Forecasting with Inadequate Data. The Piggyback Model. The Problem. The Solution. Iain Currie Heriot-Watt University. Universidad Carlos III de Madrid

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1 Forecasting with Inadequate Data CMI and company data for ages 75 & 76 The Piggyback Model Iain Currie Heriot-Watt University Universidad Carlos III de Madrid CMI: Company: CMI:75 CMI:76 User: 75 User:76 May, The Problem Forecasting with company data impossible. Forecasting with CMI data possible but biased. The Solution Use company data to adjust CMI forecast Claims : D Exposures : E D,E : CMI data

2 Claims : D Exposures : E D,E : 36 8 Company data Data splitting Social class: two levels Pension size: two levels : high status, large pension : high status, small pension : low status, large pension : low status, small pension CMI & company data CMI forecast to 2048 with 95% CI for age 75

3 CMI forecast to 2048 with 95% CI for age CMI:75 CMI:76 User: 75 User:76 Conclusions Adjust the CMI forecast to correct for the bias. Data analysis might suggest a suitable model. Log(mortality) by age CMI Gaps between CMI and user forecasts are Linear function of age

4 Log(mortality) by year CMI Gaps between CMI and user forecasts are Constant in time The Piggyback Model The Piggyback Assumption Gaps between CMI and user forecasts are Linear function of age Constant in time Very strong. Doing nothing is also an assumption!

5 Plan of Talk Forecast with the CMI data Define the piggyback model Estimate the piggyback parameters, ie, the gap functions Lee-Carter (1992) The Lee-Carter Model logµ ij = α i +β i κ j, i = 1,...,n a, j = 1,...,n y, κj = 0, κ 2 j = 1. D ij P(E ij µ ij ) Estimation: Maximum likelihood Original LC: Alpha Original LC: Beta Estimated kappa with ARIMA forecast & 95% CIs Alpha Kappa Original LC: Kappa Beta Observed LC fit Original LC: 70 Kappa

6 Lee Carter forecast for age = 50 Lee Carter forecast for age = 60 Beta in Lee Carter model Lee Carter forecast for age = Lee Carter forecast for age = 80 Beta Lee Carter model: age 53 and age 54 forecasts The Delwarde-Denuit-Eilers Model Delwarde-Denuit-Eilers (2007) smoothedβ in Lee-Carter model. Currie (2013) smoothedαas well and gave a general theory.

7 Beta in LC and DDE models DDE model: age 53 and age 54 forecasts Beta Original LC DDE DDE forecast sheet Between the Sheets 2 Mean sheet - fitted and forecast surface by age and year 4 6 Standard error sheet - SE surface by age and year

8 0.4 DDE standard error sheet 60 Company data & CMI sheet (trimmed) Company claims : D Company exposures : E CMI sheet : ˆµ ij D,E, ˆµ ij : Piggyback model D ij P(E ij µ ij ) 60 i 89, 2000 j 2007 logµ ij = ˆα i + ˆβ iˆκ j +a 0 +a 1 x i In R, glm(claims X + offset(log(exposure)) + offset(cmi.sheet), family = poisson) Piggyback forecast log ˆµ ij = ˆα i + ˆβ iˆκ j +â 0 +â 1 x i 60 i 89, 2000 j 2048

9 Log(mortality) by age Log(mortality) by age Piggyback Piggyback Gap by age in Gap by age with 95% CI Gap by age with 95% CI Gap Gap Gap Gap by age with 95% CI Gap Gap Gap by age with 95% CI

10 Log(mortality) by year Log(mortality) by year Piggyback Piggyback Gap by year in with 95% CI Forecast: age 65, Forecast: age 65, Gap Forecast: age 65, Forecast: age 65,

11 Forecast: age 75, Forecast: age 75, Forecast: age 75, Forecast: age 75, Conclusions Piggyback models allow the actuary To adjust forecasts based on standard data sets (CMI, UK population, etc) for basis risk. To make forecasts with CIs with very limited data. To classify data into business categories. Input requirements Mean and standard error sheets from CMI, UK, Spanish,... forecast. Company data by age and year split by relevant factors social class, postcode, pension size,... Output format Company mean and standard error sheets split by input factors suitable for actuarial tasks: stress testing, valuation, pricing, reserving, etc.

12 References Lee & Carter (1992) Journal of the American Statistical Association, 87, Delwarde, Denuit & Eilers (2007) Statistical Modelling, 7, Currie (2013) Statistical Modelling, 13, IDC s web page: talks and papers iain/