Regression-based algorithms for life insurance...

Regression-based algorithmsfor life insurance contractswith surrender guarantees

Anna Rita BacinelloDipartimento di Matematica Applicata ‘B. de Finetti’ - University of Trieste

Enrico BiffisFinance Group, Tanaka Business School, Imperial College, London

Pietro MillossovichDipartimento di Matematica Applicata ‘B. de Finetti’ - University of Trieste

STOCHASTIC METHODS IN FINANCE – TORINO – 3-5 JULY 2008

Options in life insurance

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Options embedded in life insurance contracts:

⊲ European stylewith possibly random maturity⇒ Titanic option[Milevsky andPosner - JRI(01)](minimum guarantees, bonus options, conversion options, . . . );


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⊲ American style: the policyholder has the right to make some well-specified actionsbefore the natural termination of the contract⇒ early termination feature.


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⊲ Most common American option is thesurrender option: the policyholder has theright to early terminate the contract and receive a cash amount, calledsurrendervalue


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⊲ Most common American option is thesurrender option: the policyholder has theright to early terminate the contract and receive a cash amount, calledsurrendervalue⇒ (non-standard) American put option on the residual contract with the surrendervalue as exercise price.


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⊲ Most common American option is thesurrender option: the policyholder has theright to early terminate the contract and receive a cash amount, calledsurrendervalue⇒ (non-standard) American put option on the residual contract with the surrendervalue as exercise price.

⊲ If mortality risk can be diversified away (by pooling), then a Titanic option can bereduced to a portfolio of European options with different maturities;this does not apply to American options⇒ valuation problem.

Approaches to valuation

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⊲ BINOMIAL TREES (AND EXTENSIONS): e.g.[Grosen and Jorgensen - IME(00)],[Bacinello - JRI(03), NAAJ(03), IME(05)], [Vannucci - wp(03)], . . .;


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⊲ FINITE DIFFERENCE METHODS: e.g.[Jensen et al. - GPRIT(01)], [Tanskanen andLukkarinen - IME(03)], [Shen and Xu - IME(05)], [Moore and Young -NAAJ(03)];


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⊲ MONTE CARLO SIMULATION : e.g.[Andreatta and Corradin wp(03)], [Baione etal. IMFI(06)], [Bacinello MAF(08)].


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⊲ Complexity of the problem involved⇒ oversimplified assumptions:

⋆ no (or not realistic) mortality risk modelling;

⋆ Restrictive hypotheses on processes of concern (constant interest rates, GBM,. . . ).


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⊲ Complexity of the problem involved⇒ oversimplified assumptions:

⋆ no (or not realistic) mortality risk modelling;

⋆ Restrictive hypotheses on processes of concern (constant interest rates, GBM,. . . ).

⊲ Monte Carlo simulation combined with LS regression (LSMC, [Carriere -IME(96)], [Longstaff and Schwarz - RFS(01)]) allows to overcome suchdrawbacks.

Target

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⊲ We exploit the flexibility of LSMC to show how a general life insurance contractembedding a surrender option can be valued even under realistic assumptions.

Target

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⊲ Mortality enters the LSMC algorithm as any other variable;

Target

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⋆ surrender only in case of survival;

⋆ underlying variable depends on mortality.

Target

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⊲ Aim at fair valuationin a frictionless and arbitrage-free market⇒ consistent withIASB proposal;

Target

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⊲ market incompleteness⇒ choice of a pricing measure;

Target

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⊲ choice of pricing measure⇒ price of early termination option (?);

Target

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⊲ choice of pricing measure⇒ price of early termination option (?);

⊲ numerical example:

⋆ unit-linked endowment insurance withterminal or cliquet guarantees;

⋆ interest rates: CIRreference portfolio: GBM+SV+Jmortality: time dependent coefficients square root+J.

Valuation framework

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⊲ Given is a filtered probability space(Ω,F , G, P ).

Valuation framework

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⊲ Policyholdertime of death(or residual lifetime)τ is aG-stopping time (s.t.).

Valuation framework

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⊲ Insurance contract (without surrender):

⋆ cumulated death benefit:Ddt = Bd

τ 1τ≤t;

⋆ cumulated survival benefit:Dst =

∫ t

01τ>udBs

u;

⋆ total cumulated benefit:Dt = Ddt + Ds

t .

Most common contracts are included in the above set-up for appropriate choices ofprocessesBd andBs.

Valuation framework

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τ 1τ≤t;


∫ t

01τ>udBs

u;


t .


⊲ Insurance contractwith surrender; for any exercise policyθ (G-s.t.):

⋆ cumulated surrender benefit:Dwt (θ) = Bw

θ 1θ≤t, θ<τ;

Valuation framework

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τ 1τ≤t;


∫ t

01τ>udBs

u;


t .


⊲ Insurance contractwith surrender; for any exercise policyθ (G-s.t.):

⋆ cumulated surrender benefit:Dwt (θ) = Bw

θ 1θ≤t, θ<τ;

⋆ total cumulated benefit:Dt∧θ + Dwt (θ)

. . . Valuation Framework

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⊲ Fix a risk neutral probabilityQ (∼ P ) under which discounted (at the risk-freerate) cumulated gain for any security is aQ-martingale.


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⊲ Very convenient if (see[Milevsky and Promislow - IME(01)], [Biffis - IME(05)],[Dahl - IME(05)], . . .)

⋆ G = F ∨ H whereH is generated byτ ;


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⋆ τ is G-Cox withF-predictableforce of mortality(µt) (⇒ easy to simulate);


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⋆ any other process of interest (e.g.Bs, Bd or Bw) is F-adapted (orpredictable).


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⊲ Value of the contract with surrender option (for fixedθ): V w0 (θ); the value of the

contract is given by theoptimal stopping problem

V w∗0 = sup

θ∈TV w

0 (θ)

whereT = set ofG-stopping times.


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⊲ Value of the contract with surrender option (for fixedθ): V w0 (θ); the value of the

contract is given by theoptimal stopping problem

V w∗0 = sup

θ∈TV w

0 (θ)

whereT = set ofG-stopping times.

⊲ one can replaceG-s.t. withF-s.t. or s.t. bounded byτ .

LSMC algorithm

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⊲ Unbundlingof the contract:V w∗

0 = V0 + W ∗0

whereV0 = value of the contract without surrender andW ∗0 = value of the

surrender option (right to receiveBw and give upV ).

LSMC algorithm

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0 = V0 + W ∗0



⊲ In order to computeV ∗0 with backward dynamic programming, the LSMC requires

⋆ discretization in the time dimension;

LSMC algorithm

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0 = V0 + W ∗0





⋆ simulation of all random processes;

LSMC algorithm

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0 = V0 + W ∗0






⋆ Approximation of thecontinuation value by regression against function ofstate variables⇒ choice ofbasis functions;

LSMC algorithm

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0 = V0 + W ∗0






⋆ Approximation of thecontinuation value by regression against function ofstate variables⇒ choice ofbasis functions;

⊲ Convergence of the whole scheme is guaranteed if state variables areMarkov, see[Clement et al. - FS(02)].

⊲ Mixed results on trade-off between basis functions and numbers of simulations andon method robustness ([Moreno and Navas - RDR(02)], [Glasserman and Yu -AAP(04)], [Stentoft - RDR(04)], [Gobet et al - AAP(05)]).

. . . LSMC algorithm

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⊲ Method I: apply the algorithm directly to

V w0 (θ) = EQ

[∫ θ

0

d(Du + Dwu (θ))

S0u

],

whereS0 is the money market account.


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V w0 (θ) = EQ

[∫ θ

0

d(Du + Dwu (θ))

S0u

],


⋆ need to simulatetimes of death; the backward algorithm starts from thesetimes.

⋆ At any time, only trajectories in which the insured is still alive enter theapproximation.


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V w0 (θ) = EQ

[∫ θ

0

d(Du + Dwu (θ))

S0u

],




⊲ Method II: exploit the Cox setting⇒ replace indicators with probabilities, i.e.discount sums at risk-adjusted rater + µ:


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V w0 (θ) = EQ

[∫ θ

0

d(Du + Dwu (θ))

S0u

],




⊲ Method II: exploit the Cox setting⇒ replace indicators with probabilities, i.e.discount sums at risk-adjusted rater + µ:

V w0 (θ) = EQ

[∫ θ

0

d(Du + Dwu )

S0u

],

where dDu = dBsu + Bd

uµudu, dDwu = Bw

u dLu(θ) with Lu = 1θ≤u, andS0 is theadjusted money-market account⇒ contract without mortality.

Numerical examples

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⊲ Focus onsingle premium unit-linked endowment insurance with maturityT > 0,individual agedx at time 0 ([Bacinello et al. - JCAM(08)]).

Numerical examples

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⊲ Price process of reference portfolio is(St).

Numerical examples

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⊲ (Cumulated) benefits:

Bst = F

gT 1t≥T Bd

t = Fgt 1t<T Bw

t = Fht 1t<T ,

with eitherTerminalor Cliquetguarantee:

Numerical examples

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Bst = F

gT 1t≥T Bd

t = Fgt 1t<T Bw

t = Fht 1t<T ,


⊲ TERMINAL GUARANTEE:

F lt = maxSt, S0e

klt, l = g, w.

Numerical examples

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Bst = F

gT 1t≥T Bd

t = Fgt 1t<T Bw

t = Fht 1t<T ,


⊲ TERMINAL GUARANTEE:

F lt = maxSt, S0e

klt, l = g, w.

⊲ CLIQUET GUARANTEE:

Ft = F lt = S0

[t]∏

u=1

max

η

(Su

Su−1− 1

)+ 1, ekl

, l = g, w

. . . Numerical example

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Financial and demographic uncertainty are independent and described by:

⊲ Interest rates (CIR): drt = ζr(θr − rt)dt + σr√

rtdW rt .


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rtdW rt .

⊲ Reference portfolio ([Bakshi et al. - JoF(97)]): S = eY , with

dYt =

(rt −

1

2Zt − λJµJ

)dt +

√Zt

(ρSZdWZ

t + ρSrdW rt

+√

1 − ρ2SZ − ρ2

SrdWSt

)+ dJt

dZt =ζZ(θZ − Zt)dt + σZ

√ZtdWZ

t

whereW = (W r, WZ , WS) is a standard B.m. inR3 independent of thecompound PoissonJ (arrival intensityλJ , Lognormal(µJ , σJ ) jumps).


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rtdW rt .

⊲ Reference portfolio ([Bakshi et al. - JoF(97)]): S = eY , with

dYt =

(rt −

1

2Zt − λJµJ

)dt +

√Zt

(ρSZdWZ

t + ρSrdW rt

+√

1 − ρ2SZ − ρ2

SrdWSt

)+ dJt

dZt =ζZ(θZ − Zt)dt + σZ

√ZtdWZ

t

whereW = (W r, WZ , WS) is a standard B.m. inR3 independent of thecompound PoissonJ (arrival intensityλJ , Lognormal(µJ , σJ ) jumps).

⊲ Stochastic mortality: left continuous version of

dµt = ζµ(m(t) − µt)dt + σµ

√µtdW

µt + dKt

wherem is a deterministic force of mortality,Wµ is a standard B.m. independentof the compound PoissonK (arrival intensityλK and exp(γK) jumps).

Stochastic Mortality

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40 50 60 70 80 90

0.00

0.02

0.04

0.06

0.08

0.10

t

Figure 1: Solid line: sample path ofµ. Dashed line: Weibull intensitym.


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⊲ T = 15, x = 40.

⊲ Average of140 sets of19000 simulations for terminal guarantee,100 sets of30000simulations for cliquet guarantee.

⊲ Forward discretization step= 1500, backward discretization step= 30.


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⊲ T = 15, x = 40.



⊲ Financial model:

⋆ r0 = 0.05, ζr = 0.6, θr = 0.05, σr = 0.03;

⋆ Z0 = 0.04, ζZ = 1.5, θZ = 0.04, σZ = 0.4;

⋆ S0 = 100, ρZS = −0.7, ρrS = 0, λJ = 0.5, µJ = 0, σJ = 0.07.


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⊲ T = 15, x = 40.



⊲ Financial model:

⋆ r0 = 0.05, ζr = 0.6, θr = 0.05, σr = 0.03;

⋆ Z0 = 0.04, ζZ = 1.5, θZ = 0.04, σZ = 0.4;

⋆ S0 = 100, ρZS = −0.7, ρrS = 0, λJ = 0.5, µJ = 0, σJ = 0.07.

⊲ Demographic model:m Weibull fitted against a SIM2001;µ0 = m(0), ζµ = 0.5, σµ = 0.03, λK = 0.1, γK = 0.01.

⊲ State variables:µ, r, S, Z (+ F for the cliquet guarantee). Basis functions:polynomials in 4 (5) variables of order 3.

⊲ Terminal guarantee:kg = k; cliquet guarantee:kg = kw = k.

Terminal Guarantee

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κ κw E1 (s.e.) A1 (s.e.) O1 E2 (s.e.) A2 (s.e.) O20% 0% 107.185 (0.047) 113.556 (0.031) 6.372 107.224 (0.046)113.577 (0.030) 6.353

2% 117.223 (0.031) 10.038 117.237 (0.031) 10.0134% 123.687 (0.031) 16.503 123.696 (0.031) 16.4726% 137.130 (0.031) 29.945 137.262 (0.030) 30.038

2% 0% 112.675 (0.045) 115.381 (0.033) 2.706 112.698 (0.044)115.324 (0.033) 2.6262% 117.551 (0.031) 4.876 117.524 (0.031) 4.8254% 123.727 (0.031) 11.052 123.738 (0.031) 11.0406% 137.327 (0.030) 24.652 137.404 (0.030) 24.706

4% 0% 122.901 (0.041) 123.087 (0.033) 0.176 122.904 (0.040)122.904 (0.033) 0.0002% 123.291 (0.033) 0.390 123.130 (0.033) 0.2264% 124.507 (0.032) 1.606 124.418 (0.032) 1.5146% 137.710 (0.030) 14.809 137.630 (0.030) 14.726

Table 1: Terminal guarantee

Terminal Guarantee

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00.01

0.020.03

0.04

0

0.02

0.04

0.0690

100

110

120

130

140

κκw

Figure 2: Terminal guarantees, Algorithm 1: value of the American contract.

Cliquet Guarantee

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κ η E1 (s.e.) A1 (s.e.) O1 E2 (s.e.) A2 (s.e.) O20% 20% 65.938 (0.004) 97.205 (0.002) 31.267 66.239 (0.004) 97.216 (0.001) 30.977

40% 89.945 (0.009) 99.496 (0.003) 9.551 90.222 (0.009) 99.651 (0.004) 9.42960% 121.902 (0.019) 122.067 (0.018) 0.165 122.136 (0.019) 122.251 (0.019) 0.115

2% 20% 69.915 (0.004) 97.598 (0.001) 27.683 70.213 (0.004) 97.609 (0.001) 27.39640% 94.938 (0.009) 100.080 (0.004) 5.142 95.210 (0.009) 100.585 (0.005) 5.37660% 128.411 (0.019) 128.470 (0.018) 0.059 128.636 (0.019) 128.698 (0.019) 0.062

4% 20% 75.343 (0.004) 98.096 (0.001) 22.753 75.635 (0.004) 98.109 (0.001) 22.47440% 100.982 (0.009) 101.959 (0.007) 0.977 101.246 (0.009) 103.107 (0.008) 1.86160% 135.952 (0.019) 135.952 (0.019) 0.000 136.165 (0.019) 136.198 (0.019) 0.033

Table 2: Cliquet guarantee

Cliquet Guarantee

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00.01

0.020.03

0.04

0.2

0.3

0.4

0.5

0.690

100

110

120

130

140

κη

Figure 3: Cliquet guarantees, Algorithm 1: value of the American contract.

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