a joint valuation of premium payment and …/media/internet/content/dateien...a resumption option,...

A Joint Valuation of Premium Payment

and Surrender Options in Participating

Life Insurance Contracts

Hato Schmeiser, Joel Wagner

Institute of Insurance EconomicsUniversity of St. Gallen, Switzerland

Singapore, July 2010

J. Wagner, A Joint Option Valuation in Participating Life Insurance Contracts, WRIEC, Singapore, July 2010 1

Abstract

• Life insurance contracts typically embed, in addition to an interestrate guarantee and annual surplus participation,

◮ a paid-up option, the right to stop payments during the contract term,◮ a resumption option, the right to resume payments, and,◮ a surrender option, the right to terminate the contract early.

• Terminal guarantees on benefits payable upon death, survival andsurrender are adapted after option exercise.

• Model framework and algorithm to jointly value the options isdeveloped.

• Standard principles of risk-neutral evaluation are applied; thepolicyholder is assumed to use an economically rational exercisestrategy.

• Option value sensitivity on different contract parameters, benefitadaptation mechanisms, and exercise behavior is analyzed numerically.


1. Introduction

Adequate risk management needed for correct valuation ofoptions and compliance with regulation rules

• Financial crisis and variable annuity products with embedded optionshave highlighted the importance of the valuation and the riskmanagement of options.→ E.g. Equitable Life (2000), The Hartford (2009), AXA (2009).

• Current and planned regulation rules prescribe risk-adequate capitaldeposits for embedded options.→ E.g. Solvency II, SST, NAIC RBC, and IFRS.

• Hypotheses on the policyholder’s exercise behavior are required toevaluate option values, in particular with regard to stress-testing ofproducts.→ How much more value can policyholders get by exercisingdifferently than what has been observed?


1. Introduction

Competitive advantage for product design and pricingwhen knowing the ”fair” price

• Separate inspection of embedded options with regard to pricing andrisk management.→ Pricing flexibility through reconfiguration of the conversionmechanism of the guaranteed benefits after exercise of the option.

• Product design and offering of products along the needs andwillingness to pay of customers.→ Possibility of offering zero-valued options, not to be paid for by theclient.

• Adequate and ”fair” pricing of embedded options since policyinception.→ Competitive advantage.


1. Introduction

Literature review of selected scientific contributions I

• Fair pricing of contracts with a guaranteed interest rate and differentannual surplus participations.→ E.g. Bacinello (2001); Hansen and Miltersen (2002); Tanskanenand Lukkarinen (2003); Ballotta et al. (2006); Branger et al. (2010).

• Surrender option, on top of guaranteed interest rate and surplusparticipation.→ E.g. Grosen and Jorgensen (2000); Jensen et al. (2001).

• Surrender option in Italian life insurance contracts with single andperiodic premiums, including mortality risk.→ E.g. Bacinello (2003a,b, 2005); Bacinello et al. (2009).

• Surrender option in French life insurance contracts.→ E.g. Albizzati and Geman (1994).

• Surrender and paid-up options with underlying stochastic interestrates and deterministic surplus participation.→ E.g. Herr and Kreer (1999).


1. Introduction

Literature review of selected scientific contributions II

• General framework analysis of surrender and paid-up options.→ E.g. Steffensen (2002).

• Actuarial approach comparison. → E.g. Linnemann (2003, 2004).

• Numerical analysis of risk-neutral valuation of embedded options andbroad overview on numerical techniques.→ E.g. Bauer et al. (2006); Bauer et al. (2008).

• Analysis of paid-up options in German government-subsidized pensionproducts based on different assumptions about the policyholder’sexercise behavior. → E.g. Kling et al. (2006).

• Option valuation through an optimal feasible exercise strategy in thecase of one Bermudan-style embedded option.→ E.g. Barraquand and Martineau (1995); Andersen (1999); Douady(2002); Kling et al. (2006).


1. Introduction

What is the contribution of this work?

• Model framework and methodology closest to the following:◮ Gatzert and Schmeiser (2008): paid-up and resumption options with

focus on the maximal risk.◮ Bacinello (2003b): surrender option pricing.◮ Andersen (1999): option valuation algorithm for one option.

• Key features:◮ Several options are valued jointly.

→ Most works consider options separately.◮ Pricing through assumptions on policyholder’s behavior is given.

→ Not only assessment of maximal risk potential.◮ Exercise through economically rational optimal strategy.

→ Feasible strategy.◮ Analysis of option values for various benefit conversion mechanisms in

a policy assets perspective.→ Perspective allows for asset transparency in comparison withreserve-linked modeling.


1. Introduction

Structure of this talk

• Introduction of the model framework◮ Basic contract B including interest guarantee and surplus participation.◮ Contract P with a paid-up option OP .◮ Contract R with a combined paid-up and resumption option OR.◮ Contract S featuring a surrender option OS .◮ Contract Q with a combined paid-up and surrender option OQ.

• Option valuation techniques and exercise behavior hypotheses◮ Maximum over all times of exercise of the option payoff.◮ Option value reached using an optimal admissible exercise strategy.◮ Upper bound of the option payoff for any exercise strategy (risk).

• Numerical results and discussion◮ Option values for different contracts and parameters.◮ Sensitivity analysis of the model.◮ Comparison with Gatzert and Schmeiser (2008) and Bacinello (2003b).


2. Model framework – Basic contract B

Basic contract B• Life insurance contract with periodic premium payments featuring

◮ an interest rate guarantee g , and,◮ an annual surplus participation, i.e. fraction α of the surplus.

• Premium payments Bt−1, t = 1, . . . ,T , are paid annually at thebeginning of the tth policy year given the insured remains alive.→ Annual premium payments Bt ≡ B are supposed constant.

• Mortality statistics: financial risk and mortality risk areuncorrelated; mortality risk is eliminated by writing a sufficiently largenumber of contracts (actuarial practice).→ Mortality risk does not contain systematic risk.

• Survival and death probabilities:◮

tpx : x-year-old policyholder survives for the next t years.◮

tqx = 1− tpx : x-year-old dies within the next t years.◮ qx+t (resp. px+t): (x + t)-year-old policyholder will die within the next

year, between t and t + 1 (resp. survive one more year, the tth year).

• Guarantees are on benefits payable upon death and survival.→ Contract B non-surrenderable and without paid-up option.



Death benefit

• In case of death during the tth year of the contract, the assignedpolicyholder’s beneficiary receives ΥB

t ≡ ΥB at the end of the year.

• Actuarial equivalence principle: the expected value of the payments tothe insured equals the expected premium payments from the insured:

BT−1∑

t=0

tpx (1 + g)−t = ΥB

(

T−1∑

t=0

tpx qx+t (1 + g)−(t+1) + Tpx (1 + g)−T

)

.

• The death benefit is given by

ΥB =B∑T−1

t=0 tpx (1 + g)−t

∑T−1t=0 tpx qx+t (1 + g)−(t+1) + Tpx (1 + g)−T

.



Survival benefit I

• In case of survival until maturity T , the insurer pays out theaccumulated policy assets AB

T .→ Including guaranteed interest and surplus participation.

• Annual premium payments B are split into◮ Premium BR

t−1 for term life insurance, used to buy insurance to coverthe difference between the guaranteed death benefit ΥB and theavailable policy assets AB

t−1. For the tth year, fair pricing implies that

BRt−1 = qx+t−1 max(ΥB − AB

t−1, 0).

◮ Savings premium BAt−1 = B − BR

t−1.→ Credited to the policy assets at the beginning of the tth year.

• ABt−1 and BA

t−1 annually earn the greater of the the guaranteedinterest rate g or a fraction α of the annual surplus (St/St−1 − 1).



Survival benefit II

• Life insurer’s investment portfolio St follows a geometric Brownianmotion given a complete, perfect, and frictionless market.For t = 1, . . . ,T , St is defined by

dSt = µStdt + σStdWPt ,

with deterministic asset drift µ, volatility σ, and a P-Brownianmotion W P .

• Under the risk-neutral unique equivalent martingale measure Q, thedrift of the process is the risk-free interest rate r , and

St = St−1 exp[

r − σ2/2 + σ(WQt −WQ

t−1)]

,

with Q-Brownian motion WQ and initial condition S0 (see,e.g., Bjork (2004)).



Survival benefit III

• Finally, the accumulated policy assets ABt , t = 1, . . . ,T , can be

calculated as follows,

ABt =

(

ABt−1 + t−1px B

At−1

)

(1 + max [g , α(St/St−1 − 1)])

=(

ABt−1 + t−1px

[

B − qx+t−1 max(ΥB − ABt−1, 0)

])

· (1 + max [g , α(St/St−1 − 1)]) ,

with AB0 = 0.



Contract payoff I

• Payoff PBT at maturity T is determined by the payments to the insured

less the premium payments, compounded with the risk-free rate r :

PBT =

T−1∑

t=0

ΥBtpx qx+t e

r(T−t−1) + Tpx ABT −

T−1∑

t=0

B tpx er(T−t).

• Let ΠB0 denote the net present value of the contract payoff PB

T at

time t = 0. With EQt denoting the conditional expected value with

respect to the probability measure Q under the information availablein t, one gets

ΠB0 = EQ

0

(

e−rTPBT

)

.



Contract payoff II

• Contract is fair if the present value of the benefits under the riskneutral martingale measure is equal to the present value of premiumspaid by the policyholder (see, e.g., Doherty and Garven (1986)):

ΠB0 = 0.

• All other parameters given, the annual surplus participation parameterα can be calibrated to obtain fair contracts.→ I.e. the two options in the basic contract, interest guarantee andsurplus participation, are covered by the annual premium payments.


2. Model framework – Contract P with paid-up option

Contract P with paid-up option and death benefit

• Contract P based on basic contract B.

• Paid-up option OP : policyholder has the right to exercise OP (once)annually until maturity. → OP is a Bermudan-style option.

• After exercising OP at time t = τ , τ = 1, . . . ,T − 1, denoted byOP(τ), and given the policyholder is still alive, the terminal benefitsare adjusted according to a mechanism described below.

• Adjusted death benefit calculated by taking the accumulated policyassets AB

τ as single premium for a new contract,

ΥP(τ) =ABτ (1 + γP

τ )∑T−1

t=τ t−τpx+τ qx+t (1 + g)−(t−τ+1) + T−τpx+τ (1 + g)−(T−τ),

with γPτ a model parameter, with default value 0.→ Parameter represents flexibility to adapt benefits.



Survival benefit and contract payoff

• Adjusted survival benefit given by AP(τ)T , where A

P(τ)t ,

t = τ, . . . ,T , denote the accumulated policy assets over time afterexercising OP at time τ :

AP(τ)t =

(

AP(τ)t−1 − t−1px qx+t−1 max(ΥP(τ) − A

P(τ)t−1 , 0)

)

· (1 + max [g , α(St/St−1 − 1)]) ,

with AP(τ)τ = AB

τ (1 + γPτ ).

• Contract payoff at maturity T , in the case the paid-up option isexercised in τ , is given by

PP(τ)T =

τ−1∑

t=0

ΥBtpx qx+t e

r(T−t−1) +T−1∑

t=τ

ΥP(τ)tpx qx+t e

r(T−t−1)

+Tpx AP(τ)T −

τ−1∑

t=0

B tpx er(T−t).



Paid-up option value

• The value at time t = 0 of the paid-up option OP(τ) exercised attime τ can be residually determined by the difference of the contractvalues of P and B:

Π0(OP(τ)) = EQ

0

[

e−rT(

PP(τ)T − PB

T

)]

= EQ0

[

e−rττpx Cτ (O

P(τ))]

,

where

Cτ (OP(τ)) =

T−1∑

t=τ

(

ΥP(τ)−ΥB

)

t−τpx+τ qx+t e−r(t−τ+1)

+(

AP(τ)T − A

BT

)

T−τpx+τ e−r(T−τ) +

T−1∑

t=τ

B t−τpx+τ e−r(t−τ)

.

• Since the basic contract is fair, we have ΠP(τ)0 = Π0(O

P(τ)).


2. Model framework – Contract R with paid-up and resumption options

Contract R with paid-up and resumption options I

• Contract R based on contract P with paid-up option.

• Combined paid-up and resumption option OR: right to stop premiumpayments at time τ = 1, . . . ,T − 1, and to resume payments at timeν = τ + 1, . . . ,T − 1.→ OR(τ,ν) denotes the combined exercise at times τ and ν.

• Death benefit

ΥR(τ,ν) =AP(τ)ν (1 + γR

ν ) + B∑T−1

t=ν t−νpx+ν (1 + g)−(t−ν)

∑T−1t=ν t−νpx+ν qx+t (1 + g)−(t−ν+1) + T−νpx+ν (1 + g)−(T−ν)

,

with γRν a model parameter, with default value 0.



Contract R with paid-up and resumption options II• Survival benefit given by A

R(τ,ν)T , with A

R(τ,ν)t , t = ν + 1, . . . ,T :

AR(τ,ν)t =

(

AR(τ,ν)t−1 + t−1px

[

B − qx+t−1 max(ΥR(τ,ν) − AR(τ,ν)t−1 , 0)

])

· (1 + max [g , α(St/St−1 − 1)]) ,

with AR(τ,ν)ν = A

P(τ)ν (1 + γRν ).

• Contract payoff at maturity T :

PR(τ,ν)T =

τ−1∑

t=0

ΥBtpx qx+t e

r(T−t−1) +ν−1∑

t=τ

ΥP(τ)tpx qx+t e

r(T−t−1)

+T−1∑

t=ν

ΥR(τ,ν)tpx qx+t e

r(T−t−1) + Tpx AR(τ,ν)T

−τ−1∑

t=0

B tpx er(T−t) −

T−1∑

t=ν

B tpx er(T−t).



Contract R with paid-up and resumption options III

• Option value at t = 0:

Π0(OR(τ,ν)) = EQ

0

[

e−rT(

PR(τ,ν)T − PB

T

)]

= EQ0

[

e−rττpx Cτ (O

R(τ,ν))]

,

where

Cτ (OR(τ,ν)) =

ν−1∑

t=τ

(

ΥP(τ) −ΥB)


+

T−1∑

t=ν

(

ΥR(τ,ν) −ΥB)


+(

AR(τ,ν)T − AB

T

)

T−τpx+τ e−r(T−τ)

+

ν−1∑

t=τ

B t−τpx+τ e−r(t−τ).


2. Model framework – Contract S with surrender option

Contract S with surrender option I

• Contract S based on basic contract B.

• Surrender option OS exercisable at time θ = 1, . . . ,T − 1.→ Contract is terminated: death benefit annihilated, surrenderbenefit paid out at t = θ.

• Surrender benefit

AS(θ)θ = AB

θ (1 + γSθ ),

where γSθ is a parameter with default value 0.

• Contract payoff at the time of surrender t = θ:

PS(θ)θ =

θ−1∑

t=0

ΥBtpx qx+t e

r(θ−t−1) + θpx AS(θ)θ −

θ−1∑

t=0

B tpx er(θ−t).


2. Model framework – Contract S with surrender option

Contract S with surrender option II


Π0(OS(θ)) = EQ

0

[

e−rθ(

PS(θ)θ − e−r(T−θ)PB

T

)]

= EQ0

[

e−rθθpx Cθ(O

S(θ))]

,

where

Cθ(OS(θ)) = −

T−1∑

t=θ

ΥBt−θpx+θ qx+t e

−r(t−τ+1)

+(

AS(θ)θ − AB

T T−θpx+θ e−r(T−θ)

)

+T−1∑

t=θ

B t−θpx+θ e−r(t−θ).


2. Model framework – Contract Q with paid-up and surrender options

Contract Q with paid-up and surrender options I

• Contract Q based on contract P with paid-up option.

• Combined paid-up and surrender option OQ: right to stop premiumpayments at time τ = 1, . . . ,T − 1, and to surrender the contract attime θ = τ + 1, . . . ,T − 1.

• Surrender benefit at time θ when contract has been paid-up att = τ :

AQ(τ,θ)θ = A

P(τ)θ (1 + γQθ ),

where γQθ is a parameter with default value 0.• Contract payoff at the time of surrender t = θ:

PQ(τ,θ)θ =

τ−1∑

t=0

ΥBtpx qx+t e

r(θ−t−1) +

θ−1∑

t=τ

ΥP(τ)tpx qx+t e

r(θ−t−1)

+θpx AQ(τ,θ)θ −

τ−1∑

t=0

B tpx er(θ−t).


2. Model framework – Contract Q with paid-up and surrender options

Contract Q with paid-up and surrender options II


Π0(OQ(τ,θ)) = E

Q0

[

e−rθ

(

PQ(τ,θ)θ − e

−r(T−θ)P

BT

)]

= EQ0

[

e−rθ

θpx Cθ(OQ(τ,θ))

]

,

where

Cθ(OQ(τ,θ)) =

θ−1∑

t=τ

(

ΥP(τ) −ΥB)

t−θpx+θ qx+t e−r(t−τ+1)

−

T−1∑

t=θ

ΥBt−θpx+θ qx+t e

−r(t−τ+1)

+(

AQ(τ,θ)θ − AB

T T−θpx+θ e−r(T−θ)

)

+

T−1∑

t=τ

B t−θpx+θ e−r(t−θ).


3. Option valuation

Option valuation I

• Valuation of options is connected to the policyholder’s exercisebehavior.

• Different ways to approach the valuation of the embedded options,e.g.,

◮ the maximum over all possible times of exercise of the expected valueof the option payoff,

◮ the option value that can be reached using an optimal admissibleexercise strategy,

◮ the upper bound of the option payoff for any exercise strategy.

• 1st and 3rd approaches suppose the policyholder to know the futurefor choosing the optimal time of exercise.→ This is not, in practice, a feasible strategy.

• 3rd approach assesses the maximal risk potential on a possible path,see, e.g., Kling et al. (2006), Gatzert and Schmeiser (2008).→ Can be helpful for stress-testing of products.


3. Option valuation – Maximum of expected option payoff

Maximum of expected option payoff

• Since the contract payoff is defined explicitly, the maximum over allexercise time(s) can be evaluated

• Contract with one option, X ∈ {P,S}:

Π0(OX (τ⋆)) = max

τ=1,...,T−1

[

Π0(OX (τ))

]

,

where τ⋆ denotes the exercise time maximizing Π0(OX (τ)).

• Contract with two options, Y ∈ {R,Q}:

Π0(OY(τ⋆,ν⋆)) = max

τ=1,...,T−1ν=τ+1,...,T−1

[

Π0(OY(τ,ν))

]

,

where τ⋆ and ν⋆ denote the exercise times maximizing Π0(OY(τ,ν)).


3. Option valuation – Optimal admissible exercise strategy

Optimal admissible exercise strategy

• Option valuation through considering a policyholder who follows anexercise strategy that maximizes the option value given theinformation available at the exercise date(s).

• Approach leads to an optimal stopping problem that can be solvedusing, for example, Monte Carlo simulation methods, as is done, e.g.,in Andersen (1999), Douady (2002) or Kling et al. (2006).

• Whenever the underlying model for the economy has only one sourceof risk, an optimal admissible strategy for exercising Bermudan-styleoptions can be found by looking only at the exercise value of theoption, see Douady (2002).→ This is apparently equivalent to considering only the current valueof the accumulated policy assets in the present framework.

• Extension of Kling et al. (2006) in a setting of a contract with twooptions.



Case of a contract with one option I

• Contract X ∈ {P,S} with one option OX which can be exercised attimes τ = 1, . . . ,T − 1.

• Accumulated assets At at times t = 1, . . . ,T , before exercise of theoption.

• Payoff stream Cτ (OX (τ)) of the option at time τ .

• Exercise strategy K = K1 × . . .× KT−1 ⊆ RT−1.

• Following this strategy, the option is exercised at time τ if and only ifthe option has not been exercised before and Aτ ∈ Kτ , i.e.,

1Kτ(Aτ ) = 1, and 1Kt

(At) = 0, ∀t = 1, . . . , τ − 1.



Case of a contract with one option II

• Value of the option under the assumption that the investor appliessome given strategy K is denoted by

ΠK0 (O

X ) = EQ0

[

e−rττpx Cτ (O

X (τ))]

,

where τ = inf {t ∈ {1, . . . ,T − 1} such that At ∈ Kt},or τ = T , if At /∈ Kt , ∀t ∈ {1, . . . ,T − 1}.→ This last case where τ = T corresponds to a strategy where theoption is never exercised, and the value of the option is zero.

• Douady (2002) and Andersen (1999) describe the so-called valuethreshold method to approximate an optimal strategy, i.e., a strategymaximizing ΠK

0 (OX ) by Monte Carlo methods.

→ Existence of an optimal strategy K = (0; k1]× . . .× (0; kT−1] forsome kt ∈ R, t = 1, . . . ,T − 1 is shown.



Case of a contract with one option III

• Method uses a backward induction algorithm to determine theoptimal values k1, . . . , kT−1.

• Calculation of a Monte Carlo approximation follows two main parts:◮ the definition of the optimal exercise strategy K (Algorithm 1), and,◮ the calculation of the approximation of the option value (Algorithm 2).

→ See Appendix.

• The value ΠK0 (O

X ), where K is the optimal strategy, gives anapproximation of the option value Π0(O

X ).



Case of a contract with two options I

• Contract Y ∈ {R,Q} with a combined option OY exercisable attimes τ = 1, . . . ,T − 1 and ν = τ + 1, . . . ,T − 1.

• Accumulated assets At at times t = 1, . . . ,T , before exercise of the

first option, and A(τ)s the assets at times s = τ, . . . ,T , where τ is the

exercise date of the first option.

• Payoff stream Cτ (OY(τ,ν)) of the combined option at time τ .

• Exercise strategy K = K1 × . . .× KT−1 ⊆ RT−1: the first option is

exercised at time τ if and only if the option has not been exercisedbefore and Aτ ∈ Kτ , i.e.,

1Kτ(Aτ ) = 1, and 1Kt

(At) = 0, ∀t = 1, . . . , τ − 1.

• For each exercise time τ = 1, . . . ,T − 1, consider a corresponding

exercise strategy L(τ) = L(τ)τ+1 × . . .× L

(τ)T−1 ⊆ R

T−τ−1 with

1L(τ)ν(A(τ)

ν ) = 1, and 1L(τ)s(A

(τ)s ) = 0, ∀s = τ + 1, . . . , ν − 1.



Case of a contract with two options II

• The value of the option under the assumption that the investorapplies some given strategy K , followed by some strategy L(τ) (for τdetermined by K ), is denoted by

ΠK ,L0 (OY) = EQ

0

[

e−rττpx Cτ (O

Y(τ,ν))]

,

where τ = inf {t ∈ {1, . . . ,T − 1} such that At ∈ Kt} ,or τ = T , if At /∈ Kt , ∀t ∈ {1, . . . ,T − 1}, and

where ν = inf{

s ∈ {τ + 1, . . . ,T − 1} such that A(τ)s ∈ L

(τ)s

}

, or

ν = T , if A(τ)s /∈ L

(τ)s , ∀s ∈ {τ + 1, . . . ,T − 1}.



Case of a contract with two options III

• Nested backward induction algorithm to determine the optimal values

kt (t = 1, . . . ,T − 1), and ℓ(t)s (t = 1, . . . ,T − 1, s = t + 1, . . . ,T ).

• The calculation of a Monte Carlo approximation is again done in twoparts,

◮ the definition of the optimal exercise strategies K and L(τ),τ = 1, . . . ,T − 1, and,

◮ the calculation of the approximation of the option value.

• The value ΠK ,L0 (OY) through the application of optimal exercise

strategies K and L(τ), gives the option value Π0(OY).


3. Option valuation – Upper bound of the option payoff for any exercise behavior

Upper bound of the option payoff for any exercise behavior• Behavioral-independent risk potential from embedded options.→ Policyholder knows the future and exercises the option at itsmaximum value.

• Upper bound for the option price pointing out the potential hazardsof offering such options, see Gatzert and Schmeiser (2008); Klinget al. (2006).→ This worst case from the product provider’s point of view does notcorrespond to a realistic strategy to expect from the clients.

• Contract with one option, X ∈ {P,S}:

Πmax0 (OX ) = EQ

0

[

maxτ=1,...,T−1

(

e−rττpx max

(

0,Cτ (OX (τ))

))

]

.

• Contract with two options, Y ∈ {R,Q}:

Πmax0 (OY) = EQ

0

maxτ=1,...,T−1

ν=τ+1,...,T−1

(

e−rττpx max

(

0,Cτ (OY(τ,ν))

))

.


4. Numerical results and discussion

Numerical simulation

• Monte Carlo simulation with antithetic variables and N = 1000 000different paths. → Variance reduction by generating negativelycorrelated variables such that large and small outputs arecounterbalanced (see, e.g., Hull (2009, p. 433)).

• Average population mortality data derived from the Bell and Miller(2002) cohort life tables for the United States.

• Reference example with the following parametrization:◮ x = 30 and x = 50 year-old policyholders in 2010 at contract inception,◮ contract with time to maturity T = 10,◮ yearly premium payments B = 1200 (currency units),◮ risk-free interest rate r = 4%,◮ guaranteed interest rate g = 3%, and,◮ insurer’s investment portfolio with volatility σ = 0.20.

• First step: calibrate the fraction α of the annual investment returnsto get fair contract conditions.→ Standard bisection method is used to determine α for given g.


4. Numerical results and discussion – Basic contract B

Policy assets ABt , death benefit YB, and risk premium BR

t

YB (x = 50)

ABt (x = 50)

YB (x = 30)

ABt (x = 30)

Time t (in years)

109876543210

15000

12500

10000

7500

5000

2500

0

(a) Evolution of accumulated assets ABt

and death benefit Y B in contract B.

BRt (x = 50)

BRt (x = 30)

Time t (in years)

Riskprem

ium

BR t

109876543210

70

60

50

40

30

20

10

0

(b) Evolution of risk premium BRt in

contract B.

Figure: Death benefit, risk premium and accumulated assets for ages x = 30, 50in basic contract B.

→ With x = 30 and x = 50, α = 24.1% and α = 29.3% respectively.→ The standard error on the values reported for AB

t resp. BRt are below

0.005% resp. 0.05%, for all t = 0, . . . ,T.J. Wagner, A Joint Option Valuation in Participating Life Insurance Contracts, WRIEC, Singapore, July 2010 37

4. Numerical results and discussion – Basic contract B

Sensitivity of fair participation coefficient α

Time to maturity T (in years)

α(in%)

302520151050

30

28

26

24

22

20

(a) α for different values T(with r = 4%, g = 3%, σ = 0.20).

Risk-free interest rate r (in %)

α(in%)

765432

50

40

30

20

10

0

(b) α for different values of r(with T = 10, g = 3%, σ = 0.20).

Guaranteed interest rate g (in %)

α(in%)

543210

50

40

30

20

10

0

(c) α for different values of g(with T = 10, r = 4%, σ = 0.20).

Volatility of investment portfolio σ

α(in%)

0.80.70.60.50.40.30.20.10

100

80

60

40

20

0

(d) α for different values of σ(with T = 10, r = 4%, g = 3%).


4. Numerical results and discussion – Contract P with paid-up option

Policy assets AP(τ)t , death benefit Y P(τ), and Π0(O

P(τ))

τ = 9

τ = 8

τ = 7

τ = 6

τ = 5

τ = 4

τ = 3

τ = 2

τ = 1τ=

8

τ=

7

τ=

6

τ=

5

τ=

4

τ=

3

τ=

2

τ=

1

Time t (in years)

109876543210

15000

12500

10000

7500

5000

2500

0

(a) Assets ABt , A

P(τ)t (solid lines), and

death benefit Y B, YP(τ) (dashed lines)in contract P for different τ = 1, . . . , 9.

γP = 0.5%

γP = 0

γP = −0.5%

γP = −1%γP = −1.5%

Year of option exercise τ

Net

presentvalueof

contractpayoff

Π0(O

P(τ

) )

987654321

60

40

20

0

-20

-40

-60

(b) Sensitivity analysis of Π0(OP(τ)) for

different γPτ = γ

P = −1.5%, . . . , 0.5%.

Figure: Analysis of contract P with paid-up option.

→ Parameters: T = 10, B = 1200, r = 4%, σ = 0.20, g = 3%, x = 30.→ γP = −1% quasi annihilates the payoff for all exercise times.



Side-by-side comparison of valuation methodsContract

ItemValuation with Valuation with Valuation with

duration Π0(OP(τ⋆)) ΠK

0 (OP ) Πmax

0 (OP )

T = 10

Value 33.5 (0.2) 33.5 (0.3) 196.4 (0.2)Time of exercise 5 5.0 (0.0) 4.2 (0.0)PVP 5 535 5 538 (0.0) 4 445 (3.2)Value / PVP 0.6% 0.6% 4.4%

T = 20


T = 30


Table: Option valuation in contract P. Reported results are for a fair underlying contract B, x = 30,

r = 4%, g = 3% and σ = 0.20. Value in parentheses indicates standard error. PVP stands for present value of exp. premiums.

→ Maximum value of payoff reached by ”optimal” exercise strategy.→ Option value ΠK

0 (OP) well below risk potential Πmax

0 (OP).→ Option is worth 1− 2% of the expected premium payments.



Sensitivity analysis on g and σ

Πmax0 (OP)

ΠK0 (O

P)

Guaranteed interest rate g (in %)

Option

valuation

43.532.521.510.50

500

400

300

200

100

0

(a) Valuation for different values of g(with T = 10, r = 4%, σ = 0.20).

Πmax0 (OP)

ΠK0 (O

P)

Volatility of investment portfolio σ

Option

valuation

0.40.350.30.250.20.150.10.050

300

250

200

150

100

50

0

(b) Valuation for different values of σ(with T = 10, r = 4%, g = 3%).

Figure: Option valuation under variation of guaranteed interest rate g andvolatility of investment portfolio σ.

→ Option value’s independence of g and σ due to the fairness parameter α



Detail: Sensitivity ”stabilized” by α

g α ΠB0 Π0(O

P(τ⋆)) ΠK0 (O

P ) Πmax0 (OP )

1% 36.2% (fair) 0.0 (0.8) 32.4 (0.3) 32.4 (0.6) 363.6 (0.3)3% 24.1% (fair) 0.0 (0.5) 33.5 (0.2) 33.5 (0.3) 196.4 (0.2)

1% 30.0% (fixed) -343.4 (0.6) 285.6 (0.5) 285.6 (0.5) 463.6 (0.3)3% 30.0% (fixed) 308.7 (0.6) 7.6 (0.1) 7.6 (0.5) 143.3 (0.1)

Table: Impact of fair and non-fair parameter α on the option valuation. Forreference, values for ΠB

0 are reported. Values reported in parentheses are thestandard errors.

→ Firstly, let g = 1% and g = 3% and report valuation results for α beingsuch that the contract B is fair.→ Secondly, fix α = 30% and report the respective values.→ α ”stabilizes” the option value.



Option valuation for different contract parametrizations

x T σ g r α ΠK0 (O

P) Value / PVP

30 10 0.20 3% 4% 24.1% 33.5 (0.2) 0.6%50 10 0.20 3% 4% 29.3% 139.7 (0.4) 2.6%30 20 0.20 3% 4% 42.8% 137.6 (1.1) 1.5%30 10 0.10 3% 4% 42.4% 33.5 (0.3) 0.6%30 10 0.20 1% 4% 36.1% 32.1 (0.6) 0.6%30 10 0.20 1% 2% 32.4% 37.7 (0.2) 0.7%

Table: Illustration of the option value, standard error and ratio of the optionvalue to the expected premium payments for the contract P, with respect todifferent values of the age of the policyholder x , the contract duration T , thevolatility of the investment portfolio σ, the guaranteed interest rate g , therisk-free interest rate r , and the determined participation rate α for the underlyingcontract B to be fair. Annual premium payments are B = 1200.

→ x and T have a high importance for the option value.→ Variations of σ, g , and r are counter-balanced by α.


4. Numerical results and discussion – Contract R with paid-up and resumption options

Option payoff Π0(OR(τ,ν)) as a function of τ and ν

τ

ν

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

10

−20

−15

−10

−5

0

5

10

15

20

25

30

(a) Π0(OR(τ,ν)) with γ

Rν = 0.0%.

τ

ν

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

10

−10

0

10

20

30

40

50

(b) Π0(OR(τ,ν)) with γ

Rν = 0.5%.

Figure: Π0(OR(τ,ν)) and sensitivity with γR

ν as a function of τ and ν.

→ Triangle ν ≤ τ is zero as resumption cannot be before paid-up.→ Boundaries τ = T, ν = T correspond to no-exercise situations.→ Times and value of maximum payoff change with γRν .


4. Numerical results and discussion – Contract R with paid-up and resumption options

Option valuation in contract R

ContractItem

Valuation with Valuation with Valuation with

duration Π0(OR(τ⋆,ν⋆)) ΠK ,L

0 (OR) Πmax0 (OR)

T = 10

Value 33.5 (0.2) 33.5 (0.2) 198.1 (0.2)Time of 1st exercise 5 5.0 (0.0) 3.9 (0.0)Times of 2nd exercise 10 10.0 (0.0) 9.6 (0.0)PVP 5 535 5 538 (0.0) 4 493 (3.1)Value / PVP 0.6% 0.6% 4.4%

Table: Option valuation in contract R.Reported results are for a fair underlying contract B, x = 30, r = 4%, g = 3% and σ = 0.20.

→ Maximal values when ν = T = 10, i.e. second option is best neverexercised (γRν = 0.0%).

→ Same values as with paid-up option only: ΠK ,L0 (OR) = ΠK

0 (OP), i.e.

resumption option value is zero.→ Swiss life insurance contract: ΠK ,L

0 (OR)|ν=min(τ+2,T )= 20.7 (0.3) withτ = 8.0, ν = 10.0. PVP is 8 386 and ratio option value / PVP is 0.3%.


4. Numerical results and discussion – Contract S with surrender option

Option valuation in contract S

ContractItem


duration Π0(OS(τ⋆)) ΠK

0 (OS) Πmax

0 (OS)

T = 10


Table: Option valuation in contract S.Reported results are for a fair underlying contract B, x = 30, r = 4%, g = 3% and σ = 0.20.

→ ΠK0 (O

S) is zero within the considered modeling frame.→ However risk situation when offering the contract S is not alike offeringa contract B since the non-zero upper bound Πmax

0 (OS).


4. Numerical results and discussion – Contract S with surrender option

Sensitivity analysis of Π0(OS(θ)) for different γS

θ

γS = 1%

γS = 0.5

γS = 0%

γS = −0.5%γS = −1%

Year of option exercise θ

Net

presentvalueof

contractpayoff

Π0(O

S(θ) )

987654321

100

50

0

-50

-100

Figure: Π0(OS(θ)) for different γS

θ = γS = −1.0%, . . . , 1.0%.

→ In practice, many contracts set γSθ = −1 for the first two years, andzero for the following years.→ Comparison with examples in Bacinello (2003b): models can becalibrated through setting the parameter γSθ . [Reserve-linked modelingcorresponds in this case to an additional (almost negligible) loading< 0.1% on the assets.]


4. Numerical results and discussion – Contract Q with paid-up and surrender options

Option payoff Π0(OQ(τ,θ)) as a function of τ and θ

τ

θ

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

10

0

5

10

15

20

25

30

(a) Π0(OQ(τ,θ)) with γ

Q

θ = 0.0%.

τ

θ

1 2 3 4 5 6 7 8 9 101

2

3

4

5

6

7

8

9

10

0

5

10

15

20

25

30

35

40

45

50

55

(b) Π0(OQ(τ,θ)) with γ

Q

θ = 0.5%.

Figure: Π0(OQ(τ,θ)) and sensitivity with γQ

θ as a function of τ and θ.

→ Π0(OQ(τ,θ)) maximal for τ = 5, θ = 10; maximum not higher than in

the contract P.→ γRν = γR to 0.5%: payoff is maximal for τ = 6, θ = 9.


4. Numerical results and discussion – Contract Q with paid-up and surrender options

Option valuation in contract Q

ContractItem


duration Π0(OQ(τ⋆,θ⋆)) ΠK ,L

0 (OQ) Πmax0 (OQ)

T = 10

Value 33.5 (0.2) 33.5 (0.2) 283.1 (0.2)Time of 1st exercise 5 5.0 (0.0) 4.7 (0.0)Times of 2nd exercise 10 10.0 (0.0) 5.7 (0.0)PVP 5 535 5 538 (0.0) 4 998 (2.9)Value / PVP 0.6% 0.6% 5.7%

Table: Option valuation in contract Q.Reported results are for a fair underlying contract B, x = 30, r = 4%, g = 3% and σ = 0.20.

→ ΠK ,L0 (OQ) = ΠK

0 (OP), hence the surrender option value is again zero.

→ R and Q are very similar since the additional option offered iszero-valued (with γRτ = γQθ = 0).→ Joint valuation is in general not equal to the sum of the single options:

• Contract P with γP = 0.0%: ΠK0 (O

P) = 33.5

• Contract S with γS = 0.5%: ΠK0 (O

S) = 45.5

• Contract Q with γQ = 0.5%: ΠK ,L0 (OQ) = 56.0


5. Conclusion

Conclusion

• Model framework dealing with fair and joint valuation of embeddedoptions; implementation of a robust numerical algorithm.

• Drivers of the option values pointed out:◮ Conversion mechanism of the guaranteed benefits.

→ Throughout the studied situations, paid-up option values are of theorder of 1− 3% of the expected premium payments.

◮ Exercise behavior of the policyholder.→ Even though an optimal exercise behavior may not be theempirically relevant case, nor the upper bound for the inherited risk bereached in practice, the practical relevance of adequate pricing is shownto be important.

• Substantial value of embedded options shows the necessity ofappropriate pricing and adequate risk management.→ Values of embedded options should be detailed carefully inparticular with regard to the offered benefits and under varioushypotheses of the client’s exercise behavior.


Further information

Further information

• Reference for this publication

H. Schmeiser, J. Wagner. A Joint Valuation of Premium Payment andSurrender Options in Particpating Life Insurance Contracts. Working Paperson Risk Management and Insurance, 77, 2010.

• Author contact information

◮ H. Schmeiser [email protected]◮ J. Wagner [email protected]

Institute of Insurance Economics

University of St. Gallen

Kirchlistrasse 2CH–9010 St. GallenPhone: +41 71 243 40 43www.ivw.unisg.ch


References

References I

M.O. Albizzati and H. Geman. Interest-rate Risk Management and Valuation of the SurrenderOption in Life-Insurance Policies. Journal of Risk and Insurance, 61(4):616–637, 1994.

L. Andersen. A Simple Approach to the Pricing of Bermudan Swaption in the Multi-Factor LiborMarket Model. Journal of Computational Finance, 3:5–32, 1999.

A.R. Bacinello. Fair Pricing of Life Insurance Participating Policies with a Minimum InterestRate Guaranteed. ASTIN Bulletin, 31(2):275–297, 2001.

A.R. Bacinello. Fair Valuation of a Guaranteed Life Insurance Participating Contract Embeddinga Surrender Option. Journal of Risk and Insurance, 70(3):461–487, 2003a.

A.R. Bacinello. Pricing Guaranteed Life Insurance Participating Policies with Annual Premiumsand Surrender Option. North American Actuarial Journal, 7(3):1–17, 2003b.

A.R. Bacinello. Endogenous Model of Surrender Conditions in Equity-Linked Life Insurance.Insurance Mathematics & Economics, 37(2):270–296, 2005.

A.R. Bacinello, E. Biffis, and P. Millossovich. Pricing Life Insurance Contracts with EarlyExercise Features. Journal of Computational and Applied Mathematics, 233(1):27–35, 2009.

L. Ballotta, S. Haberman, and N. Wang. Guarantees in With-Profit and Unitized With-ProfitLife Insurance Contracts: Fair Valuation Problem in Presence of the Default Option. Journal

of Risk and Insurance, 73(1):97–121, 2006.

J. Barraquand and D. Martineau. Numerical Valuation of High-Dimensional MultivariateAmercian Securities. Journal of Financial and Quantitative Analysis, 30(3):383–405, 1995.


References

References IID. Bauer, R. Kiesel, A. Kling, and J. Russ. Risk-Neutral Valuation of Participating Life

Insurance Contracts. Insurance: Mathematics and Economics, 39(2):171–183, 2006.

D. Bauer, D. Bergmann, and R. Kiesel. On the Risk-Neutral Valuation of Life InsuranceContracts with Numerical Methods in View. In AFIR Colloquium, Rome, 2008.

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References III

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Appendix

Algorithm 1 Definition of the optimal exercise strategy K = (0; k1]× . . .×(0; kT−1].

Initialize kϑ = 0, ϑ = 1, . . . ,T − 1. Let KT = R.Approximate kϑ, ϑ = T − 1, . . . , 1, as follows:Generate N Monte Carlo paths, denoted with superscript i , of S i

t , t = 1, . . . ,T .for times ϑ = T − 1 down to 1 do {backward induction}

for kϑ = 0 to AK in steps of aK dofor paths i = 1 to N do

Initialize τ i = 0.for t = ϑ to T do

if τ i = 0 and Ait ∈ Kτ

t = (0; kτt ] then

Let τ i = t.end if

end for {t}

Calculate e−rτ i

τ i px Cτ i (OX (τ i )).end for {i}

Calculate 1N

∑Ni=1 e

−rτ i

τ i px Cτ i (OX (τ i )) as an estimate of ΠK0 (O

X ).end for {kϑ}Choose kϑ from all tested values such that the estimate of ΠK

0 (OX ) is maximal.

end for {ϑ}


Appendix

Algorithm 2 Calculation of the Monte Carlo approximation of ΠK0 (O

X ).

Use K = K1× . . .×KT−1 determined through Algorithm 1. Let KT = R.Generate N Monte Carlo paths, denoted with superscript i , of S i

t , t =1, . . . ,T .for paths i = 1 to N do

Initialize τ i = 0.for t = 1 to T do

if τ i = 0 and Ait ∈ Kt then {exercise of option on path i under

strategy K}Let τ i = t.

end if

end for {t}

Calculate e−rτ i

τ ipx Cτ i (OX (τ i )).

end for {i}

Calculate 1N

∑Ni=1 e

−rτ i

τ ipx Cτ i (OX (τ i )) as an estimate of ΠK

0 (OX ).


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