Introduction to R
Mortality Overview
- Mortality Models
- Classical
- Stochastic
- Workshop: Stochastic Mortality Models in R
- Models Fitting
- Mortality Projection - Forecasting
- Portfolio Analysis of BEL and SCR
Case Studies
16 October 2015
Agenda
What is R?
R Highlights
- Free, open source, community based statistical software and programming language
- Main usage: Statistical and numerical computing, Data Analysis, Data Visualisation
- Under GNU GPL (General Public License) allowing commercial usage
- Power of R in its packages
- Enterprise version - Revolution Analytics (Microsoft)
“Everything that exists [in R] is an object. Everything that happens [in R] is a function call.” - John Chambers
RStudio
Most popular Environment to run R
http://www.rstudio.com/products/rstudio/
Free & Open-Source Integrated Development Environment (IDE) for R
Features:
- Built-in R console
- R, LaTeX, HTML, C++ syntax highlighting, R code completion
- Easy management of multiple projects
- Integrated R documentation
- Interactive debugger
- Package development tools
Note: R must be installed first!
Playing with R
Type in the interactive console:
3 + 3
## [1] 6
getwd()
## [1] "C:/Users/omierzwa/Documents/GitHub/Insurance"
1:10
## [1] 1 2 3 4 5 6 7 8 9 10
Playing with R
Type in the interactive console:
x <- 1:10 # "name <- value returns the value invisibly"
x
## [1] 1 2 3 4 5 6 7 8 9 10
(x <- 1:10) # creates x and prints it
## [1] 1 2 3 4 5 6 7 8 9 10
Operations in R
x + y addition
x - y substraction
x * y multiplication
x / y division
x ^ y exponention
x %% y devision remainder
x %/% y integer division
numeric vector operator numeric vector –> numeric vector
Vectors
(x <- 11:20) # exemplary vector
## [1] 11 12 13 14 15 16 17 18 19 20
x[1] # first
## [1] 11
x[length(x)] # last element using length function
## [1] 20
x[c(1, length(x), 1)] # first, last and first again
## [1] 11 20 11
Vectors
x[1000] # no such element
## [1] NA
x * 3 # multiplication
## [1] 33 36 39 42 45 48 51 54 57 60
y <- 1:10 x * y
## [1] 11 24 39 56 75 96 119 144 171 200
R in Insurance
- Main focus Non-life
- fit loss distributions and perform credibility analysis - package actuar
- estimate loss reserves - package ChainLadder
- Financial analysis
- packages YieldCurve, termstrc
- Life insurance
- handle demography data - package demography
- demography projections - package LifeTables
- actuarial and financial mathematics - package lifecontingencies
- Life models - packages ilc, LifeMetrics, StMoMo
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Mortality Overview
Basic quantities in the analysis of mortality
- Survival function
\(s_{x} = Pr (X > x)\)
- Probability that (x) will survive for another t
\(p_{xt} = s_{x+t}/s_{x}\)
- Probability that (x) will die within t years
\(q_{xt} = [s_{x} - s_{x+t}]/s_{x}\)
- Mortality intensity (hazard function or force of mortality)
\(\mu_{x} = lim_{h \to 0} 1/h * q_{xh}\)
Probability that (x) will die within h
Mortality Features
Mortality Features
Mortality Features
Mortality Features
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Mortality Models
Classic Models
Classic Models
Some special parametric laws of mortality
- De Moivre
\(\mu_{x} = 1/ (ω – x)\) subject to \(0 \leq x < ω\)
- Gompertz
\(\mu_{x} = Bc^{x}\) subject to \(x \geq 0, B>0, c>1\)
- Makeham
\(\mu_{x} = A + Bc^{x}\) subject to \(x \geq 0, B>0, c>1, A>=-B\)
- Thiele
\(\mu_{x} = B_{1}C_{1}^{-x}+B_{2}C_{2}^{[-1/2(x-k)^2]}+B_{3}C_{3}^{x}\) subject to \(x\geq 0, B_{1}, B_{2}, B_{3}>0, C_{1}, C_{2}, C_{3}>1\)
Classic Models
Advantages:
- Compact, small numbers of parameters
- Highly interpretable
- Good for comparative work
Disadvantages:
- Almost certainly “wrong”
- Too simplistic
- Struggle with a new source of mortality
Stochastic Models
Stochastic Models
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
Predictor
\(\hat{\mu}_{x}(t)\)
Predictor
\(\hat{\mu}_{x}(t) = \alpha_{x}\)
Predictor
\(\hat{\mu}_{x}(t) = \alpha_{x} + \kappa_{t}\)
Predictor
\(\hat{\mu}_{x}(t) = \alpha_{x} + \beta_{x} \kappa_{t}\)
Predictor
\(\hat{\mu}_{x}(t) = \alpha_{x} + \sum^{N}_{i=1} \beta^{i}_{x} \kappa^{i}_{t}\) \(N\) - number of age-period terms
Predictor
\(\hat{\mu}_{x}(t) = \alpha_{x} + \sum^{N}_{i=1} \beta^{i}_{x} \kappa^{i}_{t} + \gamma_{t-x}\) \(N\) - number of age-period terms
Predictor
\(\hat{\mu}_{x}(t) = \alpha_{x} + \sum^{N}_{i=1} \beta^{i}_{x} \kappa^{i}_{t} + \beta^{0}_{x} \gamma_{t-x}\) \(N\) - number of age-period terms
Cohort effect
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Models Predictor
Generalised Age-Period-Cohort Stochastic Mortality Models
- Lee-Carter (LC)
\(\hat{\mu_{xt}} = \alpha_{x} + \beta^{(1)}_{x} \kappa^{(1)}_{t}\) - Age-Period-Cohort (APC)
\(\hat{\mu_{xt}} = \alpha_{x} + \kappa^{(1)}_{t} +\gamma_{t-x}\) - Cairns-Blake Dowd (CBD)
\(\hat{\mu_{xt}} = \kappa^{(1)}_{t} + (x-\bar{x})\kappa^{(2)}_{t}\) - Renshaw and Haberman (RH)
\(\hat{\mu_{xt}} = \alpha_{x} + \beta^{(1)}_{x} \kappa^{(1)}_{t} + \gamma_{t-x}\)
Models Predictor
Generalised Age-Period-Cohort Stochastic Mortality Models
- Quadratic CBD with cohort effects M6
\(\hat{\mu_{xt}} = \kappa^{(1)}_{t} + (x-\bar{x})\kappa^{(2)}_{t} +\gamma_{t-x}\) - Quadratic CBD with cohort effects M7
\(\hat{\mu_{xt}} = \kappa^{(1)}_{t} + (x-\bar{x})\kappa^{(2)}_{t} + ((x-\bar{x})^2-\hat{\sigma^{2}_{x}}) \kappa^{3}_{t}\) - Quadratic CBD with cohort effects M8
\(\hat{\mu_{xt}} = \kappa^{(1)}_{t} + (x-\bar{x})\kappa^{(2)}_{t} + (x_{c}-x)\gamma_{t-x}\) - Plat
\(\hat{\mu_{t}} = \alpha_{x} + \kappa^{(1)}_{t} + (x-\bar{x})\kappa^{(2)}_{t} + \gamma_{t-x}\)
Model Comparison
Random Component
The numbers of deaths \(D_{xt}\) are independent
\(D_{xt}\) follows a Poisson or a Binomial distribution
Link Function
- Number of link functions possible
- Convenient to use canonical link
- Log link for Poisson \(\hat{\mu}_{xt} = log\mu_{xt}\)
- Logit link for Binomial \(\hat{\mu}_{xt} = ln[\mu_{xt}/(1-\mu_{xt})]\)
Parameter Constraints
Most stochastic mortality models not identifiable up to a transformation
Therefore parameter constraints required to ensure unique parameter estimates
Parameter constraints applied through constraint function
Stochastic Models
Models sensitive to:
Historical data
Assumptions made
Good practice
Understand and validate data
Understand models and their assumptions
Test several methods
Compare and validate results
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Case Study
Dutch mortality data from Human Mortality Database
- Look at mortality models: LC, APC, CBD, RH, M6-M8, PLAT
- Fit models
- Analyse model fit
- Project mortalities
Benchmark against AG table
BEL and SCR Projection for a portfolio
Writting own packages in R
R getting started
- Set working directory Session > Set Working Directory > Choose Directory… or press Ctrl+Shift+H and select the folder or use the console
setwd('<directory name>') # wrapped in ''
## for Windows the path uses / instead \
Exercises
- Shock in the portfolio: everybody 5 years younger
- Mortality projection based on shorter historical data: 1989 instead 1950
- Exclude RH model (problems with convergence)
References
"StMoMO: An R Package for Stochastic Mortality Modelling" A.M. Villegas et al.
"Stochastic Modelling of Mortality Risks" Frankie Gregorkiewicz
"Solvency II Glossary" CEA and the Groupe Consultatif
"R: The most powerful and most widely used statistical software" Revolution Analytics