Stochastic Models for Mortality Rate on Italian Data · Our modiﬁed Lee-Carter (see Giacometti et...

State of Art and DefinitionsModified Lee-Carter

An econometric approachA Generalization of the Milevesky-Promislow Model

Calibrating affine stochastic mortality models using term assurance premiumsA time-varying closed-form expression for the term structure of mortality rates

Conclusions and References

Stochastic Models for Mortality Rate on Italian Data

Rosella Giacometti, jointly with Marida Bertocchi, Sergio Ortobelliand Vincenzo Russo

Dipartimento di Matematica, Statistica, Informatica e Applicazioni,Universita di Bergamo

Final Workshop PRIN 2007- ” L’impatto dell’invecchiamento dellapopolazione sui mercati finanziari, intermediari e stabilita finanziaria”,

Bergamo, 27 May, 2011

Rosella Giacometti (DMSIA) Stochastic Models for Mortality Rate on Italian Data





Summary

1 State of Art and Definitions

2 Modified Lee-Carter

3 An econometric approach

4 A Generalization of the Milevesky-Promislow Model

5 Calibrating affine stochastic mortality models using term assurance premiums

6 A time-varying closed-form expression for the term structure of mortality rates

7 Conclusions and References






State of Art

Solvency II for pension funds, insurance companies ⇒ SCR ⇒need for models able to forecat the distribution of futuremortality rate which will help with both quantifiyng and pricingmortality risk.

Discrete time models of mortality rate: see Lee and Carter’s(1992) and (2000), Renshaw and Habermann (2006), Cairns(2000), Cairns et al (2006) and (2006b), Currie (2006), Plat(2009), and LifeMetrics (2007) and the references therein.

Continuos time models and Stochastic differential equationapproach: Milevesky-Promislow (2001), Biffis (2005), Ballottaand Habermann (2006), Luciano and Vigna (2009).






Mortality rate and Force of mortality

The crude mortality rate of an individual aged x at time t is

mx(t) =Dx(t)

Ex(t)

and it gives the number of death over the average populationduring calendar year t aged x.

The force of mortality µx(t) describes the instantaneous rateof death at time t of a person of age x, given survival until t.In formula,

µx(t) = lim∆t→0

=Pr(t < T ≤ t+ ∆t

∣∣T > t)

∆t= −

dlnSx(t)

dt. (1)

It is identical in concept to the failure rate or intensity ofdefault.






Mortality rate and Force of mortality

The probability of survival up to time t of an individual aged xis denoted and equal to

Sx(t) = exp(−

∫ t

0µx(u)du)

where µx(u) is the force of mortality.

The crude rate of mortality is quite close to the instantaneousdeath rate (i.e., the force of mortality) in the middle of thatinterval.

mx(t) =

∫ t+1

t

µ(u)du ≈ µx(t+ 0.5)

This approximation is given by Pollard (1973), who gives amore formal derivation.






Exposure to death and deaths

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

9

10x 10

5

0 20 40 60 80 100 1200

2

4

6

8

10

12x 10

4

19401960198020002007

19401960198020002007

Figure: source: The Human Mortality Database, University of California, Berkeley.






Crude Mortality rate

1940 1950 1960 1970 1980 1990 2000 20100

0.005

0.01

0.015

0.02

0.025Mortality rate −age 40, 50, 60

405060







Survival probability

40 50 60 70 80 90 100 110 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1survival probability over time −age 40

19401960198020002007







Requirement of a good stochastic mortality model Cairns, Blake and Dowd(2006)

The model should keep the force of mortality positive

The model should be consistent with historical data and biologicalreasonable. Mortality rate have fallen dramatically w.r.t. time at allages but increasing w.r.t age .

Long-term deviations in mortality improvements from thoseanticipated should not be mean-reverting to a pre-determined targetlevel,...meaning that faster mortality improvements will besignificantly reduced in the future.

In contrast, short-term deviations from the trend due to localenvironmental fluctuations might be mean-reverting due to annualvariations.

Robuststness of parameters, plausibility of forecast, easy toimplement and should respect parsimony.






Survival probability in a stochastic world

The probability of survival of an individual aged x up to timeT conditional on being alive at time t is denoted and equal to

Sx(t, T ) = E

[exp(−

∫ T

t

µx(u)du)|Ft

]

where Ft describes the information till time t, and the quantityµx(t) is the stochastic force of mortality or hazard rate.

In the following when we introduce a

discrete model we will use the notation µx,t

continous time model we will use the notation µx(t) or µt






Our models: an overview

Modified Lee-Carter

An econometric approach

A Generalization of the Milevesky-Promislow Model

Calibrating affine stochastic mortality models using termassurance premiums

A time-varying closed-form expression for the term structure ofmortality rates






Modified Lee-Carter

Modified Lee-Carter (see Giacometti, R., S. Ortobelli, and M.Bertocchi, [2])






Modified Lee-Carter

Lee and Carter (1992)

ln(µx,t) = αx + βxkt + ǫx,t, x = 1, . . . , A, t = 1, . . . , T

kt = kt−1 + θ + ζt

where αx and βx are age specific parameters, kt is atime-varying parameter representing a common factor risk andǫx,t is a zero mean Gaussian error with variance σ2;

Our modified Lee-Carter (see Giacometti et al. 2009) ǫx,t andζt are random errors with different variances extracted from aNormal Inverse Gaussian distribution characterized by skewnessand semi heavy tails.






Modified Lee-Carter

This model can be rewritten as

ln(µx,t) = ln(µx,t−1) + βxθ + (βxζt + ǫx,t − ǫx,t−1), (2)

x = 1, . . . , A, t = 1, . . . , T

The variance-covariance of the errors is of the form

Σ = σ2ζ .

ΨΨ′

|Ψ|2+ 2σ2

ǫ I

where Ψ = θ[β1, β2, . . . , βA]where we assume that ǫx,t and ζt are i.i.d error terms extracted byinfinitely divisible distribution different from the Gaussian one (aNIG distribution).






Modified Lee-Carter:some results

We test our distributional assumption:

H0 -Gaussian distribution- cannot be rejected for ζt.

H0 can be rejected for ǫx,t

We compare ex-post the performance we obtained modeling theLee-carter residuals either with a NIG or with Gaussiandistributions. We used 8 years (1996-2004) of Italian mortalitydata. We obtained that the absolute errors using NIG distributionare stochastically smaller than those we get with Gaussian residualwhen we move across age groups.






An econometric apporach

An econometric approach (Giacometti, R., M. Bertocchi, S.T.Rachev, and F. Fabozzi see [3])







AR(1)-ARCH(1) model (see Giacometti et al. 2010) for aspecific age x (or for a specific year t)

ln(µx,t) = px(t) + α1ln(µx,t−1) + ǫx,t,

σ2x,t = β0 + ǫ2x,t−1,

where ǫx,t is the innovation of the time series process ln(µx)(/ln(mut)), with ǫx,t = ztσx,t and zt is an i.i.d process withzero mean and constant variance and px(t) is a polynomial ofdegree n.







we fit the AR(1)-ARCH(1) model with Gaussian and tinnovation

we forecast the next 3 years using Lee-Carter and the proposedmodel

we propose an ex-post comparison based on the Theil’s Uindex, a measure of the forecast quality.

Theilt =

√∑91x=40(µx,t − µx,t)2∑91t=40(µx,t − ψx,t)2

(3)

where ψx,t is the naive forecast. The index for the Lee-Carterforecast is about 0.6 while for our model is about 0.4.






A Generalization of the Milevesky-Promislow Model

A Generalization of the Milevesky-Promislow Model(Giacometti, R., S. Ortobelli, and M. Bertocchi, see [4])






The Milevesky-Promislow Model

Milevesky-Promislow (2001) propose a process for force ofmortality that grows exponentially, exhibiting mean reversionand with variance proportional to the value of the force ofmortality:

µt = µ0egt+σYt g, σ, h0 > 0, (4)

anddYt = −bYtdt+ dBt Y0 = 0, b ≥ 0, (5)

where µt is the hazard rate for an individual aged x, Bt is aBrownian motion and Yt exhibits stronger mean reversion as bincreases.






Our proposal: a Generalization of the Milevesky-Promislow Model

Since all stochastic mortality rate models have to account longtime phenomena, it is fundamental that the model shows someflexibility along time.

Simple generalization of Milevesky-Promislow model describedby (4) and (5):

dYt = −bYtdt+ f(t)dBt Y0 = 0, b ≥ 0, (6)

where Bt is a Brownian motion and f(t) is a functiondependent on time, f(t) ∈ C

1(R+), f(t) 6= 0 (see Giacomettiet al 2011).






A Generalization of the Milevesky-Promislow Model (Cont.)

We introduce a new process K(t, lnµt) = ln µt

f(t) and applying

Ito’s lemma we obtain the following stochastic differentialequation for K(t, lnµt)

dK(t, ln µt) =

[(g + b ln µ0 + bgt)

f(t)− K(t, ln µt)(

f ′(t)

f(t)+ b)

]dt + σdBt,

(7)

By specifying f(t) we can solve the equation.

We choose f(t) = eat and equation (7) becomes

dK(t, ln µt) = −K(t, ln µt)(a+b)dt+[(g + b ln µ0 + bgt)e−at

]dt+σdBt.

(8)






Parameters’ estimation

Using the discretisation approach of Wymer(1972), it is possible torewrite (8) as

lnµt = (1 − α2) lnµ0 +α1α2

1 − α2+ α1t+ α2 lnµt−1 + ξt, (9)

whereα1 = g(1 − e−b), α2 = e−b, (10)

ξt = eatǫt (11)

V AR(ǫt) =σ2(1 − e−2(a+b))

2(a+ b). (12)






Parameters’ estimation

The estimation of (9) is obtained minimizing the squarederrors ξ2t with respect to α1, α2:

minα1,α2

2004∑

t=1930

(ln(µt)−(1−α2) lnµ0−α1α2

1 − α2−α1t−α2 lnµt−1)

2.

(13)






The Results






Parameters confidence interval






Parameters confidence interval (Cont.)

Figure: Comparison between historical logarithm of mortality rates, Generalised andMilevesky-Promislow model of logarithm of mortality rates.






Observation

This model has been used, together with a multiscale stochasticvolatility model describing the behaviour of a risky asset, to price apure endowment policy, i.e. a guaranteed minimum income benefitcontract. (see Bertocchi M.I., Giacometti R., Recchioni M.C.,Zirilli F.(2010) ”Pricing life insurance contracts as financialoptions: the endowment policy case” submitted to Insurance:Mathematics and Economics.)






Calibrating affine stochastic mortality models using term assurancepremiums

Calibrating affine stochastic mortality models using termassurance premiums (see Russo, Giacometti, R., S. Ortobelli,S. Rachev, F. Fabozzi (2011))







The proposed model involves the following three steps:

recasting the pricing function of Term assurance as CDS;

employing a bootstrapping procedure to construct themortality rates term structure;

calibrating the parameters of stochastic mortality models basedon the bootstrapped term structure.







Term assurance contracts are viewed as a swap. Consequently, thevalue of the premium is

Q(t, Tn, x) =1

∑ni=1 P (t, Ti)Dx(Ti−1, Ti)

1 +∑n−1

i=1 τ(Ti−1, Ti)P (t, Ti)Sx(t, Ti). (14)

whereP (t, Ti) is the price of a risk-free zero-coupon bond in t with maturity [Ti];τ(Ti−1, Ti) is the time between the dates Ti−1 and Ti;Dx(Ti−1, Ti) is the probability at time t that an individual aged x dies within theperiod [Ti−1, Ti] with

Dx(Ti−1, Ti) = Dx(t, Ti) − Dx(t, Ti−1); (15)

Sx(t, Ti) is the survival probability at time t of an individual aged x after time Ti

Sx(t, Ti) = 1 − Dx(t, Ti) = e−µx(t,Ti)τ(t,Ti). (16)

denoting the average mortality rate over [t, T ] by µx(t, Ti)







we consider the following models for the force of mortality

the Vasicek,

dµx(t) = k(θ − µx(t))dt+ σdW (t)

Cox-Ingersoll-Ross

dµx(t) = k(θ − µx(t))dt+ σ√µx(t)dW (t)

jump-extended Vasicek

dµx(t) = k(θ − µx(t))dt+ σdW (t) + JudNu(λu) − JddNd(λd)

where Ju and Jd are exponentially distributed random variableswith parameters ηu and ηd, respectively.







Affine stochastic mortality models imply a closed-form expressionfor the survival probabilities, (see Duffie, D., J. Pan, and K.Singleton (2000).

Sx(t, T ) = G(t, T )e−H(t,T )µx(t) (17)

As an example, in Vasicek, the survival probability can be obtainedby

Gµ(t, T ) = exp

{(θ −

σ2

2k2

)[Hµ(t, T ) − τ(t, T )

]−

σ2

4kHµ(t, T )

2}

Hµ(t, T ) =1

k

[1 − e

−kτ(t,T )]







We calibrate each model’s parameters by minimizing the sum ofsquares relative differences between mortality rates implied in thequotes and theoretical mortality rates implied by each specificaffine models.







We applied bootstrapping and calibration procedures to termassurance pure premiums of three Italian insurance companies inforce during 2010:

1. AXA MPS ASSICURAZIONI VITA S.p.A. - AXA Group

2. CATTOLICA, Societa Cattolica di Assicurazione S.C. -CATTOLICA Group

3. GENERTEL LIFE S.p.A. - GENERALI Group

More specifically, we used premiums with respect to males aged 30,40, and 50. The premiums are denominated in euros and related toan insured amount of euro 1,000. For each age, only contractswith a maturity of 5, 10, 15, 20, and 25 years are available.







b. Age = 40

AXA MPS CATTOLICA GENERTEL LIFE

VAS CIR JVAS VAS CIR JVAS VAS CIR JVAS

µ0 0,0014 0,0014 0,0009 0,0009 0,0009 0,0009 0,0014 0,0014 0,0007k - 0,1269 - 0,1269 - 0,1300 - 0,1160 - 0,1156 - 0,1159 - 0,1299 - 0,1321 - 0,1405θ 0,0005 0,0005 0,0005 0,0000 0,0000 0,0000 0,0005 0,0005 0,0000σ 0,0006 0,0130 0,0006 0,0005 0,0125 0,0005 0,0006 0,0141 0,0006

λu - - 0,0000 - - 0,0000 - - 0,0007ηu - - 0,0001 - - 0,0001 - - 0,0929λd - - 0,0000 - - 0,0000 - - 0,0000ηd - - 0,0002 - - 0,0001 - - 0,0000MSE 0,0006 0,0006 0,0006 0,0009 0,0010 0,0009 0,0005 0,0006 0,0005ECE 0,2010 0,1883 0,2274 0,2098 0,2026 0,2043 0,1792 0,1776 0,2364






A time-varying closed-form expression for the term structure of mortalityrates

A time-varying closed-form expression for the term structure ofmortality rates (see Russo, Giacometti, R., S. Rachev, F.Fabozzi)







Our goal is to introduce a time-varying formula of the termstructure as a function of a two-dimensional VAR(1) process.We provide an estimation procedure that involves the followingthree steps:

. construct the time series of the survival probabilities termstructure starting from life tables.

. fit a closed-formula, function of a set of parameters, at eachdiscrete point of the time series.

. estimate the dynamic of the parameters with a VAR(1).







We assume that the force of mortality for a fixed age x,µx(t),increase exponentially with t , and it satisfies the followingdifferential equation,

dµx(t) = kµx(t)dt, µx(0) = h, (18)

where the parameter k is constrained to be strictly positive and his constrained to be positive. It is derived from the dynamic of anaffine stochastic mortality model

dµx(t) = kµx(t)dt+ σdW, µx(0) = h, (19)

with the following two attributes: (1) is non-mean reverting and(2) the diffusion component of the model is set equal to zero.







Because of affine stochastic mortality models imply a closed-formexpression for the survival probabilities,we have that,

Sx(t, t+ n) = exp

[−H(t, t+ n)h

], (20)

where

H(t, T ) =1

kx

[1 − exp(−nkx)

]. (21)

and n = T − t. In order to introduce a stochastic dynamic in thesurvival probability formula given by (8), we assume thetime-variability of the two parameters, h and k, with respect to thereference year t,

Sx(t, t+ n) = exp

{−hx,t

kx,t

[1 − exp

(− nkx,t

)]}. (22)







The closed-form expression of the term structure of mortality ratesbecomes,

µx(t, t+ n) =hx,t

kx,tn

[1 − exp

(− nkx,t

)]. (23)

The second step in the estimation procedure involves thecalibration of the state variables, ht and kt, by means of anoptimization procedure. For a fixed age x, the vectors h and k arecalibrated for each point in time t, with t = 1, 2, ..., TFinally, estimate a VAR(1)on the time series {ht} and {kt}.






Results: A time-varying closed-form expression for the term structure ofmortality rates

Figure: Parameters estimation for an individual aged 40 from 1930 to 1980Rosella Giacometti (DMSIA) Stochastic Models for Mortality Rate on Italian Data





Results: A time-varying closed-form expression for the term structure ofmortality rates

Table 3: Parameters values

α0 α1 α2 β0 β1 β2 σh σk ρ

-0,0038 -0,1387 -1,1483 -0,0001 -0,2950 0,0060 0,0052 0,0005 0,1736






Conclusions

We have investigated the dynamics of mortality rates in differentcontexts

. relaxing Lee Carter assumption

. using a pure econometric models

. using the first stochastic model with strong mean reversion

. affine process without mean reversion

. affine process with time-varying parametres






References

BIFFIS, E., Affine processes for dynamic mortality and actuarial valuations,Insurance: Mathematics and Economics, No. 37, pp. 443–468, 2005.

BALLOTTA, L., and Habermann S., The fair valuation problem ofguaranteed annuity options: The stochastic mortality environment case,Insurance: Mathematics and Economics, No. 38, pp. 195–214,2006.

CAIRNS, A.J.G., A discussion of parameters and model uncertainty,Insurance: Mathematics and Economics, No. 27, pp. 313–330, 2000.

CAIRNS, A.J.G., D. Blake, and K. Dowd, A two factor model forstochastic mortality with parameter uncertainty: Theory and calibration,Journal of Risk and Insurance, No. 73, pp. 687–718, 2006.

CAIRNS, A.J.G., D. Blake, and K. Dowd, Pricing death: frameworks forthe valuation and securitization of mortality risk, Centre for Risk &Insurance Studies, CRIS Discussion Paper Series, The University ofNottingham, No. IV, 2006.






References (Cont.)

CURRIE, I.G., Smoothing and forecasting mortality rates with P-splines,Available on line at www.ma.hw.ac.uk/iain/research/talks.html, 2006.

GIACOMETTI, R., S. Ortobelli, and M. Bertocchi, Impact of differentdistributional assumptions in forecasting Italian mortality, InvestmentManagement and Financial Innovations, No.3, pp.65–72, 2009.

GIACOMETTI, R., M. Bertocchi, S.T. Rachev, and F. Fabozzi, Acomparison of the Lee-Carter model and AR-ARCH model for forecastingmortality rates, submitted to Insurance: Mathematics and Economics,2010.

GIACOMETTI, R., S. Ortobelli, and M. Bertocchi, A stochastic model formortality rates on Italian data, JOTA, No.149, pp.216-228, 2011,DOI:10.1007/s10957-019-9771-5.

GIROSI, F., and G. King, Understanding the Lee Carter mortalityforecasting method. Technical Report, Rand Corporation, 2007.






References (Cont.)

JPMORGAN, LifeMetrics. A toolkit for measuring and managing longevityand mortality risks, V1.0, 2007.

LEE, R.D., and Carter, L.R., Modelling and Forecasting U.S. Mortality,Journal of the American Statistical Association, No. 87, pp.659–671, 1992.

LEE, R. D., The Lee-Carter Method for Forcasting Mortality, with VariousExtensions and Applications, North American Actuarial Journal, No. 4, pp.80–93, 2000.

LUCIANO, E., and Vigna E., Non mean reverting affine processes forstochastic mortality, Belgian Actuarial Bullettin, to appear, 2009.

MILEVESKY, M. A., and Promislow, S. D., Mortality derivatives and theoption to annuitise, Insurance: Mathematics and Economics, No. 29, pp.299–318, 2001.

PLAT, R., Stochastic portfolio specific mortality and the quantification ofmortality basis risk, Available on line athttp://ssrn.com/abstract=1277803, 2009.






References (Cont.)

RENSHAW, A.E., and S. Habermann, A cohort-based extension to theLee-Carter model for mortality reduction factors, Insurance: Mathematicsand Economics, No. 38, 556–570, 2006.

RUSSO, Giacometti, R., S. Ortobelli, S. Rachev, F. Fabozzi calibratingaffine stochastic mortality models using term assurance premiums,IME,Insurance: Mathematics and Economics.Do:10.1016/j.insmatheco.2011.01.015

SARGAN, J. D., Some discrete approximations to continuous timestochastic models, Journal of the Royal Statistical Society, Series B, No.36, pp. 74–90, 1975.

SEBER, G.A.F., and Wild C.J., Non linear regression, Wiley, New York,1989.

WYMER, C. R., Econometric Estimation of Stochastic DifferentialEquation Systems, Econometrica, Vol.40, No. 3, pp. 565–577, 1972.


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