Download Normative Modelling of Retirement Decisions: An

HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Katja Hanewalda,b,c, Thomas Posta,b,c, and Helmut Gründla,b,c
a
Humboldt-Universität zu Berlin
b
Collaborative Research Center 649: Economic Risk
c
CASE - Center for Applied Statistics and Economics
-1-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Motivation
 Systematic deviations of actual mortality rates from assumed
ones: threat to the financial stability of life insurers
 Recent demographic study (Hanewald, 2009): Lee-Carter
mortality index is significantly correlated with macroeconomic
changes
 Idea: Assess the overall impact of macroeconomic fluctuations
on the financial stability of a life insurance company
-2-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Preview of Results
 Insolvency probabilities are considerably higher when
dependencies between the mortality index kt and economic
variables are taken into account
 This result is robust to variations in:
 the age of the insureds
 the insurance portfolio size
 the amount of equity capital
 the asset allocation
-3-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Contents
 Literature Review
 The Simulation Framework
 Simulation Results
 Conclusion
-4-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Literature Review
 Stochastic mortality modeling
 Status quo summarized in Cairns, Blake, and Dowd (2008)
 Lee-Carter (1992) model:
“The earliest model and still the most popular”
 Stochastic mortality in life-insurance portfolios
 Dowd, Cairns, and Blake (2006), Hári et al. (2008), and Bauer and
Weber (2008): impact of stochastic mortality on an insurer’s risk
exposure
 Gründl, Post, and Schulze (2006), Cox and Lin (2007), and Wang
et al. (2008): natural hedging opportunities
-5-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Literature Review
 The impact of macroeconomic changes on mortality
 Ruhm (2000): mortality rates in the U.S. fluctuate procyclically over
the period 1972–1991
 Similar patterns observed for:
- U.S., Spain, and Japan (Tapia Granados, 2005a, 2005b, 2008)
- Germany (Neumayer, 2004, and Hanewald, 2008)
- Sweden (Tapia Granados and Ionides, 2008)
- 23 OECD countries, 1960–1997 (Gerdtham and Ruhm, 2006)
 Especially: cardiovascular fatalities, influenza/pneunomia deaths
(Ruhm, 2004, Tapia Granados, 2008)
-6-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Literature Review
 Hanewald (2009): “Mortality modeling: Lee-Carter and the
macroeconomy”
 Relationship between the Lee-Carter mortality index kt and
changes in real GDP or unemployment rates
 Six OECD countries, 1950–2005
 Results
 Dkt significantly correlated with macroeconomic changes in
Australia, Canada, Japan, and the United States
- Structural change in that relationship at the beginning of the
1990s
-7-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Correlations between Dkt and real GDP growth, United States
Sample Period
Males
Females
1951-2005
0.285*
0.286*
1951-1970
0.400+
0.406+
1971-1990
0.367
0.321
1991-2005
-0.400
-0.113
Note: * P < 0.05, + P < 0.1
Early 1970s: Dramatic
decline in CVD mortality
1990s: Reduced mortality
from tobacco and alcohol
consumption, motor vehicle
crashes, influenza and
pneumonia
Ongoing: Substantial increase in
deaths attributable to poor diet
and lack of physical activity
-8-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Contents
 Literature Review
 The Simulation Framework
 Simulation Results
 Conclusion
-9-
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Goal: Assess the overall impact of macroeconomic fluctuations
on a life insurer’s solvency situation
 Stochastic dynamic asset-liability model
 Both sides of the balance sheet react to macroeconomic changes
 Target variable: Multi-period insolvency probability
 Compare two versions of the model
 Reduced correlation structure
Model misspecification risk
 Full correlation structure
- 10 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Newly founded life insurance company
 Writes I0 term-life contracts in t = 0
 Annual premium P
 Death benefit B
 Contract duration T
 All insureds are of age x
 Fixed proportion g of first year’s premium income raised as
equity capital E0
- 11 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Two lognormally-distributed investment opportunities
 Stocks and bonds
 Annually rebalanced asset portfolio

a  [0, 1] constant fraction of assets invested in stocks
 Fixed dividend ratio d
 Claims and reserves calculated based on the realized mortality
index
- 12 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Mortality rates
 Lee and Carter (1992): mx, t = exp(ax + bx ∙ kt)
 Stochastic drivers of the model
 Real GDP
Dln(real GDPt) = mGDP + sGDP ∙ eGDP, t
 Stock returns
rs, t = ms + ss ∙ es, t
 Bond returns
rb, t = mb + sb ∙ eb, t
 Mortality index
Dkt =  + sk ∙ ek, t
 Account for correlation structure between eGDP, t, es, t, eb, t, and ek, t
- 13 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Calibration to empirical data
 United States
 1989-2005 (Hanewald, 2009)
 Data sources
 Real GDP: U.S. Bureau of Economic Analysis
 Stock/bond returns: Morningstar (2008)
 Mortality rates: Human Mortality Database
- 14 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
The Simulation Framework
 Estimated parameters of stochastic processes
Real GDP Stock
Bond
Changes in the
growth Returns Returns mortality index kt
Mean
0.029
0.110
0.043
-0.955
Std. Deviation
0.013
0.167
0.020
0.828
Correlation Matrix
Real GDP
Stock Returns
Bond Returns
Mortality index
1.000
0.282
0.050
-0.395
1.000
0.266
-0.286
1.000
-0.195
1.000
- 15 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Contents
 Literature Review
 The Simulation Framework
 Simulation Results
 Conclusion
- 16 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
 Base scenario: term-life insurance, T = 10 years, B = $100,000,
I0 = 10,000, males, age = 40 in t = 0
0.09
reduced
full
0.08
Insolvency Prob.
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Simulation Results
0.07
0.06
0.05
Ignoring correlations between kt
and economic variables
 underestimation of insolvency
probabilities
0.04
0.03
0.02
0.01
0
1
2
3
4
5
6
7
8
9
10
Time t
- 17 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
 Vary initial age x
0.12
Increase in insolvency probabilities
from switching to the full correlation
scenario depends on bx
reduced
reduced
full
full
0.1
Insolvency Prob.
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Simulation Results
x = 30
0.08
0.06
0.04
x = 50
0.02
0
1
2
3
4
5
6
7
8
9
10
Time t
- 18 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
 Vary size I0 of the insurance portfolio
0.18
reduced
reduced
full
full
0.16
Insolvency Prob.
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Simulation Results
0.14
Underestimation risk
more severe for
larger portfolios
= + 0.015
= + 10.5%
I 0 = 5,000
0.12
0.1
0.08
I 0 = 20,000
0.06
= + 0.016
= + 53.1%
0.04
0.02
0
1
2
3
4
5
6
7
8
9
10
Time t
- 19 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
The relative increase in risk
is larger for higher initial
amounts of equity capital.
 Vary initial amount of equity E0
0.14
reduced
reduced
full
full
0.12
Insolvency Prob.
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Simulation Results
0.1
g =0
0.08
g = 0.2
0.06
0.04
0.02
0
1
2
3
4
5
6
7
8
9
10
Time t
- 20 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Larger fraction of stocks
induces higher exposure to
unfavorable dependency
between assets and liabilities
 Vary stock proportion a
0.09
reduced
reduced
full
full
0.08
Insolvency Prob.
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Simulation Results
0.07
no stocks
0.06
0.05
0.04
0.03
50% stocks
0.02
0.01
0
1
2
3
4
5
6
7
8
9
10
Time t
- 21 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Contents
 Literature Review
 The Simulation Framework
 Simulation Results
 Conclusion
- 22 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Conclusion
 Ignoring the existing dependency structure between mortality
rates and macroeconomic changes leads the insurer to
systematically underestimate true insolvency probabilities
 The relative increase in insolvency probability is higher for
insurers with:
 relatively mature insureds
 large portfolios
 a high stock exposure
 a high amount of equity capital
- 23 -
HUMBOLDT–UNIVERSITÄT ZU BERLIN
Stochastic Mortality, Macroeconomic Risks,
and Life Insurer Solvency
Conclusion
 The interaction between mortality and macroeconomic
conditions needs to be an integral part of
 life insurers’ internal risk models
 capital allocation decision making
 of solvency assessment by rating agencies and regulatory
authorities
 This will lead to
 more accurate assessments of an insurer’s risk situation
 more effective protection of policyholders’ interests
- 24 -