Title: Stochastic Portfolio Specific Mortality and the Quantification of Mortality Basis Risk
1Stochastic Portfolio Specific Mortality and the
Quantification of Mortality Basis Risk
- Richard Plat Belfast, 14th August
To appear in Insurance, Mathematics Economics
45 (2009) 123-132 Corresponding working paper
available for download at http//ssrn.com/abstra
ct1277803
2Agenda
- Introduction
- Mortality rates measured in insured amounts
- Basic model
- Fitting the model
- Application to insurance portfolios
- Numerical example 1 Value at Risk
- Numerical example 2 basis risk
- Conclusions
3Introduction (1)
- There is vast literature about stochastic
mortality models, for example - Lee and Carter (1992)
- Renshaw and Haberman (2006)
- Cairns et al (2006, 2007, 2008)
- Currie et al (2004), Currie (2006)
- Plat (2009)
- These models are tested on a long history for
large country populations, such as U.S. or U.K.
4Introduction (2)
- However, these models are generally not directly
applicable for insurance portfolios, because - In practice often not enough insurance portfolio
specific data ( years history and policies) - Insurers more interested in mortality rates
measured in insured amounts instead of measured
in number of people - Practice ? applying a deterministic portfolio
experience factor to stochastic country mortality
rates. - However, it is reasonable to assume that this
factor is also stochastic.
5Introduction (3)
- In this presentation a stochastic model is
suggested for portfolio specific mortality
experience. - This process can be combined with any stochastic
country population mortality process, leading to
stochastic portfolio specific mortality rates. - Applications
- Value at Risk (VaR) / Solvency Capital
Requirement (SCR) for mortality or longevity risk - Quantifying mortality basis risk when longevity
or mortality is hedged
6Mortality rates measured in insured amounts
- Measurement of mortality rates in insured amounts
is already used for a long time, starting with
CMI (1962). - Definition used in this paper for portfolio
(initial) mortality rate, measured in amounts
(age x, year t) - Policyholders with higher insured amounts tend to
have lower mortality rates ?
7Basic model (1)
- Aim is a stochastic model for
- where is the country population
mortality rate - It is desirable that the proposed model
- is as parsimonious as possible, because often
limited data is available - leads to an expectation of Px,t that approaches 1
for the highest ages
8Basic model (2)
- Proposed model
- Or in vector notation
9Basic model (3)
- To ensure Px,t approaches 1, we require
- The structure of X (and the corresponding ?s)
can be set in different ways, for example - Principal components analysis to derive preferred
shape of Xi - Similar structure as Nelson Siegel model for
yield curves - More simple structure, using 1 factor where
vector X is linear in age
10Fitting the model (1)
- Fitting the model requires 3 steps
- 1) Fitting the basic model
- Px,t are based on different exposures to death
and observed deaths ? Heteroskedasticity - Therefore Generalized Least Squares (GLS) is
used, with (for example) square root of
observations in group as weights - This leads to estimates for time t (with weight
matrix Wt) - This leads to a time series of vector
11Fitting the model (2)
- 2) Adding stochastic behavior
- Now a (multivariate) stochastic process can be
fitted to the time series of - Given the often limited historical period
available and requirement of parsimoniousness,
process has to be simple, for example - Correlated AR(1) or ARIMA(0,0,0) processes
- A Vector Autoregressive (1) model VAR(1)
12Fitting the model (3)
- 3) Combine with stochastic country population
model - If the historical data period is equal for the
portfolio and the country population, Seemingly
Unrelated Regression (SUR) can be used. - This will generally not be the case. Alternative
procedure is - Fit equation by equation using Ordinary Least
Squares (OLS) - Use the residuals to estimate the elements of
covariance matrix S
13Application to insurance portfolios (1)
- The model is applied to 2 portfolios
- Large portfolio 100.000 males, aged 65 or older,
collective pension - Medium portfolio 45.000 males, aged 65 or older,
annuity portfolio - For both portfolios, 14 years of historical data
is available - For these portfolios, a simple 1-factor structure
appeared to be most favourable in terms of BIC.
So the model used is a simple linear model, where
the vector X is linear in age
14Application to insurance portfolios (2)
- Example of fit to actual observations
15Application to insurance portfolios (3)
- Resulting time series of
- ARIMA(0,0,0) process led to a more favourable
BICs - Large portfolio
- Medium portfolio
16Application to insurance portfolios (4)
- This leads to the following best estimates and
99,5 / 0,5 percentiles in year 2016 for the
portfolio experience mortality factor Px,t
17Numerical example 1 Value at Risk (1)
- For stochastic county population mortality the
model of Cairns et al (2006) is used - BE and percentiles, with stoch. and deterministic
Px,t
18Numerical example 1 Value at Risk (2)
- Impact is determined for the following
definitions of VaR - 1-yr horizon, 99,5 percentile, including effect
on BE after 1 year - 10-yr horizon, 95 percentile, including effect
on BE after 10 years - Run-off of the liabilities, 90 percentile
- Impact on large portfolio
19Numerical example 1 Value at Risk (3)
- Impact on medium portfolio
- Conclusion the impact of stochastic Px,ts can
be significant, especially for medium or smaller
portfolios.
20Numerical example 2 basis risk (1)
- Interest in hedging mortality or longevity is
increasing. - Hedge derivatives are often based on country
population mortality - More transparent
- Chance of developing a liquid market for
longevity or mortality - Example q-forward (J.P. Morgan)
21Numerical example 2 basis risk (2)
- What remains is the basis risk, the risk arising
from difference in country population mortality
and portfolio mortality. - This can be quantified with the proposed model.
- A minimum variance hedge is set up for the
portfolios and the impact of the hedge on the VaR
is measured.
22Numerical example 2 basis risk (3)
- Large portfolio
- Medium portfolio
- Conclusion hedge effectiveness is less for
medium portfolio ? can be improved by
periodically adjusting.
23Conclusions
- Existing mortality models are generally not
directly applicable for insurance portfolios. - In this presentation a stochastic model is
suggested for portfolio specific mortality
experience. - Combining with a stochastic country population
mortality process, leads to stochastic portfolio
specific mortality rates. - Impact on VaR / SCR and on hedge effectiveness
can be significant.
24Stochastic Portfolio Specific Mortality and the
Quantification of Mortality Basis Risk
- Richard Plat Belfast, 14th August
To appear in Insurance, Mathematics Economics
45 (2009) 123-132 Corresponding working paper
available for download at http//ssrn.com/abstra
ct1277803