on the optimal design of participating life insurance ... · common pitfalls, the journal of risk...

UNIVERSITÀDEGLI STUDI DI TRIESTE

On the optimal designof participating life insurance contracts

Anna Rita Bacinello, Chiara Corsato and Pietro Millossovich

Department of Economics, Business, Mathematics and StatisticsUniversity of Trieste (Italy)

European Actuarial Journal ConferenceLeuven

September 10, 2018

Outline

Motivations

● Optimal insurance, an overview

A constrained maximisation problem

● Setting of the problem

● Mathematical analysis

Study of the fairness condition

An overall picture of the solutions

● Numerical results

State-of-the-art and future work

What do we do?

Framework

Policyholders and shareholders form a life insurance company, which issuesparticipating contracts.

● To maximise analytically the policyholders’ preferences.

● To consider fair valuation and solvency constraints.

● To study the life insurance demand.

● To investigate the problem numerically.

Warning: the results are very preliminary!

References - Starting pointNon life insurance, origins of the optimal insurance problem

P J. Mossin, Aspects of Rational Insurance Purchaising, Journal of Political Economy76, 533-568 (1968).

P K.J. Arrow, Essays in the Theory of Risk Bearing, Markham Publishing Co., 1971.

P A. Raviv, The Design of an Optimal Insurance Policy, American Economic Review,69, 84–96 (1979).

Life insurance, literature of reference

P E. Briys, F. de Varenne, Life Insurance in a Contingent Claim Framework: Pricingand Regulatory Implications, The Geneva Papers on Risk and Insurance Theory,19, 53–72 (1994).

P E. Briys, F. de Varenne, On the Risk of Insurance Liabilities: Debunking SomeCommon Pitfalls, The Journal of Risk and Insurance, 64(4), 673–694 (1997).

P H. Schmeiser, J. Wagner, A Proposal on How the Regulator Should Set MinimumInterest Rate Guarantees in Participating Life Insurance Contracts, The Journal ofRisk and Insurance, 82(3), 659–686 (2015).

P A. Chen, P. Hieber, Optimal Asset Allocation in Life Insurance: the Impact ofRegulation, Astin Bulletin 46(3), 605–626 (2016).

Contract featuresAt time t = 0,

Assets LiabilitiesW0 L0 = αW0

E0 = (1 − α)W0

α ∈ [0,1]: leverage ratio (policyholders’ percentage initial contribution)

At time t = T ,

Assets LiabilitiesWT LT = αW0G´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¶

guaranteed

+ δα(WT −W0G)+´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

− (αW0G −WT )+´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

default

ET =WT − LTWT WT

G ∈ [0,+∞[: guaranteed amount per euro insuredδ ∈ [0,1]: participation rate

The model follows Briys, de Varenne (1994, 1997): considered default risk, nomortality risk.

E0 = (1 − α)W0

At time t = T ,

guaranteed

+ δα(WT −W0G)+´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

− (αW0G −WT )+´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

default

ET =WT − LTWT WT

E0 = (1 − α)W0

At time t = T ,

guaranteed

+ δα(WT −W0G)+´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

− (αW0G −WT )+´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

default

ET =WT − LTWT WT

Policyholders’ final payoff profile

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

WT if WT < αW0G ,

αW0G if αW0G ≤WT <W0G ,

αW0G + δα(WT −W0G) if W0G ≤WT .

AssumptionsP historical measure.

Capital market conditions

● Deterministic risk-free interest rate r .

● Rate of assets return R normally distributed, with (under P)

µ: expected assets return per year,σ: volatility of assets return per year.

R ∼P N ((µ − σ2

2)T , σ2T) .

● µ > r , or, equivalently, risk premium λ = µ−rσ

● There exists a unique risk-neutral measure Q equivalent to P⇒ Complete and arbitrage-free market.

R ∼Q N ((r − σ2

2)T , σ2T) .

● Black-Scholes framework.

Policyholders’ preferences

u ∶ R→ R utility function: u twice differentiable, with u′ > 0, u′′ < 0 in R (homogeneouscohort of risk-averse policyholders).

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

R ∼P N ((µ − σ2

2)T , σ2T) .

R ∼Q N ((r − σ2

2)T , σ2T) .

Constraints imposed

Fairness of the contract

L0 = EQ[e−rTLT ]

or equivalently

α = αGe−rT + αδe−rTEQ[(eR −G)+] − e−rTEQ[(αG − eR)+]

or elseα = αGe−rT + αδC(1,G) − P(1, αG).

Solvency requirements

Maximum ruin probability ε ∈ [0,1] fixed:

P(WT < αW0G) ≤ ε

or equivalently

αG ≤ exp((µ − σ2

2)T +N−1(ε)σ

√T).

Constraints imposed

L0 = EQ[e−rTLT ]or equivalently

or equivalently

2)T +N−1(ε)σ

√T).

Constraints imposed

or equivalently

2)T +N−1(ε)σ

√T).

Constraints imposed

P(WT < αW0G) ≤ εor equivalently

2)T +N−1(ε)σ

√T).

The problem setup

Constrained maximisation problem

Problemmax

(α,G)∈[0,1]×[0,+∞[

EP[u(e−rTLT − L0)]

Constraints

● Fairness conditionL0 = EQ[e−rTLT ]

● Ruin probability condition

Remark: the problem is well-posed.

The problem setup

Constrained maximisation problem

Problemmax

(α,G)∈[0,1]×[0,+∞[

EP[u(e−rTLT − L0)]

Constraints

● Fairness conditionL0 = EQ[e−rTLT ]

● Ruin probability condition

Remark: the problem is well-posed.

A possible interpretation of e−rTLT − L0

The objective function is given by

EP[u(e−rTLT − L0)].

Main characters: policyholders.

Initial investment (t = 0) Final payoff (t = T )Insurance company L0 = αW0 LT

Risk-free (r) (1 − α)W0 (1 − α)W0erT

W0 LT + (1 − α)W0erT

Policyholders’ final profit discounted at t = 0:

e−rT(LT + (1 − α)W0erT) −W0 = e−rTLT + (W0 −W0) − αW0

= e−rTLT − L0.

Pareto-optimality

The maximisation problem studied is related to a Pareto-optimalityproblem, characterised by the following criteria:

Policyholders’

Insurer’s

U(α,G) = EP[u(e−rTLT − L0)]Π(α,G) = P(WT < αW0G)

Theorem

For any given maximal ruin probability ε, the solutions (α,G) of thecorresponding maximisation problem are also pareto-optima.

Pareto-optimality

The maximisation problem studied is related to a Pareto-optimalityproblem, characterised by the following criteria:

Policyholders’

Insurer’s

U(α,G) = EP[u(e−rTLT − L0)]Π(α,G) = P(WT < αW0G)

Theorem

For any given maximal ruin probability ε, the solutions (α,G) of thecorresponding maximisation problem are also pareto-optima.

Study of the fairness condition

Set of admissible solutions

The extremal case δ = 1

Theorem (δ = 1)

The solutions of the constrained maximisation problem are given by

● (α̂,0), for a unique α̂ ∈ ]0,1[, if EP[u′(W0(eR−rT − 1))(eR−rT − 1)] ≥ 0.

● (1,G), for all G ≤ exp((µ − σ2

2)T +N−1(ε)σ√T), otherwise.

The extremal case δ = 1

Theorem (δ = 1)

The solutions of the constrained maximisation problem are given by

● (α̂,0), for a unique α̂ ∈ ]0,1[, if EP[u′(W0(eR−rT − 1))(eR−rT − 1)] ≥ 0.

● (1,G), for all G ≤ exp((µ − σ2

2)T +N−1(ε)σ√T), otherwise.

The case δ ∈ [0,1[ - Preliminary results

Theorem (δ ∈ [0,1[)The only solution of the constrained maximisation problem is given by(α̃,Gδ(α̃)), for some α̃ ∈ ]0,1[.

Numerical results - 1

0.0 0.2 0.4 0.6 0.8 1.0

δ=0.80

0.0 0.2 0.4 0.6 0.8 1.0

δ=0.95

Certainty equivalent related to the policyholders’ expected utility, with δ = 0.80 (left),δ = 0.95 (right): T = 20, W0 = 100, r = 0.03, λ = 0.1, σ = 0.20 (µ = 0.05), ε = 0.05,

u(x) = (x+W0)1−ρ1−ρ , ρ = 1.5.

The ruin probability condition at the optimal solution may be binding or not!

Numerical results - 1

0.0 0.2 0.4 0.6 0.8 1.0

δ=0.80

0.0 0.2 0.4 0.6 0.8 1.0

δ=0.95

Certainty equivalent related to the policyholders’ expected utility, with δ = 0.80 (left),δ = 0.95 (right): T = 20, W0 = 100, r = 0.03, λ = 0.1, σ = 0.20 (µ = 0.05), ε = 0.05,

u(x) = (x+W0)1−ρ1−ρ , ρ = 1.5.

The ruin probability condition at the optimal solution may be binding or not!

Comparative statics (δ = 0.80)

●●

0.0 0.1 0.2 0.3 0.4 0.5 0.6

risk premium comparative statics

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400.

volatility comparative statics

Dependence of the optimal leverage ratio αon the risk premium λ, with σ = 0.20.

Dependence of the optimal leverage ratio αon the volatility σ, with λ = 0.1.

Comparative statics - 2

2 3 4 5 6 7 8

(relative) risk aversion comparative statics

Dependence of the optimal leverage ratio α on the relative risk aversion ρ, with λ = 0.1,

σ = 0.20, δ = 0.80.

To sum up

● We consider a model for the creation of a life insurance company issuingparticipating contracts.

● We maximise the policyholders’ preferences.

● We impose fair valuation and solvency constraints.

● Analytical results: if λ > 0 and G is upper unbounded, then the optimal solutioninduces the policyholders to insure (α > 0).

The properties of the optimal contracts depend on the choice of δ (δ = 1, δ < 1)and, if δ = 1, on a condition involving the policyholders’ risk aversion.

● Numerical results: according to the parameters of the contract chosen, the ruinprobability condition evaluated at the optimal solution may be binding or not.

This contradicts the assumption made in Schmeiser, Wagner (2015).

To sum up

State-of-the-art and possible developments

● The maximisation problem has been completely solved analyticallywhen δ = 1.

What can be said more on the case δ < 1 (and λ > 0)?

● The model has been constructed considering the investment risk, only.

What if the mortality risk is also taken into account?

● The risk-free rate r is assumed to be deterministic.

What if it is stochastic (Briys, de Varenne, 1994)?

● The rate of assets return R is modelled by a normal distribution.

What if R has a different type of distribution?

Thank you for your attention!

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