72.1255 73.1989.3337 73.1356 74.221221.364364 74.1515 75.2471.3969 75.1684 Mortality Assumption for female, 55, income>1k, Married with joint life @27 level =.795 *(1+%)*(1-29%)*(1-4%) =.542 21
Retention Modeling Using GLMs Some products profitability depends on lapse or conversion Improve pricing Improve profitability forecasts Improve marketing 22 Profitability Analysis Impact on portfolio of moving to theoretically correct relativities Exposure count 8 6 4 2.5 Currently profitable business.6.7.8.9 1 1.1 1.2 1.3 1.4 Currently unprofitable business 1.5 1.6 1.7 Ratio: Risk premium model/ Current rates 1.8 1.9 2 + 23
Summary GLM techniques are widely used in P&C for pricing purposes, but its application in Life Insurance may not be as well established. GLMs are robust, transparent and easy to understand. By using GLM techniques in the analysis of annuitant mortality, we were able to identify the true impact of various risk factors while allowing for the interactions between these factors. The advantage of additional knowledge of product characteristics will allow management to make better underwriting, pricing and marketing decisions to gain business advantage over competitors. 24