UTILITY,BASIS,OF,CONSUMPTION,AND,INVESTMENT,DECISIONS,IN,A,RISK,ENVIRONMENT∗

Kangping WU()

School of Economics and Management,Tsinghua University,Beijing 100084,China

E-mail:wukp@sem.tsinghua.edu.cn

Abstract Using expectations regarding utilities to make decisions in a risk environment hides a paradox,which is called the expected utility enigma.Moreover,the mystery has not been solved yet;an imagined utility function on the risk-return plane has been applied to establish the mean-variance model,but this hypothetical utility function not only lacks foundation,it also holds an internal contradiction.This paper studies these basic problems.Through risk preference VNM condition is proposed to solve the expected utility enigma.How can a utility function satisfy the VNMcondition?This is a basic problem that is hard to deal with.Fortunately,it is found in this paper that the VNM utility function can have some concrete forms when individuals have constant relative risk aversion.Furthermore,in order to explore the basis of mean-variance utility,an MV function is founded that is based on the VNM utility function and rooted in underlying investment activities.It is shown that the MV function is just the investor’s utility function on the risk-return plane and that it has normal properties.Finally,the MV function is used to analyze the laws of investment activities in a systematic risk environment.In doing so,a tool,TRR,is used to measure risk aversion tendencies and to weigh risk and return.

Key words VNM condition;relative risk aversion tendency;mean-variance utility;systematic risk

Dedicated to Professor Banghe LI on the occasion of his 80th birthday

As maximizing the expected return of an investment was negated by the St.Petersburg Paradox,it was proposed by D.Bernoulli to substitute expected utility for expected return.The utility index was further established by J.von Neumann.From this time awards,the maximization of expected utility has become the de facto practice in consumption and investment.Markovitz’s portfolio selection(1952,see [13])provides a larger application stage for the expected utility theory.In particular,the MV(mean-variance)approach has been applied broadly in investment decision and financial analysis.Research results on this has been very numerous.

In practice,Markovitz’s MV optimization may perform poorly.This phenomenon is referred to as the Markovitz optimization enigma.Lai et al.(2011,see [9])studied the enigma and explained its root causes.They proposed a new approach of flexible modelling to resolve it.Van Staden et al.(2021,see [16])found that the MV optimization can be remarkably robust for modelling misspecifi cation errors in dynamic or multi-period portfolio optimization,in sharp contrast to single-period portfolio optimization settings.They explained the causes of this surprising robustness under both the pre-commitment MV and time-consistent MV approaches.Li et al.(2022,see [10])employed model predictive control to construct a multi-period portfolio and to provide a comprehensive comparison of the models with regard to objective function choice,planning horizon and parameter estimation.They found that Markovitz portfolio optimization performs better in muti-period models than in single period ones.

In order to investigate the yield and price under uncertainty,Coyle(1999,see [5])developed a duality model of production with risk aversion by using an MV approach.The model incorporates both mean-variance preferences and expected output supply,and is tractable for empirical research.Brown(2007,see [4])gave a new application field for MV methodology.A mean-variance model is introduced in [4]to solve serial replacement problems with uncertain rewards.Liu(2022,see [11])verified that Markowitz’s asset portfolio theory is applicable for China’s A-share market by randomly selecting its four stocks and constructing a portfolio of maximum Sharpe ratio.

Based on state-dependent risk aversion and efficient dynamic programming,Rainer(2022,see [15])presented a heuristic mean-variance optimization in Markov decision processes to achieve a balance between maximizing expected rewards and minimizing risks.By using a CRRA utility function,Kassimatis(2021,see [7])examined whether mean-variance is a good proxy for portfolios,and found that MV portfolios are a poor proxy for investors with CRRA preferences.Marianil et al.(2022,see [12])proposed a measure for portfolio risk management by extending the Markowitz mean-variance approach to include the left-hand tail effects of asset returns.Two risk dimensions were captured:the asset covariance risk and the risk in left-hand tail similarity and volatility.From a simplified jump process,Khashanah et al(2022,see [8])found that mean-variance portfolios need to be enhanced by incorporating higher-order components.Andrew et al.(2022,see [1])investigated the impact of changes in the mean vector on mean-variance portfolio optimization.They found that the bounds of mean vector changes are unable to characterize portfolio sensitivity.Dai et al.(2020,see [6])proposed a dynamic portfolio choice model with MV criteria for log-returns.Their consideration conform to investment common sense;for example,rich people should invest more in risky assets.The longer the investment period,the greater the proportion of investment in risky assets.For long-term investments,investors should not short sell major stock indices whose returns are higher than the risk-free rate.

Systematic risks and their impact on investment have attracted much attention.Berk and Tutarli(2020,see [2])proposed two selection criteria for a mean variance optimization in a systematic risk environment:the beta coefficient and previous period return.In fact,the beta coefficient is a measure of systematic risk.Using these selection criteria,investors may obtain investable portfolios.Empirical analysis of the Istanbul Stock Exchange shows that the portfolio with the lowest beta coefficient is the best alternative.Bianconia et al.(2015,see [3])introduced a measure of information dissemination for the determination of systemic risk.They found that VIX volatility hasa signifi cant direct impact upon the systemic risk of financial firms under distress.They also found that consumer pessimism can also predict systemic risk,and may be dominated by the VIX.

A common feature of all of the above research is the imagining of a utility function,which is used to derive an expectation regarding utilities.However,such a practice results in a paradox that is referred to as the expected utility enigma.Meanwhile,the mean-variance utility lacks foundation and hides its own contradictions.These problems are basic and intrinsic,and hide a danger which may lead to poor decisions regarding investment and consumption.In view of these problems,this paper starts by addressing the subject of risk preference,and makes an indepth theoretical analysis.First,a VNM condition is proposed to explain the expected utility enigma.Second,the relative risk aversion tendency is used to explore the form of VNMutility functions and then to unveil the truth of the expected utility enigma.Third,a mean-variance utility function is constructed according to the VNMutility function and actual investment activities.Finally,using the MV utility function constructed,the law of investment decisions in systematic risks are revealed,and a new measure called the target rate of return is proposed in order to evaluate the risk aversion tendencies of investors.

Before starting the discussion,we explain some of the rules regarding the care of symbols in this paper.All vectors,matrices,and mappings whose values are vectors are expressed in italic bold letters.In addition,the word“function”refers to the mapping whose values are real numbers.

When individuals decide to consume or invest in an uncertain environment,they need first to know the expected utilities of their activities,and then to organize their activities according to the plan with maximal expected utility.Usually,the expected utilityEu(ξ)of a random actionξof an individual is given by the expectation thatEu(ξ)=RΩu(ξ(ω))dP(ω)is calculated from the utility functionu(x)of the individual.However,such an approach may lead to a contradiction,which is referred to as the expected utility enigma.

We can imagine a situation where a consumer chooses between two commoditiesXandY.The amounts ofXandYare denoted byxandy.Suppose thatu(x,y)=(xy)0.25andv(x,y)=(xy)0.75are the utility functions of the consumer.They display the same preference,i.e.,the following fact holds for anyx1≥0,y1≥0 andx2≥0,y2≥0:

Now suppose that the consumer is in an uncertain environment where their choice depends on the sides of a coin,while each side appears with a 50%probability.They are faced two options,AandB.

OptionA:If the positive side appears,choose(1,1);otherwise,choose(3,3).

OptionB:Always choose(2,2),no matter what side of the coin appears.

How does the consumer choose?Should they chooseAorB?Facing this situation,the consumer needs first to calculate the expected utilities of the two optionsAandB.Asu(x,y)is a utility function of the consumer,they can obtainEu(A)andEu(B)as follows:

SinceEu(A)

NowEv(A)>Ev(B)means that the consumer should choose optionA.The different answers here lead to a contradiction.Which option should be chosen?Is optionAbetter than optionBor is optionBbetter than optionA?This problem is the so-called expected utility enigma.

By this token,the underlying utility function used to calculate the expected utilities is a key factor that decides whether or not the decision is correct when individuals are in an uncertain environment.In order to solve the expected utilities enigma,the VNM condition can not be ignored;that is,the underlying utility function has to satisfy the VNMcondition.

Before explaining the meaning of the VNMcondition,let Ω denote the set of all natural states that affect the outcome of economic activities.The set Ω is called a state space.Let F denote the event field on Ω ,which is aσ-field.LetP:F→[0,1]denote the probability measure on F.The probability space( Ω ,F,P)expresses that the economic environment is one of uncertainty.

Suppose that there arelkinds of commodities on the market.Then the commodity space becomes thel-dimensional Euclidean spaceRl.The outcomes of economic activities are just vectors inRl,called commodity vectors or selection schemes.However,not all vectors inRlare available for individuals to choose,since activities are limited by some conditions.Let S denote the set of vectors from which individuals are allowed to choose.S is a subset ofRl,called outcome set.Usually,S is required to be a non-empty,convex and closed set.

For different outcomes,what is best?The answer depends on preferences.A rational individual’s preference can be expressed as a binary relationon the outcome set S obeying the following three axioms:

In the uncertain environment( Ω ,F,P),the activity of the individual is really a random vector ξ : Ω →S;its outcome ξ (ω)∈S is affected by the natural stateω∈ Ω .Let S denote the set of all random vectors on Ω ,i.e.,S={ ξ | ξ : Ω →S is a random vector}.The set S is called a risky selection set or a risk set.As any vector x∈S can be regarded as a degenerate random vector,S is contained in S,i.e.,S⊆S.

Any two activities ξ , η ∈S can becompounded into an activitypξ ⊕(1−p) η by probabilitypin such a way that the activity is ξ with probabilityp,and η with probability 1−p.pξ ⊕(1−p) η is called a compound activity.Using random events to express things,pξ ⊕(1−p) η means that the individual takes ξ ifAhappens,and takes η ifAdoesn’t happen,whereA∈F is an event with probabilityp.The compounding operation obeys obvious the following laws:

Commutative law:pξ ⊕(1−p) η =(1−p) η ⊕pξ holds for any ξ , η ∈S andp∈[0,1].

Associative law:The following formula holds for any ξ , η ∈S andα,p,q∈[0,1]:

Some general rules should be complied for the expanding of preferences from S to S.For example,when x,y∈S and x≺y,the evaluationqx⊕(1−q)y≺px⊕(1−p)y should hold for anyp,q∈[0,1]withp

These general rules for preference expansion arerecognized and admitted with the following two axioms:

The following theorem makes a clear and intuitive interpretation of the above two axioms,and shows that the preference expansion according the two axioms does conform to general rules of evaluation:

Theorem 2.1The risk preferencesatisfies the Independence and Continuity Axioms if and only ifconforms to the following five general rules:

Rule(1)( ξ ∼ η )⇔(pξ ⊕(1−p) γ ∼pη ⊕(1−p) γ )holdsfor any ξ , η , γ ∈S andp∈(0,1);

Rule(2)( ξ ≺ η )⇔(pξ ⊕(1−p) γ ≺pη ⊕(1−p) γ )holds for any ξ , η , γ ∈S andp∈(0,1);

Rule(3)( ξ ≺ η )⇔( ξ ≺pξ ⊕(1−p) η ≺ η )holds for any ξ , η ∈S andp∈(0,1);

Rule(4)(p

Rule(5)For any ξ , η , γ ∈S with ξ ≺ γ ≺ ξ ,there exists a realc∈(0,1)such that(1−c) ξ ⊕ η ∼ γ .

Furthermore,the rules(4)and(5)above imply that(1−a) ξ ⊕aη ≺ γ ≺(1−b)ξ⊕bη holds for anya∈(0,c)andb∈(c,1).

ProofIt is easy to verify thatsatisfies the Independence and Continuity Axioms ifconforms to the five general rules listed in the theorem,so we only need to prove the necessity.For this purpose,suppose thatsatisfies the Independence and Continuity Axioms.Let ξ , η , γ ∈S andp,q∈(0,1)be given arbitrarily.From the Independence Axiom,rules(1)and(2)are obviously satisfied.In the following we prove rules from(3)to(5).

Proof of rule(3).Suppose that ξ ≺ η .Note that ξ =(1−p) ξ ⊕pξ and η =pη ⊕(1−p) η .Form rule(2), ξ =(1−p) ξ ⊕pξ ≺(1−p) η ⊕pξ =pξ ⊕(1−p) η andpξ ⊕(1−p) η ≺pη ⊕(1−p) η = η hold.Rule(3)is proven.

Proof of rule(4).Here we know that ξ ≺ η .To show the necessity in rule(4),suppose thatp

Thus it can be seen that(1−p) ξ ⊕pη =(1−t) ξ ⊕tγ ≺ γ =(1−q) ξ ⊕qη holds.The necessity in rule(4)is proven.

Now we prove the sufficiency in rule(4).Suppose that(1−p) ξ ⊕pη ≺(1−q) ξ ⊕qη.The reflexivity ofimplies thatpq.Ifp>q,then from the necessity in rule(3)we have that(1−p) ξ ⊕pη ≻(1−q) ξ ⊕qη .This is a contradiction,sop>qcannot hold.Thereforep

Proof of rule(5).LetA={p∈[0,1]:(1−p) ξ ⊕pηγ }andB={p∈[0,1]:(1−p) ξ ⊕pηγ }.ThenA∪B=[0,1].The continuity Axiom tells us that bothAandBare closed subsets of interval [0,1].

As we know that ξ ≺ γ ≺ η ,it can be seen that 0∈Aand 1∈B.ThusAandBare non-empty subsets of [0,1].Now the connectivity of interval [0,1]implies thatA∩B/=Φ.Hence there exists a real numberc∈A∩B.Obviously,(1−c) ξ ⊕cη ∼ γ and 0

Furthermore,from rule(4),it can be found that(1−a) ξ ⊕aη ≺ γ ≺(1−b) ξ ⊕bη holds for anya∈(0,c)andb∈(c,1).Theorem 2.1 is proven. □

Based on the above preparations and analyses,the expected utility enigma can now be solved.Note that the expectationEu( ξ )=RΩu( ξ (ω))dP(ω)( ξ ∈S)defines a functionEu:S→R,which is an expansion of the underlying functionu:S→R,i.e.,Eu|S=u.SoEu(x)=u(x)for any x∈S.u:S→Ris said to be a utility function of the outcome preferenceif(xy)⇔(u(x)≤u(y))holds for any x,y∈S.Utility functions are invariant under strictly increasing transformations,which says that ifu:S→Ris a utility function of,thenv(x)=ϕ(u(x))(x∈S)is a utility function oftoo,whereϕ:R→Ris a strictly increasing function.

We know that in an uncertain environment( Ω ,F,P),the individual evaluates activities according to their risk preferences.If they want to use an expectationEu( ξ )to evaluate things,Eu( ξ )must be a utility function of the risk preference.This requirement is called the VNMcondition on the underlying functionu:S→R.The specific expression of this condition is as follows:

VNMcond ition:u:S→Rsatisfies that(∀ ξ , η ∈S)(( ξη )⇔(Eu( ξ )≤Eu( η ))).

Whenu:S→Rsatisfies the VNMcondition,u:S→Ris said to be a VNMfunction.It can be verified that ifu:S→Ris a VNMfunction,thenu:S→Ris a utility function of the outcome preference,so a VNMfunction is also referred to as the VNMutility function.It can be verified further that VNMfunctions are invariant under affine transformations.That is,ifu:S→Ris a VNMfunction,thenv(x)=a+bu(x)is also a VNMfunction too for anya,b∈Rwithb>0.

Up until now,the expected utility enigma hasbeen solved by asking the underlying function to be a VNMfunction.When individuals make decisions in an environment with uncertainty,they must use a VNMfunction to evaluate things.A lack of VNMfunctions will inevitably lead to incorrect decisions.

After solving the expected utility enigma,there are two important questions that follow.One is whether the VNMfunctions exist.The other is how to identify a function as a VNMfunction.Fortunately,economics gives an answer to the first question,and tells us that there exist VNMfunctions if the preferencesatisfies the following three conditions:

(1) Ω =S and {x}∈F for any x∈S;

The first condition means that every outcome can be viewed as a natural state and appears randomly.The meaning of the second condition has been explained in Theorem 2.1.Third,saying thatis measurable means that both {x∈S:xz}and {x∈S:xz}are measurable sets for any z∈S.Finally,saying thatis inherited means that for any x∈S and ξ ∈S, ξx ifP{ ξ (ω)x}=1,and ξx ifP{ ξ (ω)x}=1.Thus the judgement ξx is an inheritance from the judgement that ξ (ω)x holds almost everywhere.It is obvious that these conditions are common.Thus VNMfunctions exist in general.

In addition,VNMfunctions must be cardinal utility functions.If this were not so,the expectation of utilities could be meaningless.As a result of the cardinal meaning of VNMfunctions,there exist cardinal utility functions.The existence puzzle of cardinal utilities is now solved.

In the past,it was generally recognized that utilities are hard to measure with a ruler.When you consume a certain quantity of goods,there is no way to know how much utility you obtain.In light of this,economists abandoned the cardinal utility assumption,and used instead ordinal utility theory.However economists are contradictory.While abandoning cardinal utilities,they again used cardinal utility functions to build dynamic or multi-period models such as business cycle models,economic growth models and financial models.In particular,they use cardinal utilities but do not know whether or not cardinal utilities exist.Thus their models are built like castles in the clouds.Now,with the help of VNM utilities,a positive answer is obtained for the existence puzzle regarding cardinal utilities,and thus a foundation is added for dynamic or multi-period models.

The above discussions about the existence of VNM and of cardinal utilities can be summarized in the following theorem:

Theorem 3.1Suppose that S is a non-empty convex closed subset of spaceRl,that Ω =S and that {x}∈F for any x∈S.If the preferencesatisfies the Independence and Continuity Axioms,and is measurable and inherited,then there exist VNM functions,and there also exist cardinal utility functions of.

Now we consider the second question raised at the beginning of this section.In order to identify VNM functions,we start for analyzing risk aversion tendencies.In general,there are three kinds of attitudes towards risk:risk averse,risk love and risk neutral.An individual with risk preferenceis called a risk averter ifEξ ≻ ξ ,a risk lover if ξ ≻Eξ ,and risk neutral if ξ ∼Eξ ,for any degenerated ξ ∈S.In terms of gambling,a fair gamble is one in which the sum that is bet is equal to the expected return.Facing a fair gamble,risk avertersreject it,but risk lovers accept the gamble.Risk neutrals are indifferent to fair gambling.The reality is that most people are risk averse.Only a small portion are risk neutral.Very few are risk lovers.

Letv:S→Rbe a VNMfunction of the individual.It can be shown thatv:S→Ris concave for risk averters,and convex for risk lovers.If they are risk neutral,thenv:S→Ris a linear or one-order function.The functionv(x)=v(x1,x2,···,xl)could be assumed to be twice differentiable and have non-zero first order derivatives.Under this assumption,(i=1,2,···,l)are negative for risk averters,positive for risk lovers,and zero for risk neutrals.As a result,the decreasing marginal utility is equivalent to risk aversion,and so is verified to be a prevailing phenomenon.

With the help of the Arrow-Pratt coefficient of risk aversion,a measurement vector θ (x)is found and proposed here for multi-variate functionv(x).The vector θ (x),called a relative risk aversion tendency,is defined as follows:

The economic meaning ofθi(x)can be explained by a gambling plane.We can imagine a gambledesigned by an eventFwith probabilityp.The amount of commodityibecomesxi(1+a)ifFhappens,and becomesxi(1+b)ifFdoesn’t happen,whereaandbare percentages of changes in quantity.The amountxjof other commoditiesj(ji)remain unchanged.This gamble can be denoted by(a,b),which is a point on the planeR2called a gambling plane,as shown in Figure 1.The origin0of the coordinate means no gambling.

For convenience,letu(a)=v(x1,···,xi−1,xi(1+a),xi+1,···,xl)(a∈R).u(·)is the underlying utility function of the gamble.The expected returnERand the expected utilityEuare as follows:

Fair gambles are those(a,b)with expected returnxi,i.e.,ER(a,b)=xi,so a gamble(a,b)is fair if and only ifpa(1−p)b=0.The lineJconsisting of all fair gambles is called fair gambling line,as shown in Figure 1.

Figure 1 Risk averse acceptance set

The condition for the individual to accept a gamble(a,b)is that the expected utilityEu(a,b)is not less than the utilityEu(0,0)=u(0)=v(x)of no gambling.LetAbe the set of all gambles accepted by the individual,i.e.,A={(a,b)∈R2:pu(a)+(1−p)u(b)≥u(0)},called the acceptance set.It can be shown that the acceptance setAof risk averters is convex.Figure 1 displays the shape of the acceptance set of a risk averter,where∂Ais the boundary ofA.The boundary∂Ais determined by equationpu(a)+(1−p)u(b)=u(0).Hence∂Ais the indifference curve through the origin o;its slope at origin o is−p/(1−p),which is just the slope of the fair gambling lineJ.ThereforeJis just the tangent line of∂Aat the origin o.

Letb=ϕ(a)be the function determined by equationpu(a)+(1−p)u(b)=u(0);i.e.,b=ϕ(a)describes∂A.It can be shown thatϕ′(0)=−p/(1−p)andϕ′′(0)=−(u′′(0)/u′(0))p/(1−p)2.

Assume that the individual is a risk averter,so thatu′(0)>0 andu′′(0)<0.Thenϕ′′(0)>0.Based on curvature theory,the larger theϕ′′(0),the more curved the boundary∂A,and so the more gambles near at origin o are rejected.As we know,gambles near at origin o are all small ones.Attitudes towards small gambles can reflect best the tendencies of risk aversion.Henceϕ′′(0)measures the tendency of risk aversion.Asϕ′′(0)and−u′′(0)/u′(0)maintain a proportional relation,−u′′(0)/u′(0)can be regarded as a measure of the risk aversion tendency.

Theorem 3.2The relative risk aversion tendency θ (x)depends only on the individual,not the forms of VNMfunctions of the individual.That is,if bothu:S→Randv:S→Rare VNMfunctions of the individual,thenholds for any x∈S.

ProofBy cardinal utility,there is a rod scale or ruler in the mind of the individual used to measure the quantity of utilities.What is the scale of this ruler?This can be determined by a business plan.One foot can be defined as one meter long or 33.333 centimeters,etc.,according to the plan.Once the length is determined,the ruler cannot be deformed or broken;otherwise,the measured quantity would be inaccurate.It does not matter whether long or short;what matters is that the ruler gives the reference unit of measurement,and then is used to mark the scale on a straight line.It does not matter either where the 0 mark is;as long as the 0 point is marked,the scales can be marked to the right or left.

One can use scales of different lengths,and adopt 0 points with different positions,to mark the scales on straight lines.The correspondence between any two different scales is really an affine transformation.Like the different scales with different positions of the 0 point,the correspondence between any two cardinal utility functionsu(x)andv(x)is really an affine transformation too;that is,there exist real numbersa>0 andbsuch thatv(x)=au(x)+b.

Now suppose thatu:S→Randv:S→Rare VNMfunctions of the individual.They are cardinal utility functions,as VNMfunctions are cardinal ones.Thus there are real numbersa>0 andbsuch thatv(x)=au(x)+bfor any x∈S.This implies immediately thatholds for any x∈S andi=1,2,···,l.

For the above fact thatu(·)can be affinely transformed tov(·),we can give a strict proof.As bothu(·)andv(·)are VNMfunctions of the same individual,we have that for any ξ , η ∈S:

LetA={Eu( ξ ): ξ ∈S}andB={Ev( ξ ): ξ ∈S},whereandIt can be checked that for anyα∈[0,1],the distribution function of(1−α) ξ ⊕αη is the weighted sum(1−α)f(·)+αg(·)wheref(·)andg(·)are the distribution functions of ξ and η respectively.ThusEu((1−α) ξ ⊕αη )=(1−α)Eu( ξ )+αEu( η )for anyα∈[0,1].This implies that bothAandBare convex subsets of the real lineR.

Defi neϕ:A→Bas follows:ϕ(Eu( ξ ))=Ev( ξ )( ξ ∈S).Obviously,ϕ(·)is increasing,and

Thisshowsthatϕ:A→Bis convexly linear.Thus there exista,b∈Rwitha>0 such thatϕ(z)=az+bfor anyz∈A;i.e.,ϕ:A→Bis an affine transformation.SinceEu(x)=u(x)andEv(x)=v(x)for any x∈S⊆S,we have thatϕ(u(x))=ϕ(Eu(x))=Ev(x)=v(x)for any x∈S.Thereforev(x)=au(x)+bfor any x∈S;i.e.,u:S→Rcan be affinely transformed tov:S→R.Theorem 3.2 is proven. □

θi(x)changes with the change ofxi.What are the specific characteristics of such change?The specific answer to this is unknown,but because the stakes are in proportion,it seems thatθi(x)has nothing to do with the size ofxi.In this case,it should be a good option to assume thatθi(x)is constant.At least this option is acceptable and conforms to the principle of meeting change with constancy.

Necessity Suppose that the individual has a constant relative risk aversion tendency θ ∈Rl.Then the following formulas hold for all x∈S andi=1,2,···,l:

Notethat thepartial derivative∂vi(x)/∂xirequires that the otherxj(ji)are unchanged.Hence∂(·)/∂xiandd(·)/dxihave the same meaning,so above formula can be written as

Starting from commodity 1,we recursively deduce things from above formula.

Conclusion 1There is a constantA2relative tox1such that

Conclusion 2There is a relative constantA3such that

Recursively using similar reasoning,we can get the conclusion for commodityias follows:

Conclusion iThere is a relative constantAisuch that

Imitating the above proof,Theorem 3.3 can be generalized into a more general form,which are described in Theorem 3.4.As a preparation,here we explain the semi-orderings≤and≪onRl.For any x,y∈Rl,x≤y means thatxi≤yi(i=1,2,···,l);x

The proof of this theorem is similar to the one of Theorem 3.3.It will not be repeated here.However the vector µ is meaningful.For an individual with a constant relative risk aversion tendency, µ satisfies thatv( µ )=0 andv(x)>0 when x≫ µ .This means that µ is a package of necessities,and a starting point for lives. µ ≫0indicates a high living standard.The individual has passed through the difficult stage of robbing Peter to pay Paul,so µ signifi es entering a well-offlife.It is because of this well-offlife that the individual could have a relative constant risk aversion tendency.

The relative constant risk aversion tendencyθi(x)has another significant meaning.From its definition,it is easy to see thatθi(x)is the elasticity of marginal utility,which represents the ratio of the marginal utility decline to the consumption increase.As usual,it is said thatθi(x)is small ifθi(x)<1,large ifθi(x)>1,and appropriate ifθi(x)=1.Thusθi(x)is a measure for the sensitivity of marginal utility to consumption.

As marginal utility representsscarcity,the elasticity of marginal utilityθi(x)can be referred to as scarcity elasticity.The smaller the scarcity elasticity,the less the impact of consumption on marginal utility,and the more necessary the commodity.Therefore,a smallθi(x)implies that the commodityiis a necessity.

θi(x)also denotes the ratio of instantaneous speed to the average speed of diminishing marginal utility.When the instantaneous speed is less than the average speed,θi(x)is small.When the former is greater than the latter,θi(x)is large.When both are equal,θi(x)is appropriate.Thereforeθi(x)is also a measure for the declining intensity of marginal utility.

In a word,the relative risk aversion tendencyθi(x)has very significant meaning.It reflects both the scarcity elasticity and the declining intensity of marginal utility.The smallerθi(x),the weaker the tendency of relative risk aversion,the more necessary the commodity,and the weaker the declining intensity of marginal utility.The following theorem interprets the form of VNMutility functions of individuals with a weak tendency of relative risk aversion:

Theorems 3.3 to 3.5 have explained some relations between constant risk aversion tendency and forms of VNMfunctions,and have answered to certain extent the question of how to identify a function as a VNMfunction.

Coming back to the example of the expected utility enigma in Section 2,the truth can be eventually revealed.Two utility functions,u(x,y)=(xy)0.25andv(x,y)=(xy)0.75are given in the example for the individual to evaluate,but they get two contradict answers.Now,applying Theorem 3.5,it turns out that the truth is that if the relative risk aversion tendency is θ =(0.75,0.75),then the VNMfunction isu(x,y)rather thanv(x,y),and the individual should choose optionB.If the relativerisk aversion tendency is θ =(0.25,0.25),then the VNMfunction isv(x,y)rather thanu(x,y),and the individual should choose optionA.If neither(0.75,0.75)nor(0.25,0.25)is the relative risk aversion tendency,then neither of the evaluations from the two functions is correct.

With the help of utility theory,Markovitz’s mean-variance approach has been greatly developed.A utility function for mean and variance has been imagined,called the mean-variance utility function,or MV utility function for short.However this imagination hides two basic problems.One is similar to the enigma of expected utility without consideration of the VNMcondition.The other is the MV utility function decoupling from underlying economic behavior.The value of the MV function is confused,because different behaviors can have the same meanvariance but different utilities.This may lead to paradoxes.Now we use the VNMcondition to rebuild the MV utility function on the base of underlying behavior to solve the two problems.For this purpose,we first reexamine the mean-variance model for investment.

Let( Ω ,F,P)denote a risky environment.The outcome of an investment is its return,usually expressed in terms of monetary revenue,which is random.Letv:R→Rbe the VNMutility function of an investor.Its valuev(x)denotes the utility amount ofxunits of revenue.Usually,the investor may face two of options.One is risk-free investment,such as in monetary assets,which are safe in terms of returns.The other is risky investment,such as securities,which are usually with uncertain returns.Generally,talking about investments means dealing with risks.Letξdenote the return of an investment.It is a random variable.The mathematical expectationr=E ξexpresses its expected return.The standard deviationexpresses the risk of the investment.Whether the investor takes the investment activityξ,it depends on weighing up of riskσand returnr;after all,high(or low)risk accompanies high(or low)expected return.The mean-variance model expresses how an investor weighs risks against expected returns.

Now assumeRfandRmto be two options of an investor,whereRfis the rate of return of a risk-free investment,andRmis the rate of a risky investment.We can view the sum to invest as one unit.Note thatRfis constant,but theRmis random.Rf=ERf=rfandHigh risk accompanying high return impliesrm>rf.How much should the investor invest on the risky itemRm?It has long been said that one should not put all eggs in one basket.The investor might consider putting a proportion of their money inRm,and putting another part inRf.This proportion is the well-knownβcoefficient.

ByRβwe denote the rate of the return of a portfolio with the coefficientβ:Rβ=βRm+(1−β)Rf.The expected return isrβ=ERβ=βrm+(1−β)rf,and the standard deviation isσβ=We have thatβ=σβ/σmandrβ=rf+((rm−rf)/σm)σβ.This equation expresses the constraints on the risks and returns of portfolios,called the budget constraint of portfolios.This is displayed in Figure 2 as a straight line,called the budget line.

Figure 2 Risk-return plane

In theprevailing MV model,a utility functionU(σ,r)isimagined directly on therisk-return plane to be used to make a portfolio decision,but such an approach leaves the following basic questions unsolved:

(1)Where is the function from?What is its base?Can it be set up arbitrarily?Does it truly express the investor’s objective function?

(2)Different investment activities may have different utility levels but the same meanvariance.This means that the valueU(σ,r)is not unique,and so is confused and leads to wrong decisions.

Now these questions can be solved by using the VNMcondition.A solid theoretical foundation will be established for mean-variance modelling.

For question(1),it can be seen from the VNMcondition thatU(σ,r)can not be set casually.Otherwise,wrong decisions could result.U(σ,r)must be established on the VNMutility functionv:R→Rof the investor.Meanwhile this can not be decoupled from underlying investment activities.In order to bring outU(σ,r),it is right to calculate the expected utilityEv(ξ)of investment behaviorξ.Only in this manner of definingU(σ,r)it becomes the objective function of the investor.

Question(2)is an extremely important issue arising inevitably from the process of solving question(1).To overcome it,the behavior considered is confined within the extent of normal random variables.However,as we know,it is hard for the investment return to obey normal distributions;the problem still exists,and up until now has not been fundamentally solved.

Here we shall put forward a method to solve the question(2)satisfactorily.The idea is based on the budget line to give every point(σ,r)of the risk-return plane a portfolio investment behavior such that the behavior corresponding to(σ,r)is decided uniquely,and has meanrand varianceσ2.The specific practices are as follows:

For anyσ≥0 andr≥0,letξ(σ,r)=r+((Rm−rm)/σm)σ.It is easy to verify thatξ(σ,r)is a random variable and represents an investment activity with riskσand expected returnr;that is,we have thatE[ξ(σ,r)]=rand Var(ξ(σ,r))=σ2.

The behaviorξ(σ,r)has intuitive meanings.LetRβ=βRm+(1−β)Rf=rf+β(Rm−Rf),whereβ=σ/σm.Then the following facts are true:

Therefore,ξ(σ,r)=β(Rm+r−rβ)+(1−β)(Rf+r−rβ).This shows thatξ(σ,r)is a portfolio of the safe itemRf+r−rβand the risky itemRm+r−rβwith the coefficientβ.On the risk-return plane,asσβ=σ,the corresponding point(σ,r)ofξ(σ,r)is just the position to which the point(σβ,rβ)representing portfolioRβmove upward,as shown in Figure 2.It can be proven that the correspondence betweenξ(σ,r)and(σ,r)is one-to-one,i.e.,for any(σ1,r1)and(σ2,r2),we have that((σ1,r1)=(σ2,r2))⇔(ξ(σ1,r1)=ξ(σ2,r2)).In addition,it can be seen thatξ(σβ,rβ)=Rβholds for any proportionβ.

Note thatξ(σ,r)represents the rate of return of the investment behavior(σ,r).Its return is 1+ξ(σ,r),as the sum to invest is viewed as one unit.The expected utility isEv(1+ξ(σ,r)).Based onEv(1+ξ(σ,r)),a utility functionU(σ,r)can be defined on the risk-return planeas follows:U(σ,r)Ev(1+ξ(σ,r))for any(σ,r)∈

SuchU(σ,r)are not only based on the VNMutility functionv:R→R,but also on the underling investment activityξ(σ,r).Therefore,this is truly the most objective function in which the investor can make investment decisions.Because of this,U(σ,r)is called the meanvariance utility function,briefl y,the MV utility function or the MV function.The following two theorems explain the characteristics of the MV functions:

Theorem 4.1The MV functionU(σ,r)defined aboveis strictly concavefor risk averters,strictly convex for risk lovers,and linear or of one order for risk neutrals.

ProofLet(σ1,r1),(σ2,r2)∈andα∈(0,1)be arbitrarily given with(σ1,r1)(σ2,r2).We have that

When the investor is risk averse,their VNMutility functionv:R→Ris strictly concave,so we have thatE[αv(1+ξ(σ1,r1))+(1−α)v(1+ξ(σ2,r2))]

When the investor is a risk lever,their VNMutility functionv:R→Ris strictly convex,so we have thatE[αv(1+ξ(σ1,r1))+(1−α)v(1+ξ(σ2,r2))]>Ev(1+ξ(α(σ1,r1)+(1−α)(σ2,r2))),and henceαU(σ1,r1)+(1−α)U(σ2,r2)>U(α(σ1,r1)+(1−α)(σ2,r2));i.e.,U(σ,r)is strictly convex.

When the investor is risk neutral,their VNMutility functionv:R→Ris of one order or linear,so we have thatE[αv(1+ξ(σ1,r1))+(1−α)v(1+ξ(σ2,r2))]=Ev(1+ξ(α(σ1,r1)+(1−α)(σ2,r2))),and henceαU(σ1,r1)+(1−α)U(σ2,r2)=U(α(σ1,r1)+(1−α)(σ2,r2));and thus i.e.,U(σ,r)is of one order or linear.

The theorem is proven. □

The next theorem reveals the characteristics forU(σ,r)to reflect the phenomenon of high risks accompanying high returns.This is important for investors to weigh risks against returns.

Theorem 4.2Let the VNMfunctionv:R→Rbe twice differentiable andv′(x)>0 for allx∈R.

Proofξ(σ,r)=r+[(Rm−rm)/σm]σandU(σ,r)=Ev(1+ξ(σ,r)).Calculating the partial derivatives,we can obtain that

The theorem is proven. □

The characteristics of mean-variance utility revealed by Theorems 4.1 and 4.2 are displayed in Figure 3,where the shapes of the indifference curves are depicted separately for the risk averter,risk the lover and the risk neutral.

Figure 3 Indifference curves under mean-variance utility function

For an investor with constant relative risk aversion tendencyθ,it can be shown that they have an VNMutility function with the formv(x)=a(x1− θ−1)/(1−θ)+b,wherea>0 andbare constants.This function may be further taken asv(x)=(x1− θ−1)/(1−θ),since VNMfunctions have“invariance”under affine transformations.

Letϕm(x)be the density function of the distribution of the risky returnRm.For the investor with constant relative risk aversion tendencyθ,the MV utility functionU(σ,r)=Ev(1+ξ(σ,r))can be written as follows:

These kinds of concrete forms of mean-variance utility functions might be useful in econometrics.In particular,they are convenient for quantitative analyses.

Systematic risk refers to the fluctuation of the whole economic and financial system due to external or internal factors,which cause a series of continuous losses.No individuals are spared and anyone can suffer losses.This kind of risk can not be dispersed and can not be eliminated by investment diversification.Hence Markowitz’smethod of maximizing profi tsand minimizing risks is invalid here.

There are many factors that cause systematic risks,including political factors,policy factors,economic factors,social factors,environmental factors,and large-scale natural or manmade disasters(such as the COVID-19 epidemic),etc..Once systematic risk occurs,all kinds of investment activities will be seriously affected.This means that in the systematic risk environment,the returns of various investment activities will show the same characteristics of rise and fall.Combining different risk options can not solve the risk.Only by following the logic of the mean variance model and choosing the optimalβcoefficient can the loss be minimized.As such,it seems important to hold a certain percentage of safe assets.

Assume that the investor is in a systematic risk environment,and their VNMfunction isv(x),wherex∈R.They are faced with two options for investment activities.One option is to hold safe assetRf.This could be viewed as holding a monetary asset,whose return is constant and not affected by the systematic risks.The other option is to hold risky assetRm,which could be viewed as holding securities.The return of the risky asset is affected by systematic risks,and so is a random variable.Although risky assets are various,their return rates rise or fall simultaneously.Any one of them could be chosen as a representor.LetRmbe the chosen representor of all risky investment activities,and letRfbe the representor of safe assets.As we did in last section,RmandRfexpress the rates of returns.Rfis constant,butRmis random.The sum for the investor to invest is viewed as one unit.ThusRmandRfrepresent both rates and net returns.The investor’s MV function is thenU(σ,r)=Ev(1+ξ(σ,r))((σ,r)∈R2+),whereξ(σ,r)is defined as in last section:ξ(σ,r)=r+[(Rm−rm)/σm]σ.

No matter what happens,the object of the investor is to maximize his MV utility within the budget constraint of portfolios.Let(σ∗,r∗)be the optimal combination of risk and return;that is,(σ∗,r∗)is the solution of the following maximization problem of mean-varianceutilities:

Assume that the VNMfunctionv(x)(x∈R)is twice differentiable with positive first order derivatives.Then there is a Lagrange multiplierλsuch that(σ∗,r∗)satisfies the condition that

This condition is called the first order MV condition,and it can be also written as follows:

The first order MV condition involves two tools.One is the slopeπ=(rm−rf)/σmof the budget line.The other is the slopeρ=ρ(σ,r)=of the indifference curve.The two slopes have very significant implications for investment decision making.

(1)Actual Rate of Return(ARR)π=(rm−rf)/σm.As the slope of the budget line,πdenotes the amount by which the expected return rate of the portfolio can increase,when the risk of portfolio is increased by one unit.This amount is determined by the budget line,so is a real amount that cannot be artificially changed.Therefore,it called the ARR of risk.

(2)Target Rate of Return(TRR)ρ=ρ(σ,r)=As the slope of the indifference curve,ρdenotes the amount by which the expected rate of return should be increased to keep the utility level constant when the risk at(σ,r)is increased by one unit.This amount is the goal that investors hope to achieve for the risk-return rate.Therefore,it is called the TRR of risk.

(3)Decision Principle(DP).After increasing the risk,if the ARR reaches the TRR,the utility level will remain unchanged;if the ARR exceeds the TRR,the utility level will rise;if the ARR fails to reach the TRR,the utility level will drop.This implies the principle of investment decision which tell us that at the current point(σ,r),we should increase the risk if ARR>TRR,reduce the risk if ARR

In what follows,we use the ARR and the TRR to explain the laws of investment activities in a systematic risk environment.Before the discussion,we classify systematic risks according to their degree of infl uence.In fact,systematic risks mainly affect investor’s expectation of returns.WhenERmRf,it is viewed as heavy influence.WhenERm=Rf,it is viewed as light influence.

Case 1Serious influence of systematic risk,rm=ERm

In this case,the ARR is negative.For risk averters and risk neutrals,TRR>ARR at every point(σ,r)of the budget line;thus they do not invest in risky itemRmand will put all of their money in safe itemRf.As they make up the overwhelming majority,social investment activities will be extremely depressed.

Even for risk lovers,if TRR≥ARR at(σm,rm),they do not invest inRmeither.Instead,they will put all of their money in safe itemRf.This makes social investment situation more severe.

Case 2Heavy influence of systematic risk,rm=ERm=Rf=rf.

In this case,the ARR is zero.For risk averters,TRR>ARR at every point(σ,r)of the budget line,they will put all of their money in safe itemRf.Because risk averters are the majority,social investment activities are heavily depressed and very low.

For risk neutrals,TRR=ARR at every point(σ,r)of the budget line,and they do not care what choice they make.Some risk neutrals may invest inRm,some may not.

For risk lovers,TRR<0=ARR at every point(σ,r)of the budget line,so they will invest all of their money inRm.However,as they are very few,their investment activities can not improve the grim investment situation.

Case 3Light influence of systematic risk,rm=ERm>Rf=rf.

In this case,although systematic risk has caused adverse effects leading to a decline of expected rate of returnrm,the influence is light,so the expected rate of returnrmis still higher thanrf.Thus the ARR is still positive.

For risk neutrals and risk lovers,their TRR at every point(σ,r)of the budget line is nonpositive,and so is less than the ARR.Thus they will definitely invest all of their money inRm.

For risk averters,so long as TRR

In summary,those risk averters whose TRR at(0,rf)are less than the ARR,along with all risk lovers and risk neutrals,make up a quite large part of those who have an investment inRm.Therefore,in the case that the influence of systematic risk is light,the situation for social investment activities is not so bad;quite a few people are still engaged in investment activities.

Why do those risk averters whose TRR at(0,rf)are equal to or greater than the ARR not invest in itemRm?In order to analyze this,let us reveal another profound implication of TRR=ρ(σ,r)for risk averters.From the calculation ofin the proof of Theorem 4.2,we have that

Sincev′′(x)<0 andξ(σ,r)is positively related toRm,v′(1+ξ(σ,r))is negatively related toRm.From the Pratt Theorem(1964,see [14]),we know that the stronger the risk aversion tendency,the more concave the utility functionv(x).Obviously,the more concave thev(x)is,the faster the marginal utilityv′(x)diminishes.The faster thev′(x)diminishes,the stronger the negative correlation betweenv′(1+ξ(σ,r))andRm,the greater the−Cov(v′(1+ξ(σ,r)),Rm),the higher theρ(σ,r).Therefore,the stronger the risk aversion tendency,the higher theρ(σ,r);that is,ρ(σ,r)and the risk aversion tendency change uniformly or in the same direction.This shows thatρ(σ,r)or TRR reflects the strength of the risk aversion tendency.Hence the TRR becomes a new tool to measure the risk aversion tendency of investors.

This profound aspect of TRR implies that the stronger the investor’srisk aversion tendency,the higher the investor’s requirements for risk return rate.If you do not meet the requirements,you will not take risks,but rather consider risk-free options.

Now we can find the reason that those risk averters with TRR≥ARR at point(0,rf)have no investment inRm.At the initial point(0,rf)of the budget line,ρ(0,rf)≥πimplies that the utility at any other point of budget line is less than the utility at(0,rf),soRfis the optimal choice.In other words,the investor has a strong risk aversion tendency from the beginning so as to never invest in risky assets.

We can also find that whenρ(σm,rm)≤π,the investor never puts money intoRf,instead,they invest all of their money inRm.Thusρ(σm,rm)≤πexpresses the fact that the investor has a weak risk aversion tendency at last,so as to eventually invest all of their money inRm.This kind of behavior makes the risk averter look like a risk lover,giving others the illusion that they loves risks.

In a word,for a risk averter,ρ(0,rf)≥πmeans a strong risk aversion tendency for them to put all of their money in safe itemRf,andρ(σm,rm)≤πmeans a weak risk aversion tendency for them to put all of their money in risky itemRm.They will put their money into bothRfandRmif and only if their risk aversion tendency is neither too strong(ρ(0,rf)<π)nor too weak(ρ(σm,rm)>π).

The conclusions drawn from the above analyses can be summarized into the following theorem:

Theorem 5.1Investment decisions depend on the comparison between target rate of return(TRR)and actual rate of return(ARR).The TRR not only reflects the investor’s required rate of return for taking risks,but also reflects the risk aversion tendency.In the case that theinfl uenceof systematic risk islight(ARR>0),all risk lovers,risk neutral sand those risk averters whose risk aversion tendency is not too strong(ρ(0,rf)<π)will have investments in risky items;thus the situation for social investment activities is not so bad.Only in the cases where the influence of systematic risks is heavy or serious(ARR≤0)will social investment activities be depressed to a great extent,and most people will not make investments.

This theorem has important policy implications.It tells us that governments should focus on those systematic risks that are expected to have serious or heavy influences.Priority should be placed upon maintaining stability in politics,the economy,the environment and in people’s lives,in order to prevent macro-systemic risks from occurring.Maintaining currency stability to prevent currency itself from becoming a systematic risk factor is also important,as is maintaining the stability of the foreign exchange markets to prevent large fluctuations in exchange rates.Governments must maintain the stability of financial systems and markets to prevent the capital chain from breaking.Maintaining the continuity of policies to prevent long supply chains from breaking is crucial too.In summary,society should always pay attention to guard against the occurrence of those factors that induce heavy or serious systematic risks.