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Regret, rejoicing, and mixed insurance
Economic Modelling 58 (2016) 126–132
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Regret, rejoicing, and mixed insurance Yoichiro Fujii a, Mahito Okura b, Yusuke Osaki a,⁎ a b
Faculty of Economics, Osaka Sangyo University, 3-1-1 Nakagaito, Daito City, Osaka 574-8530, Japan Faculty of Contemporary Social Studies, Doshisha Women's College of Liberal Arts, Kodo, Kyotanabe City, Kyoto 610-0395, Japan
a r t i c l e
i n f o
Article history: Received 10 November 2015 Received in revised form 12 May 2016 Accepted 21 May 2016 Available online 8 June 2016 Keywords: Full insurance Mixed insurance Regret Rejoicing
a b s t r a c t This study examines how regret and rejoicing affect mixed insurance choice and demand. In contrast to expected utility theory, regret and rejoicing may explain why some individuals prefer to hold mixed insurance rather than term insurance. In this study, we derive the conditions under which an individual prefers to hold mixed insurance rather than term insurance. We also study demand for mixed insurance and specify the factors that influence the demand motive. The demand motive is determined by the risk effect and the rejoicing effect, when the rejoicing effect dominates the risk effect, under-insurance is optimal, and vice versa. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Term insurance is purchased to cover the specific damage suffered in a loss state. Because the damage involves risk, risk-averse individuals want to transfer that risk to insurance firms through term insurance. In other words, term insurance serves to transfer individuals' risk to insurance firms. According to the findings of traditional insurance economics studies such as Mossin (1968) and Ehrlich and Becker (1972), full coverage is optimal when insurance premiums are actuarially fair in perfect information markets. This result implies that individuals transfer all risks to the insurance firms in this case. The advantage of term insurance is that individuals can receive insurance money in a loss state. Although the realization of a loss state is unfortunate, individuals who hold term insurance might rejoice in a loss state because of the benefit of term insurance. Indeed, according to the results of the Survey of Life Protection conducted by the Japan Institute of Life Insurance in 2013, 54.5% and 28.2% of respondents purchased insurance “for medical expenses for injury or sickness” and “for unexpected death,” while only 6.7% and 2.4% purchased insurance “for living funds after retirement” and “for savings.”1 This survey results imply that a considerable proportion of individuals in Japan purchased insurance to prepare for damages in a loss state and thus might rejoice from the benefit of insurance money. On the contrary, the disadvantage of term insurance is that individuals receive no benefit in a no-loss state. Again, while the occurrence of
⁎ Corresponding author. E-mail addresses: [email protected] (Y. Fujii), [email protected] (M. Okura), [email protected] (Y. Osaki). 1 For more details, see page 70 of the following website: http://www.jili.or.jp/research/ report/pdf/FY2013_Survey_on_Life_Protection_(Quick_Report_Version).pdf.
http://dx.doi.org/10.1016/j.econmod.2016.05.026 0264-9993 © 2016 Elsevier B.V. All rights reserved.
a no-loss state is fortunate, individuals who hold term insurance might feel regret because of the perception that the payment of the term insurance premium has been wasted. According to the above-mentioned survey, 28.2% of individuals take out without-profit-type (no savings function) life insurance, while 64.7% individuals take out life insurance with a saving function.2 This survey results imply that a considerable proportion of individuals in Japan desire to receive a benefit and do not want to feel regret in a no-loss state. To attract individuals who want to rejoice or avoid regret in a no-loss state, life insurance firms in Japan sell mixed insurance, which provides two types of benefits even given the occurrence of a loss state. The first type of benefit, which is paid in a loss state, is that individuals can receive insurance benefits at the time of death. The second type of benefit, which is paid in a no-loss state, is that individuals can receive insurance benefits at the end of the policy term.3 Fig. 1 illustrates several representative insurance products in Japan. It shows that endowment insurance, a type of mixed insurance, has a certain amount of the Japanese life insurance market.4 In a loss state, both term and mixed insurance pay insurance money. The insurance premium in mixed insurance is higher than that in term insurance because it is paid in both loss and no-loss states. However, the amount of rejoicing (regret) might rise (fall) in a no-loss state 2 For more details, see page 75 of the following website: http://www.jili.or.jp/research/ report/pdf/FY2013_Survey_on_Life_Protection_(Quick_Report_Version).pdf. 3 Death and maturity proceeds are not necessarily the same except in a typical type of mixed insurance called endowment insurance. 4 It is difficult to extract the number of new mixed insurance policies since most new policies (e.g., medical insurance) do not distinguish between term insurance and mixed insurance. For example, some private medical insurance sold by life insurance firms offers not only medical protection in the case of diseases (a loss state) but also repayment in the case of no diseases (a no-loss state).
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Fig. 1. Percentage distribution of individual insurance by type (number of new policies). Source: Life Insurance Fact Book 2014 (Life Insurance Association of Japan)
through the holding of mixed insurance. Hence, the amount of rejoicing (regret) in mixed insurance is larger (smaller) than that in term insurance in a no-loss state, while the amount of rejoicing (regret) in mixed insurance is smaller (larger) than that in term insurance in a loss state. The above-mentioned suggests that rejoicing and regret affect the extent to which individuals distinguish between term and mixed insurance when choosing their holdings. Indeed, introducing rejoicing or regret might offer new insights that cannot be explained by expected utility theory, which concludes that risk-averse individuals never choose mixed insurance even if the insurance premium is actuarially fair.5 For example, because individuals are subject to a meanpreserving spread of final wealth if they purchase mixed insurance instead of term insurance, Rothschild and Stiglitz (1971) found that risk-averse individuals never want to purchase mixed insurance. Let us consider the situation in which the insurance premium is actuarially fair. In this situation, full insurance is optimal in the case of term insurance and individuals do not take any risks. By contrast, full insurance in mixed insurance exposes individuals to some risks. Instead, if individuals can choose any mixed insurance coverage rates that eliminate their risk exposure, over-insurance is chosen and expected utility is less than that under term insurance. Hence, individuals take some risks at the full insurance level in the case of mixed insurance. Thus, expected utility theory concludes that holding mixed insurance is irrational. However, as shown in Fig. 1, some individuals actually purchase mixed insurance (endowment insurance) in the Japanese insurance market, one of the largest markets globally and one that influences insurance markets in East and Southeast Asia where further economic growth is expected. Thus, why individuals purchase mixed insurance and the amounts they spend on doing so, are worthy to be pursued.
5 Another explanation for the irrationality of mixed insurance may follow from arbitrage opportunities. When mixed insurance can be replicated by using other financial instruments such as a combination of savings and term insurance, these are priced to exclude arbitrage opportunities. However, perfect replication is difficult in real situations because of differences in liquidity. For example, it is difficult to immediately withdraw money from a fixed deposit at the time of an accident.
Based on the foregoing, we examine mixed insurance choice and demand by incorporating rejoicing or regret. Both rejoicing and regret can explain why some individuals purchase mixed insurance in contrast to expected utility theory. Individuals can rejoice in a no-loss state or alleviate regret in a loss state by holding mixed insurance rather than term insurance. When this benefit is sufficiently large, individuals prefer mixed insurance to term insurance. Rejoicing theory is thus adopted to analyze mixed insurance demand since it can explain this choice under reasonable conditions. In rejoicing theory, insurance demand can be classified into the two effects, one is risk and the other is rejoicing. We find that under- (over-)insurance is optimal when the rejoicing effect dominates (is dominated by) the risk effect. The remainder of this paper is organized as follows. In Section 2, the background for this research is described. In Section 3, we explain the preference representations incorporating rejoicing and regret. In Section 4, we consider the choice problem between term and mixed insurance. In Section 5, we examine optimal demand for mixed insurance. The policy implications derived from the results of the model are shown in Section 6. Section 7 provides concluding remarks. 2. Background Bell (1982) and Loomes and Sugden (1982) introduced regret and rejoicing into preference representation. This representation can capture several paradoxical observations in expected utility theory, such as the Allais paradox (Allais, 1953) and probabilistic insurance (Kahneman and Tversky, 1979). Although Bell (1982) and Loomes and Sugden (1982) called this theory “regret theory,” to avoid confusion, we refer to it as “regret and rejoicing theory” since the representation not only includes regret but also rejoicing. The important difference between regret and rejoicing theory and prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) is that the former can incorporate ex-post state-dependent feelings into ex-ante decisions. This characteristic seems to be suitable for analyzing insurance problems because an individual who wants to purchase insurance has preference representations such as feeling ex-post regret and rejoice depending on each realized state.
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Braun and Muermann (2004) proposed regret theory, which can be viewed as a modified version of regret and rejoicing theory, and examined the effects of regret on demand for term insurance. Regret theory has also been applied to analyze insurance markets with asymmetric information by Huang et al. (2015), bank interest margins by Wong (2011) and Tsai (2012), and firms' production decisions by Wong (2012, 2014). Quiggin (1994, p.159) stated that “the original version of regret theory might, with equal justice, have been called rejoicing theory.” However, it is difficult to apply regret and rejoicing theory to most economic decisions, including insurance purchasing decisions, and thus rejoicing has been ignored in many previous studies. Given this gap in the body of knowledge on this topic, we apply regret and rejoicing theory to examine the choice problem between mixed and term insurance. Based on the presented results, we then choose rejoicing theory to analyze demand for mixed insurance.6 3. Regret and rejoicing theory According to Bell (1982) and Loomes and Sugden (1982), regret and rejoicing theory are given by U ðw; yÞ ¼ uðwÞ þ f ðuðwÞ−uðyÞÞ:
ð1Þ
Here, w denotes actual wealth and y denotes foregone wealth. The function u : R → R is called the risk function, which is assumed to be twice differentiable as well as strictly increasing, u ' N 0, and concave, u ″ ≤ 0. The function f captures regret or rejoicing by transforming the differences in utility between actual and forgone wealth. Recall that an individual feels regret (rejoicing) when actual (forgone) wealth is desirable to forgone (actual) one. Applying the theory, an individual suffers (gains) disutility (utility) by feeling regret (rejoicing) for w b y (w N y) toward his or her incorrect (correct) choice compared with a better (worse) choice. Bell (1982) and Loomes and Sugden (1982) considered the situation in which an individual compares two wealth levels. The utility function (1) cannot be applied to compare more than three wealth levels because there are more than two candidates for foregone wealth.7 Hence, to make the analysis possible, we must set a specific foregone wealth. Two suitable candidates are, say, maximum wealth and minimum wealth, which focus on regret and rejoicing, respectively. Regret theory is given by U ðw; w max Þ ¼ uðwÞ−g ðuðw max Þ−uðwÞÞ;
ð2Þ
and rejoicing theory is given by U w; w min ¼ uðwÞ þ h uðwÞ−u w min :
ð3Þ
Braun and Muermann (2004) introduced the utility function (2) for regret theory.8 Here, wmax and wmin are respectively the maximum and minimum wealth that an individual can reach in the given state (i.e., they are the ex-post maximum and minimum wealth). The risk function u is defined as above. The regret function g : R+ → R in (2) is assumed to be twice differentiable as well as strictly increasing, g ′ N 0, and convex, g ″ ≥ 0, with g(0) = 0. The rejoicing function h : R+ → R in (3) is assumed to be twice differentiable as well as strictly increasing, h' N 0, and concave, h ″ ≤0, with h(0) = 0. 6 Quiggin (1994, p.159) stated that “symmetry is a generally appealing property.” The analysis is symmetrical, whereas the result is different in the choice problem. It is arguable that we should use either regret theory or rejoicing theory (or a combination) based on both theoretical and experimental viewpoints. 7 In the case of more than three wealth levels, transitivity is violated, see Bikhchandani and Segal (2011) for more details. 8 There exist axiomatic foundations for regret theory in various settings (e.g., Sugden, 1993; Sarver, 2008).
Some explanation is needed about the convexity of the regret function and concavity of the rejoicing function. First, these shapes mean that an individual dislikes a mean-preserving spread of the utility difference between his or her actual wealth and maximum (minimum) wealth. Second, the utility functions (2) and (3) are strictly concave because g ″ ≥ 0 (h ″ ≤ 0) and u ″ ≤ 0, which guarantees that the secondorder condition (SOC) holds for decision problems.9 Third, there is experimental evidence of the convexity of the regret function (e.g., Bleichrodt et al., 2010). In regret (rejoicing) theory, an individual pays attention to maximum (minimum) wealth in decision making. Hence, regret theory should be used in cases in which maximum wealth is significant for decision making.10 By contrast, rejoicing theory should be used in cases in which minimum wealth is significant.11 Many insurance decisions are included in the latter case. Thus, although rejoicing theory has been paid little attention compared with regret theory, it is meaningful to examine the effect of rejoicing on insurance decisions in which minimum wealth is essential. 4. Choice between term and mixed insurance 4.1. Choice problem Let us consider an individual who has to choose to hold either term insurance or mixed insurance at the full insurance level. For the sake of comparison, both insurance premiums are set to be actuarially fair. An individual has initial wealth W N 0 and faces a risky loss. For simplicity, we consider that the risk is described by binary states: loss state and no-loss state. The loss size is denoted by D, with W N D N 0. The loss probability is π ∈ (0, 1) and the no-loss probability is 1 − π. If an individual holds term insurance, he or she receives coverage D in the loss state and nothing in the no-loss state. If an individual holds mixed insurance, he or she receives coverage D in the loss state and compensation C in the no-loss state, with D N C N 0.12 Since the insurance premiums are actuarially fair, the term insurance premium is given by Pt = πD and the mixed insurance premium is given by Pm = πD + (1 − π)C. If an individual holds term insurance, his or her final wealth is given by WO = W − Pt in both states. If an individual holds mixed insurance, his or her final wealth is given by WL = W − Pm in the loss state and WNL =W−Pm +C in the no-loss state. Because WO =πWL +(1−π)WNL, wealth holding mixed insurance is a mean-preserving spread of wealth holding term insurance. As a result, an individual who has a concave risk function prefers to hold term insurance rather than mixed insurance under expected utility theory. In other words, it is not possible to justify holding mixed insurance under expected utility theory, the dominant theory in insurance economics. 4.2. Regret theory Let us consider an individual whose utility function is represented by (2). Maximum wealth can be achieved by holding term insurance in the loss state, given by WO, and by holding mixed insurance in the no-loss state, given by WNL. When the individual holds term insurance, expected utility is π½uðW O Þ−g ðuðW O Þ−uðW O ÞÞ þ ð1−πÞ½uðW O Þ−g ðuðW NL Þ−uðW O ÞÞ ¼ uðW O Þ−ð1−πÞ gðuðW NL Þ−uðW O ÞÞ:
9 We assume that either g″ ≥ 0 (h″ ≤ 0) or u″ ≤ 0 holds in a strict sense. This means that the concavity of the utility functions holds in a strict sense. 10 Although it can conceivably be included when maximum wealth is huge, its probability is very low (e.g., long shots in horseracing). 11 It can conceivably be included when minimum wealth is very small and its probability is very low. 12 To enable the comparison between term and mixed insurance, coverage D in the loss state is the same in both insurance types.
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By contrast, when the individual holds mixed insurance, expected utility is π½uðW L Þ−g ðuðW O Þ−uðW L ÞÞ þ ð1−πÞ½uðW NL Þ−g ðuðW NL Þ−uðW NL ÞÞ ¼ πuðW L Þ þ ð1−π ÞuðW NL Þ−πg ðuðW O Þ−uðW L ÞÞ: The condition that the individual prefers mixed insurance to term insurance can be written as πuðW L Þ þ ð1−πÞuðW NL Þ−πg ðuðW O Þ−uðW L ÞÞ
ð4Þ
≥uðW O Þ−ð1−πÞ g ðuðW NL Þ−uðW O ÞÞ: Expected utility can be decomposed into two parts, one captured by the risk function u and the other by the regret function g. We call the former the risk effect and the latter the regret effect. Because the risk effect is simply the expected utility case, term insurance leads to higher expected utility than mixed insurance in terms of the risk effect. However, it is possible that mixed insurance leads to higher expected utility than term insurance in terms of the regret effect. In this case, the individual prefers to hold mixed insurance rather than term insurance if the regret effect dominates the risk effect. This implies that regret theory may be consistent with why mixed insurance is traded in the insurance market, although all individuals hold term insurance under expected utility theory. It is difficult to imagine a general condition in which an individual holds mixed insurance. Thus, we have to determine specific conditions. To do so, we assume that the risk function u is linear; in other words, we ignore the risk effect and focus only on the regret effect. In this setting, we reach the following result. Result 1 Suppose that the risk function is linear. If the loss probability is greater than 1/2, an individual prefers to hold mixed insurance rather than term insurance. Proof Because WO = πWL + (1 − π)WNL, condition (4) can be written as πg(WO − WL) = πg((1 − π)C) ≤ (1− π)g(πC) = (1 − π)g(WNL − WO). We need to show the following: g ðð1−πÞC Þ 1−π ≤ : gðπC Þ π Because π ≥ 1/2 ⇔ πC ≥ (1− π)C for C ≥ 0. We give a graphical proof to show the above inequality. Let us put a N 0 such that a(1 − π) = g((1− π)C). As in Fig. 2, aπC ≤ g(πC) holds since g(0) = 0 g′ N 0 and g″ N 0. Thus, we obtain g ðð1−πÞC Þ að1−π ÞC ð1−πÞC 1−π ≤ ¼ ¼ : gðπC Þ aπC πC π
Fig. 2. The effect of convexeity of the regret function.
Result 1 means that although an individual is slightly risk-averse, that is, u″ is slightly negative, he or she holds mixed insurance when the loss probability is greater than 1/2. 4.3. Rejoicing theory We now turn to the argument for the case of rejoicing theory. Because the analysis is similar to that of regret theory, we do not repeat it all. The worst wealth is WL in the loss state and WO in the no-loss state. When an individual holds term insurance, expected utility is uðW O Þ þ πhðuðW O Þ−uðW L ÞÞ: When an individual holds mixed insurance, expected utility is πuðW L Þ þ ð1−π ÞuðW NL Þ þ ð1−πÞhðuðW NL Þ−uðW O ÞÞ: As in the regret theory case, an individual may prefer to hold mixed insurance rather than term insurance in rejoicing theory. When the risk function is linear, we obtain the following result. Result 2 Suppose that the risk function is linear. If the loss probability is less than 1/2, an individual prefers to hold mixed insurance rather than term insurance. A similar intuition of Result 1 can be applied to Result 2. It appears more natural to consider the loss probability being less than 1/2 in the insurance context. We thus adopt rejoicing theory to analyze demand for mixed insurance in the following section. However, the parallel analysis can be applied by regret theory. 5. Optimal insurance demand
We provide an intuition why an individual prefers to hold mixed insurance to term insurance. Expected wealth is the same when an individual holds either type of insurance. Since the risk function is assumed to be linear, it is sufficient to compare regret from holding both mixed and term insurance. Let us assume that an individual holds term insurance. Regret is represented by the multiplication of the no-loss probability by the disutility from feeling regret. These two components have the opposite effects on regret as the loss probability is increasing: the noloss probability is decreasing, whereas the disutility is increasing. Since the regret function is convex, disutility is rapidly increasing by holding term insurance when the loss probability is sufficiently large. As a result, when the loss probability is sufficiently large, regret by holding term insurance is sufficiently large, meaning that an individual prefers to hold mixed insurance to term insurance.
In this section, we examine how rejoicing influences demand for mixed insurance compared with the full insurance level, which is optimal for expected utility maximizers. Before proceeding with the analysis, we review the optimality of full insurance in our setting under expected utility theory following Mossin (1968). 5.1. Expected utility and term insurance In this section, we consider the case where a risk-averse individual purchases term insurance under expected utility theory. As in the previous section, the insurance premium is assumed to be actuarially fair. This assumption keeps our analysis simple and is common when full insurance is set to be a benchmark.
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case has no rejoicing effect and degenerates into expected utility theory. By contrast, the latter case has no risk effect. In the first case, the objective function is written as
Expected utility is given by max V ðα Þ ¼ πu W TL ðα Þ þ ð1−π Þu W TNL ðα Þ :
α∈½0;1
Here, WTL (α) is wealth in the loss state when the individual purchases term insurance whose coverage rate is α ∈ [0, 1], and is given by WTL(α) = W − D − α(Pt − D). WTNL(α) is wealth in the no-loss state, and is given by WTNL(α) = W − αPt. Over-insurance is prohibited by law, and thus α N 1 is not included in the interval. By applying Pt = πD, the first-order condition (FOC) can be written as n 0 o 0 0 ¼ 0: V ðα Þ ¼ πð1−πÞD u W TL ðα Þ −u W TNL ðα Þ
ð5Þ
The SOC is also satisfied because u″ b 0. We omit the argument for the SOC below because it is easy to confirm. According to (5), u′(WTL (α)) must be equal to u′(WTNL(α)); that is, α satisfies the condition WTL(α) = WTNL(α). Thus, α⁎ = 1 holds, which means that full insurance is optimal. We denote that α⁎ is the optimal coverage. This is because that there is no risk on final wealth when holding term insurance at the full insurance level. We note that this full insurance is an interior solution and serves as a benchmark in the remainder of the analysis. We should distinguish between interior and corner solutions at the full insurance level. For convenience, the former is termed full insurance and the latter is termed over-insurance hereafter.
max V ðα Þ ¼ πuðW L ðα ÞÞ þ ð1−πÞuðW NL ðα ÞÞ:
α∈½0:1
The FOC at α =1 is written as n 0 o 0 0 V ð1Þ ¼ πð1−πÞðD−CÞ u ðWL Þ−u ðWNL Þ :
ð9Þ
Because u is strictly concave and WL b WNL, (9) is strictly positive. This means that over-insurance is optimal. An intuition for this result is as follows. In contrast to the term insurance case, wealth holding mixed insurance remains some risks at the full insurance level. Thus, the individual is eager to hold mixed insurance beyond the full insurance level to decrease the risks that are remained in wealth. We summarize this argument in the following result. Result 3 If an individual has no rejoicing function, the optimal mixed insurance holding is over-insurance. Next, consider the special case where the risk function is linear. The objective function is written as max V ðα Þ ¼ π½W L ðα Þ þ hðα ðD−P m ÞÞ þ ð1−πÞ
α∈½0;1
5.2. Rejoicing theory and mixed insurance
½W NL ðα Þ þ hðð1−α ÞðP m −C ÞÞ:
Let us now consider an individual under rejoicing theory. The individual chooses an optimal mixed insurance coverage rate α⁎ to maximize the following objective function: max V ðα Þ ¼ π ½uðW L ðα ÞÞ þ hfuðW L ðα ÞÞ−uðW L ð0ÞÞg þ ð1−πÞ
ð10Þ
Here, W L (α) − W L (0) = α(D − P m ) and W NL (α) − W NL (1) = (1 − α)(Pm − C). The FOC (10) at α = 1 is given by 0 0 0 V ð1Þ ¼ πð1−πÞðD−CÞ h ðD−Pm Þ−h ð0Þ :
ð11Þ
α∈½0;1
½uðW NL ðα ÞÞ þ hfuðW NL ðα ÞÞ−uðW NL ð1ÞÞg: Here, WL(α) represents wealth in the loss state when the individual purchases mixed insurance with coverage rate α, and is given by WL(α) = w − D − α(Pm − D). WNL(α) is wealth in the no-loss state, and is given by WNL(α) = w − α(Pm − C). By applying Pm = πD + (1 − π)C, the FOC can be written as 0 0 V ðα Þ ¼ πð1−πÞðD−CÞ½u0 ðWL ðα ÞÞ 1 þ h fuðWL ðα ÞÞ−uðWL ð0ÞÞg ð6Þ 0 −u0 ðWNL ðα ÞÞ 1 þ h fuðWNL ðα ÞÞ−uðWNL ð1ÞÞg ¼ 0:
To examine the optimality of full insurance in this setting, we determine the sign of (6) when α⁎ = 1: 0 0 0 V ð1Þ ¼ πð1−πÞðD−C Þ u0 ðW L Þ 1 þ h ðUDL Þ −u0 ðW NL Þ 1 þ h ð0Þ : ð7Þ For ease of notation, we write WL(1) = WL, WNL(1) = WNL, and UDL =u(WL) − u(WL(0)). The sign of (7) coincides with that of 0 0 u0 ðWL Þ 1 þ h ðUDL Þ −u0 ðWNL Þ 1 þ h ð0Þ ;
ð8Þ
from π(1 − π)(D − C) N 0. Because WL b WNL and u′ is decreasing, u′(WL) ≥ u ′ (WNL ). Because UDL N 0 and h' is decreasing, h′(UL) ≤ h ′ (0). From this observation, we see that the sign of (8)—and thus that of (7)—is indeterminate. This means that it is possible that optimal demand for mixed insurance is either over- or under-insurance. That is, full insurance occurs in a knife-edge case.
Because h′(D − Pm) b h ′ (0), and the other elements are strictly positive, (11) is strictly negative. This means that under-insurance is optimal. An intuition for this result is as follows. At the full insurance level, an individual gains rejoicing in the loss state, while he or she does not gain this in the no-loss state. For a marginal decrease in his or her mixed insurance holding, he or she loses some rejoicing in the loss state, but gains some rejoicing in the no-loss state. The latter dominates the former because of the concavity of the rejoicing function. As a result, the optimal mixed insurance holding is under-insurance. We summarize this argument in the following result. Result 4 If an individual has a linear risk function, the optimal mixed insurance holding is under-insurance. From the results of these two special cases, we gain the following intuition: the risk effect drives over-insurance and the rejoicing effect drives under-insurance, while optimal insurance is determined by which effect outweighs the other. We confirm this intuition formally in the following subsection. 5.4. Risk and rejoicing effects on demand for mixed insurance We examine the effect of the risk and rejoicing functions. Let us consider the risk function û and rejoicing function ĥ such that full insurance is optimal, that is, h i 0 0 0 0 V 0 ð1Þ ¼ π ð1−π ÞðD−C Þ û ðW L Þ 1 þ ĥ ðUDL Þ −û ðW NL Þ 1 þ ĥ ð0Þ ¼ 0: ð12Þ
5.3. Decomposition of risk and rejoicing effects (12) can be rewritten as To examine which factors determine demand for mixed insurance under rejoicing theory, we consider two special cases: one with no rejoicing function and the other with a linear risk function. The first
0 û0 ðW L Þ 1 þ ĥ ð0Þ ¼ : 0 0 û ðW NL Þ 1 þ ĥ ðUDL Þ
Y. Fujii et al. / Economic Modelling 58 (2016) 126–132
The following result states that the risk and rejoicing effects are characterized as the degrees of the concavity of the risk function u and of the rejoicing function h to determine demand for mixed insurance. Result 5 Let us consider the risk function û and rejoicing function ĥ such that full insurance is optimal. Over- (Under-)insurance is optimal if either of the following conditions is satisfied: • The risk function u is more (less) concave than the risk function û, in the sense that
−
u″ ðwÞ û″ ðwÞ ≥ ð ≤ Þ− 0 ; 0 u ðwÞ û ðwÞ
h″ ðwÞ 0
1 þ h ðwÞ
≤ ð ≥ Þ−
Both degrees of concavity have appeared in the literature. The degree of the concavity of the risk function (13) is the same as the absolute Arrow–Pratt aversion in expected utility theory. The degree of the concavity of the rejoicing function (14) appeared in Gee (2012).14 Finally, we examine the effect of compensation in the no-loss state on demand for mixed insurance. We denote by C ̂ compensation such that full insurance is optimal. Result 6 Let us consider compensation C ̂ such that full insurance is optimal. Over- (Under-)insurance is optimal if compensation is more (less) than C ̂, that is, C ≥ð≤ÞC ̂.
ð13Þ Proof From a straightforward calculation, the signs of
• The rejoicing function h is less (more) concave than the rejoicing function ĥ, in the sense that
−
131
ĥ″ ðwÞ : 0 1 þ ĥ ðwÞ
ð14Þ
0 0 1 þ h ð0Þ u ðW L ðC ÞÞ ∂ ∂ 0 0 u ðW NL ðC ÞÞ 1 þ h ðUDL ðC ÞÞ and ∂C ∂C are determined as the following, respectively:
Proof Let us first consider that inequality (13) holds. The condition that 0 0 over- (under-)insurance is optimal is written as u ðW L Þð1 þ ĥ ðUDL ÞÞ−
00
0
00
0
0
ð1−πÞh ðUDL ðC ÞÞu ðW L ðC ÞÞ≤0:
0
u ðW NL Þð1 þ ĥ ð0ÞÞ≥ð≤Þ0: 0 Inequality (13) is equivalent to u0 ðwÞ=û ðwÞ decreasing (increasing) in w. Thus, for every wh , wl with wh N wl, 0
0
0
0
u0 ðwl Þ u ðw Þ u ðw Þ û ðw Þ ≥ð≤Þ 0 h ⟺ 0 l ≥ð≤Þ 0 l 0 u ðwh Þ û ðwl Þ û ðwh Þ û ðwh Þ holds. By setting wh = WNL and wl = WL, we obtain the following13: ! 0 0 0 u ðW L Þ û ðW L Þ 1 þ ĥ ð0Þ ≥ð≤Þ 0 ¼ 0 0 u ðW NL Þ û ðW NL Þ 1 þ ĥ ðUDL Þ 0 0 0 0 ⟺u ðW L Þ 1 þ ĥ ðUDL Þ −u ðW NL Þ 1 þ ĥ ð0Þ ≥ ð ≤ Þ0: This indicates that over- (under-)insurance is optimal when the individual's risk function u is more (less) concave than û in the sense that inequality (13) holds. We next consider whether inequality (14) holds. From an argument similar to that above, we have 0 0 0 û ðW L Þ 1 þ ĥ ð0Þ 1 þ h ð0Þ 0 ¼ ⟺û ðW L Þ ≥ð≤Þ 0 0 0̂ û ðW NL Þ 1 þ h ðUDL Þ 1 þ h ðUDL Þ 0
0
0
1 þ h ðUDL Þ −û ðW NL Þ 1 þ h ð0Þ ≥ ð ≤ Þ0:
This indicates that over- (under-)insurance is optimal when the individual's rejoicing function h is less (more) concave than ĥ in the sense that inequality (14) holds. □ An intuition of Result 5 can be obtained from Results 3 and 4 in the previous section. From these results, we know that the risk effect strengthens demand for mixed insurance, while the rejoicing effect weakens it at the full insurance level. Since the risk and rejoicing effects can be captured by the degree of the concavity of their functions, over- (under-)insurance is optimal when the risk (rejoicing) function is more concave compared with the full insurance case. 13
00
−ð1−πÞu ðW L ðC ÞÞu ðW NL ðC ÞÞ−πu ðW NL ðC ÞÞu ðW L ðC ÞÞ≥0;
The risk function is preserved under affine transformations, meaning that we can set UDL ¼ uðW L Þ−uðW L ð0ÞÞ ¼ ûðW L Þ−ûðW L ð0ÞÞ.
Here, compensation C is expressed as the argument of the function. From the above, we have the following inequalities for compensation C such that C ≥ð≤ÞC :̂ 0 0 u W L Ĉ u ðW L ðC ÞÞ ; ≥ð≤Þ u0 ðW NL ðC ÞÞ u0 W Ĉ NL
0
0
1 þ h ð0Þ
1 þ h ð0Þ : ≤ð≥Þ 0 0 1 þ h ðUDL ðC ÞÞ 1 þ h UDL C ̂
By combining these inequalities with 0 0 u W L Ĉ 1 þ h ð0Þ ¼ ; 0 u0 W NL C ̂ 1 þ h UDL C ̂ we obtain 0
1 þ h ð0Þ 0
1 þ h ðUDL ðC ÞÞ
0
≤ð≥Þ
u ðW L ðC ÞÞ u0 ðW NL ðC ÞÞ
0 0 0 0 ⇔u ðW L ðC ÞÞ 1 þ h ðUDL ðC ÞÞ −u ðW NL ðC ÞÞ 1 þ h ð0Þ ≥ ð ≤ Þ0: □ From the above analysis, we know that demand for mixed insurance is determined by the risk and rejoicing effects. The risk effect strengthens such demand, whereas the rejoicing effect weakens it. If compensation is increasing, an individual feels less rejoicing. Thus, the risk effect dominates (is dominated by) the rejoicing effect when compensation is relatively large (small). As a result, an individual has a motive to hold more (less) mixed insurance than the full insurance level. 14 Gee (2012) axiomatized the regret form and characterized the degree of the concavity of the regret function as second-order regret aversion. A similar argument can be applied to the rejoicing form.
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6. Policy implications By substituting α⁎ = 1 and C = 0 to (6), we have 0 0 πð1−πÞDu0 ðW L Þ h ðUDL Þ−h ð0Þ b 0:
This result implies that the optimal coverage rate of term insurance in rejoicing theory is under-insurance if the rejoicing function is not linear. It also means that this coverage rate is always smaller than that under expected utility theory since the full insurance is optimal under expected utility theory. Furthermore, from the combination of Results 3 and 4, we find that the optimal coverage rate of mixed insurance in rejoicing theory is always smaller than that under expected utility theory because the rejoicing effect lowers the optimal coverage rate. In the case of term insurance, individuals do not choose full insurance in rejoicing theory and this means that the optimal risk allocation (i.e., that a risk-neutral insurance firm undertakes all risks) cannot be realized. To realize the optimal risk allocation in rejoicing theory, raising the optimal coverage rate by lowering the insurance premium seems to be effective. One way of lowering the insurance premium is to create a more competitive insurance market. For example, if the number of insurance firms increases, the insurance premium lowers through fiercer competition and the optimal coverage rate finally rises. Theoretically speaking, the insurance premium approaches the actuarially fair level if a perfectly competitive market is realized and the optimal coverage rates in the model are actually realized. However, the insurance premium is never below the expected insurance money even if a perfectly competitive market is realized. Thus, to keep the insurance premium below the expected insurance money, subsidies and tax deductions are necessary to reduce the insurance premium. For example, some life insurance premiums in Japan can be tax deductions from an individual's income. Under the present tax system in Japan, 120,000 JPY (about 1128 USD) and 70,000 JPY (about 658 USD) are the maximum deductions in income and resident tax, respectively.15 The Life Insurance Association of Japan (LIAJ) was requested to stabilize the tax deduction.16 Then, it requested to expand the deduction in income tax from 120,000 to 150,000 JPY (about 1410 USD)17 in order to incentivize individuals to raise their coverage rates. However, according to expected utility theory, we might conclude that the coverage rate is unchanged because individuals have already achieved full insurance, especially in the case of mixed insurance. In other words, expected utility theory cannot explain the reason why the LIAJ has expanded the tax deduction in a competitive insurance market. By contrast, in rejoicing theory, under-insurance is achieved in term insurance and, possibly, in mixed insurance. In other words, the coverage rate could be raised by expanding the tax deduction. This also means that the risk allocation becomes more desirable because an insurance firm undertakes more risk when the tax deduction is expanded. From this viewpoint, rejoicing theory can provide a more rational explanation of the actions of the LIAJ. Moreover, we find that a further tax deduction is needed in rejoicing theory than in expected utility theory to realize a more desirable risk allocation in the insurance market.
15
1 USD = 106.35 JPY on April 29, 2016. For more details, see the website of the LIAJ: http://www.seiho.or.jp/english/news/ 2013/0621.html (requests on Tax Reform for FY2014). 17 For more details, see the following websites in LIAJ: http://www.seiho.or.jp/english/ news/2014/0718.html (requests on Tax Reform for FY2015) and http://www.seiho.or.jp/ english/news/2015/0717.html (requests on Tax Reform for FY2016). 16
7. Concluding remarks This study examines how regret and rejoicing theory affect mixed insurance choice and demand. By using these theories, we derive the conditions under which an individual prefers to hold mixed insurance rather than term insurance. We also examine which factors determine demand for mixed insurance (under-insurance, full insurance, overinsurance), by decomposing the risk and rejoicing effects. Braun and Muermann (2004) opened up the possibility that regret theory is a promising tool in insurance economics. This study broadens its possibility to the analysis of mixed insurance as well as sheds light on rejoicing, which receives equal weight to regret. Future work could examine other insurance problems by using either regret theory or rejoicing theory or both. In general, behavioral approaches, not limited to regret and rejoicing, will become more important in insurance economics in the future. Acknowledgments We are grateful to the three anonymous reviewers whose comments and suggestions helped us improve this paper greatly. We would like to thank the seminar participants at the 2014 Asia-Pacific Risk and Insurance meeting, the 2015 World Risk and Insurance Economic Congress, and the second East Asia RMI insurance workshop for their helpful comments on earlier versions of this paper. All remaining errors are, of course, ours. This study was partly supported by JSPS KAKENHI Grant numbers 24730362 and 26705004 and Kampo-Zaidan Grants. References Allais, M., 1953. Le comportement de l'homme rationnel devant le risque; critique des postulats et axiomes de l'ecole Americaine. Econometrica 21, 503–546. Bell, D.E., 1982. Regret in decision making under uncertainty. Oper. Res. 30, 961–981. Bikhchandani, S., Segal, U., 2011. Transitive regret. Theor. Econ. 6, 95–108. Bleichrodt, H., Cillo, A., Diecidue, E., 2010. A quantitative measurement of regret theory. Manag. Sci. 56, 161–175. Braun, M., Muermann, A., 2004. The impact of regret on the demand for insurance. J. Risk Insur. 71, 737–767. Ehrlich, I., Becker, G.S., 1972. Market insurance, self-insurance, and self-protection. J. Polit. Econ. 80, 623–648. Gee, C., 2012. A behavioral model of regret aversion. Working Paper (Available at https:// sites.google.com/site/econgee/research). Huang, R.J., Muermann, A., Tzeng, L.Y., 2015. Hidden regret in insurance markets. J. Risk Insur. 83, 181–216. Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291. Loomes, G., Sugden, R., 1982. Regret theory: an alternative theory of rational choice under uncertainty. Econ. J. 92, 805–824 (1982). Mossin, J., 1968. Aspects of rational insurance purchasing. J. Polit. Econ. 76, 553–568. Quiggin, J., 1994. Regret theory with general choice sets. J. Risk Insur. 8, 153–165. Rothschild, M., Stiglitz, J.E., 1971. Increasing risk II: its economic consequences. J. Econ. Theory 3, 66–84. Sarver, T., 2008. Anticipating regret: why fewer options may be better. Econometrica 76, 263–305. Sugden, R., 1993. An axiomatic foundation for regret theory. J. Econ. Theor. 60, 159–180. Tsai, J.Y., 2012. Risk and regret aversions on optimal bank interest margin under capital regulation. Econ. Model. 29, 2190–2197. Tversky, A., Kahneman, D., 1992. Advances in prospect theory: cumulative representation of uncertainty. J. Risk Insur. 5, 297–323. Wong, K.P., 2011. Regret theory and the banking firm: the optimal bank interest margin. Econ. Model. 28, 2483–2487. Wong, K.P., 2012. Production and insurance under regret aversion. Econ. Model. 29, 1154–1160. Wong, K.P., 2014. Regret theory and the competitive firm. Econ. Model. 36, 172–175.
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Regret, rejoicing, and mixed insurance Yoichiro Fujii a, Mahito Okura b, Yusuke Osaki a,⁎ a b
Faculty of Economics, Osaka Sangyo University, 3-1-1 Nakagaito, Daito City, Osaka 574-8530, Japan Faculty of Contemporary Social Studies, Doshisha Women's College of Liberal Arts, Kodo, Kyotanabe City, Kyoto 610-0395, Japan
a r t i c l e
i n f o
Article history: Received 10 November 2015 Received in revised form 12 May 2016 Accepted 21 May 2016 Available online 8 June 2016 Keywords: Full insurance Mixed insurance Regret Rejoicing
a b s t r a c t This study examines how regret and rejoicing affect mixed insurance choice and demand. In contrast to expected utility theory, regret and rejoicing may explain why some individuals prefer to hold mixed insurance rather than term insurance. In this study, we derive the conditions under which an individual prefers to hold mixed insurance rather than term insurance. We also study demand for mixed insurance and specify the factors that influence the demand motive. The demand motive is determined by the risk effect and the rejoicing effect, when the rejoicing effect dominates the risk effect, under-insurance is optimal, and vice versa. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Term insurance is purchased to cover the specific damage suffered in a loss state. Because the damage involves risk, risk-averse individuals want to transfer that risk to insurance firms through term insurance. In other words, term insurance serves to transfer individuals' risk to insurance firms. According to the findings of traditional insurance economics studies such as Mossin (1968) and Ehrlich and Becker (1972), full coverage is optimal when insurance premiums are actuarially fair in perfect information markets. This result implies that individuals transfer all risks to the insurance firms in this case. The advantage of term insurance is that individuals can receive insurance money in a loss state. Although the realization of a loss state is unfortunate, individuals who hold term insurance might rejoice in a loss state because of the benefit of term insurance. Indeed, according to the results of the Survey of Life Protection conducted by the Japan Institute of Life Insurance in 2013, 54.5% and 28.2% of respondents purchased insurance “for medical expenses for injury or sickness” and “for unexpected death,” while only 6.7% and 2.4% purchased insurance “for living funds after retirement” and “for savings.”1 This survey results imply that a considerable proportion of individuals in Japan purchased insurance to prepare for damages in a loss state and thus might rejoice from the benefit of insurance money. On the contrary, the disadvantage of term insurance is that individuals receive no benefit in a no-loss state. Again, while the occurrence of
⁎ Corresponding author. E-mail addresses: [email protected] (Y. Fujii), [email protected] (M. Okura), [email protected] (Y. Osaki). 1 For more details, see page 70 of the following website: http://www.jili.or.jp/research/ report/pdf/FY2013_Survey_on_Life_Protection_(Quick_Report_Version).pdf.
http://dx.doi.org/10.1016/j.econmod.2016.05.026 0264-9993 © 2016 Elsevier B.V. All rights reserved.
a no-loss state is fortunate, individuals who hold term insurance might feel regret because of the perception that the payment of the term insurance premium has been wasted. According to the above-mentioned survey, 28.2% of individuals take out without-profit-type (no savings function) life insurance, while 64.7% individuals take out life insurance with a saving function.2 This survey results imply that a considerable proportion of individuals in Japan desire to receive a benefit and do not want to feel regret in a no-loss state. To attract individuals who want to rejoice or avoid regret in a no-loss state, life insurance firms in Japan sell mixed insurance, which provides two types of benefits even given the occurrence of a loss state. The first type of benefit, which is paid in a loss state, is that individuals can receive insurance benefits at the time of death. The second type of benefit, which is paid in a no-loss state, is that individuals can receive insurance benefits at the end of the policy term.3 Fig. 1 illustrates several representative insurance products in Japan. It shows that endowment insurance, a type of mixed insurance, has a certain amount of the Japanese life insurance market.4 In a loss state, both term and mixed insurance pay insurance money. The insurance premium in mixed insurance is higher than that in term insurance because it is paid in both loss and no-loss states. However, the amount of rejoicing (regret) might rise (fall) in a no-loss state 2 For more details, see page 75 of the following website: http://www.jili.or.jp/research/ report/pdf/FY2013_Survey_on_Life_Protection_(Quick_Report_Version).pdf. 3 Death and maturity proceeds are not necessarily the same except in a typical type of mixed insurance called endowment insurance. 4 It is difficult to extract the number of new mixed insurance policies since most new policies (e.g., medical insurance) do not distinguish between term insurance and mixed insurance. For example, some private medical insurance sold by life insurance firms offers not only medical protection in the case of diseases (a loss state) but also repayment in the case of no diseases (a no-loss state).
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Fig. 1. Percentage distribution of individual insurance by type (number of new policies). Source: Life Insurance Fact Book 2014 (Life Insurance Association of Japan)
through the holding of mixed insurance. Hence, the amount of rejoicing (regret) in mixed insurance is larger (smaller) than that in term insurance in a no-loss state, while the amount of rejoicing (regret) in mixed insurance is smaller (larger) than that in term insurance in a loss state. The above-mentioned suggests that rejoicing and regret affect the extent to which individuals distinguish between term and mixed insurance when choosing their holdings. Indeed, introducing rejoicing or regret might offer new insights that cannot be explained by expected utility theory, which concludes that risk-averse individuals never choose mixed insurance even if the insurance premium is actuarially fair.5 For example, because individuals are subject to a meanpreserving spread of final wealth if they purchase mixed insurance instead of term insurance, Rothschild and Stiglitz (1971) found that risk-averse individuals never want to purchase mixed insurance. Let us consider the situation in which the insurance premium is actuarially fair. In this situation, full insurance is optimal in the case of term insurance and individuals do not take any risks. By contrast, full insurance in mixed insurance exposes individuals to some risks. Instead, if individuals can choose any mixed insurance coverage rates that eliminate their risk exposure, over-insurance is chosen and expected utility is less than that under term insurance. Hence, individuals take some risks at the full insurance level in the case of mixed insurance. Thus, expected utility theory concludes that holding mixed insurance is irrational. However, as shown in Fig. 1, some individuals actually purchase mixed insurance (endowment insurance) in the Japanese insurance market, one of the largest markets globally and one that influences insurance markets in East and Southeast Asia where further economic growth is expected. Thus, why individuals purchase mixed insurance and the amounts they spend on doing so, are worthy to be pursued.
5 Another explanation for the irrationality of mixed insurance may follow from arbitrage opportunities. When mixed insurance can be replicated by using other financial instruments such as a combination of savings and term insurance, these are priced to exclude arbitrage opportunities. However, perfect replication is difficult in real situations because of differences in liquidity. For example, it is difficult to immediately withdraw money from a fixed deposit at the time of an accident.
Based on the foregoing, we examine mixed insurance choice and demand by incorporating rejoicing or regret. Both rejoicing and regret can explain why some individuals purchase mixed insurance in contrast to expected utility theory. Individuals can rejoice in a no-loss state or alleviate regret in a loss state by holding mixed insurance rather than term insurance. When this benefit is sufficiently large, individuals prefer mixed insurance to term insurance. Rejoicing theory is thus adopted to analyze mixed insurance demand since it can explain this choice under reasonable conditions. In rejoicing theory, insurance demand can be classified into the two effects, one is risk and the other is rejoicing. We find that under- (over-)insurance is optimal when the rejoicing effect dominates (is dominated by) the risk effect. The remainder of this paper is organized as follows. In Section 2, the background for this research is described. In Section 3, we explain the preference representations incorporating rejoicing and regret. In Section 4, we consider the choice problem between term and mixed insurance. In Section 5, we examine optimal demand for mixed insurance. The policy implications derived from the results of the model are shown in Section 6. Section 7 provides concluding remarks. 2. Background Bell (1982) and Loomes and Sugden (1982) introduced regret and rejoicing into preference representation. This representation can capture several paradoxical observations in expected utility theory, such as the Allais paradox (Allais, 1953) and probabilistic insurance (Kahneman and Tversky, 1979). Although Bell (1982) and Loomes and Sugden (1982) called this theory “regret theory,” to avoid confusion, we refer to it as “regret and rejoicing theory” since the representation not only includes regret but also rejoicing. The important difference between regret and rejoicing theory and prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) is that the former can incorporate ex-post state-dependent feelings into ex-ante decisions. This characteristic seems to be suitable for analyzing insurance problems because an individual who wants to purchase insurance has preference representations such as feeling ex-post regret and rejoice depending on each realized state.
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Braun and Muermann (2004) proposed regret theory, which can be viewed as a modified version of regret and rejoicing theory, and examined the effects of regret on demand for term insurance. Regret theory has also been applied to analyze insurance markets with asymmetric information by Huang et al. (2015), bank interest margins by Wong (2011) and Tsai (2012), and firms' production decisions by Wong (2012, 2014). Quiggin (1994, p.159) stated that “the original version of regret theory might, with equal justice, have been called rejoicing theory.” However, it is difficult to apply regret and rejoicing theory to most economic decisions, including insurance purchasing decisions, and thus rejoicing has been ignored in many previous studies. Given this gap in the body of knowledge on this topic, we apply regret and rejoicing theory to examine the choice problem between mixed and term insurance. Based on the presented results, we then choose rejoicing theory to analyze demand for mixed insurance.6 3. Regret and rejoicing theory According to Bell (1982) and Loomes and Sugden (1982), regret and rejoicing theory are given by U ðw; yÞ ¼ uðwÞ þ f ðuðwÞ−uðyÞÞ:
ð1Þ
Here, w denotes actual wealth and y denotes foregone wealth. The function u : R → R is called the risk function, which is assumed to be twice differentiable as well as strictly increasing, u ' N 0, and concave, u ″ ≤ 0. The function f captures regret or rejoicing by transforming the differences in utility between actual and forgone wealth. Recall that an individual feels regret (rejoicing) when actual (forgone) wealth is desirable to forgone (actual) one. Applying the theory, an individual suffers (gains) disutility (utility) by feeling regret (rejoicing) for w b y (w N y) toward his or her incorrect (correct) choice compared with a better (worse) choice. Bell (1982) and Loomes and Sugden (1982) considered the situation in which an individual compares two wealth levels. The utility function (1) cannot be applied to compare more than three wealth levels because there are more than two candidates for foregone wealth.7 Hence, to make the analysis possible, we must set a specific foregone wealth. Two suitable candidates are, say, maximum wealth and minimum wealth, which focus on regret and rejoicing, respectively. Regret theory is given by U ðw; w max Þ ¼ uðwÞ−g ðuðw max Þ−uðwÞÞ;
ð2Þ
and rejoicing theory is given by U w; w min ¼ uðwÞ þ h uðwÞ−u w min :
ð3Þ
Braun and Muermann (2004) introduced the utility function (2) for regret theory.8 Here, wmax and wmin are respectively the maximum and minimum wealth that an individual can reach in the given state (i.e., they are the ex-post maximum and minimum wealth). The risk function u is defined as above. The regret function g : R+ → R in (2) is assumed to be twice differentiable as well as strictly increasing, g ′ N 0, and convex, g ″ ≥ 0, with g(0) = 0. The rejoicing function h : R+ → R in (3) is assumed to be twice differentiable as well as strictly increasing, h' N 0, and concave, h ″ ≤0, with h(0) = 0. 6 Quiggin (1994, p.159) stated that “symmetry is a generally appealing property.” The analysis is symmetrical, whereas the result is different in the choice problem. It is arguable that we should use either regret theory or rejoicing theory (or a combination) based on both theoretical and experimental viewpoints. 7 In the case of more than three wealth levels, transitivity is violated, see Bikhchandani and Segal (2011) for more details. 8 There exist axiomatic foundations for regret theory in various settings (e.g., Sugden, 1993; Sarver, 2008).
Some explanation is needed about the convexity of the regret function and concavity of the rejoicing function. First, these shapes mean that an individual dislikes a mean-preserving spread of the utility difference between his or her actual wealth and maximum (minimum) wealth. Second, the utility functions (2) and (3) are strictly concave because g ″ ≥ 0 (h ″ ≤ 0) and u ″ ≤ 0, which guarantees that the secondorder condition (SOC) holds for decision problems.9 Third, there is experimental evidence of the convexity of the regret function (e.g., Bleichrodt et al., 2010). In regret (rejoicing) theory, an individual pays attention to maximum (minimum) wealth in decision making. Hence, regret theory should be used in cases in which maximum wealth is significant for decision making.10 By contrast, rejoicing theory should be used in cases in which minimum wealth is significant.11 Many insurance decisions are included in the latter case. Thus, although rejoicing theory has been paid little attention compared with regret theory, it is meaningful to examine the effect of rejoicing on insurance decisions in which minimum wealth is essential. 4. Choice between term and mixed insurance 4.1. Choice problem Let us consider an individual who has to choose to hold either term insurance or mixed insurance at the full insurance level. For the sake of comparison, both insurance premiums are set to be actuarially fair. An individual has initial wealth W N 0 and faces a risky loss. For simplicity, we consider that the risk is described by binary states: loss state and no-loss state. The loss size is denoted by D, with W N D N 0. The loss probability is π ∈ (0, 1) and the no-loss probability is 1 − π. If an individual holds term insurance, he or she receives coverage D in the loss state and nothing in the no-loss state. If an individual holds mixed insurance, he or she receives coverage D in the loss state and compensation C in the no-loss state, with D N C N 0.12 Since the insurance premiums are actuarially fair, the term insurance premium is given by Pt = πD and the mixed insurance premium is given by Pm = πD + (1 − π)C. If an individual holds term insurance, his or her final wealth is given by WO = W − Pt in both states. If an individual holds mixed insurance, his or her final wealth is given by WL = W − Pm in the loss state and WNL =W−Pm +C in the no-loss state. Because WO =πWL +(1−π)WNL, wealth holding mixed insurance is a mean-preserving spread of wealth holding term insurance. As a result, an individual who has a concave risk function prefers to hold term insurance rather than mixed insurance under expected utility theory. In other words, it is not possible to justify holding mixed insurance under expected utility theory, the dominant theory in insurance economics. 4.2. Regret theory Let us consider an individual whose utility function is represented by (2). Maximum wealth can be achieved by holding term insurance in the loss state, given by WO, and by holding mixed insurance in the no-loss state, given by WNL. When the individual holds term insurance, expected utility is π½uðW O Þ−g ðuðW O Þ−uðW O ÞÞ þ ð1−πÞ½uðW O Þ−g ðuðW NL Þ−uðW O ÞÞ ¼ uðW O Þ−ð1−πÞ gðuðW NL Þ−uðW O ÞÞ:
9 We assume that either g″ ≥ 0 (h″ ≤ 0) or u″ ≤ 0 holds in a strict sense. This means that the concavity of the utility functions holds in a strict sense. 10 Although it can conceivably be included when maximum wealth is huge, its probability is very low (e.g., long shots in horseracing). 11 It can conceivably be included when minimum wealth is very small and its probability is very low. 12 To enable the comparison between term and mixed insurance, coverage D in the loss state is the same in both insurance types.
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By contrast, when the individual holds mixed insurance, expected utility is π½uðW L Þ−g ðuðW O Þ−uðW L ÞÞ þ ð1−πÞ½uðW NL Þ−g ðuðW NL Þ−uðW NL ÞÞ ¼ πuðW L Þ þ ð1−π ÞuðW NL Þ−πg ðuðW O Þ−uðW L ÞÞ: The condition that the individual prefers mixed insurance to term insurance can be written as πuðW L Þ þ ð1−πÞuðW NL Þ−πg ðuðW O Þ−uðW L ÞÞ
ð4Þ
≥uðW O Þ−ð1−πÞ g ðuðW NL Þ−uðW O ÞÞ: Expected utility can be decomposed into two parts, one captured by the risk function u and the other by the regret function g. We call the former the risk effect and the latter the regret effect. Because the risk effect is simply the expected utility case, term insurance leads to higher expected utility than mixed insurance in terms of the risk effect. However, it is possible that mixed insurance leads to higher expected utility than term insurance in terms of the regret effect. In this case, the individual prefers to hold mixed insurance rather than term insurance if the regret effect dominates the risk effect. This implies that regret theory may be consistent with why mixed insurance is traded in the insurance market, although all individuals hold term insurance under expected utility theory. It is difficult to imagine a general condition in which an individual holds mixed insurance. Thus, we have to determine specific conditions. To do so, we assume that the risk function u is linear; in other words, we ignore the risk effect and focus only on the regret effect. In this setting, we reach the following result. Result 1 Suppose that the risk function is linear. If the loss probability is greater than 1/2, an individual prefers to hold mixed insurance rather than term insurance. Proof Because WO = πWL + (1 − π)WNL, condition (4) can be written as πg(WO − WL) = πg((1 − π)C) ≤ (1− π)g(πC) = (1 − π)g(WNL − WO). We need to show the following: g ðð1−πÞC Þ 1−π ≤ : gðπC Þ π Because π ≥ 1/2 ⇔ πC ≥ (1− π)C for C ≥ 0. We give a graphical proof to show the above inequality. Let us put a N 0 such that a(1 − π) = g((1− π)C). As in Fig. 2, aπC ≤ g(πC) holds since g(0) = 0 g′ N 0 and g″ N 0. Thus, we obtain g ðð1−πÞC Þ að1−π ÞC ð1−πÞC 1−π ≤ ¼ ¼ : gðπC Þ aπC πC π
Fig. 2. The effect of convexeity of the regret function.
Result 1 means that although an individual is slightly risk-averse, that is, u″ is slightly negative, he or she holds mixed insurance when the loss probability is greater than 1/2. 4.3. Rejoicing theory We now turn to the argument for the case of rejoicing theory. Because the analysis is similar to that of regret theory, we do not repeat it all. The worst wealth is WL in the loss state and WO in the no-loss state. When an individual holds term insurance, expected utility is uðW O Þ þ πhðuðW O Þ−uðW L ÞÞ: When an individual holds mixed insurance, expected utility is πuðW L Þ þ ð1−π ÞuðW NL Þ þ ð1−πÞhðuðW NL Þ−uðW O ÞÞ: As in the regret theory case, an individual may prefer to hold mixed insurance rather than term insurance in rejoicing theory. When the risk function is linear, we obtain the following result. Result 2 Suppose that the risk function is linear. If the loss probability is less than 1/2, an individual prefers to hold mixed insurance rather than term insurance. A similar intuition of Result 1 can be applied to Result 2. It appears more natural to consider the loss probability being less than 1/2 in the insurance context. We thus adopt rejoicing theory to analyze demand for mixed insurance in the following section. However, the parallel analysis can be applied by regret theory. 5. Optimal insurance demand
We provide an intuition why an individual prefers to hold mixed insurance to term insurance. Expected wealth is the same when an individual holds either type of insurance. Since the risk function is assumed to be linear, it is sufficient to compare regret from holding both mixed and term insurance. Let us assume that an individual holds term insurance. Regret is represented by the multiplication of the no-loss probability by the disutility from feeling regret. These two components have the opposite effects on regret as the loss probability is increasing: the noloss probability is decreasing, whereas the disutility is increasing. Since the regret function is convex, disutility is rapidly increasing by holding term insurance when the loss probability is sufficiently large. As a result, when the loss probability is sufficiently large, regret by holding term insurance is sufficiently large, meaning that an individual prefers to hold mixed insurance to term insurance.
In this section, we examine how rejoicing influences demand for mixed insurance compared with the full insurance level, which is optimal for expected utility maximizers. Before proceeding with the analysis, we review the optimality of full insurance in our setting under expected utility theory following Mossin (1968). 5.1. Expected utility and term insurance In this section, we consider the case where a risk-averse individual purchases term insurance under expected utility theory. As in the previous section, the insurance premium is assumed to be actuarially fair. This assumption keeps our analysis simple and is common when full insurance is set to be a benchmark.
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case has no rejoicing effect and degenerates into expected utility theory. By contrast, the latter case has no risk effect. In the first case, the objective function is written as
Expected utility is given by max V ðα Þ ¼ πu W TL ðα Þ þ ð1−π Þu W TNL ðα Þ :
α∈½0;1
Here, WTL (α) is wealth in the loss state when the individual purchases term insurance whose coverage rate is α ∈ [0, 1], and is given by WTL(α) = W − D − α(Pt − D). WTNL(α) is wealth in the no-loss state, and is given by WTNL(α) = W − αPt. Over-insurance is prohibited by law, and thus α N 1 is not included in the interval. By applying Pt = πD, the first-order condition (FOC) can be written as n 0 o 0 0 ¼ 0: V ðα Þ ¼ πð1−πÞD u W TL ðα Þ −u W TNL ðα Þ
ð5Þ
The SOC is also satisfied because u″ b 0. We omit the argument for the SOC below because it is easy to confirm. According to (5), u′(WTL (α)) must be equal to u′(WTNL(α)); that is, α satisfies the condition WTL(α) = WTNL(α). Thus, α⁎ = 1 holds, which means that full insurance is optimal. We denote that α⁎ is the optimal coverage. This is because that there is no risk on final wealth when holding term insurance at the full insurance level. We note that this full insurance is an interior solution and serves as a benchmark in the remainder of the analysis. We should distinguish between interior and corner solutions at the full insurance level. For convenience, the former is termed full insurance and the latter is termed over-insurance hereafter.
max V ðα Þ ¼ πuðW L ðα ÞÞ þ ð1−πÞuðW NL ðα ÞÞ:
α∈½0:1
The FOC at α =1 is written as n 0 o 0 0 V ð1Þ ¼ πð1−πÞðD−CÞ u ðWL Þ−u ðWNL Þ :
ð9Þ
Because u is strictly concave and WL b WNL, (9) is strictly positive. This means that over-insurance is optimal. An intuition for this result is as follows. In contrast to the term insurance case, wealth holding mixed insurance remains some risks at the full insurance level. Thus, the individual is eager to hold mixed insurance beyond the full insurance level to decrease the risks that are remained in wealth. We summarize this argument in the following result. Result 3 If an individual has no rejoicing function, the optimal mixed insurance holding is over-insurance. Next, consider the special case where the risk function is linear. The objective function is written as max V ðα Þ ¼ π½W L ðα Þ þ hðα ðD−P m ÞÞ þ ð1−πÞ
α∈½0;1
5.2. Rejoicing theory and mixed insurance
½W NL ðα Þ þ hðð1−α ÞðP m −C ÞÞ:
Let us now consider an individual under rejoicing theory. The individual chooses an optimal mixed insurance coverage rate α⁎ to maximize the following objective function: max V ðα Þ ¼ π ½uðW L ðα ÞÞ þ hfuðW L ðα ÞÞ−uðW L ð0ÞÞg þ ð1−πÞ
ð10Þ
Here, W L (α) − W L (0) = α(D − P m ) and W NL (α) − W NL (1) = (1 − α)(Pm − C). The FOC (10) at α = 1 is given by 0 0 0 V ð1Þ ¼ πð1−πÞðD−CÞ h ðD−Pm Þ−h ð0Þ :
ð11Þ
α∈½0;1
½uðW NL ðα ÞÞ þ hfuðW NL ðα ÞÞ−uðW NL ð1ÞÞg: Here, WL(α) represents wealth in the loss state when the individual purchases mixed insurance with coverage rate α, and is given by WL(α) = w − D − α(Pm − D). WNL(α) is wealth in the no-loss state, and is given by WNL(α) = w − α(Pm − C). By applying Pm = πD + (1 − π)C, the FOC can be written as 0 0 V ðα Þ ¼ πð1−πÞðD−CÞ½u0 ðWL ðα ÞÞ 1 þ h fuðWL ðα ÞÞ−uðWL ð0ÞÞg ð6Þ 0 −u0 ðWNL ðα ÞÞ 1 þ h fuðWNL ðα ÞÞ−uðWNL ð1ÞÞg ¼ 0:
To examine the optimality of full insurance in this setting, we determine the sign of (6) when α⁎ = 1: 0 0 0 V ð1Þ ¼ πð1−πÞðD−C Þ u0 ðW L Þ 1 þ h ðUDL Þ −u0 ðW NL Þ 1 þ h ð0Þ : ð7Þ For ease of notation, we write WL(1) = WL, WNL(1) = WNL, and UDL =u(WL) − u(WL(0)). The sign of (7) coincides with that of 0 0 u0 ðWL Þ 1 þ h ðUDL Þ −u0 ðWNL Þ 1 þ h ð0Þ ;
ð8Þ
from π(1 − π)(D − C) N 0. Because WL b WNL and u′ is decreasing, u′(WL) ≥ u ′ (WNL ). Because UDL N 0 and h' is decreasing, h′(UL) ≤ h ′ (0). From this observation, we see that the sign of (8)—and thus that of (7)—is indeterminate. This means that it is possible that optimal demand for mixed insurance is either over- or under-insurance. That is, full insurance occurs in a knife-edge case.
Because h′(D − Pm) b h ′ (0), and the other elements are strictly positive, (11) is strictly negative. This means that under-insurance is optimal. An intuition for this result is as follows. At the full insurance level, an individual gains rejoicing in the loss state, while he or she does not gain this in the no-loss state. For a marginal decrease in his or her mixed insurance holding, he or she loses some rejoicing in the loss state, but gains some rejoicing in the no-loss state. The latter dominates the former because of the concavity of the rejoicing function. As a result, the optimal mixed insurance holding is under-insurance. We summarize this argument in the following result. Result 4 If an individual has a linear risk function, the optimal mixed insurance holding is under-insurance. From the results of these two special cases, we gain the following intuition: the risk effect drives over-insurance and the rejoicing effect drives under-insurance, while optimal insurance is determined by which effect outweighs the other. We confirm this intuition formally in the following subsection. 5.4. Risk and rejoicing effects on demand for mixed insurance We examine the effect of the risk and rejoicing functions. Let us consider the risk function û and rejoicing function ĥ such that full insurance is optimal, that is, h i 0 0 0 0 V 0 ð1Þ ¼ π ð1−π ÞðD−C Þ û ðW L Þ 1 þ ĥ ðUDL Þ −û ðW NL Þ 1 þ ĥ ð0Þ ¼ 0: ð12Þ
5.3. Decomposition of risk and rejoicing effects (12) can be rewritten as To examine which factors determine demand for mixed insurance under rejoicing theory, we consider two special cases: one with no rejoicing function and the other with a linear risk function. The first
0 û0 ðW L Þ 1 þ ĥ ð0Þ ¼ : 0 0 û ðW NL Þ 1 þ ĥ ðUDL Þ
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The following result states that the risk and rejoicing effects are characterized as the degrees of the concavity of the risk function u and of the rejoicing function h to determine demand for mixed insurance. Result 5 Let us consider the risk function û and rejoicing function ĥ such that full insurance is optimal. Over- (Under-)insurance is optimal if either of the following conditions is satisfied: • The risk function u is more (less) concave than the risk function û, in the sense that
−
u″ ðwÞ û″ ðwÞ ≥ ð ≤ Þ− 0 ; 0 u ðwÞ û ðwÞ
h″ ðwÞ 0
1 þ h ðwÞ
≤ ð ≥ Þ−
Both degrees of concavity have appeared in the literature. The degree of the concavity of the risk function (13) is the same as the absolute Arrow–Pratt aversion in expected utility theory. The degree of the concavity of the rejoicing function (14) appeared in Gee (2012).14 Finally, we examine the effect of compensation in the no-loss state on demand for mixed insurance. We denote by C ̂ compensation such that full insurance is optimal. Result 6 Let us consider compensation C ̂ such that full insurance is optimal. Over- (Under-)insurance is optimal if compensation is more (less) than C ̂, that is, C ≥ð≤ÞC ̂.
ð13Þ Proof From a straightforward calculation, the signs of
• The rejoicing function h is less (more) concave than the rejoicing function ĥ, in the sense that
−
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ĥ″ ðwÞ : 0 1 þ ĥ ðwÞ
ð14Þ
0 0 1 þ h ð0Þ u ðW L ðC ÞÞ ∂ ∂ 0 0 u ðW NL ðC ÞÞ 1 þ h ðUDL ðC ÞÞ and ∂C ∂C are determined as the following, respectively:
Proof Let us first consider that inequality (13) holds. The condition that 0 0 over- (under-)insurance is optimal is written as u ðW L Þð1 þ ĥ ðUDL ÞÞ−
00
0
00
0
0
ð1−πÞh ðUDL ðC ÞÞu ðW L ðC ÞÞ≤0:
0
u ðW NL Þð1 þ ĥ ð0ÞÞ≥ð≤Þ0: 0 Inequality (13) is equivalent to u0 ðwÞ=û ðwÞ decreasing (increasing) in w. Thus, for every wh , wl with wh N wl, 0
0
0
0
u0 ðwl Þ u ðw Þ u ðw Þ û ðw Þ ≥ð≤Þ 0 h ⟺ 0 l ≥ð≤Þ 0 l 0 u ðwh Þ û ðwl Þ û ðwh Þ û ðwh Þ holds. By setting wh = WNL and wl = WL, we obtain the following13: ! 0 0 0 u ðW L Þ û ðW L Þ 1 þ ĥ ð0Þ ≥ð≤Þ 0 ¼ 0 0 u ðW NL Þ û ðW NL Þ 1 þ ĥ ðUDL Þ 0 0 0 0 ⟺u ðW L Þ 1 þ ĥ ðUDL Þ −u ðW NL Þ 1 þ ĥ ð0Þ ≥ ð ≤ Þ0: This indicates that over- (under-)insurance is optimal when the individual's risk function u is more (less) concave than û in the sense that inequality (13) holds. We next consider whether inequality (14) holds. From an argument similar to that above, we have 0 0 0 û ðW L Þ 1 þ ĥ ð0Þ 1 þ h ð0Þ 0 ¼ ⟺û ðW L Þ ≥ð≤Þ 0 0 0̂ û ðW NL Þ 1 þ h ðUDL Þ 1 þ h ðUDL Þ 0
0
0
1 þ h ðUDL Þ −û ðW NL Þ 1 þ h ð0Þ ≥ ð ≤ Þ0:
This indicates that over- (under-)insurance is optimal when the individual's rejoicing function h is less (more) concave than ĥ in the sense that inequality (14) holds. □ An intuition of Result 5 can be obtained from Results 3 and 4 in the previous section. From these results, we know that the risk effect strengthens demand for mixed insurance, while the rejoicing effect weakens it at the full insurance level. Since the risk and rejoicing effects can be captured by the degree of the concavity of their functions, over- (under-)insurance is optimal when the risk (rejoicing) function is more concave compared with the full insurance case. 13
00
−ð1−πÞu ðW L ðC ÞÞu ðW NL ðC ÞÞ−πu ðW NL ðC ÞÞu ðW L ðC ÞÞ≥0;
The risk function is preserved under affine transformations, meaning that we can set UDL ¼ uðW L Þ−uðW L ð0ÞÞ ¼ ûðW L Þ−ûðW L ð0ÞÞ.
Here, compensation C is expressed as the argument of the function. From the above, we have the following inequalities for compensation C such that C ≥ð≤ÞC :̂ 0 0 u W L Ĉ u ðW L ðC ÞÞ ; ≥ð≤Þ u0 ðW NL ðC ÞÞ u0 W Ĉ NL
0
0
1 þ h ð0Þ
1 þ h ð0Þ : ≤ð≥Þ 0 0 1 þ h ðUDL ðC ÞÞ 1 þ h UDL C ̂
By combining these inequalities with 0 0 u W L Ĉ 1 þ h ð0Þ ¼ ; 0 u0 W NL C ̂ 1 þ h UDL C ̂ we obtain 0
1 þ h ð0Þ 0
1 þ h ðUDL ðC ÞÞ
0
≤ð≥Þ
u ðW L ðC ÞÞ u0 ðW NL ðC ÞÞ
0 0 0 0 ⇔u ðW L ðC ÞÞ 1 þ h ðUDL ðC ÞÞ −u ðW NL ðC ÞÞ 1 þ h ð0Þ ≥ ð ≤ Þ0: □ From the above analysis, we know that demand for mixed insurance is determined by the risk and rejoicing effects. The risk effect strengthens such demand, whereas the rejoicing effect weakens it. If compensation is increasing, an individual feels less rejoicing. Thus, the risk effect dominates (is dominated by) the rejoicing effect when compensation is relatively large (small). As a result, an individual has a motive to hold more (less) mixed insurance than the full insurance level. 14 Gee (2012) axiomatized the regret form and characterized the degree of the concavity of the regret function as second-order regret aversion. A similar argument can be applied to the rejoicing form.
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6. Policy implications By substituting α⁎ = 1 and C = 0 to (6), we have 0 0 πð1−πÞDu0 ðW L Þ h ðUDL Þ−h ð0Þ b 0:
This result implies that the optimal coverage rate of term insurance in rejoicing theory is under-insurance if the rejoicing function is not linear. It also means that this coverage rate is always smaller than that under expected utility theory since the full insurance is optimal under expected utility theory. Furthermore, from the combination of Results 3 and 4, we find that the optimal coverage rate of mixed insurance in rejoicing theory is always smaller than that under expected utility theory because the rejoicing effect lowers the optimal coverage rate. In the case of term insurance, individuals do not choose full insurance in rejoicing theory and this means that the optimal risk allocation (i.e., that a risk-neutral insurance firm undertakes all risks) cannot be realized. To realize the optimal risk allocation in rejoicing theory, raising the optimal coverage rate by lowering the insurance premium seems to be effective. One way of lowering the insurance premium is to create a more competitive insurance market. For example, if the number of insurance firms increases, the insurance premium lowers through fiercer competition and the optimal coverage rate finally rises. Theoretically speaking, the insurance premium approaches the actuarially fair level if a perfectly competitive market is realized and the optimal coverage rates in the model are actually realized. However, the insurance premium is never below the expected insurance money even if a perfectly competitive market is realized. Thus, to keep the insurance premium below the expected insurance money, subsidies and tax deductions are necessary to reduce the insurance premium. For example, some life insurance premiums in Japan can be tax deductions from an individual's income. Under the present tax system in Japan, 120,000 JPY (about 1128 USD) and 70,000 JPY (about 658 USD) are the maximum deductions in income and resident tax, respectively.15 The Life Insurance Association of Japan (LIAJ) was requested to stabilize the tax deduction.16 Then, it requested to expand the deduction in income tax from 120,000 to 150,000 JPY (about 1410 USD)17 in order to incentivize individuals to raise their coverage rates. However, according to expected utility theory, we might conclude that the coverage rate is unchanged because individuals have already achieved full insurance, especially in the case of mixed insurance. In other words, expected utility theory cannot explain the reason why the LIAJ has expanded the tax deduction in a competitive insurance market. By contrast, in rejoicing theory, under-insurance is achieved in term insurance and, possibly, in mixed insurance. In other words, the coverage rate could be raised by expanding the tax deduction. This also means that the risk allocation becomes more desirable because an insurance firm undertakes more risk when the tax deduction is expanded. From this viewpoint, rejoicing theory can provide a more rational explanation of the actions of the LIAJ. Moreover, we find that a further tax deduction is needed in rejoicing theory than in expected utility theory to realize a more desirable risk allocation in the insurance market.
15
1 USD = 106.35 JPY on April 29, 2016. For more details, see the website of the LIAJ: http://www.seiho.or.jp/english/news/ 2013/0621.html (requests on Tax Reform for FY2014). 17 For more details, see the following websites in LIAJ: http://www.seiho.or.jp/english/ news/2014/0718.html (requests on Tax Reform for FY2015) and http://www.seiho.or.jp/ english/news/2015/0717.html (requests on Tax Reform for FY2016). 16
7. Concluding remarks This study examines how regret and rejoicing theory affect mixed insurance choice and demand. By using these theories, we derive the conditions under which an individual prefers to hold mixed insurance rather than term insurance. We also examine which factors determine demand for mixed insurance (under-insurance, full insurance, overinsurance), by decomposing the risk and rejoicing effects. Braun and Muermann (2004) opened up the possibility that regret theory is a promising tool in insurance economics. This study broadens its possibility to the analysis of mixed insurance as well as sheds light on rejoicing, which receives equal weight to regret. Future work could examine other insurance problems by using either regret theory or rejoicing theory or both. In general, behavioral approaches, not limited to regret and rejoicing, will become more important in insurance economics in the future. Acknowledgments We are grateful to the three anonymous reviewers whose comments and suggestions helped us improve this paper greatly. We would like to thank the seminar participants at the 2014 Asia-Pacific Risk and Insurance meeting, the 2015 World Risk and Insurance Economic Congress, and the second East Asia RMI insurance workshop for their helpful comments on earlier versions of this paper. All remaining errors are, of course, ours. This study was partly supported by JSPS KAKENHI Grant numbers 24730362 and 26705004 and Kampo-Zaidan Grants. References Allais, M., 1953. Le comportement de l'homme rationnel devant le risque; critique des postulats et axiomes de l'ecole Americaine. Econometrica 21, 503–546. Bell, D.E., 1982. Regret in decision making under uncertainty. Oper. Res. 30, 961–981. Bikhchandani, S., Segal, U., 2011. Transitive regret. Theor. Econ. 6, 95–108. Bleichrodt, H., Cillo, A., Diecidue, E., 2010. A quantitative measurement of regret theory. Manag. Sci. 56, 161–175. Braun, M., Muermann, A., 2004. The impact of regret on the demand for insurance. J. Risk Insur. 71, 737–767. Ehrlich, I., Becker, G.S., 1972. Market insurance, self-insurance, and self-protection. J. Polit. Econ. 80, 623–648. Gee, C., 2012. A behavioral model of regret aversion. Working Paper (Available at https:// sites.google.com/site/econgee/research). Huang, R.J., Muermann, A., Tzeng, L.Y., 2015. Hidden regret in insurance markets. J. Risk Insur. 83, 181–216. Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291. Loomes, G., Sugden, R., 1982. Regret theory: an alternative theory of rational choice under uncertainty. Econ. J. 92, 805–824 (1982). Mossin, J., 1968. Aspects of rational insurance purchasing. J. Polit. Econ. 76, 553–568. Quiggin, J., 1994. Regret theory with general choice sets. J. Risk Insur. 8, 153–165. Rothschild, M., Stiglitz, J.E., 1971. Increasing risk II: its economic consequences. J. Econ. Theory 3, 66–84. Sarver, T., 2008. Anticipating regret: why fewer options may be better. Econometrica 76, 263–305. Sugden, R., 1993. An axiomatic foundation for regret theory. J. Econ. Theor. 60, 159–180. Tsai, J.Y., 2012. Risk and regret aversions on optimal bank interest margin under capital regulation. Econ. Model. 29, 2190–2197. Tversky, A., Kahneman, D., 1992. Advances in prospect theory: cumulative representation of uncertainty. J. Risk Insur. 5, 297–323. Wong, K.P., 2011. Regret theory and the banking firm: the optimal bank interest margin. Econ. Model. 28, 2483–2487. Wong, K.P., 2012. Production and insurance under regret aversion. Econ. Model. 29, 1154–1160. Wong, K.P., 2014. Regret theory and the competitive firm. Econ. Model. 36, 172–175.