Lokaali bilineaarinen (lokaali Lee-Carter) kuolevuusmalli

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Aleksi Hukkanen
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1 Lokaali bilineaarinen (lokaali Lee-Carter) kuolevuusmalli Aktuaariyhdistyksen kuolevuusseminaari Vesa Ronkainen Vesa Ronkainen

2 Stochastic modeling of financing longevity risk in pension insurance, E44:2012* Johdantoluku 1 käsittelee tutkimuksen taustaa ja sovelluksia Luvussa 2 kehitetään malli osaketuotoille Luvussa 3 esitetään malli keskipitkien (5 v) obligaatioiden tuotoille Luvussa 4 kehitetään lokaali kuolevuusmalli (kohorttimalli) Luvussa 5 esitetään riippuvuusmalli em. mallien aggregoimiseksi simulointia ja skenaariotyöskentelyä varten Luvussa 6 sovelletaan simulointimallia esimerkkilaskelmiin: Esim. 1 Eläkevakuutus eri kokoisille kohorteille iässä 65 Esim. 2 Eläkevakuutus eri ikäisille suurille kohorteille Esim. 3 Eläkevakuutus asiakkaan näkökulmasta pitkällä aikavälillä Luku 7: Discussion *Kirjan pdf-versio löytyy osoitteesta: Vesa Ronkainen 2

3 Tieteellisen mallinnusprosessin vaiheet (esim. Lee-Carter) SPECIFICATION Problem specification Objectives specification Literature review Market best practice Expert knowledge PRELIMINARIES Earlier experience Background theory Tools Data collection MODEL BUILDING INITIAL DATA ANALYSIS (IDA) (summary statistics, tables, graphs etc.) MODEL FORMULATION MODEL ESTIMATION MODEL VALIDATION (diagnostics / residual analysis, comparison with other models etc.) MODEL CHOICE MODEL USE Vesa Ronkainen 3

4 Riippuvuuksien mallinnus (Luku 5) 1-q q (shokkivuoden tn) Osaketuottojen jakauma: ~N(.) ~Ga(.) (<0) Prosessien innovaatioiden korrelaatiot: rij r'ij i=1: osaketuottoprosessi i=2: bondituotto - - i=3: kuolevuus Vesa Ronkainen 4

5 Bilineaarinen Lee-Carter malli (1992) ln( m ) a b k x, t x x t x, t mx,t on kuolevuus iässä x vuonna t ax kuvaa keskimääräistä kuolevuuden tasoa iässä x kt on kuolevuuden suhteellista tasoa kuvaava indeksi vuonna t bx on painokerroin iässä x on normaalijakautunut virhetermi x,t Ennuste perustuu parametrin kt aikasarjaan k t kt k kt 1 c et, c T 1 ˆ 1 missä c on vakio (drift) ja et normaalijakautunut virhetermi Vesa Ronkainen 5

6 1. Introduction In the literature there have been numerous attempts to improve the model of Lee &Carter (1992) (see e.g. the review in Booth (2008)). The goal of Chapter 4 in Ronkainen (2012) was to generalize the Lee- Carter (L-C) model to gain more flexibility in modeling different ages (cf. the L-C validation figure below) This generalization, a Local Bilinear model (LBL), was developed in steps: 1. L-C model was estimated using the standard SVD method, but it was done locally for each age x. The optimal bandwidth was chosen by cross-validation. 2. Model parameters were estimated and validated. 3. Random walk with drift models were fitted for each age and forecasts of the model were generated and compared to the L-C forecasts. 4. Finally, the preferred model for the simulations was chosen Vesa Ronkainen 6

7 1. Yearly difference of log-mortality for ages 30-49(solid), 50-64(dashed), 65-79(dotted), 80-99(longdashed) in Vesa Ronkainen 7

8 2. Model specification and bandwidth selection The data cover the Finnish population for t=1955,...,2008. We specify the L-C model locally for each age x=10,11,...,90, using different bandwidths W x, h [x - h, x h], h 1,...,10 When y W x, h, the model for the death rate in year t is ln( m ) y, t a y b x, y k x, t x, y, t where the terms have the same interpretation as in L-C model except for the added indeces. When the model is extended to cover the first and last h values of the age range, we use the first or last available value for the middle term Vesa Ronkainen 8

9 2. Model specification and bandwidth selection The optimal bandwidth selection is based on least squares crossvalidation: 1. We estimate the model separately for each age x using the punctured interval in the SVD-estimation. W x,h \{x} 2. We leave out age x in order to use it in the leave-one-out crossvalidation. is estimated locally as b x, y, y 1 x, y 2 ( b x b 1) / 3. We then compare the estimated results to the actual data, and choose the value of h which minimizes the Sum of Squared Errors. We find that the optimal bandwidth is Vesa Ronkainen 9

10 3. Estimation and validation The local bilinear model is estimated as in the previous section, except that now h=5 is fixed and we include the whole interval when running the SVD algorithm for age x. We validate our model by computing the terms b x, ykx, t and b k x t in the local bilinear model and in the Lee-Carter model, respectively, and taking their differences. The resulting surface is shown in the following figure. We observe that the LBL model fits the data better. The differences LBL minus L-C for the most part seem to be rather random. However, a notable exception is the cohort born around year 1940, i.e. during the wartime, for which the LBL model gives lower mortality Vesa Ronkainen 10

11 3. Difference of LBL and Lee-Carter bilinear terms 11

12 4. Deterministic forecasting and model comparison Forecasting is based on the Random Walk with Drift (RWD) model as in the Lee-Carter case, but now we have a different model for each age. Deterministic mortality forecasts are given by the formula m x,0 a x tc x which is a sum of the last data point (year 2008) in the log-mortality data matrix, the subtracted mean value and the forecast term. c x Coefficients in the RWD models for each age are the differences between the last and the first observation divided by the number of data points minus 1. Finally we de-transform the results by the exp function. In the following figure we plot an LBL forecast together with the observed values Vesa Ronkainen 12

13 4. Observed and forecasted mortality for age 30 in LBL Vesa Ronkainen 13

14 4. Deterministic forecasting and model comparison The final step in our deterministic calculation is to form a life table for the LBL and L-C models. Using the actuarial estimator and the familiar life table formulas we get the expected life time of infant in the forecasts. The results for the LBL model and the difference to L-C model are shown in the tables below. We note that for long forecasting horizons the LBL model gives slightly lower mortality forecasts than the L-C model does. The differences though are minor and therefore (by the principle of parsimony) our simulations were based on a Bayesian MCMC simulation algorithm for the Lee-Carter model. Note however that in some applications the cohort effect may be significant Vesa Ronkainen 14

15 4. Expected lifetime of infant in the LBL model Vesa Ronkainen 15

16 4. Difference of expected lifetimes of infant in the LBL and the L-C forecast Vesa Ronkainen 16

17 References Booth, H. & Tickle, L. (2008), Mortality modelling and forecasting: A review of methods, ADSRI Working Paper No.3. Lee, R. & Carter, L. (1992), Modeling and Forecasting U.S. Mortality, Journal of the American Statistical Association, 87, Ronkainen, V. (2012), Stochastic modeling of financing longevity risk in pension insurance, Helsinki: Bank of Finland, E44: Vesa Ronkainen 17

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