Measure of riskiness based on RDEU model

Guo Chuanfeng, Du Xinze, Wu Qinyu, Mao Tiantian*

1. School of Data Science, University of Science and Technology of China, Hefei 230026, China;2. School of Management, University of Science and Technology of China, Hefei 230026, China

Abstract: Motivated by References[3,4], we introduce a new measure of riskiness based on the rank-dependent expected utility (RDEU) model. The new measure of riskiness is a generalized class of risk measures which includes the economic index of riskiness of Reference[3] and the operational measure of riskiness of Reference[4] as special cases. We probe into the basic properties as a measure of riskiness such as monotonicity, positive homogeneity and subadditivity. We study its applications in comparative risk aversion as well. In addition, we present a simulation to illustrate the results.

Keywords: measure of riskiness; RDEU model; distortion function

1 Introduction

People want to quantify the risk of a decision, far beyond the expectation or variance. Even though many contributions have been made in this area[1,2], no one ever constructed a perfect risk measure both satisfying all the desired properties and applicable in economics. Reference[3] introduced a new measure of riskiness and proved several desired properties. They defined the risk of a specific gamble, which yields both positive and negative outcomes, with the same measurement unit as gambles. However, the risk, according to them, is totally based on the distribution of a gamble about which one may doubt whether others take the gamble as serious as him.

Almost all the measures of riskiness are objective, without the interference of decision-makers[4]. However, a risky asset may be taken as riskless to someone but, as, at the same time, too risky to be accepted by others. In this paper, based on the rank-dependent expected utility (RDEU) model, we propose a measure of riskiness of ``gambles″ (risky assets) that is subjective: it depends on both the gamble and the one who is considering investing. Even though we have a huge step up, the subjective measure is still ideal to use. Meanwhile, the measure is applicable to all the bounded gambles, making the comparison of different gambles easier.

The RDEU theory is proposed by References[5,6] in which the expectation can be defined as rank-dependent, which permits the analysis of phenomena associated with the distortion of subjective probability and applies better in real than simply weighted expectations, according to References[7,8].

For the discussion of the distortion function, named after the intuition that the expectation is “distorted”, it can be concave: this rank-dependent way of modeling pessimism and optimism was suggested before by Reference[5]. It was described in full by Reference[6], which can be convex, and even a mixed pattern of both[8].

In this paper, we apply a new model to the index of riskiness and obtain desired properties. It is natural that some of them are no longer satisfied. However, after assuming the distortion function to be concave, almost all of the properties still hold. Besides, we extend the definition of risks to nearly all gambles, even with no loss, without loss of desired properties.

The rest of the paper is organized as follows. In Section 2, we introduce some preliminaries, such as the RDEU model. In Section 3, we introduce the new risk measure of riskiness based on the RDEU model. Section 4 shows our main results. In Section 5, we present a simulation to illustrate the results.

2 Preliminaries

2.1 The RDEU model

The rank-dependent expected utility (RDEU) model is one of classic models in the economical behavior theory introduced by References[5,6]. A decision decision-maker behaves in accordance with the RDEU model if the decision-maker is characterized by an increasing and continuous utility functionu:R→R and a probability-perception functionh: [0,1]o[0,1], that is,his increasing withh(0)=0 andh(1)=1. Such a decision-maker prefers the random variableYto the random variableXif and only if

Vu,h(Y)≥Vu,h(X),

whereVu,h(Y), also denoted byVu,h(G), is the RDEU functional or the Quiggin-Yaari functional ofY[5,6]given by

(1)

HereGis the cumulative distribution function (CDF) ofY. Any decision maker in the REDU model makes decision according to the RDEU functionalVu,his denoted by a (u,h)-decision-maker.

It is well-known that the RDEU model has the classic expected utility (EU) model and the Yarri′s dual theory as its special case. Specifically, if the probability-perception functionh(s)=s,s∈[0,1], then the RDEU functional reduces to the classic expected utility functional.

Throughout the paper, the utility functionuis the von Neumann-Morgenstern utility function for money. We confine the utility functionuin the following set

If the utility function is the linear function (identical function), that is,u(x)=x,x∈R, then the RDEU functional reduces to the Yarri’s dual utility

(2)

One important property of distorted expectation is that Eh(aX+b)=aEh(X)+bholds for real numbersa≥0 andb∈R. However, Eh(X+Y)=Eh(X)+Eh(Y) may not be true for random variablesXandY, and it will hold ifXandYare comonotonic. In the bivariate setting, a random vector (X;Y) is comonotonic if there exists increasing functionsfandgsuch thatX=f(X+Y) andY=g(X+Y) almost surely. This fact leads to problems when dealing with the portfolio investment, a linear combination of different risky assets. Thus, additional presumptions are necessary for the distortion functionh. Throughout the paper, the distortion functionhis assumed in the following set

H={h:[0,1]→[0,1]|h(0)=0,h(1)=1,

his concave and has no jump at zero}.

We will also denoteh′ by the left derivative of distortion functionh. There are some examples of distortion functions, and Figure 1 presents a more intuitive picture.

Figure 1. (a) A concave distortion function; (b) A distortion function concave for small probability and convex for moderate and high probabilities.

Figure 2. The scale function for (a) with and (b) with a linear h. The riskiness is 2.79 for (a) and is 1.69 for (b).

(Ⅰ) Proportional hazard transform function[10], with the distortion function

(Ⅱ) Dual-power function[11], with the distortion function

h(x)=1-(1-x)ν,ν≥1.

(Ⅲ) Wang’s transform weighting function, known as the WT weighting function[12], it is applied widely into the pricing of financial derivatives for its fine properties. It is usually represented by

h(x)=Φ(Φ-1(x)+α),α∈R,

whereΦ(x) is the cumulative distribution function of a standard normal distribution.

2.2 Comparative risk aversion

Risk aversion is an important concept in the decision theory. We use the notation Reference[3] to describe the comparative risk aversion. Agentsiandjare going to decide whether to accept or reject such a gamble.

Definition 2.1(Ⅰ) A (u,h)-decision maker in the RDEU model accepts gambleXat the wealth levelwif

Vu,h(w+X)>u(w).

3 Measure of riskiness based on the RDEU model

In this section, we will introduce a new measure of riskiness based on the RDEU model. To this end, we confine the gambles to some subsets of the family of all gambles.

Definition 3.1For a given distortion functionhwe define

G={X:Xis bounded and P(X=0)<1}

and

Gh={X|Eh(X)>0, P(X<0)>0}.

The condition Eh(X)>0 is due to that people will not hesitate to reject a gamble that they think would be nonprofitable, while violating the condition P(X<0)>0 means that the gamble brings no loss at all.

For a utility functionu, a distortion functionhand a gambleX, we define

fu,h,X(α):=f(α)=Eh(u(αX))

on [0,∞], which is called a scale function throughout the paper. Then,f∈C2[0,∞] which meansfis second order continuously differentiable. In the following, we state some basic properties for the scale function.

Theorem 3.1Suppose that the utility functionu∈U has an upper bound andh∈H. ForX∈Gh, the scale functionf(α)=Eh(u(αX)) is concave on [0,∞) withf(0)=0. Moreover, there exists a real numberρu,h(X)>0 uniquely determined by

f(1/ρu,h(X))=Ehu(X/ρu,h(X))=0

(3)

ProofIt is clear thatf(0)=Eh[u(0)]=0. Note thatf′(0)=Eh(Xu′(0))=Eh(X)>0 because ofX∈Gh. Hence, there exists anαsmall enough such thatf(α)>0. Meanwhile,f″(α)=Eh(X2u″(αX))≤0 meansfis concave on the positive axis.

Assuming now thatp0=P(X<-)>0 and P(|X|≤M)=1, letX0be a gamble that yieldsMwith probability 1-p0and -withp0. It is obvious thatFX0(x)≤FX(x) for allx. Then

h(p0)u(-α)+(1-h(p0))u(αM).

Thus, the scale functionf(α) becomes non-positive forαlarge enough sinceuhas an upper bound. Up to now, we obtain three observations of the functionf:

(Ⅰ)f(0)=0 andf′(0)>0;

(Ⅱ)fis concave on [0,∞);

(Ⅲ) There exists anαlarge enough such thatf(α)<0.

Hence, there exists a uniqueρu,h(X)>0 such that equation (3) holds.

Here, we setu(x)=1-e-xfor allx∈R. LetXyield 6 with probability 0.2, 2 with probability 0.3 and -1 with probability 0.5. Checking that it meets all the requirements, for differenth, we draw its scale function in Figure 2.

Figure 3. Estimated riskiness of distribution X1 (wave line) with real riskiness (beeline).

Figure 4. Estimated riskiness of distribution X2 (wave line) with real riskiness (beeline).

Figure 5. Estimated riskiness of distribution X3 (wave line) with real riskiness (beeline).

Definition 3.2Foru∈U andh∈H, the measure of riskiness based on (u,h)-RDEU model is defined by a functionalρu,h:G→[0,+∞] as the following way

For the case Ehu(X)≤0, a decision maker with distortion functionhwon’t take it. Thus, we set its riskiness to be +∞. For another case that P(X<0)=0, people accepts the gamble violating the condition with absolutely no loss. For this one, we set its riskiness to be 0 because nobody would be afraid of it for any reasons.

Example 1(Ⅰ) Ifhis the identical function, i.e.,h(p)=pforp∈[0,1] andu(x)=1-exp(-x) forx∈R. The measure of riskinessρu,hreduces to the case introduced in Reference[3].

(Ⅱ) Ifhis the identical function andu(x)=log(1+x) forx∈R. The measure of riskinessρu,hreduces to the case introduced in Reference[4].

(Ⅲ) Ifu(x)=x-1 forx∈R, we haveρu,h(X)=Eh(X) for allX∈Gh.

The third one of the above examples illustrates that the constraint ofuin Theorem 3.1 is not necessary to guarantee that equation (3) has an unique solution. For some feasible utility functionu,ρu,hcan be the Yarri’s dual utility.

4 Main results

It follows directly from the definition that two axiomatic characterizations are identical to those of Reference[3]. Hence, the similar results are also obtained for the distorted riskiness.

4.1 Basic properties for the measure of riskiness

Definition 4.1For any two lotteries with cumulative distribution functionsFandG, respectively. We sayFfirst-order stochastic dominatesG, denoted byF1G, if for any increasing functionu

Proposition 4.1Foru∈U with a upper bound andh∈H, the measure of riskiness based on RDEU model has following properties:

(Ⅰ) Monotonicity with respect to the first-order stochastic dominance: ForX,Y∈G, ifX?1Y, thenρu,h(X)≥ρu,h(Y).

(Ⅱ) Positive Homogeneity:ρu,h(λX)=λ{(lán)ρu,h(X) forλ>0 andX∈G;

(Ⅲ) Subdilution:ρu,h(Xp)≥ρu,h(X) holds forp∈(0,1] andX∈G, whereXpis a compound gamble that yieldsXwith probabilitypand 0 with probability 1-p;

ProofWe first consider the properties ofρu,hon Gh. ForX,Y∈Ghsuch thatX?1Y, we have 0=Ehu(X1/ρu,h(X))≤Ehu(Y/ρu,h(X)). Recall the properties of scale function in Theorem 3.1, we obtain 1/ρu,h(Y)≥1/ρu,h(X), and hence,ρu,h(X)≥ρu,h(Y). The positive homogeneity is trivial by the definition ofρu,h.

To prove the subdilution, first note thatX∈GhimpliesXp∈Ghforp∈(0,1]. We use the third form of the rank-dependent expectation for this part. i.e.

whereMis the bound ofX. For the diluted gambleXp, one writes the CDFFXp(x) aspFX(x)+(1-p)I[0,∞](x). Thus,

fXp(α)=Ehu(αXp)=

(1-p)I[0,∞](x))du(αx)≤

(1-p)h(I[0,∞](x))du(αx)=

pfX(α).

For gambles on G, one can easily verify the monotonicity, positive homogeneity and subdilution ofρu,hafter classification discussions.

4.2 Measure of riskiness for CARA utility function

The exponential utility function is the only one class of utility functions such that the Arrow-Pratt coefficient is constant, that is, the utility function with constant absolute risk aversion[13]. In this section, we set the utility functionu(x) to be 1-exp(-x). For convenience, we denote byRhthe measure of riskiness in this case, i.e.

Rh(X):=ρu,h(X).

In the following, more properties ofRhwill be found. To present the result, we need the following lemma which is coming from Reference[14].

(Ⅱ)Φ(q)≥ 0,q∈[0,1].

Proposition 4.2The following two properties hold for gamblesX1,X2∈G.

(Ⅰ) Subadditivity:Rh(X1+X2)≤Rh(X1)+Rh(X2), ifX1+X2∈G;

(Ⅱ) Convexity:Rh(λX1+(1-λ)X2)≤λRh(X1)+(1-λ)Rh(X2) ifλX1+(1-λ)X2∈G.

To this end, we turn to prove the next two inequalities hold

(4)

(5)

Note that formula (4) is equivalent to

Sinceh′(q) is non-negative andex-1≥xfor allx∈R, to prove formula(4), we only need to show that

or, equivalently,

Next assume thatX1,X2∈G andX1+X2∈G. There are just two kinds of potential violation ofRh(X1+X2)≤Rh(X1)+Rh(X2):

(ⅰ)Rh(X1+X2)=∞ butRh(X1) andRh(X2) are positive and finite;

(ⅱ)Rh(X1)=0 andRh(X1+X2)>Rh(X2).

Here we define G*={X:Xis bounded}. For the first one, note that the mapping Eh: G*→R satisfies the subadditivity, i.e., Eh(X1+X2)≥Eh(X1)+Eh(X2) for allX1,X2∈G*(see e.g., Theorem 2.2 in Reference[15]. SinceRh(X1) andRh(X2) are positive and the finite it follows that Eh(X1),Eh(X2)>0, we have Eh(X1+X2) is finite. The second case can’t happen sinceRhis monotonic with respect to first-order stochastic dominance.

(Ⅱ)The convexity follows immediately from thatRhis positively homogeneous.

4.3 The necessary of concavity of distortion function

As someone doubts whether the presumptions of distortion functionhcan be revised, we claim that concavity is necessary for subadditivity property. We prove it in the following part that there is some violation of subadditivity unlesshis concave on [0,1].

Proposition 4.3Suppose the distortion function has no jump at zero, the validity of subadditivity forceshto be concave.

where -x1<-δ<0<δ

Ehu(αX)=h(p1)u(-αx1)+

(1-h(p2))u(αx2)+

Meanwhile, the derivative offat zero is

f′(0)=-x1h(p1)+x2(1-h(p2))-

As we can see, the marginal distributions of bothX1andX2are the same as that ofX, so they have the same riskinessesr. Noting that

One can calculate the riskiness ofX1+X2with the similar method. The distorted expectation ofX1+X2isu(-2x1).h(p1)+u(2x2)(1-h(p2)). With a useful fact thatu(x)≤xfor allxonR, we findu(-2x1)h(p1)+u(2x2)(1-h(p2))≤-2x1h(p1)+2x2(1-h(p2))=0, which indicates that gambleX1+X2has infinite riskiness. We can conclude thatRh(X1+X2)=∞ >2r=2Rh(X)=Rh(X1)+Rh(X2), a violation of subadditivity.

4.4 Application in comparative risk aversion

Mentioned by Reference[3], duality implies that less risk-averse agents accept riskier gambles. Once they share the same distortion function, duality holds for the two agents.

To prove Theorem 4.1, we denoteρ(w) by the Arrow-Pratt coefficient of absolute risk aversion for an agent with the utility functionuat wealth levelw, i.e.,ρ(w)=-u″(w)/u′(w). Besides, we need some extra lemmas, some of which are the direct results in Reference[3].

Lemma 4.2(Lemma 2 in Reference[3]) For someδ>0, suppose thatρi(w)>ρj(w) at eachwwith |w|<δ, thenui(w)

From Lemma 4.2, we can immediately get the following corollary.

Corollary 4.1(Corollary 3 in Reference[3]) Ifρi(w)≤ρj(w) for allw, thenui(w)≥uj(w) for allw.

Given the changes compared with the definitions by Reference[3], some lemmas also need to be generalized.

Lemma 4.3Ifρi(0)>ρj(0), there is a gambleXthat agentjaccepts at 0 but agentirejects at 0.

ProofBy the precondition that utility function is twice continuously differentiable,ρ(w) is continuous. There existsδ>0 such thatρi(w)>ρj(w) for all |w|<2δ. For -δ≤x≤δ,letXXbe a gamble yieldingx-δandx+δwith probabilityp0∈(0,1) and 1-p0, respectively, wherep0satisfiesh(p0)∈(0,1). By lemma 4.2, we can getui(w)≤uj(w) for all |w|<2δ, where the equation is satisfied if and only ifw=0. Then denote Ehkuk(XX)-uk(0) bygk(x) fork=i,j. One can compute fork=i,jthat

h(p0)uk(x-δ)+(1-h(p0))uk(x+δ).

Under the condition thatui(w)≤uj(w), inequalitygi(x)0 andgk(-δ)=h(p0)uk(-2δ)<0. Thus, it follows from the continuity and monotonic ofgkthat there exists somex0between -δandδsuch thatgi(x0)≤0

Lemma 4.4Ifρi(wi)>ρj(wj), then there is a gambleXsuch that agentjaccepts atwjbut agentirejects atwi.

Now we assumeiaccepts the gambleX1, then we need to show thatjaccepts the gambleX2. By definition, Ehui(X1)>0, thus

resulting inβi<α1. Then one getsβj≤βi<α1≤α2. Hence,βj<α2. The following inequality holds

Thus, the duality axiom is satisfied whenX1,X2∈Gh. Suppose nowRh(X2)=0, so thatX2∈G andP(X2<0)=0. It is obvious thatEhuj(X2)>uj(0)=0. Finally, we will show thatiacceptsX1at 0 andRh(X1)≥Rh(X2) implyRh(X2)<∞. Otherwise, we haveRh(X1)=∞, which impliesX1∈G with Eh(X1)≤0. However, it follows from Jessen′s inequality that

Ehui(X1)≤ui(Eh(X1))≤0.

This meansirejectsX1at 0, yielding a contradiction.

5 Simulation

We can check with computer that their risk value are respectively 2.77, ∞ and 0. Figures 3,4 and 5 show how the calculated riskinesses approaches the real riskinesses.

Figure 6. Estimated riskiness of normal distribution with mean 1 and standard deviation 1 under a linear distortion function.

Figure 7. Estimated riskiness of normal distribution with mean 1 and standard deviation 1 under the WT distortion function.

However, it is computational expensive to solve such an equation with a largenin practice for a continuously distributed random variable. By the enlightment of the generalized method of moments, computing the numeric solution can be seen as an optimization problem.

whereWis the inverse of var(Y), estimated by

Suppose a continuous distribution is a normal distribution with mean 1 and standard deviation 1, we can then compute that the riskiness of such an random varible is 0.5 with a linear distortion function and 1 with the WT distortion function. Figures 6 and 7 show how the calculated estimators approach the real riskiness.