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Some new results about optimal insurance demand under uncertainty
Finance Research Letters 17 (2016) 280–284
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Some new results about optimal insurance demand under uncertainty Baoan Huang a,b,∗, Jianjun Miao a, Zongliang Zhang b, Dianbo Zhao c a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, No169 Shengtai West Road, Nanjing 211106, Jiangsu China b Linyi University, Linyi 276000, Shandong, China c Shandong Urban Constructional College, Jinan 250103, Shandong, China
a r t i c l e
i n f o
Article history: Received 2 March 2016 Accepted 28 March 2016 Available online 31 March 2016 JEL classification: D11 D81 G22
a b s t r a c t The aim of this paper is to investigate the optimal insurance demand of a risk-averse agent who is faced with background uncertainty. The preferences of the agent are represented by two-moment, mean-standard deviation utility functions. By the comparative statics, we find that under the assumption of decreasing absolute risk aversion (DARA), the changes of background uncertainty have effects on optimal insurance demand. © 2016 Elsevier Inc. All rights reserved.
Keywords: Background uncertainty Decreasing absolute risk aversion (μ,σ ) preferences Optimal insurance demand
1. Introduction The topic of how insurance demand responds to the change of risk has attracted much interest of many researchers in the fields of economics and finance since long (see e.g., Briys et al. (1993); Broll et al. (1995); Broll and Eckwert (1999)). And the optimal insurance demand problem is especially hot (see e.g., Demers and Demers, 1991; Hadar and Seo (1990) and Dionne and Gollier (1992)). More recently the comparative statics about optimal insurance demand in (μ, σ )−space have attracted much more interest of many researchers. According to Battermann et al. (2002), a risk-averse agent reduces his demand for insurance upon an increase in the risk of an insurable wealth loss if and only if the elasticity of his risk aversion with respect to the standard deviation of wealth is greater than unity. Bonilla and Ruiz (2014) have recently argued that the risk-averse individuals will always reduce their demand for insurance when the expected value of the insurable loss decreases. Eichner and Wagener (2014) argue that insurance demand goes down when the expected size of insurable losses decreases or insurance premia increases if the elasticity of risk aversion with respect to expected wealth exceeds −1. The results all of the above only considered the effects of the changes of random loss itself, but didn’t consider other factors. In this paper, we define a new model, which is different to the model in Battermn et al. (2002). In fact, decision makers always face many kinds of background uncertainty, such as economic background, policy environment etc. These factors often have ∗
Corresponding author. Tel.: +8613869957121. E-mail addresses: [email protected] (B. Huang), [email protected] (J. Miao), [email protected] (Z. Zhang), [email protected] (D. Zhao).
http://dx.doi.org/10.1016/j.frl.2016.03.026 1544-6123/© 2016 Elsevier Inc. All rights reserved.
B. Huang et al. / Finance Research Letters 17 (2016) 280–284
281
impacts on decision makers. So, in this paper, we extend the model of Bonilla and Ruiz (2014) by adding a background uncertainty risk. This kind of model is especially useful in the analysis of optimal demand for price index insurance. Our study proceeds as follows: Section 2 presents the new model and discuss the comparative statics effects of some parameter changes. Section 3 presents the main results of the paper. Section 4 concludes with some brief remarks. 2. The model Like in Bonilla and Ruiz (2014), we define Yi as a random variable which belongs to the random set Y. As in Sinn (1983) and Meyer (1987), we also suppose that each element of Y differs from one another only by location and scale Y −μ parameters. Hence, each Yi has a finite mean and standard deviation denoted by μi and σ i respectively. Let X = i σ i , then i every Yi has the same distribution function as μi + σi X, no matter which was selected to define X. Therefore, the expected utility from Y for any agent with utility function u( · ) can be denoted by
Eu(Y ) =
b
a
u(μ + σ x )dF(x ) =: φ (μ, σ ).
(1)
The interval [a, b]⊆R with a < b is the support of X, and the distribution function of X is F. According to the results of Sinn (1983) and Meyer (1987), it is true that φ u > 0, φ σ < 0, φ μu < 0, where the subscripts denote the partial derivatives. For the convenience of later proof, we list a series of equivalence between φ and u (see Meyer, 1987 or Wagener, 2003):
u ( y ) > 0 ⇔ φμ ( μ , σ ) > 0 ;
(1.1)
u (y ) < 0 ⇔ φσ (μ, σ ) < 0;
(1.2)
u (y ) > 0 ⇔ φμσ (μ, σ ) > 0;
(1.3)
2 u (y ) < 0 ⇔ φ is strictly concave: φμμ < 0, φσ σ < 0, and φμμ φσ σ − φμσ > 0;
(1.4)
u (y ) − u (y )
< 0 ⇔ φμ φμσ − φσ φμμ > 0.
(1.5)
Now we consider the following model about insurance demand, which is different to the model in Eichner and Wagener (2014), and Bonilla and Ruiz (2014). Assume that a risk averse agent has an initial wealth w, and faces an insurable risky loss L and a background uncertainty ɛ. The mean and the standard deviation of the loss L are represented by μL and σ L . The mean and the standard deviation of the background uncertainty is zero and σ ɛ respectively. Let α be the coinsurance rate. In this coinsurance problem we allow for arbitrary correlations between insurable loss L and background uncertainty ɛ. The loss L can be insured at constant marginal costs (1 + λ )μL , where λ > 0 is a fixed loading factor. Then the final wealth equals to y(α ) = − (1 − α )L − (1 + λ )αμL + ε. Hence, the mean μy and variance σy2 of y(α ) amount to − (1 + αλ )μL , (1 − α )2 σL2 + σε2 − 2(1 − α )cov(L, ε ) respectively. In (μ, σ )-space, the maximum utility problem about coinsurance rate for the agent can be represented as
max φ (μy (α ), σy (α )), α
where μy (α ) = − (1 + αλ )μL , σy (α ) =
(2)
(1 − α )2 σL2 + σε2 − 2(1 − α )cov(L, ε ). Now, for later reference, we list some formulas about σ (α )
∂σ (α ) (α − 1 )σL2 + cov(L, ε ) = ; ∂α σ (α ) ∂σ (α ) (1 − α )2 σL = > 0; ∂ σL σ (α ) ∂σ (α ) σε = > 0; ∂ σε σ (α )
(2.1)
(2.2) (2.3)
∂ 2 σ (α ) σL2 σε2 (1 − ρ 2 (L, ε )) = > 0; ∂ α2 σ 3 (α )
(2.4)
∂ 2 σ (α ) (α − 1 )σL [(α − 1 )2 σL2 + 3(α − 1 )cov(L, ε ) + 2σε2 ] = ; ∂ α∂ σL σ 3 (α )
(2.5)
σε [(α − 1 )σL2 + cov(L, ε )] ∂ 2 σ (α ) =− ; ∂ α∂ σε σ 3 (α )
(2.6)
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∂ 2 σ (α ) σ 2 + (α − 1 )cov(L, ε ) = ε . ∂ α∂ cov(L, ε ) σ 3 (α )
(2.7)
The first order condition of the maximum problem (2) can be implicitly characterized by the following equation:
F (μL , σL , σε , α ) := φμ (−λμL ) + φσ
∂σ (α ) = 0. ∂α
(3)
We denote the solution of (3) by α ∗. Due to the strictly concavity of φ , the second order condition for a maximum will always be satisfied at α ∗. Hence, from Eq (3), it is true that α ∗ should satisfy the following condition:
∂σ (α ) (α − 1 )σL2 + cov(L, ε ) = < 0. ∂α σ (α )
(4)
3. Results In this paper, we mainly focus on the study of the effects of the changes in the background uncertainty on optimal coinsurance rate α ∗. Three factors are to be studied: change in the standard deviation of the loss σ L , change in the covariance between the loss and the background uncertainty, and change in the standard deviation σ ɛ of the background uncertainty. For the simplicity of the note in proof, we will omit the asterisk on α ∗ at some places from now. This means that α is the same as α ∗ from now. Result 1 Assume the risk aversion individual belongs to the decreased risk absolute aversion (DARA), then the optimal σ (α ) insurance demand α ∗ increases upon an increase in σ L , if ∂∂ α∂ σL < 0, and ρ (L, ε ) ≤ 2
8/9.
Proof. From the Eq. (3), the comparative statics of α ∗ with respect to σ L can be evaluated by
∂ α∗ ∂ F /∂ σL ∂σ (α ) ∂ 2 σ (α ) 1 ∂σ (α ) =− =− −λμL φμ σ + φσ σ + φσ ∂ σL ∂ F /∂ α ∂ F /∂ α ∂ σL ∂α ∂ α∂ σL 2 φ 1 ∂σ (α ) ∂ σ (α ) =− −λμL . φμ σ − φσ σ μ + φ σ ∂ F /∂ α ∂ σL φσ ∂ α∂ σL ∗
It is sufficient to prove ∂∂ασ to be positive. Here, ∂ F/∂ α < 0 because of the second-order condition. Then, since φ σ < 0 L
2 σ (α ) ∗ and ∂σ∂ σ(α ) > 0, ∂∂ασ is positive if the second round-bracketed expression is negative and ∂∂ α∂ σL ≤ 0 . L L On the one hand, because φ μσ > 0 > φ σ σ , then, as in Wagener (2003), it is true that (1.4) and (1.5) together ensure 2 /φ thatφμ φμσ /φσ < φμμ < φμσ σ σ , which leads to φμ σ − φμ φσ σ /φσ < 0. On the other hand, because 0 < α < 1 and
∂ 2 σ (α ) (α − 1 )σL [(α − 1 )2 σL2 + 3(α − 1 )cov(L, ε ) + 2σε2 ] = . ∂ α∂ σL σ 3 (α ) 2 σ (α ) Then ∂∂ α∂ σL is non-positive if the square-bracketed expression therein is non-negative. This can be sure ifcov(L, ε ) ≤ 0. If cov(L, ε ) > 0 we rewrite the square-bracket term in (2.5) by
√
(α − 1 )σL + 2σε
2
√ + (α − 1 )σL σε (3ρ (L, ε )) − 2 2 ).
∂ 2 σ (α )
Then ∂ α∂ σ is non-positive if (5) is non-negative. In fact, if ρ (L, ε ) ≤ L
(5)
8/9, then (5) is surely non-negative.
According to the Result 1, DARA ensures that α ∗ reacts positively upon an increase in the standard deviation of the back-
σ (α ) ∂σ (α ) 8/9, which ensures that φσ ∂∂ α∂ σL is non-negative, is over strict for ∂ σL . to be positive. And notice that the condition ρ (L, ε ) ≤ 8/9 = 0.943 includes most of the states of the relation between the random loss and the background risk. So, we can infer that under the DARA assumption, optimal insurance demand increases with an increase of the standard deviation of the random loss if the random loss and the background uncertainty risk are not too positively correlated. Now, we present the following result which is about the effect of the increase of the covariance between the random loss L and the background uncertainty ɛ on optimal insurance demand.
ground risk. However the condition ρ (L, ε ) ≤
2
Result 2 Assume that α ∗ ≥ 0, for the DARA preferences, the optimal insurance demand α ∗ decreases upon an increase in cov(L, ε ) if cov(L, ε ) ≤ 0. Proof. From Eq. (3), the implicit function gives
1 ∂ α∗ ∂σ (α ) ∂ 2 σ (α ) α−1 =− −λμL φμ σ + φσ σ + φσ . ∂ cov(L, ε ) ∂ F /∂ α σ (α ) ∂α ∂ α∂ cov(L, ε )
(6)
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From the proof in the Result 1, the round-bracketed in the first term in the square-bracketed of (6) is positive under the DARA assumption. Hence, the first term in the square-bracketed of (6) is negative because α < 1. It is true that ∂ F/∂ α < 0 from the second-order condition. Then because φ σ < 0, if, but not only if, ∂ 2 σ (α )/∂α∂ cov(L, ε ) > 0 then ∂ α ∗ /∂ cov(L, ε ) < 0. From (2.7) and (4), we may find that becausecov(L, ε ) ≤ 0,then ∂ 2 σ (α )/∂α∂ cov(L, ε ) = [σε2 + (α − 1 )cov(L, ε )]/σ 3 (α ) > ∂ α∗ 0.So, ∂ cov (L,ε ) is negative. From Result 2, we know that the covariance between the random loss and the background uncertainty also has an impact on insurance demand. Next we consider how the optimal insurance demand α ∗ reacts on the change in the background uncertainty. Firstly, for the convenience of the proof, we introduce a utility function as
W (μ, ν ) := φ (μ,
√
ν ), Where ν = σ 2 .
Result 3 Suppose W(μ, ν ) is concave, and φ satisfy DARA (1.5), then optimal insurance demand increases upon an increase in σ ɛ . Proof. Implicit differentiation of Eq. (3) yields
1 ∂ α∗ ∂ F /∂ σε ∂σ (α ) ∂ 2 σ (α ) ∂σ (α ) =− =− −λμL φμ σ + φσ σ + φσ . ∂ σε ∂ F /∂ α ∂ F /∂ α ∂ σε ∂α ∂ α∂ σε
(7)
Because W(μ, v)is concave, like in the Fact 5 of Wagener (2003), we can have
Wνν = [1/4σ 2 (α )](φσ σ − (φσ /σ (α )), Wμν = φμσ /(2σ (α )) > 0,
Wμμ = φμμ < 0.
Since W(μ, ν ) is concave, we can have
φσ 2 0 < WμμWvv − Wμv = − φμσ . φμμ φσ σ − σ (α ) 4σ 2 ( α ) 2
1
(8)
Because Wμμ < 0, (8) is true only if WVV < 0. By (2.6) and (2.1), we can have
∂ 2 σ (α ) σε ∂σ (α ) =− 2 . . ∂ α∂ σε σ (α ) ∂α
(9)
Using (9) and Eq. (3) and (2.3), we rewrite (7) as
φμ 1 λμL σε ∂ α∗ φ =− −φμ σ + . φσ σ − σ ∂ σε ∂ F /∂ α σ (α ) φσ σ (α )
From (8) and (10), we can have
(10)
φσ φμμ φσ ∂ α ∗ = − 1 λμL σε φμ ∂ σε ∂ F/∂ α σ (α ) φμμ φσ −φμ σ φμ + φμμ φσ σ − σ (α )
φμ φσ 2 L σε > − ∂ F/1∂ α λμ > 0. σ (α ) φμμ φσ −φμσ + φμμ φσ σ − σ (α )
Where the first inequality is true because of the DARA in (1.5), and the second inequality is true because of (8).
(11)
Notice that there is no restriction on the sign of cov(L, ε ) for Result 3 to be true. It means that more uncertain environment may force the risk-averse agent to buy more insurance under the assumption of the Result 3. 4. Conclusion In this study, we have analyzed a new optimal insurance demand model which includes a risk named background uncertainty ɛ. By comparative statics, we obtained several new results. Optimal insurance demand may change according to the changes of several factors, such as the standard deviation of the loss, the covariance between the loss and the background uncertainty, and the standard deviation of the background uncertainty. Moreover the kind of preference which the agent belongs to can also affect optimal insurance demand. For further research one can add another kind of risk variable to the model to analyze the effect of background risk on the decision makers, and also can change the assumption of the DARA restrictions. Acknowledgement The authors thank for the fund support by the Scientific Innovation Research of College Graduate in Jiangsu Province, China (KYZZ15_0105).
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References Battermann, H., Broll, U., Wahl, J.E., 2002. Insurance demand and the elasticity of risk aversion. Oper. Res. Spectrum 24, 145–150. Bonilla, C.A., Ruiz, J.L., 2014. Insurance demand and first order risk increases under (μ,σ )-preferences. Finan. Res. Lett. 11 (3), 219–223. Briys, E., Couchy, M., Schlesinger, H., 1993. Optimal hedging in a futures market with back-ground noise and basis risk. Euro. Econ. Rev. 37, 949–960. Broll, U., Wahl, J.E., Zilcha, I., 1995. Indirect hedging of exchange rate risk. J. Int. Money Finance 14, 667–678. Broll, U., Eckwert, B., 1999. Exchange rate volatility and international trade. South. Econ. J. 66, 178–185. Demers, F., Demers, M., 1991. Increases in risk and the optimal deductible. J. Risk Uncertainty 51, 670–699. Dionne, G., Gollier, Ch., 1992. Comparative statics under multiple sources of risk with applications to insurance demand. Geneva Pap. Risk Ins. Theory 17, 21–33. Eichner, T., Wagener, A., 2014. Insurance demand and first order risk increases under (μ,σ )-preferences revisited. Finan. Res. Lett. 11 (4), 326–331. Hadar, J., Seo, T., 1990. The effects of shifts in a return distribution on optimal portfolios. Int. Econ. Rev. 31, 721–736. Meyer, J., 1987. Two-moment decision models and expected utility maximization. Am. Econ. Rev. 77, 421–430. Sinn, H.-W., 1983. Economic Decisions under Uncertainty. Physica-Verlag Heidelberg, North-Holland, Amsterdam. Wagener, A., 2003. Comparative statics under uncertain: the case of mean-variance preferences. Euro. J. Oper. Rese. 151, 224–232.
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Some new results about optimal insurance demand under uncertainty Baoan Huang a,b,∗, Jianjun Miao a, Zongliang Zhang b, Dianbo Zhao c a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, No169 Shengtai West Road, Nanjing 211106, Jiangsu China b Linyi University, Linyi 276000, Shandong, China c Shandong Urban Constructional College, Jinan 250103, Shandong, China
a r t i c l e
i n f o
Article history: Received 2 March 2016 Accepted 28 March 2016 Available online 31 March 2016 JEL classification: D11 D81 G22
a b s t r a c t The aim of this paper is to investigate the optimal insurance demand of a risk-averse agent who is faced with background uncertainty. The preferences of the agent are represented by two-moment, mean-standard deviation utility functions. By the comparative statics, we find that under the assumption of decreasing absolute risk aversion (DARA), the changes of background uncertainty have effects on optimal insurance demand. © 2016 Elsevier Inc. All rights reserved.
Keywords: Background uncertainty Decreasing absolute risk aversion (μ,σ ) preferences Optimal insurance demand
1. Introduction The topic of how insurance demand responds to the change of risk has attracted much interest of many researchers in the fields of economics and finance since long (see e.g., Briys et al. (1993); Broll et al. (1995); Broll and Eckwert (1999)). And the optimal insurance demand problem is especially hot (see e.g., Demers and Demers, 1991; Hadar and Seo (1990) and Dionne and Gollier (1992)). More recently the comparative statics about optimal insurance demand in (μ, σ )−space have attracted much more interest of many researchers. According to Battermann et al. (2002), a risk-averse agent reduces his demand for insurance upon an increase in the risk of an insurable wealth loss if and only if the elasticity of his risk aversion with respect to the standard deviation of wealth is greater than unity. Bonilla and Ruiz (2014) have recently argued that the risk-averse individuals will always reduce their demand for insurance when the expected value of the insurable loss decreases. Eichner and Wagener (2014) argue that insurance demand goes down when the expected size of insurable losses decreases or insurance premia increases if the elasticity of risk aversion with respect to expected wealth exceeds −1. The results all of the above only considered the effects of the changes of random loss itself, but didn’t consider other factors. In this paper, we define a new model, which is different to the model in Battermn et al. (2002). In fact, decision makers always face many kinds of background uncertainty, such as economic background, policy environment etc. These factors often have ∗
Corresponding author. Tel.: +8613869957121. E-mail addresses: [email protected] (B. Huang), [email protected] (J. Miao), [email protected] (Z. Zhang), [email protected] (D. Zhao).
http://dx.doi.org/10.1016/j.frl.2016.03.026 1544-6123/© 2016 Elsevier Inc. All rights reserved.
B. Huang et al. / Finance Research Letters 17 (2016) 280–284
281
impacts on decision makers. So, in this paper, we extend the model of Bonilla and Ruiz (2014) by adding a background uncertainty risk. This kind of model is especially useful in the analysis of optimal demand for price index insurance. Our study proceeds as follows: Section 2 presents the new model and discuss the comparative statics effects of some parameter changes. Section 3 presents the main results of the paper. Section 4 concludes with some brief remarks. 2. The model Like in Bonilla and Ruiz (2014), we define Yi as a random variable which belongs to the random set Y. As in Sinn (1983) and Meyer (1987), we also suppose that each element of Y differs from one another only by location and scale Y −μ parameters. Hence, each Yi has a finite mean and standard deviation denoted by μi and σ i respectively. Let X = i σ i , then i every Yi has the same distribution function as μi + σi X, no matter which was selected to define X. Therefore, the expected utility from Y for any agent with utility function u( · ) can be denoted by
Eu(Y ) =
b
a
u(μ + σ x )dF(x ) =: φ (μ, σ ).
(1)
The interval [a, b]⊆R with a < b is the support of X, and the distribution function of X is F. According to the results of Sinn (1983) and Meyer (1987), it is true that φ u > 0, φ σ < 0, φ μu < 0, where the subscripts denote the partial derivatives. For the convenience of later proof, we list a series of equivalence between φ and u (see Meyer, 1987 or Wagener, 2003):
u ( y ) > 0 ⇔ φμ ( μ , σ ) > 0 ;
(1.1)
u (y ) < 0 ⇔ φσ (μ, σ ) < 0;
(1.2)
u (y ) > 0 ⇔ φμσ (μ, σ ) > 0;
(1.3)
2 u (y ) < 0 ⇔ φ is strictly concave: φμμ < 0, φσ σ < 0, and φμμ φσ σ − φμσ > 0;
(1.4)
u (y ) − u (y )
< 0 ⇔ φμ φμσ − φσ φμμ > 0.
(1.5)
Now we consider the following model about insurance demand, which is different to the model in Eichner and Wagener (2014), and Bonilla and Ruiz (2014). Assume that a risk averse agent has an initial wealth w, and faces an insurable risky loss L and a background uncertainty ɛ. The mean and the standard deviation of the loss L are represented by μL and σ L . The mean and the standard deviation of the background uncertainty is zero and σ ɛ respectively. Let α be the coinsurance rate. In this coinsurance problem we allow for arbitrary correlations between insurable loss L and background uncertainty ɛ. The loss L can be insured at constant marginal costs (1 + λ )μL , where λ > 0 is a fixed loading factor. Then the final wealth equals to y(α ) = − (1 − α )L − (1 + λ )αμL + ε. Hence, the mean μy and variance σy2 of y(α ) amount to − (1 + αλ )μL , (1 − α )2 σL2 + σε2 − 2(1 − α )cov(L, ε ) respectively. In (μ, σ )-space, the maximum utility problem about coinsurance rate for the agent can be represented as
max φ (μy (α ), σy (α )), α
where μy (α ) = − (1 + αλ )μL , σy (α ) =
(2)
(1 − α )2 σL2 + σε2 − 2(1 − α )cov(L, ε ). Now, for later reference, we list some formulas about σ (α )
∂σ (α ) (α − 1 )σL2 + cov(L, ε ) = ; ∂α σ (α ) ∂σ (α ) (1 − α )2 σL = > 0; ∂ σL σ (α ) ∂σ (α ) σε = > 0; ∂ σε σ (α )
(2.1)
(2.2) (2.3)
∂ 2 σ (α ) σL2 σε2 (1 − ρ 2 (L, ε )) = > 0; ∂ α2 σ 3 (α )
(2.4)
∂ 2 σ (α ) (α − 1 )σL [(α − 1 )2 σL2 + 3(α − 1 )cov(L, ε ) + 2σε2 ] = ; ∂ α∂ σL σ 3 (α )
(2.5)
σε [(α − 1 )σL2 + cov(L, ε )] ∂ 2 σ (α ) =− ; ∂ α∂ σε σ 3 (α )
(2.6)
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B. Huang et al. / Finance Research Letters 17 (2016) 280–284
∂ 2 σ (α ) σ 2 + (α − 1 )cov(L, ε ) = ε . ∂ α∂ cov(L, ε ) σ 3 (α )
(2.7)
The first order condition of the maximum problem (2) can be implicitly characterized by the following equation:
F (μL , σL , σε , α ) := φμ (−λμL ) + φσ
∂σ (α ) = 0. ∂α
(3)
We denote the solution of (3) by α ∗. Due to the strictly concavity of φ , the second order condition for a maximum will always be satisfied at α ∗. Hence, from Eq (3), it is true that α ∗ should satisfy the following condition:
∂σ (α ) (α − 1 )σL2 + cov(L, ε ) = < 0. ∂α σ (α )
(4)
3. Results In this paper, we mainly focus on the study of the effects of the changes in the background uncertainty on optimal coinsurance rate α ∗. Three factors are to be studied: change in the standard deviation of the loss σ L , change in the covariance between the loss and the background uncertainty, and change in the standard deviation σ ɛ of the background uncertainty. For the simplicity of the note in proof, we will omit the asterisk on α ∗ at some places from now. This means that α is the same as α ∗ from now. Result 1 Assume the risk aversion individual belongs to the decreased risk absolute aversion (DARA), then the optimal σ (α ) insurance demand α ∗ increases upon an increase in σ L , if ∂∂ α∂ σL < 0, and ρ (L, ε ) ≤ 2
8/9.
Proof. From the Eq. (3), the comparative statics of α ∗ with respect to σ L can be evaluated by
∂ α∗ ∂ F /∂ σL ∂σ (α ) ∂ 2 σ (α ) 1 ∂σ (α ) =− =− −λμL φμ σ + φσ σ + φσ ∂ σL ∂ F /∂ α ∂ F /∂ α ∂ σL ∂α ∂ α∂ σL 2 φ 1 ∂σ (α ) ∂ σ (α ) =− −λμL . φμ σ − φσ σ μ + φ σ ∂ F /∂ α ∂ σL φσ ∂ α∂ σL ∗
It is sufficient to prove ∂∂ασ to be positive. Here, ∂ F/∂ α < 0 because of the second-order condition. Then, since φ σ < 0 L
2 σ (α ) ∗ and ∂σ∂ σ(α ) > 0, ∂∂ασ is positive if the second round-bracketed expression is negative and ∂∂ α∂ σL ≤ 0 . L L On the one hand, because φ μσ > 0 > φ σ σ , then, as in Wagener (2003), it is true that (1.4) and (1.5) together ensure 2 /φ thatφμ φμσ /φσ < φμμ < φμσ σ σ , which leads to φμ σ − φμ φσ σ /φσ < 0. On the other hand, because 0 < α < 1 and
∂ 2 σ (α ) (α − 1 )σL [(α − 1 )2 σL2 + 3(α − 1 )cov(L, ε ) + 2σε2 ] = . ∂ α∂ σL σ 3 (α ) 2 σ (α ) Then ∂∂ α∂ σL is non-positive if the square-bracketed expression therein is non-negative. This can be sure ifcov(L, ε ) ≤ 0. If cov(L, ε ) > 0 we rewrite the square-bracket term in (2.5) by
√
(α − 1 )σL + 2σε
2
√ + (α − 1 )σL σε (3ρ (L, ε )) − 2 2 ).
∂ 2 σ (α )
Then ∂ α∂ σ is non-positive if (5) is non-negative. In fact, if ρ (L, ε ) ≤ L
(5)
8/9, then (5) is surely non-negative.
According to the Result 1, DARA ensures that α ∗ reacts positively upon an increase in the standard deviation of the back-
σ (α ) ∂σ (α ) 8/9, which ensures that φσ ∂∂ α∂ σL is non-negative, is over strict for ∂ σL . to be positive. And notice that the condition ρ (L, ε ) ≤ 8/9 = 0.943 includes most of the states of the relation between the random loss and the background risk. So, we can infer that under the DARA assumption, optimal insurance demand increases with an increase of the standard deviation of the random loss if the random loss and the background uncertainty risk are not too positively correlated. Now, we present the following result which is about the effect of the increase of the covariance between the random loss L and the background uncertainty ɛ on optimal insurance demand.
ground risk. However the condition ρ (L, ε ) ≤
2
Result 2 Assume that α ∗ ≥ 0, for the DARA preferences, the optimal insurance demand α ∗ decreases upon an increase in cov(L, ε ) if cov(L, ε ) ≤ 0. Proof. From Eq. (3), the implicit function gives
1 ∂ α∗ ∂σ (α ) ∂ 2 σ (α ) α−1 =− −λμL φμ σ + φσ σ + φσ . ∂ cov(L, ε ) ∂ F /∂ α σ (α ) ∂α ∂ α∂ cov(L, ε )
(6)
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From the proof in the Result 1, the round-bracketed in the first term in the square-bracketed of (6) is positive under the DARA assumption. Hence, the first term in the square-bracketed of (6) is negative because α < 1. It is true that ∂ F/∂ α < 0 from the second-order condition. Then because φ σ < 0, if, but not only if, ∂ 2 σ (α )/∂α∂ cov(L, ε ) > 0 then ∂ α ∗ /∂ cov(L, ε ) < 0. From (2.7) and (4), we may find that becausecov(L, ε ) ≤ 0,then ∂ 2 σ (α )/∂α∂ cov(L, ε ) = [σε2 + (α − 1 )cov(L, ε )]/σ 3 (α ) > ∂ α∗ 0.So, ∂ cov (L,ε ) is negative. From Result 2, we know that the covariance between the random loss and the background uncertainty also has an impact on insurance demand. Next we consider how the optimal insurance demand α ∗ reacts on the change in the background uncertainty. Firstly, for the convenience of the proof, we introduce a utility function as
W (μ, ν ) := φ (μ,
√
ν ), Where ν = σ 2 .
Result 3 Suppose W(μ, ν ) is concave, and φ satisfy DARA (1.5), then optimal insurance demand increases upon an increase in σ ɛ . Proof. Implicit differentiation of Eq. (3) yields
1 ∂ α∗ ∂ F /∂ σε ∂σ (α ) ∂ 2 σ (α ) ∂σ (α ) =− =− −λμL φμ σ + φσ σ + φσ . ∂ σε ∂ F /∂ α ∂ F /∂ α ∂ σε ∂α ∂ α∂ σε
(7)
Because W(μ, v)is concave, like in the Fact 5 of Wagener (2003), we can have
Wνν = [1/4σ 2 (α )](φσ σ − (φσ /σ (α )), Wμν = φμσ /(2σ (α )) > 0,
Wμμ = φμμ < 0.
Since W(μ, ν ) is concave, we can have
φσ 2 0 < WμμWvv − Wμv = − φμσ . φμμ φσ σ − σ (α ) 4σ 2 ( α ) 2
1
(8)
Because Wμμ < 0, (8) is true only if WVV < 0. By (2.6) and (2.1), we can have
∂ 2 σ (α ) σε ∂σ (α ) =− 2 . . ∂ α∂ σε σ (α ) ∂α
(9)
Using (9) and Eq. (3) and (2.3), we rewrite (7) as
φμ 1 λμL σε ∂ α∗ φ =− −φμ σ + . φσ σ − σ ∂ σε ∂ F /∂ α σ (α ) φσ σ (α )
From (8) and (10), we can have
(10)
φσ φμμ φσ ∂ α ∗ = − 1 λμL σε φμ ∂ σε ∂ F/∂ α σ (α ) φμμ φσ −φμ σ φμ + φμμ φσ σ − σ (α )
φμ φσ 2 L σε > − ∂ F/1∂ α λμ > 0. σ (α ) φμμ φσ −φμσ + φμμ φσ σ − σ (α )
Where the first inequality is true because of the DARA in (1.5), and the second inequality is true because of (8).
(11)
Notice that there is no restriction on the sign of cov(L, ε ) for Result 3 to be true. It means that more uncertain environment may force the risk-averse agent to buy more insurance under the assumption of the Result 3. 4. Conclusion In this study, we have analyzed a new optimal insurance demand model which includes a risk named background uncertainty ɛ. By comparative statics, we obtained several new results. Optimal insurance demand may change according to the changes of several factors, such as the standard deviation of the loss, the covariance between the loss and the background uncertainty, and the standard deviation of the background uncertainty. Moreover the kind of preference which the agent belongs to can also affect optimal insurance demand. For further research one can add another kind of risk variable to the model to analyze the effect of background risk on the decision makers, and also can change the assumption of the DARA restrictions. Acknowledgement The authors thank for the fund support by the Scientific Innovation Research of College Graduate in Jiangsu Province, China (KYZZ15_0105).
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