new directions in the modelling of longevity risk andrew
TRANSCRIPT
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New Directions
in the Modelling of Longevity Risk
Andrew Cairns
Heriot-Watt University,
and The Maxwell Institute, Edinburgh
Joint work (in progress!) with:
George Mavros, Torsten Kleinow, George Streftaris
Mexico City, 2 October 2012
2
Plan
• Genealogy
• New directions in modelling
• Numerical illustrations
3
Development of New Models
• Many new stochastic mortality models since
Lee-Carter
• Are they fit for purpose?
• Are they robust?
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GENEALOGY: 1st GENERATION MODELS
--Time
Lee-Carter (M1)1992
Eilers/MarxP-splines
��1Currie/Richards (M4)
2-D P-splines2002, ...
CBD-1 (M5)2006
5
Improvements + more complexity
--Time
APC model (M3) - APC model (M3)
Booth et al.
Hyndman et al.
Lee-Carter (M1) -������
����
��3
Currie/Richards (M4)2-D P-splines
Eilers/MarxP-splines
��1-
DDE
��
����
-
CBD-1 (M5) -QQQs
JJJJJ
Renshaw-Haberman (M2)�����
CBD-2 (M6)
CBD-3 (M7)
CBD-4 (M8)
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More improvements + even more complexity
--Time
APC model (M3) - APC model (M3)
Booth et al.
Hyndman et al.
Lee-Carter (M1) -������
����
��3
@@@ -
Currie/Richards (M4)2-D P-splines
Eilers/MarxP-splines
��1-
DDE
��
����
-
CBD-1 (M5) -QQQs
JJJJJ
Renshaw-Haberman (M2)�����
CBD-2 (M6)
CBD-3 (M7)
CBD-4 (M8)
���*
�����
Plat -���@@R
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Multiple population modelling
--Time
APC model (M3) - APC model (M3)
Booth et al.
Hyndman et al.
Lee-Carter (M1) -������
����
��3
@@@ -
Currie/Richards (M4)2-D P-splines
Eilers/MarxP-splines
��1-
DDE
��
����
-
CBD-1 (M5) -QQQs
JJJJJ
Renshaw-Haberman (M2)�����
CBD-2 (M6)
CBD-3 (M7)
CBD-4 (M8)
���*
�����
Plat -���@@R
@@ ����- Multi-
population
- Multi-population
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Why do we need complexity?
1970 1980 1990 2000
6065
7075
8085
90
M1
Lee−Carter Model
1970 1980 1990 2000
6065
7075
8085
90
M7
CBD Model + Cohort Effect
Black ⇒ model over-estimates m(x, t) death rate
Gray ⇒ model under-estimates m(x, t) death rate
LC: non-random clusters + errors are too big
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Issues on complexity
• Lee-Carter, CBD-1: simple and robustBUT underlying assumptions are violated:
A: Deaths, D(x, t) are cond. Poisson(m(x, t)E(x, t)
)B: Death counts in neighbouring (x, t) cells are independent
• More complexity e.g. CBD-1 → CBD-3 → Plat ...
– Underlying assumptions now okay
– But excessive complexity ⇒ less robust forecasts???
• Dowd et al. (2010a,b): out-of-sample backtesting
Models that fit much better in sample
are not obviously better at out-of-sample forecasting
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Issues on complexity
• More complex ⇒ More random processes
• More random processes ⇒MUCH more difficult to model multiple populations
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A Possible Way Forward
Single-population models
• Paradigm shift away from independent Poisson model
• Focus on small number of key drivers
⇒ much easier to extend to multi-populations
• Focus on greater robustness of forecasts
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Case Study: CBD/Plat Revisited
logm(x, t) = β(x) + κ1(t) + κ2(t)(x− x)
1975 1980 1985 1990 1995 2000 2005
5060
7080
90
Year
Age
D(x,t): Actual vs Expected Deaths
Red ⇒ actual deaths > expected deaths
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CBD/Plat Revisited: Key Idea: Possible responses
logm(x, t) = β(x) + κ1(t) + κ2(t)(x− x)
Add:
• Cohort effect, γ(t− x)
• Extra age-period effects
• Do something new ......
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Key Idea: CBD/Plat Revisited
Underlying log m(x, t) =
• β(x) + κ1(t) + κ2(t)(x− x): two key drivers
PLUS
R(x, t) Residuals
• Assume: vector R(t) → R(t + 1) mean reverting
process
⇒ long term risk depends on two key drivers
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Specific Model
logm(x, t) = β(x) + κ1(t) + κ2(t)(x− x) +R(t, x)
•(κ1(t), κ2(t)
): bivariate random walk
• R(t) = (nx × 1 vector) VAR(2), reverting to 0
R(t) = AR(t− 1) +BR(t− 2) + Z(t)
• Z(x, t) i.i.d. ∼ N(0, σ2Z)
• A = A1 + A2 and B = −A2A1
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VAR matrices A1 and A2
Ai =
ai 0 0 · · ·ci di 0 0 · · ·
di/2 ci di/2 0 0 · · ·0 di/2 ci di/2 0 0 · · ·0 0 di/2 ci di/2 0 0 · · ·...
. . . . . . . . . . . . . . .
ai =AR terms for new members;
ci = cohort persistence;
di = diffusion coeff.
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Further details
• Deaths: D(x, t) ∼ Poisson (m(x, t)E(x, t))
• Bayesian approach:
posterior density = likelihood × prior
• Upcoming results: mode of posterior density
• Further work: Bayesian parameter uncertainty
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1980 2000 2020 2040 2060
0.00
20.
005
0.01
00.
020
0.05
00.
100
0.20
0
Age 55
Age 65
Age 75
Age 85
FULL
R(x,t) = 0
Underlying trend, R(x,t)=0
Year
log
Dea
th R
ate
England and Wales, Males 1971−2008
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Cohort-type effects
1980 2000 2020 2040 2060
0.00
20.
005
0.01
00.
020
0.05
00.
100
0.20
0
FULL
R(x,t) = 0
Year
log
Dea
th R
ate
England and Wales, Males 1971−2008
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1975 1980 1985 1990 1995 2000 2005
5060
7080
90
Year
Age
R(x,t): i.i.d. residuals Z(x,t)
Red ⇒ Z(x, t) > 0
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1975 1980 1985 1990 1995 2000 2005
5060
7080
90
Year
Age
D(x,t): Actual vs Expected Deaths
Red ⇒ actual > expected
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Comparison with related models
logm(x, t) = β(x) + κ1(t) + κ2(t)(x− x)
logm(x, t) = β(x) + κ1(t) + κ2(t((x− x) + γ(t− x)
logm(x, t) = β(x) + κ1(t) + κ2(t)(x− x) +R(x, t)
R(t) = AR(t− 1) +BR(t− 2) + Z(t) (A, B as specified earlier)
logm(x, t) = β(x) + κ1(t) + κ2(t((x− x) +R(x, t)
R(t) = AR(t− 1) +BR(t− 2) + Z(t) (simplified A, B)
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1980 2000 2020 2040
0.00
20.
005
0.01
00.
020
Age 60
Year
log
Dea
th R
ate
1980 2000 2020 2040
0.01
0.02
0.05
0.10
Age 75
Year
log
Dea
th R
ate
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Conclusions: Model Comparisons
• Long term underlying trends (κ(t)) are reasonably
consistent
• Model risk more evident in the mean reverting R(x, t)
Further work
• Bayesian parameter uncertainty
• Multiple populations: focus on underlying κ(t)
⇒ less complexity
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Multipopulations
Borrow from multifactor asset models: e.g.
• Asset i return: Ri = αi + βi1F1 + βi2F2 + ϵi
• F1, F2 are systematic risk factors
• ϵi = idiosyncratic risks
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Multipopulations
Mortality – version 1:
• Population, P , specific κ(P )i (t) correlated
• R(P )(x, t): assume independent
Mortality – version 2:
• All populations have the same κi(t)
• R(P )(x, t): assume independent
• Greater role for R(x, t) as country specific effect
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Other models for R(x, t)
1. R(x, t) = ϕR(x− 1, t− 1) + ZR(x, t)
2. R(x, t) = ϕR(x− 1, t− 1) + diffusion + ZR(x, t)
3. Smooth underlying period effects, κ1(t), κ2(t)
plus annual shocks
e.g. R(1), R(2), . . . are i.i.d. vectors, correlated
across ages