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NTU Management Review Vol. 36 No. 1 Apr. 2026

Arrow (1971) proves that, for a risk-averse individual who maximizes his/her expected
utility, the optimal insurance contract is a straight deductible D ∈ (0,L) such that I(x)
*
*
= max (0,x – D ) for all x ∈ [0,L]. Under the deductible design, for a loss x ≤ D , the
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*
individual obtains no indemnity and takes on all the loss. For a loss x > D , the individual
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obtains the partial indemnity x – D and suffers a loss limited to D . In the absence
*
of ambiguity, this result holds for risk-averse individuals with different ambiguity
preferences.

In the presence of ambiguity, we examine the optimality of a straight deductible.
Since a risk-averse and ambiguity-neutral individual is insensitive to ambiguity, the
loss , the individual obtains the partial indemnity and suffers a loss limited to . In the
loss , the individual obtains the partial indemnity and suffers a loss limited to . In the
∗ ∗
∗ ∗
∗ ∗
straight deductible D is still optimal in the presence of ambiguity. The result is described
*
absence of ambiguity, this result holds for risk-averse individuals with different ambiguity preferences.
absence of ambiguity, this result holds for risk-averse individuals with different ambiguity preferences.
in the following proposition:
In the presence of ambiguity, we examine the optimality of a straight deductible. Since a risk-averse
In the presence of ambiguity, we examine the optimality of a straight deductible. Since a risk-averse
and ambiguity-neutral individual is insensitive to ambiguity, the straight deductible is still optimal in
and ambiguity-neutral individual is insensitive to ambiguity, the straight deductible is still optimal in
Proposition 1: Suppose that ambiguity preferences can be described by an α-maxmin
∗ ∗
the presence of ambiguity. The result is described in the following proposition:
the presence of ambiguity. The result is described in the following proposition:
model. Under Assumptions 1–3, for a risk-averse and ambiguity-neutral
individual, the optimally straight deductible D ∈ [0,L] in the absence of
*
Proposition 1. Suppose that ambiguity preferences can be described by an -maxmin model. Under
Proposition 1. Suppose that ambiguity preferences can be described by an -maxmin model. Under
Assumptions 1–3, for a risk-averse and ambiguity-neutral individual, the optimally
Assumptions 1–3, for a risk-averse and ambiguity-neutral individual, the optimally
ambiguity such that I(x) = max (0,x – D ) for all x ∈ [0,L] remains optimal
*
∗ ∗
after the introduction of ambiguity for all x ∈ [0,L].
straight deductible ∈ (0, in the absence of ambiguity such that ( max(0,
straight deductible ∈ (0, in the absence of ambiguity such that ( max(0,
for all ∈ 0, remains optimal after the introduction of ambiguity for all ∈ 0, . .
∗ ∗
for all ∈ 0, remains optimal after the introduction of ambiguity for all ∈ 0,

Proof: The proof is explained as follows.
Proof. The proof is explained as follows.
Proof. The proof is explained as follows.

1
This result can be obtained directly from the objective function (3) with α = ; thus, the
� �
This result can be obtained directly from the objective function (3) with ; thus, the loss distribution
This result can be obtained directly from the objective function (3) with ; thus, the loss distribution
2
*
loss distribution is G(x;π ) under Assumption 2. Following the standard approach to
� �
is ( under Assumption 2. Following the standard approach to solve the optimal insurance contract
is ( under Assumption 2. Following the standard approach to solve the optimal insurance contract
∗ ∗
solve the optimal insurance contract (e.g., Arrow, 1971; Gollier, 2014), the optimum is
(e.g., Arrow, 1971; Gollier, 2014), the optimum is characterized by three first-order conditions (FOCs):
(e.g., Arrow, 1971; Gollier, 2014), the optimum is characterized by three first-order conditions (FOCs):
characterized by three first-order conditions (FOCs):
( ( + ( ( ≤ ( + ( , for all ∈ 0, ,
( ( + ( ( ≤ ( + ( , for all ∈ 0, ,
∗ ∗
∗ ∗
∗ ∗
� �
, (4)
∗ ∗ ∗ (4) (4)
with an equality when ( 0, 0,
with an equality when I(x) > 0,
with an equality when (
� �
(5)
� ( ( + ( ( , (5)
� ( ( + ( ( , (5)
∗ ∗
∗ ∗
� �
� �
� �
(6)
( + � ( ( ( , (6)
( + � ( ( ( , (6)
∗ ∗
∗ ∗
∗ ∗
∗
� � ∗
where is the Lagrange multiplier of the premium constraint (1
where is the Lagrange multiplier of the premium constraint (1). ).
where λ is the Lagrange multiplier of the premium constraint (1).
For a risk- and ambiguity-averse individual in the presence of ambiguity, we first show that, under
For a risk- and ambiguity-averse individual in the presence of ambiguity, we first show that, under
For a risk- and ambiguity-averse individual in the presence of ambiguity, we first
the optimal insurance contract, not all the losses will be reimbursed by the insurer when they occur. We
the optimal insurance contract, not all the losses will be reimbursed by the insurer when they occur. We
present the result in the following proposition:
present the result in the following proposition:
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Proposition 2. Suppose that ambiguity preferences can be described by an -maxmin model. Under
Proposition 2. Suppose that ambiguity preferences can be described by an -maxmin model. Under
Assumptions 1–3, for a risk- and ambiguity-averse individual, there exists at least one
Assumptions 1–3, for a risk- and ambiguity-averse individual, there exists at least one
∈ 0, such that ( 0 under the optimal insurance contract after the introduction
∈ 0, such that ( 0 under the optimal insurance contract after the introduction
of ambiguity for all ∈ 0, . .
of ambiguity for all ∈ 0,

Proof. See Appendix A.
Proof. See Appendix A.

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