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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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 Berlinb Collaborative Research Center 649: Economic Riskc CASE - Center for Applied Statistics and Economics

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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

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Contents

Literature Review

The Simulation Framework

Simulation Results

Conclusion

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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

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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)

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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

kt significantly correlated with macroeconomic changes in

Australia, Canada, Japan, and the United States

- Structural change in that relationship at the beginning of the 1990s

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The Simulation Framework

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

Correlations between kt andreal GDP growth, United States

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

Note: * P < 0.05, + P < 0.1

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Contents

Literature Review

The Simulation Framework

Simulation Results

Conclusion

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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

Full correlation structure

Model misspecification risk

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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 of first year’s premium income raised as equity capital E0

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The Simulation Framework

Two lognormally-distributed investment opportunities

Stocks and bonds

Annually rebalanced asset portfolio

  [0, 1] constant fraction of assets invested in stocks

Fixed dividend ratio d

Claims and reserves calculated based on the realized mortality index

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The Simulation Framework

Mortality rates

Lee and Carter (1992): mx, t = exp(ax + bx ∙ kt)

Stochastic drivers of the model

Real GDPln(real GDPt) = GDP + GDP ∙ GDP, t

Stock returns rs, t = s + s ∙ s, t

Bond returns rb, t = b + b ∙ b, t

Mortality index kt = + k ∙ k, t

Account for correlation structure between GDP, t, s, t, b, t, and k, t

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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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

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The Simulation Framework

Real GDP growth

Stock Returns

Bond Returns

Changes in the 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 1.000 0.282 0.050 -0.395

Stock Returns 1.000 0.266 -0.286

Bond Returns 1.000 -0.195

Mortality index 1.000

Estimated parameters of stochastic processes

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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Contents

Literature Review

The Simulation Framework

Simulation Results

Conclusion

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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Simulation Results

Base scenario: term-life insurance, T = 10 years, B = $100,000, I0 = 10,000, males, age = 40 in t = 0

0

0.01

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0.03

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0.05

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0.07

0.08

0.09

1 2 3 4 5 6 7 8 9 10

Time t

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reducedfull

Ignoring correlations between kt

and economic variables underestimation of insolvency probabilities

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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Simulation Results

Vary initial age x

0

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0.1

0.12

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Time t

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x = 30

x = 50

Increase in insolvency probabilities from switching to the full correlation scenario depends on bx

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Vary size I0 of the insurance portfolio

Simulation Results

= + 0.015= + 10.5%

= + 0.016= + 53.1%

0

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I 0 = 5,000

I 0 = 20,000

Underestimation risk more severe for larger portfolios

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Simulation Results

Vary initial amount of equity E0

The relative increase in risk is larger for higher initial amounts of equity capital.

0

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= 0

= 0.2

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

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Simulation Results

Vary stock proportion

0

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no stocks

50% stocks

Larger fraction of stocks induces higher exposure to unfavorable dependency between assets and liabilities

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H U M B O L D T – U N I V E R S I T Ä T Z U B E R L I N

- 22 -

Contents

Literature Review

The Simulation Framework

Simulation Results

Conclusion

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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

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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