partial splitting of longevity and financial risks: the ...bensusan/bensusan-axa-fev2011-final.pdfh....

Interest rate risk transferMortality and longevity modeling

Quantative analysis

PARTIAL SPLITTING OF LONGEVITY AND FINANCIAL RISKS:THE LIFE NOMINAL CHOOSING SWAPTIONS

H. Bensusan (Société Générale)

joint work with N. El Karoui, S. Loisel and Y. Salhi

Longevity and Pension Funds

February 4th, 2011

Thanks to C. Michel and all the R & D team of CACIB

H. Bensusan (Société Générale) The Life Nominal Choosing Swaptions


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OUTLINE OF THE TALK

1 INTEREST RATE RISK TRANSFER

2 MORTALITY AND LONGEVITY MODELING

3 QUANTATIVE ANALYSIS



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INTRODUCTION

I Improvements in longevity are bringing new issues and challenges at variouslevels: social, political, economic and regulatory.

I Need of MORE AND MORE CAPITAL to face this long-term risk

I Hedging longevity risk is now an important element of risk management formany organisations



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LONGEVITY RISK: RISK MANAGEMENT SUBJECT FOR INSURANCECOMPAGNIES

The insurance industry is also facing some specific challenges related tolongevity risk.

I Accurate longevity projections are delicate (prospective life tables)

I Modeling the embedded risk (such long term interest rate risk) remainschallenging

I Important to find a suitable and efficient way to cross-hedge or totransfer part of the longevity risk to reinsurers or to financial markets



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HETEROGENEITY

I Longevity patterns and longevity improvements are very different fordifferent countries, and different geographic area.

I Factors affecting the mortality

socio-economic level (occupation, income, education...)

gender

marital status

living environment (pollution, nutritional standards, hygienic...)

wealth



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BASIS RISK I

Difference between the national mortality data and the one from an insured portfolio.I Insurance companies have much more detailed information

They know the exact ages at death and not only the year of death (time continuousdata)

Cause of death are specified

Characteristics of the policyholders : socio economic level, living conditions ...

I BUTlimited size of their portfolios (in comparison to national populations : 700 000individuals from 19 different insurance companies)

small range of the observation period

Furthermore, insurers are tending to select individuals (given their health andmedical history for example).

This heterogeneity is very important for longevity risk transfer based on NATIONAL

INDICES: for too important basis risk, the hedge would be too imperfect



Quantative analysisHedging of life-insurance products

OUTLINE







FINANCIAL RISK TRANSFER

Insurance companies are exposed to interest rate riskI Annuities at rate k

I Investments in interest rate products (Bonds,...)

Looking for a product that transfers interest rate risk while keeping longevity risk:I Insurance can manage longevity risk

I Banks can manage intereste rate risk

⇒ Product with decorrelation of both risks




YES BUT HOW?

Difficulties for launching a pure longevity product:

I No longevity market (No shared reference)

I Information asymmetry

I Modelling and pricing (evaluation in historical probability)

⇒ Pure interest rate products




REAL PORTFOLIO

FIGURE: Age distribution of policyholders




PORTFOLIO EVOLUTION

FIGURE: Survival extreme scenarii of policyholders

Using our longevity modelH. Bensusan (Société Générale) The Life Nominal Choosing Swaptions



CHALLENGES

Interest rate hedging of life-insurance products :

I Static hedge of interest rate risk⇒ Swaps

I Dynamic hedge of forward risk⇒ Swaptions

I Longevity⇒ Swaptions with variable nominal

I Stochastic evolution of longevity⇒ Choice of the nominal profile




LIFE NOMINAL CHOOSING SWAPTION

I Swaption on a swap with variable nominal Nt with strike k :

Pswaption

δB(0, T)= EQT

[(k − SVT(T0, TN , δ,Nt))

+N∑

i=1

B(T, Ti)NTi

]

I Choice of αT ∈ [0, 1] by the insurer at date T (with available information)

⇒ Hedge on the nominal series NαTt = αT N−t + (1− αT)N+

t

I Forward swap rate with variable nominal SVT(T0, TN , δ, αT ,N−t ,N+t ) for the

series Nt determined by αT

I Evaluation of "Life Nominal Choosing Swaption" (LNCS) at strike k :

PLNCS

δB(0, T)= EQT

max0≤αT≤1

(k − SVT (T0, TN , δ, αT , N−, N+))

+N∑

i=1

B(T, Ti)(αT N−Ti+ (1− αT )N+

Ti)

∼ EQT

max0≤l≤n

(k − SVT (T0, TN , δ,l

n, N−, N+

))+

N∑i=1

B(T, Ti)(l

nN−Ti

+ (1−l

n)N+

Ti)




CHOICE OF αT (EXAMPLE)

FIGURE: Choice of the parameter αT in 2019




CHOICE OF αT (EXAMPLE)

FIGURE: Choice of the parameter αT in 2029



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OUTLINE






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CAIRNS, BLAKE AND DOWD (CBD) MODEL (2006)

The 2-factor CBD model gives the dynamics of the annual mortality rate qt(x) at agex during the year t:

logitqt(x) = A1(t) + xA2(t).

where the function logit is defined by logit(x) = ln(

x1−x

).



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INDIVIDUAL MORTALITY MODEL

AIM: Reduce the basis risk by estimating the deviation of the "individual mortality"from the average mortality (as given by mortality tables).

I Find individual characteristics (such as socio-economic level or income) thatcan explain mortality

I Take them into account in a stochastic mortality model

The mortality model by age and trait (feature) is calibrated on national mortality dataand on specific data (i.e. with information on individual characteristics)



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MARITAL STATUS INFLUENCE AT INITIAL DATE (MALES)

FIGURE: Logit of mortality rate for French males in 2007 with different marital status



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MORTALITY PROJECTIONS IN 2017 (MALES)

FIGURE: Logit of mortality rate for French males in 2017 with different marital status



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MICRO/MACRO MODELLING

Macroscopic models:

I Evolution of population with macroscopic information

I Whole population

I Average evolution

Microscopic models:

I Evolution at the level of the individual using accurate information

I Sample of the population

I Interaction between individuals

I Uncertainty⇒ Scenarii for population evolution



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INDIVIDUAL BASED MODELS IN POPULATION DYNAMICS

Microscopic models in ecology:I Microscopic model for a trait-structured population (N. Fournier et S. Méléard

2004)

I Extension to an age-structured population (C. Tran Viet 2006)

Suggested extensions for a human population:

I Rate of trait evolution e (marriage rate, divorce rate,...)

I Evolution rates depending on time and and beging random : addition ofstochastic exogeneous factors Yb

t , Ybt and Ye

t

I Immigration modelling



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MICROSCOPIC EVOLUTION PROCESS

Occurence of an event at time Tn:1 At time T−n , selection of a person in the population

2 age and traits of individuals allow to calculate probabilities:b(x, a, Yb

Tn) birth intensity

d(x, a, YdTn) death intensity

e(x, a, YbTn) trait evolution intensity

3 A random variable is drawn to select an eventDeath: the person is removed fromš the population

Birth: a person is added in the population with an age a = 0 and a trait x′

drawn in adistribution kb(x, a, x

′)

Trait evolution: the person with trait x is replaced by a person with trait x′

drawn ina distribution ke(x, a, x

′) and with the same age a

Nothing: an auxiliary event occurs



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MICRO/MACRO LINK C. TRAN VIET (2006)

I McKendrick (1926) and VonFoerster (1959) : Equation in demography(∂g∂t

+∂g∂a

)(a, t) = −d(a)g(a, t), g(0, t) =

∫ ∞0

b(a)g(a, t)da

I Approximation (a.s) for large populations:(∂g∂t

+∂g∂a

)(ω, x, a, t) = −

[d(

x, a, Ydt (ω)

)+ e (x, a, Ye

t (ω))]

g(ω, x, a, t)

+

∫χ

e(x′, a, Ye

t )ke(x′, a, x)g(ω, x

′, a, t)P(dx

′)

g(ω, x, 0, t) =∫χ̃

b(

x′, a, Yb

t (ω))

kb(x′, a, x)g(ω, x

′, a, t)P(dx

′)da

g(ω, x, a, 0) = g0(ω, x, a),



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EVOLUTION SCENARII IN 2097

FIGURE: Evolution Scenarii of the size of the population in 2097 with a mortality scenario



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PROJECTION SCENARII

I Double stochasticity (mortality process and evolution process)

⇒ various scenarii

I Approximation to large propulation⇒ relevant behaviour

I Average scenario is realistic and close to INSEE projections

⇒ Identification of the scenarii panel (extreme or not) generated by the model



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Basis riskPricing of LNCS

OUTLINE






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USE OF THE LONGEVITY MODEL

Survival scenarii of a specific group (N persons):

Standard method (no modelling of population dynamics):

I Mortality model⇒ 1 mortality scenario

I N survival scenarii (for each person)

⇒ provides 1 evolution scenario of a group (N persons)

Suggested modelling :

I Individual mortality model⇒ 1 mortality scenario

I Microscopic model for population dynamics

⇒ provides 1 evolution scenario of a group (N persons)

⇒ Efficient modelling numerically



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BASIS RISK: IMPACT OF MARITAL STATUS (MALES)

FIGURE: Central scenario of the annuities for a portfolio with 60-years old French men withdifferent marital status



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LIFE NOMINAL CHOOSING SWAPTIONS

Product on maximum: impact of rates correlation

I 2-factor HJM calibrated on European rate curve and on specific swaptions(maturity and tenor adapted)

I Assume constant correlation between successive forward rates:

ρ = correl(f (., T), f (., T + dt))

I Focus on price impact of k and ρ



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INITIAL RATE CURVE

FIGURE: European rate curve as of 8 April 2009



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NOTION OF COST ON ANNUITIES

I Annuities:Pannuities = k × LVL(α),

where LVL(α) =∑N

i=1(αN−Ti+ (1− α)N+

Ti)B(T, Ti)

I Life Nominal Choosing Swaption:

PLNCS = c× LVL(α)

⇒ c = PLNCSLVL(α) called cost on annuities

I Interpretation :Without product : annuities at rate k without hedging interest rate risk

With product : annuities at rate k + c with hedging interest rate risk



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CORRELATION IMPACT

Successive rate Swap correlation Price Price Price Price Cost oncorrelation 10Y/12Y α = 0 α = 0.5 α = 1 LNCS Annuitiesρ = 0.998 99.6% 3015 bp 2656 bp 2244 bp 3020 bp 0.87%ρ = 0.997 99.3% 2954 bp 2580 bp 2215 bp 2960 bp 0.853%ρ = 0.996 99.1% 2886 bp 2529 bp 2183 bp 2897 bp 0.835%ρ = 0.995 98.8% 2813 bp 2474 bp 2147 bp 2828 bp 0.815%ρ = 0.994 98.6% 2732 bp 2412 bp 2107 bp 2751 bp 0.793%ρ = 0.993 98.3% 2641 bp 2342 bp 2061 bp 2667 bp 0.769%ρ = 0.992 98% 2540 bp 2264 bp 2009 bp 2574 bp 0.742%

TABLE: Evolution of the price at k = 4.3% depending on successive rate correlation

When ρ decreases

I Price of swaptions with variable nominal decreases

I Price of the switch option increases from 3020− 3015 = 5bp to2574− 2540 = 34bp

⇒ exotic nature increases but remains low



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CONCLUSIONS

I Estimation of basis risk: portfolio heterogeneity

I Pure interest rate product suited to expectations of banks/insurance companies

I Could use more sophisticated model for interest rate



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Thank you for your attention



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ARTICLES/PROJECTS

I Understanding, Modelling and Managing Longevity Risk: Key issues and MainChallenge (Longevity group)

I Risques de taux et de longévité : modélisation dynamique et applications auxproduits dérivés d’assurance-vie (PhD thesis, H. Bensusan)

I Detection of changes in longevity trends (N. El Karoui, S. Loisel, C. Mazza)

I Microscopic modelling of population dynamic: an analysis of longevity risk (H.Bensusan, N. El Karoui)

I On inter-age correlations in stochastic mortality models (S. Loisel and Y. Salhi)

I Pricing longevity securities ( N. El Karoui, S. Loisel, C. Hillairet, Y. Salhi)

I Growth optimal portfolio and Long term interest rate (I. Camilier, N. El Karoui,C. Hillairet)

I Variables annuities (S. Loisel, T.L. Nguyen)


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