The twomoment decision model with additive risks
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Abstract
With multiple additive risks, the mean–variance approach and the expected utility approach of risk preferences are compatible if all attainable distributions belong to the same location–scale family. Under this proviso, we survey existing results on the parallels of the two approaches with respect to risk attitudes, the changes thereof, and the comparative statics for simple, linear choice problems under risks. In mean–variance approach all effects can be couched in terms of the marginal rate of substitution between mean and variance. We provide some simple proofs of some previous results. We apply the theory we stated or developed in our paper to study the behavior of banking firm and study risktaking behavior with background risk in the mean–variance model.
Keywords
Mean–variance model Location–scale family Background risk Multiple additive risks Expected utility approachJEL Classification
C0 D81 G11Introduction
In mean–variance (MV) or \((\mu , \sigma )\)analysis, preferences over random distributions of, say, consumption or wealth are represented by functions that depend only on the mean and the variance (or standard deviation) of consumption or wealth. In addition to being an intuitive tool in the analysis of decision making under uncertainty, MV preferences are a perfect substitute for the classical expected utility (EU) approach if all attainable distributions belong to a location–scale family (Meyer 1987). Then, risk attitudes (such as risk aversion, prudence) originally formulated in the EU approach have convenient analogues in terms of MV preferences (see, e.g., Meyer 1987; LajeriChaherli 2002, 2005; Eichner and Wagener 2003b). Moreover, as argued by Meyer (1987) and others, the location–scale property is satisfied in a wide range of univariate economic decision problems. Such problems, encompassing portfolio selection (Fishburn and Porter 1976), competitive firm behavior (Sandmo 1971), coinsurance (Meyer 1992), export production (Broll et al. 2006), bank (Broll et al. 2015), and others, can then be studied equivalently both in terms of the EU and the MV approach.
In their simplest form, preferences and choices under risk are analyzed under the assumption that there is only a single source of uncertainty, a “direct” risk. The—probably more relevant—case of multiple risks has only recently found more attention in MV analysis. Inspired by studies on the effects of (additive) background risks on risktaking under the EU hypothesis (see, e.g., Eeckhoudt et al. 1996; Caballé and Pomansky 1997), Wong and Ma (2008) or Eichner and Wagener (2003a, 2009) analyze quasilinear decision problems where the MV decision maker has faced both a direct, controllable risk and an exogenous background risk. Eichner and Wagener (2011a) study linear portfolio choices with several risky assets. In these studies, the different risks are additive, i.e., final wealth or consumption emerges as a linear combination of multiple random variables.^{1}
In this paper we survey previous studies on MV preferences in the presence of several additive risks (capturing, but not being confined to, the case of a direct risk plus a background risk). Such a linear setting is particularly suited to draw parallels between the EU and the MV approach since the location–scale property often prevails and MV and EU approach can be considered as perfect substitutes. Compared to the EU approach, where the analysis of background uncertainty is quite complex, MV analysis with its simple twoparameter utility functions has the advantage that all risk attitudes or comparative statics can be couched in terms of marginal rates of substitution between risk and return, represented, respectively, by the variance/standard deviation and the mean.
A key feature of the twoparameter structure (in combination with additively connected risks) is that it ensures that risk attitudes that have been studied for univariate sources of risks are bequeathed also to the multiplerisks scenario. We demonstrate this in the following way: we start from formal parallels between EU and MV approach, relating, e.g., to absolute and relative risk aversion, prudence, temperance, and their monotonicity properties in univariate settings (“MV preferences and EU approach” section) and then show that the attending MV concepts (in terms of marginal rates of substitution between risk and return) are preserved with several additive risks (“Additive risks and risk attitudes” section). In “Additive risks and risk attitudes” section, we apply these results to study the comparative statics of optimal risktaking in the presence of (dependent) background risks.
Most studies on additive (background) risks both in the EU and the MV framework suppose that all risks are independently distributed (for exceptions, see Tsetlin and Winkler 2005, or Eichner and Wagener 2012). A particular advantage of MV preferences is, however, that risk attitudes and comparative statics with dependent random variables can be dealt with relatively easily, due to the fact that the variance (or standard deviation) as a measure of riskiness reduces—and limits—all dependence structures to just linear ones. As we show, background risks do not pose significant analytical problems for the MV approach within its linear confines, neither for risk attitudes nor for comparative statics of changes in the distribution and even in the dependence structure of direct and background risks. In “Additive risks and risk attitudes” and “Optimal decisions with additive risks” sections we fully characterize these features. Moreover, with the help of the analogies between MV and EU approach reported in “MV preferences and EU approach” section all MV features can be related to results for EU preferences. Although the results in “Optimal decisions with additive risks” section are actually Propositions 1 and 2 in Eichner and Wagener (2009), our contribution here is to simplify the related proofs and embed them into a comprehensive framework to make them easier to understand. As a new illustration, we apply the results obtained in “Optimal decisions with additive risks” section to study the risktaking behavior of a banking firm with background risk in the MV model.
A frequent source of concern with respect to MV analysis is the role of higherorder derivatives of the \((\mu , \sigma )\)function, the attending indifference maps or, once compatibility with the EU approach is assumed, of the underlying von Neumann–Morgenstern (vNM) utility index; such derivatives of general order n will also appear in “MV preferences and EU approach” section. For the EU approach, studies on higherorder moments and on higherorder risk measures indeed reveal close relations between highorder risk changes or dominance relations and higherorder derivatives of vNM utility (see Chan et al. 2016 or Niu et al. 2017 for surveys). The MV framework is, by construction, confined to changes in the first two moments. Still, concepts of (vNM) risk preferences that involve higherorder derivatives can, in many ways, be translated into twoparameter parlance; this is simply due to the fact that the signs of higherorder derivatives (and their combinations) convey more and different things than just preferences towards highorder changes in risk. For example, already Lajeri and Nielsen (2000) show that for MV analysis with normal distributions, the corresponding utility function is concave if and only if the agent has decreasing prudence. LajeriChaherli (2002) presents an economic interpretation for the quasiconcavity of a MV utility function and finds that quasiconcavity plus decreasing risk aversion is equivalent to proper risk aversion, as coined by Pratt and Zeckhauser (1987) in the expected utility framework. Wagener (2002) demonstrates how prudence, risk vulnerability, temperance, and some related concepts can be meaningfully formulated in terms of twomoment, mean–standard deviation preferences. Eichner and Wagener (2003a) show the equivalence of decreasing absolute prudence in the expected utility framework and the concavity of utility as a function of mean and variance. Wagener (2003) finds that in the twoparameter approach, a number of plausible comparative statics already emerges under the assumption of decreasing absolute risk aversion. Moreover, risk vulnerability, temperance, and standardness imply, appropriately transferred to the MV framework, the plausible effect that risktaking will be reduced if background risks increase. LajeriChaherli (2004) assumes that the agent expects two independent, risky incomes in the future and focusses on his precautionary saving motive or equivalently consumption behavior at time zero. She finds that this framework allows for the definition of new concepts, called proper prudence, standard prudence, and precautionary vulnerability. Eichner and Wagener (2004) show that relative risk aversion being smaller than one and relative prudence being smaller than two emerge as preference restrictions that fully determine the optimal responses of decisions under uncertainty to certain shifts in probability distributions. They characterize the magnitudes of relative risk aversion and relative prudence in terms of the twoparameter approach. They also demonstrate that this characterization is instrumental in obtaining comparative static results in the twoparameter setting. Eichner (2008) transfers the concept of risk vulnerability to mean variance preferences, showing that it is equivalent to the slope of the MV indifference curve being decreasing in mean and increasing in variance. He also shows that MV vulnerability links the concepts of decreasing absolute risk aversion, risk vulnerability, properness, and standardness. These concepts are characterized in terms of MV indifference curve properties and in terms of absolute risk measures. The general equivalences presented in “MV preferences and EU approach” section are instrumental in deriving these and potentially other relations between EU and MV preferences (without leaving the linear domain).
The remainder of the paper is organized as follows: “MV preferences and EU approach” section sets up the formal framework of MV preferences and their relations to the EU approach. In that framework, “Additive risks and risk attitudes” section then studies the impact of additive risks on the shapes of indifference curves and measures for risk attitudes. “Optimal decisions with additive risks” section analyzes the comparative statics of changes in risk parameters in a generic linear decision problem with additive background uncertainty. An application to the banking firms’ risktaking behavior is also given in this section. “Concluding remarks” section concludes.
MV preferences and EU approach
General
For a location–scale family \({\mathcal{D}} \subset {\mathcal{D}}_{Y_0},\) denote by \(M \subseteq {\mathbb{R}}_{++} \times {\mathbb{R}}\) with \(M=\{(\sigma , \mu )  \mu + \sigma Y_0 \in {\mathcal{D}} \}\) the set of attending distribution parameters.
Parallels

if \(u^{\prime \prime}(y)< 0 < u^{\prime}(y)\) for all y, then \((\sigma , \mu )\)indifference curves are strictly convex upward in \((\sigma , \mu )\)space: the compensation in term of μ needed for an increase in uncertainty is always positive and increases in the level of uncertainty (risk aversion);

if \(u^{\prime}(y), u^{\prime \prime}(y) >0\) for all y, then \((\sigma , \mu )\)indifference curves are concave downward: μ needs to be reduced to compensate for an increase in uncertainty, and this reduction decreases in the level of uncertainty (riskseeking attitude);

if \(u^{\prime}(y) >0=u^{\prime \prime}(y)\) for all y, then \((\sigma , \mu )\)indifference curves are parallel to the σaxis (risk neutrality).
Similarly interpretations arise for \(n>1.\) E.g., for \(n=2,\) a prudent and riskaverse decision maker (\(u'''>0>u^{\prime \prime}\)) faced with an increase in uncertainty will require an increase in μ to keep his marginal utility from μ constant.
Interestingly, analogies extend to monotonicity properties as well:
Result 2.1
In the case \(n=1,\) the equivalences in (12) and (12) mean that risk aversion for \((\sigma , \mu )\)utility functions (as measured by \(S_1\)) (i) decreases [increases] in μ if the underlying vNMindex exhibits decreasing [increasing] absolute risk aversion and (ii) increases [decreases] in σ if the vNMindex exhibits increasing [decreasing] partial relative risk aversion. As Menezes and Hanson (1970) argue, if one wants \(R_1\) to be monotone in z everywhere, then this is only compatible with \(A_1>0\) if \(R_1\) strictly increases. Hence, \(\partial S/\partial \sigma <0\) can then at most be a local property. Moreover, decreasing absolute risk aversion (\(A(y)> 0 > A^{\prime}(y)\)) implies that \(S(\sigma , \mu )\) is decreasing in σ (for details, see Eichner and Wagener 2005).
The cases \(n>1\) are analogous to \(n=1,\) lifting relationships between partial relative measures of risk attitudes for vNMfunctions and to higher orders.
Additive risks and risk attitudes
General
How does the addition of risks (e.g., via background uncertainty in one’s investment) affect risk attitudes? Specifically, if an additive uncertainty B, also measured in terms of final wealth, changes returns on a risky activity from X to \(Y=X+B,\) how are risk preferences affected? To ensure transferability to the EU approach, we require that the location–scale framework still applies and make the following
Assumption 3.1
Let \(X_0\) and \(B_0\) be two seed variables with attending location–scale families \({\mathcal{D}}_{X_0}\) and \({\mathcal{D}}_{B_0}.\) Then the set of all \(Y = X+B\) with \(X \in {\mathcal{D}}_{X_0}\) and \(B \in {\mathcal{D}}_{B_0}\) forms a location–scale family \({\mathcal{D}}_{Y_0}\) with seed \(Y_0.\)
We note that in Assumption 3.1 \(Y_0\) may not be equal to \(X_0+B_0.\) Under Assumption 3.1 we have \(Y= X+B = \mu _X + \mu _B + \sigma _X X_0 + \sigma _B B_0,\) implying that \(\mu _Y = \mu _X + \mu _B\) and \(\sigma _Y = \sqrt{\sigma _X^2 + \sigma _B^2 + 2 \sigma _{XB}},\) where the covariance between X and B, \(\sigma _{XB} = \rho \sigma _X \sigma _B\) measures the linear dependence of X and B; \(\rho \in (1,1)\) denotes (Pearson’s) correlation coefficient. Denote by \(F_{XB}(x,b)\) the joint distribution of (X, B).
Assumption 3.1 will, e.g., be satisfied if \(X_0\) is equal in distribution as \(B_0,\) both are independent, and \(X_0\) adheres to a stable distribution; \(X_0+B_0\) then even inherits the type of distribution. Moreover, if both \(X_0\) and \(B_0\) are elliptically distributed (but not necessarily identically or independently), then so is their sum (Fang et al. 1990, Theorem 2.16).^{5} This encompasses, e.g., that \(X_0, B_0 \sim N(0,1)\) such that \(X+B \sim N \big (\mu _X+\mu _B, \sqrt{Var(X+B) \big )}\); the same holds if \(X_0\) and \(B_0\) are gammadistributed with equal scale parameter.
Assumption 3.1 allows for dependence between the two random variables. While independence is routinely assumed in the EU literature on background risks, the MV approach can quite easily cater for dependent background risks. In fact, if we were assuming independence, then for elliptical distributions Assumption 3.1 essentially confines the analysis to X and B both being Gaussian (Fang et al. 1990, Theorem 4.11).
The impact of (greater) additive uncertainty on risk attitudes can now be studied by help of our previous observations. Assumption 3.1 essentially implies that all risk attitudes (and their monotonicity properties) in the absence of background uncertainty remain unchanged if background risks are added.
Changes in location parameters
Taking the partial derivative with respect to \(\mu _B\) or \(\mu _X\) captures the effects of a shift in risks. They are identical to the standard wealth or income effects that arise when some exogenous, nonrisky wealth changes. In particular, as a straightforward implication from Result 2.1, Eq. (12), we obtain
Corollary 3.1
Hence, a higher expected return on any risk makes decision makers more [less] riskaverse if absolute risk aversion is increasing [decreasing] in income (\(n=1\)). It makes them more [less] prudent if absolute prudence rises [diminishes] with income (\(n=2\)); and similar for higher degrees.
Changes in scale parameters
Hence, the effect of an increase in \(\sigma _k\) (for \(k=X,B\)) on risk attitudes depends on (i) how that change affects overall riskiness \(\sigma _Y\) and (ii) the risk attitude proper.
As for (i), an increase in either \(\sigma _X\) or \(\sigma _B\) does not necessarily increase \(Var(X+B)\); increases in marginal risks may well be beneficial in the MV framework. This reflects that for the variance (or standard deviation) as a risk measure increases in the marginal riskordering for that measure are not preserved under linear combinations of dependent random variables. Increases in \(\sigma _k\) will only raise \(\sigma _Y\) if \(\sigma _{XB} >\sigma^2_k.\) This is the case if (but not only if) X and B are independent or positively correlated.^{6}
As for effect (ii) in (17), condition (13) in Result 2.1 applies:
Corollary 3.2
This observation conveys that, if a greater marginal riskiness makes total wealth riskier, this renders decision makers more [less] riskaverse if relative risk aversion is increasing [decreasing] in income (\(n=1\)). It makes them more [less] prudent if relative prudence rises [diminishes] with income (\(n=2\)); and similar for higher degrees of n. In case a greater marginal riskiness makes total wealth safer, the results are reversed.
Changes in the dependence structure
With the decomposition (16), an increase in ρ represents that X and B move more closely together (with invariant marginals). An increase in ρ is detrimental to utility as it also increases \(\sigma _Y.\) Hence, we can directly apply (13) from Result 2.1 again:
Corollary 3.3
For the interpretation of (19), we can refer to the discussion of Corollary 3.2 above.
Compensatory changes in risks
For \(n=2,\) (8) captures that if a decision makers wishes to keep the marginal utility from wealth unchanged in the wake of a more pronounced riskiness, this requires an increase in μ if he is either riskaverse and prudent (\(u^{\prime \prime}(y)<0<u'''(y)\) for all y) or riskloving and imprudent (\(u^{\prime \prime}(y)>0>u'''(y)\) for all y)—and a decrease in μ otherwise.
Optimal decisions with additive risks
Setup
It is well known that changes in risk attitudes do not necessarily lead to the intuitively expected changes in decision maker’s behavior. For example, somebody who becomes more riskaverse upon a change in risk does not necessarily engage in less risky activities upon that change in risk. Against that backdrop, it is informative to see how additive risks affect risky choices in a generic decision problem.
The signs of the comparative statics with respect to the distribution parameters in \(\theta\) are obtained by applying the implicit function theorem to (21), taking into account that the SOC for \(\alpha^{\ast}(\theta )\) requires that the derivative of the lefthand side of (21) is negative. A common intuition for the comparative statics to come can be gained from interpreting (21) geometrically: it defines the optimal choice, α, as a situation where the slope, \(S_1,\) of a decision maker’s \((\sigma _Y, \mu _Y)\)indifference curve is equal to the slope, given by \(\mu _X/(\partial \sigma _Y/\partial \alpha),\) of the “opportunity locus,” which defines the marginal tradeoff between the increases in return and in risk to which the choice problem (20) exposes the decision maker. Whether and into what direction the optimal choice drifts when a parameter of the choice problem varies then depends on whether the marginal rate of substitution between risk and return varies relatively more strongly than the slope of the opportunity locus. This gives rise to the elasticity considerations in Eqs. (23)–(27) below. It also explains why the comparative statics with respect to parameters related to the “endogenous” direct risk differ qualitatively from those for the exogenous background risk: the exposure to the former is a chosen one (via α), the exposure to the latter cannot be avoided (but at best be indirectly reduced, via a covariance effect). In essence, this makes the comparative statics with respect to the background risk simpler—which is in marked contrast to the EU framework.
In full detail, the elasticity intuition for comparative statics in the MV framework is developed in Eichner and Wagener (2009, pp. 1145ff), which also includes a discussion of the differences between studying background risk in the MV model and in the conventional expected utility model.
Changes in the background risk
Starting with the background risk B, we get
Result 4.1
Proof
From (23), a riskaverse decision maker increases risktaking upon a shift in the location of a dependent background risk if and only if his preferences exhibit decreasing absolute risk aversion (also cf. (15)).
He will reduce risktaking in response to an increase in the scale of the background risk if the elasticity of his risk aversion with respect to the riskiness of final wealth is larger than one. Comparing (24) with (18) (for \(n=1\) and \(k=B\)) we observe: in order that a greater background risk reduces risktaking (\(\partial \alpha^{\ast}/\partial \sigma _B <0\)), it does not suffice that the decision maker gets more riskaverse; \(\partial S_1/\partial \sigma _B\) being positive is necessary, but not sufficient for \(\partial \alpha^{\ast}/\partial \sigma _B\) to be negative.
Changes in the direct risk
The comparative statics with respect to the direct risk X are slightly more difficult to characterize. They can be framed, however, in terms of the concepts of risk attitudes introduced in “MV preferences and EU approach” section:
Result 4.2
Proof
From (25) the decision maker will increase risktaking in response to an increase in the expected return of his activity if the elasticity of his risk aversion with respect to expected wealth is smaller than one. This condition has an expected utility analogue, too. As shown in Eichner and Wagener (2014), if EU and MV approach are compatible, then the wealth elasticity of MV risk aversion being smaller than one is equivalent to the index of partial relative risk aversion, \(R_1(a,ya)\) (cf. (10)) being smaller than one for all \(a>0.\) Hadar and Seo (1990) and Dionne and Gollier (1992) have shown that this condition characterizes the comparative static effects for firstorder stochastic dominance shifts in the returns to a risky activity—of which an increase in \(\mu _X\) is the MV analogue.
Condition (26) says that the decision maker will decrease risktaking in response to an increase in the variance of his activity if the elasticity of his risk aversion with respect to wealth risk is larger than \(1.\) Again this condition—which originally was derived in Battermann et al. (2002) and Broll et al. (2006)—has an EU analogue, viz. that the index of partial relative risk prudence, \(R_2(a,ya)= (ya) \frac{u'''(y)}{u^{\prime \prime}(y)}\) (again cf. (10)) being smaller than 2 for all \(a>0\) (Eichner and Wagener 2005). Ormiston and Schlee (2001) identify this as the condition that a meanpreserving spread in the returns to a risky activity tempers risktaking—of which an increase in \(\sigma _X\) is the MV analogue here.
Changes in the dependence between the direct risk and the background risk
Now we turn to study the comparative statics with respect to the dependence between the direct risk and the background risk. It can be framed in terms of the concepts of risk attitudes introduced in “MV preferences and EU approach” section:
Result 4.3
Proof
Condition (27) says that the decision maker will reduce risktaking in response to an increase in the covariance of the two risks if the elasticity of his risk aversion with respect to wealth risk is larger than 0. Again, this condition has an EU analogue, viz. that the index of partial relative risk prudence, \(R_2(a,ya)= (ya) \frac{u'''(y)}{u^{\prime \prime}(y)}\) (again cf. (10)) being smaller than 1 for all \(a>0.\)
For Results 4.1–4.3, which can actually be found as Propositions 1 and 2 in Eichner and Wagener (2009), our contribution is to simplify the proofs and make them easier to access.
Application: a risktaking bank with background risk
Recently, Broll et al. (2015) have investigated the banking firm and risktaking in a twomoment decision model. In this section, we add a background risk to this problem and apply the results presented above to its comparative statics.
Consider a bank that decides on how many and which fiscal assets to hold. The bank has the following balance sheet: \(\alpha = K + D,\) where α is the amount of financial assets, D is the quantity of deposits, and K is the stock of equity capital. We assume that short sales of the asset are forbidden, i.e., \(\alpha \ge 0.\) Moreover, there is a capital requirement, imposing that \(K \ge k \cdot \alpha\) for some \(k \in (0,1).\) The risky return on financial assets is given by random variable \(\tilde{r}.\)
The bank’s shareholders contribute equity capital with a required rate of return, \(r_K,\) on their investment. The supply of deposits is perfectly elastic at an exogenous deposit rate, \(r_D.\) We suppose that \(r_K > r_D,\) implying that the capital requirement will bite: \(k \alpha = K.\) Moreover, the bank’s weighted average cost of capital (WACC) is then given by \(r_c := (1  k) r_D + k r_K.\) There are no fixed costs; the bank’s operating cost, \(C(\alpha ),\) is increasing and convex, that is, \(C(0)=0,\) \(C'>0,\) and \(C''\ge 0\) for all α. There is some additive background risk B (e.g., from operations off the balance sheet).
For changes in the background risk, conditions (23), (24), and (27) apply: the bank will take in more risky assets in response to a higher expected background income if its preferences exhibit decreasing absolute risk aversion; its response to an increase in the risk of background income or in the correlation between the risks on financial and other incomes depends on the magnitude of the elasticity of its risk aversion with respect to \(\sigma _Y.\)
For changes in the direct financial risk, conditions (25) and (26) apply^{7}: the magnitude of the elasticity of the bank’s risk aversion with respect to \(\mu _Y\) and \(\sigma _Y\) determines whether the bank holds more financial assets when, respectively, their expected return or their riskiness increases.
The interpretations of the above conditions are similar to the general cases and thus are omitted here. By adopting the MV approach, the effects of dependent background risk on the banking firm’s risktaking can be easily structured and clearly studied.
Concluding remarks
With multiple additive risks, the MV approach and the expected utility approach of risk preferences are compatible if all attainable distributions belong to the same location–scale family. For such scenarios, this paper presents parallels of the two approaches with respect to risk attitudes, the changes thereof, and the comparative statics for simple, linear choice problems under risks.
Given that the preference functional in the MV approach only depends on mean and variance, all effects depend on the monotonicity properties either of the utility function itself or of the attending marginal rate of substitution between the two parameters. This once again highlights the simplicity and convenience of the MV approach: all effects can be framed in terms of riskreturn tradeoffs.
The MV approach provides a genuine and surprisingly rich framework for the economic modeling of preferences and choice under risk. Still, many extensions can be envisioned, both within and beyond the location–scale framework where equivalence with the EU approach prevails. Starting from the discussion offered in this paper, nonadditive background risks or Sshaped vNM utilities appear to be promising topics. Last, we note that after establishing a theoretical model, the next step is to develop an estimation and/or hypothesis testing (see, for example, Leung and Wong 2008) for the model. We leave the estimation and testing of the model we developed in our paper in the future study.
There are many applications of the theory developed in this paper and other papers. For example, recently, Broll and Mukherjee (2017) examine the optimal production and trade decisions of a domestic firm facing uncertainties owing to exchange rate volatility under MV preferences. Extending their analysis to situations with background risk is an interesting and important problem.
Footnotes
 1.
EU studies with multiplicative background risks include, e.g., Franke et al. (2006).
 2.
 3.
Also see Eichner and Wagener (2005). For (smooth) functions f and integers \(n \in {\mathbb{N}}_0,\) \(f^{(n)}(y)\) denotes the nth order derivative of f(y); by convention \(f^{(0)}(y) \equiv f(y).\) In multivariate functions, subscripts denote partial derivatives.
 4.
We only report results where the curvature of vNMfunctions is uniform. Recent advances in decision theory under risk focus on Sshaped or reverse Sshaped utility functions (Levy and Levy 2004; Wong and Chan 2008). Broll et al. (2010) or Egozcue et al. (2011) studied the properties of \((\mu , \sigma )\)indifference curves with reverse Sshaped utility functions.
 5.
Chamberlain (1983) argues that this is the only relevant case such that mean–variance approach and expected utility are isomorphic.
 6.
For convex risk measures (such as the variance), this is a simple application of the condition of X and B being “conditionally increasing” in Mueller and Scarsini (2001).
 7.
The proof requires a slight modification. Differently from (25), for the condition (25) observe that \(\mu _Y\ge \mu _X\alpha C(\alpha )\ge (\mu _XC'(\alpha ))\alpha .\) The second inequality is true since \(C(\alpha )\) is convex and \(C(0)=0.\) When \(C(\alpha )\) is linear, the second equality always holds.
Notes
Acknowledgements
The authors are grateful to Professor Ira Horowitz for his valuable comments that have significantly improved this manuscript. The third author would like to thank Professors Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. The authors are grateful to the Editor and two anonymous referees for constructive comments and suggestions that led to a significant improvement of an early manuscript. This research has been partially supported by the Fundamental Research Funds for the Central Universities, grants from the National Natural Science Foundation of China (11626130, 11601227, 11671042), Natural Science Foundation of Jiangsu Province, China (BK20150732), Asia University, Hang Seng Management College, Lingnan University, Shanghai University of International Business and Economics, Hong Kong Baptist University, the Research Grants Council (RGC) of Hong Kong (project numbers 12502814 and 12500915), and Ministry of Science and Technology (MOST), R.O.C.
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