Hedging Longevity Risk in Life Insurance Companies

(1)

1

The Optimal Product Mix for Hedging

Longevity Risk in Life Insurance Companies

H.C. Huang, Sharon S. Yang, Jennifer L. Wang, Jeff T. Tsai

National Cheng-chi U. /Soochow U./ National Cheng-chi U./National Taiwan U.

(2)

2

Outline

_Introduction

_{Literature review}

_{Immunization Strategy}

_{Modeling Longevity Risk}

_{Numerical Illustration}

(3)

3

Introduction

Hedging longevity risks has taken on an increasingly important role for life insurance companies.

_{According to the concept of natural hedging, life}

insurance can serve as a dynamic hedge vehicle against unexpected mortality risk.

_{To help life insurers achieve a better natural hedging}

effect, we propose an immunization model that incorporates a stochastic mortality to calculate the optimal level of a product mix to effectively reduces longevity risks for insurance companies.

(4)

4

Introduction-Con’t

Research Purpose

_{Using US mortality experience, we}

demonstrates that our proposed model can

lead to calculate the optimal product mix

and thus effectively reduce longevity risks

for insurance companies.

(5)

5

Literature Review

Mortality risk and pricing issues for annuity

products

Friedman and Warshawsky (1990)

_{Frees, Carriere, and Valdez (1996)}

Mortality derivatives and survival bonds

Blake and Burrows (2001)

_{Charupat and Milevsky (2001)} Lin and Cox (2005 )

_{Dowd, Blake, Cairns, Dawson (2006)} Denuit, Devolder, Godernaiaux (2007)

(6)

6

Literature Review-Con’t

Stochastic mortality

Lee and carter (1992)

Marceau and Gaillardetz (1999)

Lee (2000) and Yang (2000)

Milevsky and Promislow (2001, 2002)

Renshaw and Haberman (2003)

Pitacco (2004)

Cairns et. al. (2006)

Schrager (2006)

Natural hedging

(7)

7

No discussion in the insurance literature so

far addresses product strategies for natural

hedging.

This paper attempts to fill this gap.

(8)

8

_{The total liability V of the insurer equals}

the sum of the liabilities for different

business

_{To achieve that the effect of changing}

mortality on total liability is immunized.

Immunization Strategy

life annuity

V

=

V

+

V

0 dV dμ =

(9)

9

_{Under the assumption of constant force of mortality,}

we define mortality duration for insurance and annuity as follows

Immunization Strategy-Con’t

1

life life life

dV

D

d

V

μ

= −

_μ

⋅

1

annuity annuity annuity

dV

D

d

V

μ

= −

_μ

⋅

(10)

10

Immunization Strategy-Con’t

_{Therefore, we can achieve by setting}

the mortality duration of total liability equal to 0.

where . 0 dV dμ = life annuity life annuity

D

_μ

=

D

_μ

⋅

ω

+

D

_μ

⋅

ω

0 D

_μ

=

(11)

11

The optimal product mix of liability proportions:

Immunization Strategy-Con’t

.

a n n u ity

life _{a n n u ity} _life

u

D

μ

ω

=

−

(12)

12

Immunization Strategy-Con’t

_{The concept of duration employs assumptions of}

constant cash flows, flat yield curves, and parallel shifts in interest rates. However, these assumptions may not be realistic in practice.

_{Kalotay, Williams, and Fabozzi (1993), David,}

Merrill, and Panning (1997), and Gajek, Ostaszewski, and Zwiesler (2005) propose effective duration as an alternative risk measure, which also applies to

(13)

13

Immunization Strategy-Con’t

Effective Mortality Duration

2

life life life eu _life

V

D

V

μ

−

₋

+

=

×

× Δ

2 annuity annuity annuity eu _annuity V V D V μ − ₋ + = × × Δ

(14)

14

(15)

15

Stochastic Mortality Model

_{Lee Carter Model}

: the central death rate for age x in year t

exp(α_x)：the general shape of the mortality schedule

β_x : rates decline rapidly and which slowly over time in response to change in κ_t. κ_t : is a stochastic process , , s.t 1 0

ln

x t x t x t x x t x t and

m

β κ

α

β κ

ε

= =

=

+

∑

, x t m

(16)

16

Practical Issues

_{Mortality experience for life insurance product}

is different to that for annuity product.

Mortality experience is different to countries.

Model risk and parameter risk are important in

dealing with natural hedging.(Melnikov and

Romaniuk 2006; Koissi, Shapiro and Hognas

2006)

(17)

17

Mortality Investigation

US mortality experience obtained from HMD data base.

Data period: US aged 25–100 from 1959 to 2002

Trend of Probabilities of Death for 10-Year Age Cohorts, (left: male; right: female)

1960 1965 1970 1975 1980 1985 1990 1995 2000 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Calander year D eat h r a te age group 50-59 age group 60-69 age group 70-79 age group 80-90 1960 1965 1970 1975 1980 1985 1990 1995 2000 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Calander year De a th ra te age group 50-59 age group 60-69 age group 70-79 age group 80-90

(18)

18

Model Fitting

_{SVD method (Estimation method has been discussed in} Parameter Estimates of α_x and β_x _{in Lee-Carter model}

Female Male age α_x β_x age αx βx 25 -7.35118 0.00938 25 -6.36892 0.00712 26 -7.31421 0.00910 26 -6.36916 0.00613 27 -7.27519 0.00881 27 -6.35958 0.00576 28 -7.20874 0.00883 28 -6.32864 0.00532 … … … … … … 98 -1.25084 0.00338 98 -1.14248 0.00188 99 -1.18752 0.00306 99 -1.09254 0.00163 100 -1.12742 0.00276 100 -1.04501 0.00141 101 -1.07054 0.00247 101 -0.99989 0.00120

(19)

19

Model Fitting-Con’t

Parameter Estimates of κ_t in Lee-Carter model

Female Male Year t κ_t κ_t 1959 26.64288 19.19884 1960 27.19024 19.59866 1961 23.80692 16.70521 … … … 1999 -23.11967 -29.11703 2000 -24.34012 -30.69297 2001 -24.70004 -32.10778 2002 -25.53304 -32.83798

(20)

20

Model Fitting-Con’t

Female Male MAPE 0.0682 0.0893 t t 1 ˆ X : forecasted value X : observed value ˆ 1 n t t t _t where X X MAPE n = X − =

∑

(21)

21

Forecasting Survival Probability

Estimated Confidence Interval of Simulated Survival Rate (left: male, right: female)

30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ages (x -25)P 2 5

HMD historical survival rate Stochastic mortality model 95% confidence interval 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Ages (x -25)P 2 5

HMD historical survival rate Stochastic mortality model 95% confidence interval

(22)

22

(23)

23

Assumptions of Numerical Settings

Product Deferred Annuity Life insurance

Age of insured 25 25

Gender Female Female

Coverage/payout benefit US$10,000 (per year) US$1,000,000

Coverage /payout benefit

period (years) Whole life Whole life Method of paying premium Single Single

Interest rate 4% 4%

Deferred period 30 None

Pricing mortality basis HMD, 2002 HMD, 2002

Forecasted mortality basis Stochastic mortality

model

Stochastic mortality model

(24)

24

Insurance Premiums of illustrated insurance

products with different mortality estimate

Pricing Mortality Basis Forecasted Mortality Basis by Lee-Carter (expected) Forecasted Mortality with 10% shock (unexpected) Coverage /payout benefit period 30-year Deferred Annuity Life 30-year Deferred Annuity Life 30-year Deferred Annuity Life 20-year term 37,026 12,836 37,043 11,959 37,495 10,772 30-year term 44,149 24,027 44,212 22,848 45,001 20,606 Whole life 46,749 129,328 46936 128,121 48,105 122,667

(25)

25

Optimal Product Mix Ratio

The K-ratio implies that if an insurance company sells one unit of an annuity policy, it should sell K units of life insurance policy to achieve the hedging effect and immunize itself against longevity risk. annuity life annuity e annuity annuity life life e life P D P w K w P D P μ μ = ⋅ = − ⋅

(26)

26

Product Mix Proportion and K-Ratio:

(

Women, single premium

)

life ω Coverage /payout benefit period 20-Year Term Life Whole Life 11.6% 22.9% (0.380) (0.085) 16.0% 30.1% (0.660) (0.150) 20.7% 37.1% (0.950) (0.210) Whole life annuity

30-year term annuity 20-year term annuity

(27)

27

Product Mix Proportion and K-Ratio:

(

Men, single premium

)

(

* In parentheses represent the K-ratios.

Coverage /payout benefit period 20-Year Term Life Whole Life 16.0% 31.0% (0.260) (0.096) 20.7% 38.1% (0.420) (0.150) 24.1% 42.8% (0.530) (0.190) Whole life annuity

30-year term annuity 20-year term annuity

(28)

28

Product Mix Proportion and K-Ratio:

(

Deferred Period)

Product Mix Males Females

31.8% 26.8% 0.220 0.230 42.8% 37.1% 0.190 0.210 55.5% 49.5% 0.150 0.180 Whole life Whole life annuity

(deferred 40 years) Whole life Whole life annuity

(deferred 30 years) Whole life Whole life annuity

(deferred 20 years)

(29)

29

Product Mix Proportion and K-Ratio:

(

Different issued Age)

* In parentheses represent the K-ratios.

42.8% 37.1% (0.190) (0.210) 56.8% 50.2% (0.170) (0.190) 71.3% 64.7% (0.130) (0.150) Whole life Whole life annuity

(issued at age of 45) Whole life Whole life annuity

(issued at age of 35) Whole life Whole life annuity

(30)

30

Product Mix Proportion and K-Ratio:

(

Between Gender and Age)

56.8% 49.8% (0.170) (0.160) 55.6% 48.9% (0.250) (0.230) 44.5% 34.9% (0.528) (0.467) 57.3% 50.2% (0.210) (0.190) 56.0% 49.3% (0.300) (0.280) 44.5% 34.9% (0.532) (0.472) Whole life (Female, 35)

Whole life annuity

(issued at age of 55, Attend age 65) Whole life

(Female 35)

Deferred life annuity

(Female, 35)

(Male, 35)

Whole life annuity

(Male, 35)

(31)

31

Product Mix Proportion and K-Ratio:

(with mortality shift 25%)

* The proportions have little difference to 10%-shift, because of the convexity to the productions are not significant. We have a good approximation in linear hedging.

20-Year Term Life Difference in Mortality Curve Shift Whole Life Difference in Mortality Curve Shift 11.6% 0% 22.7% 0% (0.380) 0.076 (0.084) -0.018 16.0% 0% 29.9% 0% (0.660) 0.185 (0.150) -0.014 20.9% 0% 37.1% 0% (0.960) 0.395 (0.210) 0.018 Whole life annuity 30-year term annuity 20-year term annuity