Abstract
It is known that the zero utility principle under Cumulative Prospect Theory can be uniquely extended from the family of all ternary risks. On the other hand, the extension from the family of all binary risks need not be unique. We establish a characterization of those zero utility principles which coincide on the family of binary risks. The characterization is expressed in terms of relations between the systems of generators of the principles.
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1 Introduction
The zero utility principle, introduced by Bühlmann [2], is a method of insurance contract pricing based on a utility function. According to the principle, an insurance premium for a risk X, represented by a non-negative essentially bounded random variable on a given probability space, is defined implicitly as a unique real number \(H_{u}(X)\) satisfying the equation
where \(u:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous strictly increasing utility function such that \(u(0)=0\). For more details concerning the properties of the principle defined by (1) we refer e.g. to [1, 2, 4, 10, 14]. In this paper we deal with the zero utility principle under the Cumulative Prospect Theory, one of the behavioral models of decision-making under risk. The principle in this setting was introduced by Kałuszka and Krzeszowiec [11, 12]. In their approach a premium for a risk X is defined through the equation
where, for any essentially bounded random variable Y,
is the Choquet integral with respect to the probability weighting functions g for gains and h for losses, that is non-decreasing functions mapping [0, 1] into [0, 1] and satisfying the conditions \(g(0)=h(0)=0\) and \(g(1)=h(1)=1\). It was proved in [3] that, if g and h are continuous, then the premium is uniquely determined by (2) for every risk X if and only if
In [5, 7] and [11, 12] a series of results concerning characterizations of some important properties of the premium defined by (2) were proved. Furthermore, in [6] and [8], the extension problem for such a premium was investigated. In particular, it was proved in [6] that, if g and h are strictly increasing and continuous, then the zero utility principle defined by (2) can be uniquely extended from the family of all ternary risks. However (cf. e.g. the case \(w=0\) in [6, Example 3.5] and [8, Example 1]) the extension from the family of all binary risks need not be unique. The aim of this paper is to establish a characterization of the zero utility principles coinciding on the family of all binary risks. A crucial role in our considerations is played by the solutions of the multiplicative Pexider equation on an open and connected subset of \((0, \infty )^2\).
In the next section we present basic facts concerning the zero utility principle for binary risks. The main result is formulated and proved in the third section. In the last section some consequences of the main result are discussed.
2 Preliminary results
Assume that \((\Omega , {\mathcal {F}},P)\) is a non-atomic probability space, \(g,h:[0,1]\rightarrow [0,1]\) are continuous probability weighting functions satisfying (4) and \(u:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous utility function such that \(u(0)=0\). Let \({\mathcal {X}}\) be a family of all risks, that is the non-negative essentially bounded random variables on \((\Omega , {\mathcal {F}},P)\). By
we denote a family of all binary risks on the space \((\Omega , {\mathcal {F}},P)\). According to (3), for any \(X=x\cdot 1_{A}\in {\mathcal {X}}^{(2)}\) Eq. (2) becomes
Thus, we get
Moreover, it follows from (5) that the premium for a risk \(X\in {\mathcal {X}}^{(2)}\) depends only on a distribution of X. Note also that, as the probability space \((\Omega , {\mathcal {F}},P)\) is non-atomic, we have \(\{P(A):A\in {\mathcal {F}}\}=[0,1]\). Therefore, in view of (5), for every \(x\in [0,\infty )\) and \(p\in [0,1]\), we obtain
where \(H_{(u,g,h)}(x;p)\) denotes a premium for any risk \(x\cdot 1_{A}\in {\mathcal {X}}^{(2)}\) such that \(P(A)=p\). In view of (6), we have
Remark 2.1
Assume that, for \(i\in \{1,2\}\), \(g_i\) and \(h_i\) are continuous probability weighting functions satisfying the condition
and \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function with \(u_i(0)=0\). If
then according to (7), for \(i\in \{1,2\}\), we get
Lemma 2.2
Assume that, for \(i\in \{1,2\}\), \(g_i\) and \(h_i\) are continuous probability weighting functions satisfying (9) and \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function such that \(u_i(0)=0\). Let
and
If (10) holds then
and
Proof
Assume that (10) holds. Then, according to Remark 2.1, (11) is valid for \(i\in \{1,2\}\). Suppose that \(p_{g_1}\ne p_{g_2}\), say \(p_{g_1}<p_{g_2}\) and fix a \(p\in (1-p_{g_2},1-p_{g_1})\). Then \(1-p\in (p_{g_1},p_{g_2})\) and so \(g_2(1-p)=0<g_1(1-p)\). Hence, making use of (11) for \(i=2\) and \(x=1\), we get \(h_2(p)u_2(H(1;p)-1)=0\). Furthermore, in view of (9), we have \(h_2(p)>0\). Thus, \(u_2(H(1;p)-1)=0\) and so \(H(1;p)=1\). Therefore, applying (11) again, this time for \(i=1\) and \(x=1\), we obtain
which yields a contradiction. In this way we have proved (12). The proof of (13) is similar. Note that, according to (9) and (12)–(13), we get
and so
which implies (14). Finally, (12)–(13) imply (15). \(\square \)
Lemma 2.3
Assume that, for \(i\in \{1,2\}\), \(g_i\) and \(h_i\) are continuous probability weighting functions satisfying (9) and \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function with \(u_i(0)=0\). Let \(\psi _i:(p_h,1-p_g)\rightarrow (0,1)\) for \(i\in \{1,2\}\) be given by
If (10) holds then, for every \(x\in [0,\infty )\), \(p\in (p_h,1-p_g)\) and \(i\in \{1,2\}\), we have
Moreover, for every \(p,q\in (p_h,1-p_g)\) the following conditions are pairwise equivalent:
-
(i)
\(\psi _1(p)=\psi _1(q)\);
-
(ii)
\(\psi _2(p)=\psi _2(q)\);
-
(iii)
\(H(x;p)=H(x;q) \;\;\; \text{ for } \;\;\; x\in [0,\infty )\).
Proof
Assume that (10) holds. Then it follows from Remark 2.1 that (11) is valid for \(i\in \{1,2\}\). Dividing both sides of (11) by \(h_i(p)+g_i(1-p)\), in view of (9) and (16), we conclude that (17) holds for every \(x\in [0,\infty )\), \(p\in [0,1]\) and \(i\in \{1,2\}\).
Let \(p,q\in (p_h,1-p_g)\). If (i) holds then, applying (17) with \(i=1\), for every \(x\in (0,\infty )\), we get
Since \(u_1\) is strictly increasing, this implies (iii). Conversely, if (iii) holds then making use of (17), we obtain
Hence, as \(u_1\) is strictly increasing, we have (i). This proves the equivalence of (i) and (iii). In a similar way one can show that (ii) and (iii) are equivalent.\(\square \)
Lemma 2.4
Assume that, for \(i\in \{1,2\}\), \(g_i\) and \(h_i\) are continuous probability weighting functions satisfying (9), \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function with \(u_i(0)=0\) and \(\psi _i:(p_h,1-p_g)\rightarrow (0,1)\) is defined by (16). Let \({\overline{\psi }}:(0,1)\rightarrow {\mathbb {R}}\) and \({\overline{H}}:(0,\infty )\times (0,1)\rightarrow {\mathbb {R}}\) be given by
and
respectively. If (10) holds then
and
Furthermore, for every \(p\in (p_h,1-p_g)\), a function
is strictly increasing and continuous, with
Proof
Assume that (10) holds. Applying Lemma 2.3 and making use of (18), for every \(p\in (p_h,1-p_g)\), we get
which gives (20). Furthermore, taking Lemma 2.3 into account, in view of (19), for every \(x\in (0,\infty )\) and \(p\in (p_h,1-p_g)\), we obtain
Thus, making use of (17) and (20), for every \(x\in (0,\infty )\) and \(p\in (p_h,1-p_g)\), we get
and
Moreover, it follows from (16) that \(\psi _1\) is continuous, with
and
Hence \(\{\psi _1(p):p\in (p_h,1-p_g)\}=(0,1)\) and so, from (25) and (26) we derive (21) and (22), respectively.
Fix a \(p\in (p_h,1-p_g)\) and suppose that the function given by (23) is not strictly increasing. Then there exist \(x,y\in (0,\infty )\) such that \(x<y\) and \({\overline{H}}(y,p)\le {\overline{H}}(x,p)\). Hence \({\overline{H}}(y,p)-y<{\overline{H}}(x,p)-x\) and so, making use of (21), we get
which yields a contradiction. Therefore, the function defined by (23) is strictly increasing. Let \(z\in (0,\infty )\). Then the limits \(l(z):=\lim _{x\rightarrow z^-}{\overline{H}}(x,p)\) and \(r(z):=\lim _{x\rightarrow z^+}{\overline{H}}(x,p)\) exist and they are finite. Furthermore, as \(u_1\) is continuous, passing in (21) to the limit with \(x\rightarrow z^-\) and then with \(x\rightarrow z^+\), we obtain
Hence
Since \(u_1\) is strictly increasing, this implies that \(l(z)=r(z)\) and proves the continuity of the function given by (23). Finally note that, letting in (21) \(x\rightarrow 0^+\), we get (24). \(\square \)
Remark 2.5
It follows from Lemma 2.4 that, for every \(p\in (0,1)\), there exists a limit
and
Let \(u_1(-\infty ):=\lim _{x\rightarrow -\infty }u_1(x)\) and \(u_1(\infty ):=\lim _{x\rightarrow \infty }u_1(x)\). Then, letting in (21) \(x\rightarrow \infty \), we conclude that either \(u_1(-\infty )=-\infty \) and \(c(p)=\infty \) for \(p\in (0,1)\), or \(-\infty< u_1(-\infty )< 0\) and
Thus, in view of (28), for every \(p\in (0,1)\), we have
with the convention \(a\cdot (-\infty )=\infty \) for \(a\in (-\infty ,0)\).
3 Main result
The following theorem is the main result of the paper.
Theorem 3.1
Assume that, for \(i\in \{1,2\}\), \(g_i\) and \(h_i\) are continuous probability weighting functions satisfying (9) and \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function with \(u_i(0)=0\). Then (10) holds if and only if there exist \(\alpha ,\beta ,r\in (0,\infty )\) such that
and
Proof
Assume that (10) holds. Then, making use of (21) and (22), we get
and
respectively. Hence, for every \(x\in (0,\infty )\) and \(p\in (0,1)\), we have
and so, taking
we obtain
Replacing in this equality p by \(\frac{s}{s+1}\), for every \(s\in (0,\infty )\) and \(y\in \left\{ u_1\left( {\overline{H}}\left( x,\frac{s}{s+1})\right) \right) :x\in (0,\infty )\right\} \), we get
Furthermore, in view of (29), for every \(s\in (0,\infty )\), we have
Therefore, we get
where
Note that D is a non-empty, open and connected subset of \((0,\infty )^2\), with
and
Moreover, in view of (32), f is a strictly increasing continuous function. Hence, applying [13, Theorem 13.1.6, p. 349] and [15, Corollary 2], we obtain that there exist \(a,b,r\in (0,\infty )\) such that
and
It follows from (33) that
Hence, making use of (16) and (20), we get
Furthermore, in view of (12)–(13), the last equality trivially holds also for \(p\in [0,1]\setminus (p_h,1-p_g)\). Therefore, taking (32) and (34)–(35) into account, we obtain (30)–(31) with \(\alpha :=ab\) and \(\beta :=b\).
In order to prove the converse implication, assume that (30)–(31) hold with some \(\alpha ,\beta ,r\in (0,\infty )\). Fix an \(x\in (0,\infty )\) and a \(p\in [0,1]\). If \(g_1(1-p)=0\) then from (9) and (30) we deduce that \(g_2(1-p)=0\) and \(h_1(p)h_2(p)>0\). Hence, as \(u_i\) for \(i\in \{1,2\}\) is strictly increasing, with \(u_i(0)=0\), in view of (7), we get
If \(g_1(1-p)>0\), then applying (6) and (30)–(31), we obtain
On the other hand, according to (7), we have
Thus
which, in view of (7), implies that \(H_{(u_1,g_1,h_1)}(x;p)=H_{(u_2,g_2,h_2)}(x;p)\). Hence, (10) holds. \(\square \)
4 Applications
In this section we show that some known results concerning the zero utility principle in various settings can be directly derived from Theorem 3.1. We begin with the following two simple observations.
Remark 4.1
Let g and h be continuous probability weighting functions satisfying (4) and let \(u:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be a strictly increasing continuous function such that \(u(0)=0\). Assume that the zero utility principle defined by (2) coincides on the family of all binary risks with the net premium, that is
Let \(g_0,h_0:[0,1]\rightarrow [0,1]\) be given by \(g_0(p)=h_0(p)=p\) for \(p\in [0,1]\) and let \(u_0:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be of the form \(u_0(x)=x\) for \(x\in {\mathbb {R}}\). Then, for any \(X=x\cdot 1_{A}\in {\mathcal {X}}^{(2)}\), we have
Thus, taking (5) into account, we conclude that \(H_{(u_0,g_0,h_0)}(X)=E[X]\) for \(X\in {\mathcal {X}}^{(2)}\). Hence, in view of (36), we get
Therefore, applying Theorem 3.1, we obtain [5, Theorem 3.1], which says that (36) holds if and only if there exist \(\alpha ,\beta ,r\in (0,\infty )\) such that
and
Remark 4.2
Assume that g and h are continuous probability weighting functions satisfying (4), \(u:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function with \(u(0)=0\) and the zero utility principle defined by (2) coincides on the family of all binary risks with the exponential premium, i.e.
where \(c\in (0,\infty )\) is fixed. Let \(g_0\) and \(h_0\) be as in Remark 4.1 and let \(u_0:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be given by \(u_0(x)=1-e^{-cx}\) for \(x\in {\mathbb {R}}\). Then, for every \(X=x\cdot 1_{A}\in {\mathcal {X}}^{(2)}\), we obtain
Hence, in view of (5), we have \(H_{(u_0,g_0,h_0)}(X)=\frac{1}{c}\ln E[e^{cX}]\) for \(X\in {\mathcal {X}}^{(2)}\) and so making use of (39), we get (37). Thus, applying Theorem 3.1, we obtain that (39) holds if and only if there exist \(\alpha ,\beta ,r\in (0,\infty )\) such that (38) is valid and
Therefore, [5, Theorem 4.1] is a particular case of Theorem 3.1.
The next result shows that, if the probability weighting functions are fixed, then the zero utility principle can be uniquely extended from the family of all binary risks onto the family of all risks.
Corollary 4.3
Assume that g and h are continuous probability weighting functions satisfying (4) and \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) for \(i\in \{1,2\}\) is a strictly increasing continuous function with \(u_i(0)=0\). If
then there exists an \(\alpha \in (0,\infty )\) such that
Conversely, if (41) holds with some \(\alpha \in (0,\infty )\), then
Proof
Assume that (40) is valid. Then, according to Theorem 3.1, there exist \(\alpha ,\beta ,r\in (0,\infty )\) such that (31) holds and
Thus
Suppose that \(r\ne 1\). Then from (43) we derive that
Since g and h are continuous, letting in this equality \(p\rightarrow p_h^+\), we obtain \(g(1-p_h)=0\). Hence \(h(p_h)+g(1-p_h)=0\), which contradicts (4). Therefore, \(r=1\) and so, in view of (43), we get \(\alpha =\beta \). Thus, (31) becomes (41).
If (41) is valid with some \(\alpha \in (0,\infty )\), then applying the positive homogeneity of the Choquet integral and making use of (2), for every \(X\in {\mathcal {X}}\), we obtain
which implies (42). \(\square \)
Remark 4.4
Recall that the zero utility principle under the Expected Utility model is defined by (1), which is a particular case of (2) with \(g=h=id_{[0,1]}\). Hence, Corollary 4.3 is a generalization of [4, Theorem 6].
It is worth mentioning that Heilpern [9] introduced and studied the zero utility principle under the Rank-Dependent Utility model. In this setting, the premium for \(X\in {\mathcal {X}}\) is defined as a solution of the equation
where w is a fixed real number, \(u:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function such that \(u(0)=0\) and, for any essentially bounded random variable Y,
is the Choquet integral with respect to a probability weighting function g. The following result, being a consequence of Theorem 3.1, characterizes those zero utility principles defined by (44) which coincide on the family of all binary risks.
Corollary 4.5
Let \(w\in {\mathbb {R}}\). Assume that, for \(i\in \{1,2\}\), \(g_i\) is a continuous probability weighting functions and \(u_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a strictly increasing continuous function with \(u_i(0)=0\). Then
if and only if there exist \(a,b,r\in (0,\infty )\) such that
and
where
Proof
According to (44), we have
Since the Choquet integral given by (45) is translative, in view of (3), Eq. (47) can be rewritten in the following form
where \(v_i:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and \(h_i:[0,1]\rightarrow [0,1]\) for \(i\in \{1,2\}\) are given by
and
respectively. Thus
and so, (46) is equivalent to
Note that, for \(i\in \{1,2\}\), \(h_i\) is a continuous probability weighting function and \(v_i\) is strictly increasing and continuous, with \(v_i(0)=0\). Therefore, applying Theorem 3.1, we obtain that (46) holds if and only if there exist \(\alpha ,\beta ,r\in (0,\infty )\) such that (30) is valid and
Hence, taking (48)–(49) into account and using the fact that \(u_i(0)=0\) for \(i\in \{1,2\}\), after a standard computation we get the assertion. \(\square \)
Remark 4.6
Corollary 4.5 is a generalization of [8, Theorem 3], where the case \(w\in [0,\infty )\) was considered. Let us note that the strict monotonicity of g, although not explicitly assumed, was used in the proof of [8, Theorem 3]. Moreover, there is a missprint in the characterization presented in [8, Theorem 3]. Namely, in formula (30), on the right hand side, in the denominator instead of \(a(1-g(p)^r)\) there should be \(a(1-g(p))^r\).
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The paper is dedicated to the 70th birthdays of Professors Maciej Sablik and László Székelyhidi.
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Chudziak, J., Chudziak, M. Zero utility principles coinciding on binary risks. Aequat. Math. 97, 1245–1258 (2023). https://doi.org/10.1007/s00010-023-01016-2
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DOI: https://doi.org/10.1007/s00010-023-01016-2