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On the Optimal Designof Participating Life Insurance Contracts
Pietro MillossovichCass Business School, City, University of London
and DEAMS, University of Trieste(joint with Chiara Corsato and Anna Rita Bacinello)
OICA Conference 2020
1
Aim
• Investigate how policyholders and shareholders contribute to theformation of a life insurance company
• Focus on stylized, participating contracts
• policyholders max their preferences over
B contribution rateB minimum guaranteed
for given participation rate
• role of financial and (systematic) longevity risk ⇒ focus on a largeportfolio
• discuss insurance demand - role of regulatory environment (SII)constraints
B fair pricingB solvency
2
Aim
• Investigate how policyholders and shareholders contribute to theformation of a life insurance company
• Focus on stylized, participating contracts
• policyholders max their preferences over
B contribution rateB minimum guaranteed
for given participation rate
• role of financial and (systematic) longevity risk ⇒ focus on a largeportfolio
• discuss insurance demand - role of regulatory environment (SII)constraints
B fair pricingB solvency
2
Aim
• Investigate how policyholders and shareholders contribute to theformation of a life insurance company
• Focus on stylized, participating contracts
• policyholders max their preferences over
B contribution rateB minimum guaranteed
for given participation rate
• role of financial and (systematic) longevity risk ⇒ focus on a largeportfolio
• discuss insurance demand - role of regulatory environment (SII)constraints
B fair pricingB solvency
2
References - Optimal Insurance
• Non Life Insurance (. . .)
• (Participating) Life insurance
B Briys and de Varenne, GPRIT (1994), JRI (1997) ⇒ extended in manydirections by Jorgensen, Le Courtois, Quittard-Pinon, A. Chen,Bernard, . . .⇒ stochastic interest rates, continuous regulatorymonitoring, surrender, . . .
B Bacinello et al., EAJ (2018)B Schmeiser and Wagner, JRI (2015), Braun et al. JRF (2019)B Chen and Hieber, Astin (2016)B Gatzert et al., JRI (2012)B Huang et al., JRI (2008)
3
References - Optimal Insurance
• Non Life Insurance (. . .)
• (Participating) Life insurance
B Briys and de Varenne, GPRIT (1994), JRI (1997) ⇒ extended in manydirections by Jorgensen, Le Courtois, Quittard-Pinon, A. Chen,Bernard, . . .⇒ stochastic interest rates, continuous regulatorymonitoring, surrender, . . .
B Bacinello et al., EAJ (2018)B Schmeiser and Wagner, JRI (2015), Braun et al. JRF (2019)B Chen and Hieber, Astin (2016)B Gatzert et al., JRI (2012)B Huang et al., JRI (2008)
3
Contract features - finite portfolio
• Insurer’s capital structure at t = 0
Assets LiabilitiesW0 L0 = αW0 = global premium
E0 = (1− α)W0 = equityholders’ contribution
W0 W0
• Pure endowment with maturity T
• 0 ≤ α ≤ 1: leverage ratio
• G ≥ 0: guaranteed amount per £ insured
• 0 ≤ δ ≤ 1: participation rate
4
Contract features - finite portfolio
• Insurer’s capital structure at t = 0
Assets LiabilitiesW0 L0 = αW0 = global premium
E0 = (1− α)W0 = equityholders’ contribution
W0 W0
• Pure endowment with maturity T
• 0 ≤ α ≤ 1: leverage ratio
• G ≥ 0: guaranteed amount per £ insured
• 0 ≤ δ ≤ 1: participation rate
4
Contract features - finite portfolio
• N0 initial (homogeneous) policyholders
• N =∑N0
i=1 1Ei survivors at T (Ei = ‘policyholder i is alive at T ’)
• WT = W0eR, assets at maturity, R: log-return
• Guaranteed global payment
GT =αW0
N0GN
• Global liability at T : following (Briys and de Varenne, 1994, 1997),on {N > 0}
LT = GT︸︷︷︸guaranteed
+ δ(αWT −GT
)+
︸ ︷︷ ︸bonus
−(GT −WT
)+
︸ ︷︷ ︸default
5
Contract features - finite portfolio
• N0 initial (homogeneous) policyholders
• N =∑N0
i=1 1Ei survivors at T (Ei = ‘policyholder i is alive at T ’)
• WT = W0eR, assets at maturity, R: log-return
• Guaranteed global payment
GT =αW0
N0GN
• Global liability at T : following (Briys and de Varenne, 1994, 1997),on {N > 0}
LT = GT︸︷︷︸guaranteed
+ δ(αWT −GT
)+
︸ ︷︷ ︸bonus
−(GT −WT
)+
︸ ︷︷ ︸default
5
Contract features - finite portfolio
• N0 initial (homogeneous) policyholders
• N =∑N0
i=1 1Ei survivors at T (Ei = ‘policyholder i is alive at T ’)
• WT = W0eR, assets at maturity, R: log-return
• Guaranteed global payment
GT =αW0
N0GN
• Global liability at T : following (Briys and de Varenne, 1994, 1997),on {N > 0}
LT = GT︸︷︷︸guaranteed
+ δ(αWT −GT
)+
︸ ︷︷ ︸bonus
−(GT −WT
)+
︸ ︷︷ ︸default
5
Pricing assumptions
• P historical measure
B R has a distribution with support RB Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with
P(Ei|Π) = Π
B R independent of Π, (Ei)
• P ∼ P, insurer’s pricing measure
B Risk neutrality condition: E[eR] = erT , r = risk-free interest rate
B Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with
P(Ei|Π) = Π
B R independent of Π, (Ei)
• Π, Π: (systematic) longevity risk
6
Pricing assumptions
• P historical measure
B R has a distribution with support RB Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with
P(Ei|Π) = Π
B R independent of Π, (Ei)
• P ∼ P, insurer’s pricing measure
B Risk neutrality condition: E[eR] = erT , r = risk-free interest rate
B Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with
P(Ei|Π) = Π
B R independent of Π, (Ei)
• Π, Π: (systematic) longevity risk
6
Pricing assumptions
• P historical measure
B R has a distribution with support RB Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with
P(Ei|Π) = Π
B R independent of Π, (Ei)
• P ∼ P, insurer’s pricing measure
B Risk neutrality condition: E[eR] = erT , r = risk-free interest rate
B Conditionally on 0 < Π < 1 the Ei’s are i.i.d. with
P(Ei|Π) = Π
B R independent of Π, (Ei)
• Π, Π: (systematic) longevity risk
6
Pricing measure P vs historical measure P
• risk premium for financial risk
P(R) �st P(R)
⇒ E[eR] ≥ erT
• loading for (systematic) longevity risk
P(Π) �st P(Π)
⇒ P(Ei) ≥ P(Ei)
7
Pricing measure P vs historical measure P
• risk premium for financial risk
P(R) �st P(R)
⇒ E[eR] ≥ erT
• loading for (systematic) longevity risk
P(Π) �st P(Π)
⇒ P(Ei) ≥ P(Ei)
7
Large portfolio
• Global liability in a finite portfolio: on {N > 0}
LT = GT︸︷︷︸guaranteed
+ δ(αWT −GT
)+
︸ ︷︷ ︸bonus
−(GT −WT
)+
︸ ︷︷ ︸default
• Assumption: as N0 → +∞
W0
N0→ w0(> 0)
w0 = assets per individual contract in a large portfolio
• LLN for exchangeable sequences: as N0 → +∞
N
N0→ Π a.s. under P (→ Π a.s. under P)
8
Large portfolio
• Global liability in a finite portfolio: on {N > 0}
LT = GT︸︷︷︸guaranteed
+ δ(αWT −GT
)+
︸ ︷︷ ︸bonus
−(GT −WT
)+
︸ ︷︷ ︸default
• Assumption: as N0 → +∞
W0
N0→ w0(> 0)
w0 = assets per individual contract in a large portfolio
• LLN for exchangeable sequences: as N0 → +∞
N
N0→ Π a.s. under P (→ Π a.s. under P)
8
Large portfolio
• Global liability in a finite portfolio: on {N > 0}
LT = GT︸︷︷︸guaranteed
+ δ(αWT −GT
)+
︸ ︷︷ ︸bonus
−(GT −WT
)+
︸ ︷︷ ︸default
• Assumption: as N0 → +∞
W0
N0→ w0(> 0)
w0 = assets per individual contract in a large portfolio
• LLN for exchangeable sequences: as N0 → +∞
N
N0→ Π a.s. under P (→ Π a.s. under P)
8
Large portfolio
• Individual liability in a large portfolio: on Ei
`(i) = limN0→+∞
LTN
= αw0G︸ ︷︷ ︸guaranteed
+ δαw0
(eR
Π−G
)+
︸ ︷︷ ︸bonus
−w0
(αG− eR
Π
)+
︸ ︷︷ ︸default
a.s. under P
• Same holds under P with Π replaced by Π
9
Large portfolio
• Individual liability in a large portfolio: on Ei
`(i) = limN0→+∞
LTN
= αw0G︸ ︷︷ ︸guaranteed
+ δαw0
(eR
Π−G
)+
︸ ︷︷ ︸bonus
−w0
(αG− eR
Π
)+
︸ ︷︷ ︸default
a.s. under P
• Same holds under P with Π replaced by Π
9
Individual liability profile
10
The Policyholders’ Problem
• Policyholders’ preferences
u : VNM utility function, u′ > 0, u′′ < 0
• (representative) Policyholder’s decision problem
maxα,G
E[u(
e−rT `(i) − αw0︸ ︷︷ ︸NPV
)]
• Regulatory constraintsB maximum guaranteedB fair pricingB solvency based capital allocation criterion
11
The Policyholders’ Problem
• Policyholders’ preferences
u : VNM utility function, u′ > 0, u′′ < 0
• (representative) Policyholder’s decision problem
maxα,G
E[u(
e−rT `(i) − αw0︸ ︷︷ ︸NPV
)]
• Regulatory constraintsB maximum guaranteedB fair pricingB solvency based capital allocation criterion
11
The Policyholders’ Problem
• Policyholders’ preferences
u : VNM utility function, u′ > 0, u′′ < 0
• (representative) Policyholder’s decision problem
maxα,G
E[u(
e−rT `(i) − αw0︸ ︷︷ ︸NPV
)]
• Regulatory constraintsB maximum guaranteedB fair pricingB solvency based capital allocation criterion
11
The Policyholders’ Problem
• Regulatory constraintsB Regulatory cap on minimum guaranteed
G ≤ G
B Fairness conditionαw0 = E
[e−rT `(i)
]B Solvency constraint, 0 < ε < 1
P(w0eR < w0αGΠ
)≤ ε
12
The Policyholders’ Problem
• Regulatory constraintsB Regulatory cap on minimum guaranteed
G ≤ G
B Fairness conditionαw0 = E
[e−rT `(i)
]
B Solvency constraint, 0 < ε < 1
P(w0eR < w0αGΠ
)≤ ε
12
The Policyholders’ Problem
• Regulatory constraintsB Regulatory cap on minimum guaranteed
G ≤ G
B Fairness conditionαw0 = E
[e−rT `(i)
]B Solvency constraint, 0 < ε < 1
P(w0eR < w0αGΠ
)≤ ε
12
The Policyholders’ Problem
• Policyholder’s decision problem
maxα,G
E[u(e−rT `(i) − αw0
)]
• Solution (α∗, G∗)
• can be seen as optimal allocation problem
• optimal contracts include Pareto efficient contracts (criteria: expectedutility for ph, ruin prob for insurer)
13
The Policyholders’ Problem
• Policyholder’s decision problem
maxα,G
E[u(e−rT `(i) − αw0
)]
• Solution (α∗, G∗)
• can be seen as optimal allocation problem
• optimal contracts include Pareto efficient contracts (criteria: expectedutility for ph, ruin prob for insurer)
13
Regulatory constraints (δ < 1)
α
G
g(0)=G
g(α)
0 1
αG=xε
G
erT
E~(Π~)
14
Preliminaries
• Which factors drive the insurance demand: α∗ = 0 or α∗ > 0?
• Distinguish partial participation δ < 1 from full participation δ = 1
Lemma
if P(R) = P(R) then α∗ = 0,
hence assumeP(R) ≺st P(R)
15
Preliminaries
• Which factors drive the insurance demand: α∗ = 0 or α∗ > 0?
• Distinguish partial participation δ < 1 from full participation δ = 1
Lemma
if P(R) = P(R) then α∗ = 0,
hence assumeP(R) ≺st P(R)
15
Full Participation (δ = 1)
• Fairness: α = 0 or G = 0 or α = 1
Theorem
α∗ > 0 alwaysα∗ = 1 (mutual company) iff
E[u′(w0(J − 1))(J − 1)Π− u′(−w0)(1−Π)
]≥ 0,
J = eR−rT
Π
• E.g., CARA, α∗ < 1 when risk aversion is small
16
Full Participation (δ = 1)
• Fairness: α = 0 or G = 0 or α = 1
Theorem
α∗ > 0 alwaysα∗ = 1 (mutual company) iff
E[u′(w0(J − 1))(J − 1)Π− u′(−w0)(1−Π)
]≥ 0,
J = eR−rT
Π
• E.g., CARA, α∗ < 1 when risk aversion is small
16
Partial participation (δ < 1)
• Fairness: G = g(α) for α > 0
Theorem: δ = 0 (traditional policy)
When E[Π] < E[Π], there exists G′ ≥ erT /E[Π] st α∗ = 0 for all G ≤ G′
Theorem: 0 < δ < 1
There exists d(G) ↓ G with d(G) > 0 iff G < erT /E[Π] st
1 if 0 < δ ≤ d(G) then α∗ = 0,
2 if δ > d(G) and
g(0)E[Π]
+ δE[(eR − g(0)Π)+
]> erT , (∗)
then α∗ > 0
17
Partial participation (δ < 1)
• Fairness: G = g(α) for α > 0
Theorem: δ = 0 (traditional policy)
When E[Π] < E[Π], there exists G′ ≥ erT /E[Π] st α∗ = 0 for all G ≤ G′
Theorem: 0 < δ < 1
There exists d(G) ↓ G with d(G) > 0 iff G < erT /E[Π] st
1 if 0 < δ ≤ d(G) then α∗ = 0,
2 if δ > d(G) and
g(0)E[Π]
+ δE[(eR − g(0)Π)+
]> erT , (∗)
then α∗ > 0
17
. . . Partial participation (δ < 1)• Fairness: G = g(α) for α > 0• when does the condition hold?
Corollary
1 There exists 0 < δ′ < 1 st α∗ > 0 for all δ > δ′
2 For δ > d(G) and Π “close” to Π, then α∗ > 0
Corollary: binding constraints
1 If for G > 0 and d(G) < δ < 1 condition (∗) holds, there exists ε′ > 0 st forall 0 < ε ≤ ε′,
P(w0eR < w0α∗G∗Π) = ε,
2 If for 0 < δ < 1 condition (∗) holds , there exists G′> g(0) st
G∗ = g(α∗) = G
for all g(0) < G < G′.
18
. . . Partial participation (δ < 1)• Fairness: G = g(α) for α > 0• when does the condition hold?
Corollary
1 There exists 0 < δ′ < 1 st α∗ > 0 for all δ > δ′
2 For δ > d(G) and Π “close” to Π, then α∗ > 0
Corollary: binding constraints
1 If for G > 0 and d(G) < δ < 1 condition (∗) holds, there exists ε′ > 0 st forall 0 < ε ≤ ε′,
P(w0eR < w0α∗G∗Π) = ε,
2 If for 0 < δ < 1 condition (∗) holds , there exists G′> g(0) st
G∗ = g(α∗) = G
for all g(0) < G < G′.
18
. . . Partial participation (δ < 1)• Fairness: G = g(α) for α > 0• when does the condition hold?
Corollary
1 There exists 0 < δ′ < 1 st α∗ > 0 for all δ > δ′
2 For δ > d(G) and Π “close” to Π, then α∗ > 0
Corollary: binding constraints
1 If for G > 0 and d(G) < δ < 1 condition (∗) holds, there exists ε′ > 0 st forall 0 < ε ≤ ε′,
P(w0eR < w0α∗G∗Π) = ε,
2 If for 0 < δ < 1 condition (∗) holds , there exists G′> g(0) st
G∗ = g(α∗) = G
for all g(0) < G < G′.
18
Conclusions & Extensions
• Main features:
B Analytical yet stylized model for participating life officesB General policyholder’s preferencesB effect of regulator (G, solvency, ruin) or corporate (δ, σ) on insurance
demandB effect of (systematic) longevity risk
• Extensions:
B Stochastic interest ratesB Dynamic modelB Deferred (guaranteed) annuities vs pure endowment?B Asymmetric information
19
Conclusions & Extensions
• Main features:
B Analytical yet stylized model for participating life officesB General policyholder’s preferencesB effect of regulator (G, solvency, ruin) or corporate (δ, σ) on insurance
demandB effect of (systematic) longevity risk
• Extensions:
B Stochastic interest ratesB Dynamic modelB Deferred (guaranteed) annuities vs pure endowment?B Asymmetric information
19