Evaluating fair premiums of equity-linked policies with surrender option in a bivariate model

Abstract

We tackle the problem of computing fair periodical premiums of an equity-linked policy with a maturity guarantee and an embedded surrender option. We consider the policy as a Bermudan-style contingent claim that can be exercised at the premium payment dates. The evaluation framework is based on a discretization of a bivariate model that considers the joint evolution of the equity value with stochastic interest rates. To deeply reduce the computational complexity of the pricing problem we use the singular points framework that allows us to compute accurate upper and lower estimates of the policy premiums.

Full text

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IntroductionEquity linked policies The bivariate modelComputation of the fair premiumNumerical results
Evaluating fair premiums of equity-linked
policies with surrender option in a bivariate
model
M.Costabile1
I.Massabò1
M.Gaudenzi2
A.Zanette2
1Università della Calabria,Italy
2Università di Udine, Italy
Third Conference on Numerical Methods in
Finance,ENPC,2009

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IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Outline
1
Introduction
Introduction
2
Equity linked policies
Equity linked policies
Previous literature
3
The bivariate model
The model
Discretization of the bivariate model
4
Computation of the fair premium
Singular points algorithm
5
Numerical results

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IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Introduction
Introduction
We tackle the problem of computing the fair periodical
premium of an equity-linked policy with a surrender option
and a minimum guarantee

Page 4

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premium Numerical results
Introduction
Introduction
We tackle the problem of computing the fair periodical
premium of an equity-linked policy with a surrender option
and a minimum guarantee
We consider the policy as a Bermudan-style contingent
claims.

Page 5

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Introduction
Introduction
We tackle the problem of computing the fair periodical
premium of an equity-linked policy with a surrender option
and a minimum guarantee
We consider the policy as a Bermudan-style contingent
claims.
A bivariate model describes the evolution of the equity
price and of stochastic interest rate.
Life insurance policies are usually long term contracts, the
assumption of interest rates constant during the whole
policy lifetime is to restrictive.

Page 6

IntroductionEquity linked policiesThe bivariate model Computation of the fair premiumNumerical results
Introduction
Introduction
We tackle the problem of computing the fair periodical
premium of an equity-linked policy with a surrender option
and a minimum guarantee
We consider the policy as a Bermudan-style contingent
claims.
A bivariate model describes the evolution of the equity
price and of stochastic interest rate.
Life insurance policies are usually long term contracts, the
assumption of interest rates constant during the whole
policy lifetime is to restrictive.
We propose a tree algorithm : the framework of the
singular points (Gaudenzi-Lepellere- Zanette [2007]) in
order to estimate the fair periodical premium.

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IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider an equity-linked policy whose payoff depends
on the performance of a portfolio (reference fund).

Page 8

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider an equity-linked policy whose payoff depends
on the performance of a portfolio (reference fund).
The reference fund is generated by investing a fixed
contribution, D, to acquire equities at the beginning of each
year until the maturity T (years).

Page 9

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider an equity-linked policy whose payoff depends
on the performance of a portfolio (reference fund).
The reference fund is generated by investing a fixed
contribution, D, to acquire equities at the beginning of each
year until the maturity T (years).
The payoff function depends on the value of underlying
asset S(ti) at given contribution times ti= i, i = 0,...,T −1.

Page 10

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider an equity-linked policy whose payoff depends
on the performance of a portfolio (reference fund).
The reference fund is generated by investing a fixed
contribution, D, to acquire equities at the beginning of each
year until the maturity T (years).
The payoff function depends on the value of underlying
asset S(ti) at given contribution times ti= i, i = 0,...,T −1.
At time t0= 0, with the first contribution D, the insurer buys
D
S(0)equities. At each contribution time ti, the insurer buys
D
S(ti)equities.

Page 11

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider an equity-linked policy whose payoff depends
on the performance of a portfolio (reference fund).
The reference fund is generated by investing a fixed
contribution, D, to acquire equities at the beginning of each
year until the maturity T (years).
The payoff function depends on the value of underlying
asset S(ti) at given contribution times ti= i, i = 0,...,T −1.
At time t0= 0, with the first contribution D, the insurer buys
D
S(0)equities. At each contribution time ti, the insurer buys
D
S(ti)equities.
The payoff function depends on a minimum guarantee.

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IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
The reference fund at time t, is
F(0) = D,
F(t) = S(t)
j0
?
j=0
D
S(tj),
t > 0
where j0is the largest index j such that tj< t.
The minimum guarantee at time t, is
G(t) =
j0
?
j=0
De(t−tj)δ,
G(T) = DeδeTδ− 1
eδ− 1,
where δ > 0 is the minimum guarantee interest rate.
The payoff function is therefore
φ(F(T),G(T)) = max{F(T),G(T)}.

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IntroductionEquity linked policies The bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider a fixed term policy with periodical premiums P
to be paid at the beginning of each year but not at maturity.

Page 14

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider a fixed term policy with periodical premiums P
to be paid at the beginning of each year but not at maturity.
An amount D of the premium P is invested in the reference
fund.

Page 15

IntroductionEquity linked policiesThe bivariate modelComputation of the fair premiumNumerical results
Equity linked policies
Equity linked policies
We consider a fixed term policy with periodical premiums P
to be paid at the beginning of each year but not at maturity.
An amount D of the premium P is invested in the reference
fund.
A surrender option is embedded into the policy: this allows
to the policyholder to escape out of the contract at the
beginning of each year, just before the premium payment.