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

Appendices

Appendix A: Proof of Proposition 2

Due to ambiguity aversion, the first-order conditions (FOCs) (4) and (5) become:

( ( )+ ( )) ( ) + (1 ) ( )] ≤ (1 + ) ( )
∗
∗
�
�

for all ], with an equality when ( ) ,
(A1)
and
�
∗
�
� ( ( ) + ( )) � ( ) + (1 ) ( )� = ,
�
�
�
(A2)
while Condition (6) is unchanged.
Suppose by contradiction that the insurer provides ( ) for all ] . Then,
�
∗
�
Condition (A1) holds at the equality. This means that � ( ( )+
�
( )) � ( ) + (1 ) ( )� = (1 + ), which contradicts Condition (A2) since
�
�
. As a result, there exists at least one ] that will not be reimbursed by the insurer when it
occurs.
Q.E.D.

Appendix B: Proof of Proposition 3

Let = and = + , where is positive and small. Then, the degree of
∗
∗
�
�
ambiguity is represented by the set + ]. When = , this is the case without
∗
∗
ambiguity, and Proposition 1 states that is optimal satisfying Conditions (4) to (6).
∗
In the presence of ambiguity, . The optimal insurance contract should satisfy Conditions
(A1) and (A2). By the continuity of the optimal insurance contract, when , we know that
∗
Conditions (A1) and (A2) hold. Accordingly, there exists a straight deductible such that ( ) =
�
max( ) for all ] which is optimal.
∗
�
Q.E.D.

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