# New Robust Reward-Risk Ratio Models with CVaR and Standard Deviation

Authors: Lijun Xu; Yijia Zhou
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we present two robust reward-risk ratio optimization models. Two new models contain the worst case of not only conditional value-at-risk (CVaR), but also standard deviation (SD). Using properties of reward measure, CVaR measure, and standard deviation measure, new models can be proved to equivalent to min-max problems. When the uncertainty set is an ellipsoid, new models can be further rewritten as second-order cone problems step by step. Finally, we implement new models to portfolio problems. It shows that new models are robust and comparable with mean-CVaR ratio model. Since considering standard deviation, allocation decision obtained by new models can give reasonable rewards according to personal preferences.


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## Body

## 1. Introduction

In the 1950s, Markowitz [1] proposed the mean-variance optimization (MVO) model. This model is regarded as leading the studies on financial investment. It is also well known as one of classical models in portfolio optimization. The mean-variance portfolio theory is considered as the pioneering work which is a practice tool and a main content of the financial investment theory study. Due to risk-aversion in market, investors hope receiving the maximum return under certain risk, or minimizing the risk to obtain expected profit. In [2], Markowitz presented the typical methodology on MVO model in portfolio selection.In the early 1990s, Morgan presented value-at-risk (VaR) to measure risk. Under the fluctuation of normal markets and a certain level of probability, within a given confidence interval, VaR is the maximum possible loss of the financial assets in the future. But there are some shortcomings of VaR [3]. Later, Rockafeller and Uryasev [4–6] proposed conditional VaR (CVaR) to measure expected loss, which has many good properties and has been used widely.One of the basic problems of portfolio is to study reward-risk optimization model [7–9]. There are two questions: (1) How to deal with the case that the total volatility of reward is not expected too high? (2) How to define the bound of a better selection in terms of return or risk? For the former, the variance should be added to the constraints like MVO model. For the latter, Sharpe presented a kind of models based on MVO model, called Sharpe ratio [10–12]. Considering the above two questions, we will propose a new kind of reward-risk ratio models by introducing both CVaR and standard deviation constraint simultaneously, where standard deviation is considered as the volatility.In most reward-risk portfolio problems, either CVaR or standard deviation would be considered as risk measure in models. CVaR is the extended risk measure of value-at-risk(VaR) that quantifies the average loss over a specified time period of unlikely scenarios beyond the confidence level. However, when we get the same mean reward and maximum loss under several portfolios, how to evaluate which one is better? This is, the smaller the variance, the better the portfolio because the rewards are all near the mean. It follows that it is necessary to study the deviation of the random variables. In addition, CVaR measure actually restricts the fat tail of risk and standard deviation measures its volatility. Combining CVaR with standard deviation can hopefully help investors to obtain both high and stable rewards. So both CVaR and standard deviation are considered in this paper. In fact, CVaR measure represents the predicated maximum loss which is not a positive functional. Commonly, the measurement function is convex, so we set up a risk measure containing a sum of CVaR and standard deviation. The risk measure could be assumed strictly positive in Proposition3 and Proposition 3. In this way, the proposed optimization models are computationally tractable.However, many practical problems have uncertain information. For example, Ayse Ozmen et al. [13,14] studied robust versions of multivariate adaptive regression spline (Mars) under polyhedral uncertainty. For portfolio optimization under data uncertainty, Jörg Fliege and Ralf Werner [15] investigated and illustrated the treatment to the uncertainty by a robust multiobjective formulation. Guray Kara et al. [16] used robust conditional value-at-risk (RCVaR) under parallelepiped uncertainty to find more robust portfolio allocation and reduce the risk. In addition, more and more theories and methods with incomplete messages have been studied recently in various applications (see [17–29] for example). In this paper, we also focus on a kind of robust models under some uncertainty set, which refer to robust ratio models with CVaR and standard deviation.This paper is outlined as follows. In next section, some incomplete distributions are introduced and then robust reward-risk models with CVaR and standard deviation under those incomplete distributions are proposed. According to the properties of robust optimization problems, the equivalent forms of ratio optimization models are proposed in section3. Next, we prove the ratio optimization models with the ellipsoidal uncertainty constraint set can be rewritten as tractable second-order conic problems. Finally, we validate new models are robust and comparable by numerical experiments to asset allocation problems.


## 2. Models of Robust Reward-Risk Optimization

Firstly, some incomplete distribution sets are introduced and then two new robust models in general are proposed. Letx∈X⊆Rn be the decision variable, β∈Rn be the random variable of rewards. Since considering to involve standard deviation into the proposed models, we denote random vector σβ as standard deviation of β. Suppose that distributions of σβ and β are partially given. In other words, the density pσβ only lies in a given set of probability distribution P, namely, pσβ∈P.


### 2.1. Typical Incomplete Distributions and Robust Reward-Risk Definition

We first introduce some typical distribution sets so as to define the incomplete distribution sets related toβ and σβ.Definition 1.
(mixture distribution [30]). Denote(1)PM=∑i=1lγipi∗:∑i=1lγi=1,γi≥0,i∈L=1,…,lwhere pi∗ and l are the i -th known distribution function and total number of distributions, respectively. PM is called a mixture distribution.Definition 2.
(discrete distribution). For thei -th known distribution, let βi be the sample space and σβi be its standard deviation, then denote a discrete distribution set as(2)PD=β1,σβ1,β2,σβ2,…,βm,σβm,with Prβi,σβi=πi and ∑i=1mπi=1, πi≥0i=1,…,m.
Throughout this article, we will focus on one type of discrete distribution, ellipsoidal uncertainty set. It can be described as follows:(3)π∈PπE=π:π=π0+Aη,π0+Aη≥0,eTAη=0,η≤1,where η=ηTη, π0>0 is a known distribution and can be regarded as the center of the ellipsoid, A∈Rm×n is a scaling matrix of the ellipsoid, and e represents the vector with all ones. The constraints eTAη=0 and π0+Aη≥0 guarantee that π satisfies the properties of probability distribution. For more details. refer to [8].
Now we introduce the definition of robust reward and robust risk. The definition of robust reward is the same as the general ones in [31]. Compared with robust risk measure in paper [31], we present a special robust risk measure with respect to both standard deviation and CVaR.Definition 3.
DenoteX=Xx,β,σβ as a variable with respect to x, random variable β, and standard deviation σβ. Given a distribution p of β and σβ, the corresponding reward measure, CVaR measure, and standard deviation measure are denoted as μp, CVaRp, and Dp, respectively. Let P be a given set of distributions where p lies in. Then robust reward μWx and robust risk measured by CVaR and standard deviation can be denoted as(4)μWx≡infp∈PμpX,CVaRx+DxW≡supp∈PCVaRpX+DpX,where the superscript W stands for “worst case”.


### 2.2. Common Models with CVaR and Standard Deviation

In order to reduce volatility and minimize risk value, motivated by the classical robust reward-risk model and MVO model, we present two new optimization models by combining standard deviation and CVaR in following two ways. LetS∗ and S∗ be two positive constants, one way is to maximize robust reward with a constraint on robust CVaR and standard deviation and the other way is to minimize robust risk function related to robust CVaR and standard deviation with a robust reward constraint.Model 1.
Maximize the robust reward:(5)maxxμWxs.t.CVaRx+DxW≤S∗,x∈X.Model 2.
Minimize the robust risk:(6)minxCVaRx+DxWs.t.μWx≥S∗,x∈X.
Here we consider the box decision set:(7)X=x:eTx=1,Lb≤Bx≤Ub,with given lower bound Lb, upper bound Ub and matrix B.
The models protect against the worst-case simultaneously happened in reward and risk measure, which may be a conservative assumption such like models in [30, 32].
In the models above, the constantsS∗ and S∗ should be given at first. But in fact, it may be hard for investors to determine minimum acceptable expected returns, or maximum risk-aversion levels with confidence. So how to choose a suitable level becomes a considerable question. Under this situation, we will propose ratio optimization models to solve portfolio problems efficiently in the next section.


## 2.1. Typical Incomplete Distributions and Robust Reward-Risk Definition

We first introduce some typical distribution sets so as to define the incomplete distribution sets related toβ and σβ.Definition 1.
(mixture distribution [30]). Denote(1)PM=∑i=1lγipi∗:∑i=1lγi=1,γi≥0,i∈L=1,…,lwhere pi∗ and l are the i -th known distribution function and total number of distributions, respectively. PM is called a mixture distribution.Definition 2.
(discrete distribution). For thei -th known distribution, let βi be the sample space and σβi be its standard deviation, then denote a discrete distribution set as(2)PD=β1,σβ1,β2,σβ2,…,βm,σβm,with Prβi,σβi=πi and ∑i=1mπi=1, πi≥0i=1,…,m.
Throughout this article, we will focus on one type of discrete distribution, ellipsoidal uncertainty set. It can be described as follows:(3)π∈PπE=π:π=π0+Aη,π0+Aη≥0,eTAη=0,η≤1,where η=ηTη, π0>0 is a known distribution and can be regarded as the center of the ellipsoid, A∈Rm×n is a scaling matrix of the ellipsoid, and e represents the vector with all ones. The constraints eTAη=0 and π0+Aη≥0 guarantee that π satisfies the properties of probability distribution. For more details. refer to [8].
Now we introduce the definition of robust reward and robust risk. The definition of robust reward is the same as the general ones in [31]. Compared with robust risk measure in paper [31], we present a special robust risk measure with respect to both standard deviation and CVaR.Definition 3.
DenoteX=Xx,β,σβ as a variable with respect to x, random variable β, and standard deviation σβ. Given a distribution p of β and σβ, the corresponding reward measure, CVaR measure, and standard deviation measure are denoted as μp, CVaRp, and Dp, respectively. Let P be a given set of distributions where p lies in. Then robust reward μWx and robust risk measured by CVaR and standard deviation can be denoted as(4)μWx≡infp∈PμpX,CVaRx+DxW≡supp∈PCVaRpX+DpX,where the superscript W stands for “worst case”.


## 2.2. Common Models with CVaR and Standard Deviation

In order to reduce volatility and minimize risk value, motivated by the classical robust reward-risk model and MVO model, we present two new optimization models by combining standard deviation and CVaR in following two ways. LetS∗ and S∗ be two positive constants, one way is to maximize robust reward with a constraint on robust CVaR and standard deviation and the other way is to minimize robust risk function related to robust CVaR and standard deviation with a robust reward constraint.Model 1.
Maximize the robust reward:(5)maxxμWxs.t.CVaRx+DxW≤S∗,x∈X.Model 2.
Minimize the robust risk:(6)minxCVaRx+DxWs.t.μWx≥S∗,x∈X.
Here we consider the box decision set:(7)X=x:eTx=1,Lb≤Bx≤Ub,with given lower bound Lb, upper bound Ub and matrix B.
The models protect against the worst-case simultaneously happened in reward and risk measure, which may be a conservative assumption such like models in [30, 32].
In the models above, the constantsS∗ and S∗ should be given at first. But in fact, it may be hard for investors to determine minimum acceptable expected returns, or maximum risk-aversion levels with confidence. So how to choose a suitable level becomes a considerable question. Under this situation, we will propose ratio optimization models to solve portfolio problems efficiently in the next section.


## 3. New Robust Reward-Risk Ratio Models and Analysis

Referring to the idea in [30], we also consider ratio optimization models to deal with the difficulty on choosing suitable parameters. According to the Sharpe ratio model, two types of ratio optimization models are presented as follows:(8)minx∈XCVaRx+DxWμWx=minx∈Xsupp∈PCVaRpx+Dpxinfp∈Pμpx,(9)maxx∈XμWxCVaRx+DxW=maxx∈Xinfp∈Pμpxsupp∈PCVaRpx+Dpx.The models above should lead to a set of solutions on the efficient frontier, which are then given to the decision-maker to decide from. This is the most important and fundamental principle of Markowitz’ original mean-variance model. Using a ratio objective can solve the problem of determining the correct trade-off between the competing objectives of risk and reward. We take model (8) for example.Proposition 1.
If portfoliox is a solution of model (8) with normal distribution p∈P, then it is robust mean-variance efficient.Proof.
If the optimal solutionx is not robust mean-variance efficient [33], there would be a x^∈X satisfying μpx≤μpx^ and Dpx≥Dpx^, and at least one inequality is strictly true, for example,(10)μpx≤μpx^.
LetF⋅ be distribution function of standard normal and f⋅ be density of standard normal distribution, we have [34](11)CVaRp=TDp−μp,where T=fF−1α/1−α, in which α is a confidence level. We have(12)CVaRpx+Dpx>CVaRpx^+Dpx^.
From (10) and (12), we have(13)μWx<μWx^,CVaRx+DxW>CVaRx^+Dx^W.
Hence,(14)μWxCVaRαx+DxW<μWx^CVaRαx^+Dx^W,which contradict the assumption of x. So the optimal solution of model (8) is robust efficient.
Therefore, models (8) and (9) are reasonable in economic meanings. Moreover they should be solvable at optimization point of view. For this purpose, we first give an assumption on reward, CVaR, and standard deviation in terms of robust optimization theorem.Assumption 1.
Letμp⋅, CVaRp⋅, and Dp⋅ define measures of the general reward, CVaR, and standard deviation with incomplete distribution p, respectively. Suppose measures μp⋅ and CVaRp⋅ satisfy:(1)
μp⋅ is a strictly positive functional and satisfies positive homogeneity and concavity:(15)μptX=tμpX,∀t>0μptX1+1−tX2≥tμpX1+1−tμpX2,∀t∈0,1.(2)
CVaR measure is a functional with positive homogeneity and subadditivity:(16)CVaRptX=tCVaRpX,∀t>0,CVaRpX1+X2≤CVaRpX1+CVaRpX2.
Suppose rewards of assets are approximate linear correlation in this paper and standard deviation measureDp⋅ satisfies: it is positive homogeneity and subadditivity:(17)DptX=tDpX,∀t>0,DpX1+X2≤DpX1+DpX2.
Assumption1 is easy satisfied for many reward and risk measures. We can choose the expectation of reward μX=ExTβ as a reward and the classic CVaR measure as a risk measure function, where β denotes the return vector of assets. Denote Γβ=AσβATσβ as the covariance matrix and the standard deviation as DX=xTAσβ. We have (18)DpX1+X2=x1+x2TApσβ≤x1TApσβ+x2TApσβ=DpX1+DpX2,DptX=txTApσβ=txTApσβ=tDpX,∀t>0.
Throughout the paper, the distributionP is assumed to be convex and compact. Thus, the robust measures retain inherent characteristic of return, CVaR and standard deviation measures.Proposition 2.
Suppose that the reward, standard deviation, and CVaR measures with convex and compact distribution setP satisfy Assumption 1, then μW and CVaR+DW satisfy(19)μWtx=tμWx,t>0,CVaRtx+DtxW=tCVaRx+DxW,t>0.Proof.
Since compactness ofP, we can replace “inf” and “sup” with “min” and “max,” respectively. And the conclusion of μW can refer to Proposition 1 in [31]. In a similar way, for robust risk, with assumption 1, we have(20)CVaRtx+DtxW=supp∈PCVaRptX+DptX=maxp∈PtCVaRpX+tDpX=tmaxp∈PCVaRpX+DpX=tVaRx+DxW.
DenoteR++ as all real numbers α such that α>0 and X as the convex decision set in (6). Then there are some convex and concave properties of worst-case ratio functions as follows.Proposition 3.
Letμp:X⊂Rn⟶R++ and CVaRp+Dp:X⊂Rn⟶R++ with convex and compact distribution set P where p∈P. Assuming the reward, CVaR, and standard deviation measures satisfy Assumption 1, then μW is concave and CVaR+DW is convex. Further, the following two ratio functions(21)CVaRx+DxW/μWx:X⟶R++,μWx/CVaRx+DxW:X⟶R++,are quasiconcave and quasiconvex, respectively. In addition, the optimal decisions of two ratios are the same:(22)argmaxx∈XμWxCVaRx+DxW=argminx∈XCVaRx+DxWμWx.Proof.
We can easily obtain the concavity ofμp and the convexity of CVaRp+Dp with uncertainty distribution p from Assumption 1. Thus, the concavity of μW and convexity of CVaR+DW are obtained similarly from Proposition 1 in [31]. The following conclusion on two ratios can be obviously derived referring to Proposition 2 in [35].
Under the above assumption and propositions, we can reformulate ratio model (8) as follows.Proposition 4.
Suppose the reward measureμp, CVaR measure CVaRp, and standard deviation measure Dp satisfy Assumption 1 for an incomplete distribution p∈P, where P is a convex and compact distribution set. Then the robust optimization model (8) is equivalent to(23)maxw,tμWws.t.CVaRw+DwW≤1,eTw=t,tLb≤Bw≤tUb,t≥0.
Ifw∗,t∗ is a solution of model (23), then x∗=w∗/t∗ solves model (8), and μWw∗=μWx∗/CVaRx∗+Dx∗W. Conversely, if x∗ solves model (8), denote t∗=1/CVaRx∗+Dx∗W, then the pair t∗x∗,t∗ is a solution of model (23).Proof.
The proof proceeds in two stages. First, we show that model (8) is equivalent to(24)maxw,tμWws.t.CVaRw+DwW=1,eTw=t,tLb≤Bw≤tUb,t≥0.
Denotet=1/CVaRx+DxW>0. For model (8), suppose x is a feasible point, in terms of the positive homogeneity of μp, CVaRp, and Dp, we have that w,t is feasible for model (24) where w=tx. Hence,(25)maxx∈μWxCVaRx+DxW≤maxw,t∈X1μWw,where X and X1 are feasible sets for models (8) and (24), respectively. Conversely, assume w,t is feasible for (24), and it is easy to know x=w/t is feasible for (8) and(26)maxx∈XμWxCVaRx+DxW≥maxw,t∈X1μWw
Hence,(27)maxx∈XμWxCVaRx+DxW≥maxw,t∈X1μWwand if x∗ solves model (8), t∗x∗,t∗ is a solution corresponding to model (24), where t∗=1/CVaRx∗+Dx∗W. Similarly, if w∗,t∗ solves model (24), then x∗=w∗/t∗ is a solution of model (8).
Next, we focus on two problems (24) and (23). In (23), we should know that the constraint on risk measurement has been relaxed. It will be proven that ifw∗,t∗ is a solution of (23), then CVaRw∗+Dw∗W=1. In other words, the first inequality in (23) becomes equality at a solution point. Suppose for contradiction that there is a solution w0,t0 of (23) such that(28)CVaRw0+Dw0W<1.
We choose a constanta satisfying(29)CVaRw0+Dw0W−1≥a>1
In terms of the positive homogeneity in Proposition3, it implies that(30)aCVaRw0+Dw0W=CVaRaw0+Daw0W≤1.
It is obvious thataw0,at0 satisfies all constraints in (23), that is, aw0,at0 is a feasible point of (23).
Considering the objective function of (23) at point the aw0,at0 where a>1, we have(31)μWaw0=aμWw0>μWw0,which contradicts our assumption that w0,t0 is an optimal solution of model (23). So the risk inequality is satisfied by equality at an optimal point.
In all, the conclusion is obtained.
As before, the reward measurement is assumed to be strictly positive. It should be noted that portfolios in decision set could not always make positive rewards. So we suppose that all considerablew in the decision set satisfies μw>0. In addition, the risk measurement is also assumed to be a positive functional. For more details, refer to [35]. For model (32), we can also obtain the following similar conclusion.Proposition 5.
Under the assumption in Proposition3, the robust optimization model (32) is equivalent to(32)minw,tCVaRw+DwWs.t.μWw≥1,eTw=t,tLb≤Bw≤tUb,t≥0.
If the pairw∗,t∗ is a solution of (32), then x∗=w∗/t∗ solves model (9), and CVaRw∗+Dw∗W=CVaRx∗+Dx∗W/μWx∗. Conversely, if x∗ solves model (9), then the pair t∗x∗,t∗ is a solution of model (27), where t∗=1/μWx∗.
We omit the proof of Proposition3 because it is similar to that of Proposition 3. In the next section, we prefer to consider models (18) and (27) from optimal viewpoint since they retain the convex property according to Proposition 3. We will focus on the analysis, reformulation, and solution on two models.


## 4. Reformations of Models under Probability Distribution SetPπE

Models (47) and (27) optimize the worst case of rewards and risk, respectively. As Definition 3, (4) and (27) actually are complicated max-min and min-max problems. In this section, we focus on the models under ellipsoidal probability distribution set PπE and reformulate them to tractable forms by optimization techniques.Firstly, as in [31], the reward and CVaR function are specialized as follows:(33)μX=μxTβ=xTEβ,CVaRX=minzz+11−α∑k=1mπkfx,βk−z+,where fx,β is the loss function and α denotes confidence level. Here, we suppose the rewards of assets are approximate linear correlation with correlate coefficient θ=1, and then the covariance matrix is Γβ=σβσβT and the standard deviation function is(34)DX=xTEσβ.DenoteΞ=β1,β2,…,βm and Σβ=σβ1,σβ2,…,σβm. Each vector βi (or σβi) i=1,…,m has the same dimension with x.Under the ellipsoid uncertainty setPπE, model (23) becomes a specific model as follows:(35)maxw,tminπ∈PπEwTEπβs.t.maxπ∈PπEminzz+11−α∑k=1mπkfw,βk−z++wTEπσβ≤1,which is also called “counterpart” in robust optimization.Let(36)RTw=wTΞ,φTw,u=11−αuT+wTΣβ,we will simplify the objective and constraints to restate the above model (35).Proposition 6.
Max-min model (35) can be reformulated to the following maximization problem:(37)maxw,t,τ,v,δ,θRTwπ0−θ−π0Tτs.t.v+ATτ+ATeδ=ATRw,v≤θ,τ≥0,Φw≤1,tLb≤Bw≤tUb,eTw=t,t≥0,where τ,v,δ,θ∈Rm×Rn×R×R and ε,ω,ζ,ξ∈Rn×Rm×R×R and Φw is the optimal objective value of the minimization problem(38)minz,u,ε,ω,ζ,ξz+φTw,uπ0+ζ+π0Tωs.t.ε≤ζ,ω≥0,−ε−ATω+ATeξ=ATφw,u,uk≥fw,βk−z,uk≥0,k=1,2,…,m.Proof.
First, we consider the robust objective function. It follows fromRTw=wTΞ and π=π0+Aη that(39)wTEπβ=wTΞπ=wTΞπ0+Aη=RTwπ0+RTwAη.
By substituting (39) into minπ∈PπEwTEπβ, the problem minπ∈PπEwTEπβ becomes(40)minηRTwπ0+RTwAηs.t.eTAη=0,π0+Aη≥0,η≤1.
From conic duality theorem [36], (40) can be rewritten as its dual form(41)maxτ,v,δ,θRTwπ0−θ−π0Tτs.t.v+ATτ+ATeδ=ATRw,v≤θ,τ≥0,where τ,v,δ,θ∈Rm×Rn×R×R. Obviously, the primal problem (40) is bounded below and strictly feasible. By dual theorem, the dual problem (41) is solvable and has the same optimal objective value with (40).
Second, we focus on the max-min constraint function(42)maxπ∈PπEminzz+11−α∑k=1mπkfw,βk−z++wTEπσβ=minzmaxπ∈PπEz+11−α∑k=1mπkfw,βk−z++wTEπσβ=minz,u∈Umaxπ∈PπEz+11−αuTπ+wTΣβπ,where U=u∈Rm:uk≥fw,βk−z,uk≥0,k=1,2,…,m, and the first “ = ” is founded because z+1/1−α∑k=1mπkfw,βk−z++wTEπσβ is convex in z for any given π and concave in π for any given z referred to Lemma 1 in [37].
It follows from denotationsφTw,u=1/1−αuT+wTΣβ and π=π0+Aη that(43)z+11−αuTπ+wTΣβπ=z+11−αuTπ0+Aη+wTΣβπ0+Aη=z+φTw,uπ0+φTw,uAη.
Hence, the problemmaxπ∈PπEz+1/1−αuTπ+wTΣβπ can be written as(44)maxηz+φTw,uπ0+φTw,uAηs.t.eTAη=0,π0+Aη≥0,η≤1.
Similarly to the proof in the first part, we have that dual problem (45) of (44)(45)minε,ω,ζ,ξz+φTw,uπ0+ζ+π0Tωs.t.−ε−ATω+ATeξ=ATφw,u,ε≤ζ,ω≥0where ε,ω,ζ,ξ∈Rn×Rm×R×R is solvable and the two problems have the same optimal objective values.
By substituting (41) and (45) into model (35), model (37) is obtained.
Note that constraints in model (37) contain a subproblem which also has a very complex form. Now continue to simplify the model. Denote(46)Q=w,t,τ,v,δ,θ,z,u,ζ,ε,ω,ξ,q=w,t,τ,v,δ,θ.
We can get the following model:(47)maxQRTwπ0−θ−π0Tτs.t.v+ATτ+ATeδ=ATRw,z+φTw,uπ0+ζ+π0Tω≤1,−ε−ATω+ATeξ=ATφw,u,‖ε‖≤ζ,ω≥0,‖v‖≤θ,τ≥0,uk≥fw,βk−z,uk≥0,k=1,2,…,m,tLb≤Bw≤tUb,eTw=t,t≥0,where τ,v,δ,θ∈Rm×Rn×R×R, ε,ω,ζ,ξ∈Rn×Rm×R×R. Obviously, it is easier to solve simple mathematical programming problem (47) than model (37). In fact, model (47) is a second-order cone programming (SOCP) with totally 3m+3n+6 variables. So we give the following proposition and its proof referring to Proposition 4.1 in [8].Proposition 7.
Assume that the random variableβ and the standard deviation σβ obey the discrete distribution with ellipsoidal uncertainty set PπE. If Q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗,z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ is a solution of model (47), then q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ solves model (37). Conversely, if q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ is a solution of model (37), then Q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗,z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ solves model (47), where z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ is a solution of the problem (38) with w=w∗.Proof.
LetQ∗=w∗,t∗,τ∗,v∗,δ∗,θ∗,z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ be a solution of model (47), we will prove that q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ can solve model (37). First, the feasibility of q∗=w∗,t∗,τ∗, v∗,δ∗,θ∗ is proved as follows.
SinceQ∗=w∗,t∗,τ∗,v∗,δ∗,θ∗,z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ is a solution of model (47), we have(48)z∗+φTw∗,u∗π0+ζ∗+π0Tω∗≤1,−ε∗−ATω∗+ATeξ∗=ATφw∗,u∗,ε∗≤ζ∗,ω∗≥0,uk∗≥fw∗,βk−z∗,uk∗≥0,k=1,2,…,m.
Hence,Φw∗≤1. Combined with other constraints in (47), it is easily to know q∗ is feasible for model (11).
Next, we prove thatq∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ is an optimal solution of (37). If q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ is not an optimal solution, suppose for contradiction that there exists the other optimal solution q^=w^,t^,τ^,v^,δ^,θ^. Let ϕw,τ,θ=RTwπ0−θ−π0Tτ, then(49)ϕw^,τ^,θ^>ϕw∗,τ∗,θ∗.
So, there is a solution of the problem (38) with w=w^ denoted as z^,u^,ζ^,ε^,ω^,ξ^ such that Q^=w^,t^,τ^,v^,δ^,θ^,z^,u^,ζ^,ε^,ω^,ξ^ is feasible to model (47). Because Q∗ solves model (47), we have that ϕw^,τ^,θ^≤ϕw∗,τ∗,θ∗, which contradicts (49). So q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ is an optimal point solving (37).
Conversely, letq∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ be a solution of model (37), we need to prove Q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗,z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ is a solution of model (47), where z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ is a solution of the problem (38) with w=w∗.
IfQ∗=w∗,t∗,τ∗,v∗,δ∗,θ∗,z∗,u∗,ζ∗,ε∗,ω∗,ξ∗ is not a solution of model (47), suppose for contradiction that there is another solution Q0=w0,t0,τ0,v0,δ0,θ0,z0,u0,ζ0,ε0,ω0,ξ0. From the proof of the first part in this proposition, we know that w0,t0,τ0,v0,δ0,θ0 is a solution of model (11) and ϕw0,τ0,θ0>ϕw∗,τ∗,θ∗, which contradicts that q∗=w∗,t∗,τ∗,v∗,δ∗,θ∗ is a solution of model (37).
Hence, the conclusion is obtained.
The analysis to model (32) can be made in the same way. We give the following propositions, and their proofs are omitted.Proposition 8.
Model (32) under ellipsoid uncertainty set PπE can be reformulated to(50)minw,t,z,u,ε,γ,ϖ,κs.t.z+φTw,uπ0+ε+π0TϖΨw≥1,−γ−ATϖ+ATeκ=ATφw,u,γ≤ε,ϖ≥0,uk≥fw,βk−z,uk≥0,k=1,2,…,m,where m,j,s,l∈Rm×R×Rn×R, γ,ϖ,κ,ε∈Rn×Rm×R×R and Ψw is the maximum value of the problem:(51)maxm,s,q,lRTwπ0−l−π0Tms.t.s≤l,m≥0,s+ATm+ATeq=ATRw.Proposition 9.
Assume that the random variableβ and the standard deviation σβ obey the discrete distribution with ellipsoidal uncertainty set PπE. If H∗=w∗,t∗,z∗,u∗,ε∗,γ∗,ϖ∗,κ∗,m∗,s∗,j∗,l∗ is a solution of(52)minw,t,z,u,ε,γ,ϖ,κ,m,s,j,lz+φTw,uπ0+ε+π0Tϖs.t.RTwπ0−l−π0Tm≥1,s+ATm+ATej=ATRw,−γ−ATϖ+ATeκ=ATφw,u,then h∗=w∗,t∗,z∗,u∗,ε∗,γ∗,ϖ∗,κ∗ solves model (50), where m,j,s,l∈Rm×R×Rn×R, γ,ϖ,κ,ε∈Rn×Rm×R×R. Conversely, if h∗=w∗,t∗,z∗,u∗,ε∗,γ∗,ϖ∗,κ∗ is a solution of model (50), then H∗=w∗,t∗,z∗,u∗,ε∗,γ∗,ϖ∗,κ∗,m∗,s∗,j∗,l∗ solves model (52), where m∗,s∗,j∗,l∗ is a solution of the problem (51) with w=w∗.
Letfw,βk be linear on w, we should know it is actually a second-order cone programming with 3m+3n+6 variables as model (47). Thus, it can be solved efficiently by many methods, for example, interior point methods. There also exist many efficient solvers even for large-scale problems (including MOSEK, SeDuMi, SDPT3, GUROBI, CPLEX).


## 5. Application in Allocation of Assets

In this section, we apply the new robust reward-risk ratio model (47) with CVaR and standard deviation to solve portfolio problems. Firstly, we test on the power market to indicate the robustness of our new model compared with deterministic models. Secondly, we conduct two experiments on real data of power market and financial assets to compare with the popular Mean-CVaR ratio (also known as STARR) model and Sharpe ratio model.


### 5.1. Verification on Robustness of the New Model

#### 5.1.1. Experimental Data and Parameters

In the portfolio problem, the assets indicate the electric energy. For testing the model, the experimental data are given as follows. Ellipsoidal uncertaintyPπE in (3) is chosen as the probability distribution set, and denote(53)fw,βk=0.1wT5βk−e,where βk is the reward expectation of β at the k th period. The known parameters in (47) are listed as follows:(54)Lb=0,0T,Ub=15,20T,π0=0.2,0.5,0.3T,α=0.95,A=374614,B=224526,The history data referring to [30] is as follows:(1)
Two types trade data in the power market: spot and future markets(2)
The periods of reward sample: 3 periods(3)
The size of reward sample points of each contract: 1000 sample points(4)
The possible distribution: normal distribution for each market(5)
Spot contract: 200 samples in the first period have mean reward $37.52/MWh and standard deviation 10.86; the second 500 ones with the mean reward $33.96/MWh and standard deviation 11.35; the rest 300 ones satisfy that the mean is $35.01/MWh and standard deviation is 10.24(6)
Future contract: the first 200 samples with mean reward $3.57/MWh and standard deviation 1.53; the second 500 ones satisfy with the mean $3.79/MWh and standard deviation 1.24; the rest 300 ones satisfy that the mean is $3.38/MWh and standard deviation is 1.89


#### 5.1.2. Results Analysis

First, we choose the following different scaling matrices:(55)A1=3k7k4614,A2=374k6k14,k∈1,5,to show different results on model (47). From the results (Figure 1) responding to A1 and A2, we can clearly see that the different scaling matrices A in the ellipsoidal uncertainty set will obtain different rewards. Specifically, (1) when the first row of A1 becomes larger, the rewards obtained by solving model (47) becomes lower and lower but higher than $14.9/MWh; (2) when the second row of the scaling matrix A2 becomes larger, the reward curve first decreases and then increases. Numerical results indicate that a proper probability distribution set is important to obtain higher reward.Figure 1
Rewards over differentA in model (47). (a) Rewards over scaling matrices A1. (b) Rewards over scaling matrices A2.
(a)(b)Second, to compare with deterministic reward-risk ratio model, that is the caseη=0 in model (35). The optimal reward obtained from the deterministic reward-risk ratio model is $23.08/MWh, which is larger than the worst-case optimal reward $15.06/MWh (when k=1, see Figure 1), obtained from the robust model (47). This indicates that the new model is robust.


### 5.2. Comparison with Mean-CVaR Ratio Model and Sharpe Ratio Model

In this section, we compare our model (47) with two conventional robust reward-risk models.Robust Mean-CVaR ratio model [30]: refer to proposition (8), the initial robust Mean-CVaR ratio model can be equivalent to(56)maxw,tμWwTβ−trbs.t.CVaRWwTβ−trb≤1,eTw=t,tLb≤Bw≤tUb,t≥0,in terms of two different kinds of history rewards. Model (53) with the risk-free reward rb only considers CVaR constraint. To compare two models fairly, the risk-free reward rb is set to 0 in (56), that is, we replace model (53) by the following model with the ellipsoid uncertainty set PπE in (3).(57)maxw,tminπ∈PπEμπwTβs.t.maxπ∈PπECVaRπwTβ≤1,eTw=t,tLb≤Bw≤tUb,t≥0.We should note that (57) is actually a special model of robust STARR model.Robust Sharpe ratio model [12]: refer to Proposition (8), the robust Sharpe ratio model can be written as(58)maxw,tminπ∈PπEμπwTβs.t.maxπ∈PπEDπwTβ≤1,eTw=t,tLb≤Bw≤tUb,t≥0.


#### 5.2.1. Test on Trade Data in the Power Market

The three models are applied to two types history trade data in power market in Section5.1.1. The allocations of assets obtained by solving robust Mean-CVaR ratio model (57), Robust Sharpe ratio model (58), and our model (47) are (0.286, 0.714), (0.611, 0.389), and (0.370, 0.630), respectively.Now we focus on some rewards on spot and future markets in the future (45 transactions), respectively, as shown in Figure2. In terms of the three portfolios above, we compute the corresponding expected rewards in the future 45 transactions. Figure 3 plots the expected rewards based on model (57), (58), and our model (47).Figure 2
Rewards of future 45 transactions on two markets.Figure 3
The expected rewards obtained by three models (57), (58), and (47).We see from Figure3 that rewards under the optimal portfolio in our model (47) are better than ones in the reward-risk ratio model (57), but less than Sharpe ratio model (58). In fact, robust Sharpe ratio model gives higher rewards due to the data of spot contract have more lager mean return and Sharpe ratio. However, the standard deviation of (57), (58), and (47) is 0.8497, 2.8182, and 1.2231, respectively. From Figure 3, it is obvious that the results of Sharpe ratio model have more larger volatility. In other words, our portfolio can result in relatively higher rewards with acceptable volatility, which is more attractive to investors. So our model (47) is a preferable choice in certain cases when the investors would pursue better rewards within a reasonable volatility level.In addition, we note that the historical rewards in two markets in Section5.1.1 differ apparently in magnitude. In other words, the rewards in future market are much lower than the ones in spot market. In this case, even though spot contract possess highly volatile rewards, investors would most likely invest heavily in this market. This is the other explanation of Figure 3.


#### 5.2.2. Test on Portfolios of the Two Hang Seng Index

In contrary, we consider the following case without too large difference of returns in magnitude. Referring to [28,31], we test three models (57), (58), and (47) on portfolios of the two Hang Seng Index. We consider the day returns of these assets. 1000 samples of returns of the two assets are collected from the time period (2017.01.03–2021.01.22).(1)
HSNF: the numbers of samples in three periods are 200, 500 and 300, respectively. Their mean returns and standard deviations are (1.1989, 8.1898), (0.0878, 11.5249), and (0.1231, 14.2933), respectively.(2)
HSNU: the same numbers of samples in three periods with HSNF. Their mean returns and standard deviations are (0.4854, 6.1943), (0.0898, 7.7243), and (0.06677, 12.2633), respectively.(3)
The other information is the same as that in Section5.1.1.Based on these messages, the allocation solved by robust Mean-CVaR ratio model (57) is (0.676, 0.323) and Sharpe ratio model (58) and our model (47) give the same allocation (0.416, 0.584). We consider returns on HSNF and HSNU in the next 45 days as shown in Figure 4, compute the corresponding expected returns with respective allocations and plot them in Figure 5.Figure 4
Returns of future 45 days on two financial assets.Figure 5
The expected returns of three models in future 45 days.As shown, both of the mean returns on the two allocations fluctuate around 0.7. However, the volatility of returns corresponding to model (57) which only involves CVaR is apparently larger. On the other hand, the worst expected return with our portfolio is lightly higher than that with the allocation solved by model (57). It should be noted that samples in this test have similar mean returns and standard deviations. With the known parameters in Section 5.1.1, the new model (47) obtains the same results with robust Sharpe ratio model (58). In all, for conservative investors, the portfolio solved by our model would be a better choice resulting in an acceptable reward with a smaller volatility because of the advantage of involving standard deviation.


## 5.1. Verification on Robustness of the New Model

### 5.1.1. Experimental Data and Parameters

In the portfolio problem, the assets indicate the electric energy. For testing the model, the experimental data are given as follows. Ellipsoidal uncertaintyPπE in (3) is chosen as the probability distribution set, and denote(53)fw,βk=0.1wT5βk−e,where βk is the reward expectation of β at the k th period. The known parameters in (47) are listed as follows:(54)Lb=0,0T,Ub=15,20T,π0=0.2,0.5,0.3T,α=0.95,A=374614,B=224526,The history data referring to [30] is as follows:(1)
Two types trade data in the power market: spot and future markets(2)
The periods of reward sample: 3 periods(3)
The size of reward sample points of each contract: 1000 sample points(4)
The possible distribution: normal distribution for each market(5)
Spot contract: 200 samples in the first period have mean reward $37.52/MWh and standard deviation 10.86; the second 500 ones with the mean reward $33.96/MWh and standard deviation 11.35; the rest 300 ones satisfy that the mean is $35.01/MWh and standard deviation is 10.24(6)
Future contract: the first 200 samples with mean reward $3.57/MWh and standard deviation 1.53; the second 500 ones satisfy with the mean $3.79/MWh and standard deviation 1.24; the rest 300 ones satisfy that the mean is $3.38/MWh and standard deviation is 1.89


### 5.1.2. Results Analysis

First, we choose the following different scaling matrices:(55)A1=3k7k4614,A2=374k6k14,k∈1,5,to show different results on model (47). From the results (Figure 1) responding to A1 and A2, we can clearly see that the different scaling matrices A in the ellipsoidal uncertainty set will obtain different rewards. Specifically, (1) when the first row of A1 becomes larger, the rewards obtained by solving model (47) becomes lower and lower but higher than $14.9/MWh; (2) when the second row of the scaling matrix A2 becomes larger, the reward curve first decreases and then increases. Numerical results indicate that a proper probability distribution set is important to obtain higher reward.Figure 1
Rewards over differentA in model (47). (a) Rewards over scaling matrices A1. (b) Rewards over scaling matrices A2.
(a)(b)Second, to compare with deterministic reward-risk ratio model, that is the caseη=0 in model (35). The optimal reward obtained from the deterministic reward-risk ratio model is $23.08/MWh, which is larger than the worst-case optimal reward $15.06/MWh (when k=1, see Figure 1), obtained from the robust model (47). This indicates that the new model is robust.


## 5.1.1. Experimental Data and Parameters

In the portfolio problem, the assets indicate the electric energy. For testing the model, the experimental data are given as follows. Ellipsoidal uncertaintyPπE in (3) is chosen as the probability distribution set, and denote(53)fw,βk=0.1wT5βk−e,where βk is the reward expectation of β at the k th period. The known parameters in (47) are listed as follows:(54)Lb=0,0T,Ub=15,20T,π0=0.2,0.5,0.3T,α=0.95,A=374614,B=224526,The history data referring to [30] is as follows:(1)
Two types trade data in the power market: spot and future markets(2)
The periods of reward sample: 3 periods(3)
The size of reward sample points of each contract: 1000 sample points(4)
The possible distribution: normal distribution for each market(5)
Spot contract: 200 samples in the first period have mean reward $37.52/MWh and standard deviation 10.86; the second 500 ones with the mean reward $33.96/MWh and standard deviation 11.35; the rest 300 ones satisfy that the mean is $35.01/MWh and standard deviation is 10.24(6)
Future contract: the first 200 samples with mean reward $3.57/MWh and standard deviation 1.53; the second 500 ones satisfy with the mean $3.79/MWh and standard deviation 1.24; the rest 300 ones satisfy that the mean is $3.38/MWh and standard deviation is 1.89


## 5.1.2. Results Analysis

First, we choose the following different scaling matrices:(55)A1=3k7k4614,A2=374k6k14,k∈1,5,to show different results on model (47). From the results (Figure 1) responding to A1 and A2, we can clearly see that the different scaling matrices A in the ellipsoidal uncertainty set will obtain different rewards. Specifically, (1) when the first row of A1 becomes larger, the rewards obtained by solving model (47) becomes lower and lower but higher than $14.9/MWh; (2) when the second row of the scaling matrix A2 becomes larger, the reward curve first decreases and then increases. Numerical results indicate that a proper probability distribution set is important to obtain higher reward.Figure 1
Rewards over differentA in model (47). (a) Rewards over scaling matrices A1. (b) Rewards over scaling matrices A2.
(a)(b)Second, to compare with deterministic reward-risk ratio model, that is the caseη=0 in model (35). The optimal reward obtained from the deterministic reward-risk ratio model is $23.08/MWh, which is larger than the worst-case optimal reward $15.06/MWh (when k=1, see Figure 1), obtained from the robust model (47). This indicates that the new model is robust.


## 5.2. Comparison with Mean-CVaR Ratio Model and Sharpe Ratio Model

In this section, we compare our model (47) with two conventional robust reward-risk models.Robust Mean-CVaR ratio model [30]: refer to proposition (8), the initial robust Mean-CVaR ratio model can be equivalent to(56)maxw,tμWwTβ−trbs.t.CVaRWwTβ−trb≤1,eTw=t,tLb≤Bw≤tUb,t≥0,in terms of two different kinds of history rewards. Model (53) with the risk-free reward rb only considers CVaR constraint. To compare two models fairly, the risk-free reward rb is set to 0 in (56), that is, we replace model (53) by the following model with the ellipsoid uncertainty set PπE in (3).(57)maxw,tminπ∈PπEμπwTβs.t.maxπ∈PπECVaRπwTβ≤1,eTw=t,tLb≤Bw≤tUb,t≥0.We should note that (57) is actually a special model of robust STARR model.Robust Sharpe ratio model [12]: refer to Proposition (8), the robust Sharpe ratio model can be written as(58)maxw,tminπ∈PπEμπwTβs.t.maxπ∈PπEDπwTβ≤1,eTw=t,tLb≤Bw≤tUb,t≥0.


### 5.2.1. Test on Trade Data in the Power Market

The three models are applied to two types history trade data in power market in Section5.1.1. The allocations of assets obtained by solving robust Mean-CVaR ratio model (57), Robust Sharpe ratio model (58), and our model (47) are (0.286, 0.714), (0.611, 0.389), and (0.370, 0.630), respectively.Now we focus on some rewards on spot and future markets in the future (45 transactions), respectively, as shown in Figure2. In terms of the three portfolios above, we compute the corresponding expected rewards in the future 45 transactions. Figure 3 plots the expected rewards based on model (57), (58), and our model (47).Figure 2
Rewards of future 45 transactions on two markets.Figure 3
The expected rewards obtained by three models (57), (58), and (47).We see from Figure3 that rewards under the optimal portfolio in our model (47) are better than ones in the reward-risk ratio model (57), but less than Sharpe ratio model (58). In fact, robust Sharpe ratio model gives higher rewards due to the data of spot contract have more lager mean return and Sharpe ratio. However, the standard deviation of (57), (58), and (47) is 0.8497, 2.8182, and 1.2231, respectively. From Figure 3, it is obvious that the results of Sharpe ratio model have more larger volatility. In other words, our portfolio can result in relatively higher rewards with acceptable volatility, which is more attractive to investors. So our model (47) is a preferable choice in certain cases when the investors would pursue better rewards within a reasonable volatility level.In addition, we note that the historical rewards in two markets in Section5.1.1 differ apparently in magnitude. In other words, the rewards in future market are much lower than the ones in spot market. In this case, even though spot contract possess highly volatile rewards, investors would most likely invest heavily in this market. This is the other explanation of Figure 3.


### 5.2.2. Test on Portfolios of the Two Hang Seng Index

In contrary, we consider the following case without too large difference of returns in magnitude. Referring to [28,31], we test three models (57), (58), and (47) on portfolios of the two Hang Seng Index. We consider the day returns of these assets. 1000 samples of returns of the two assets are collected from the time period (2017.01.03–2021.01.22).(1)
HSNF: the numbers of samples in three periods are 200, 500 and 300, respectively. Their mean returns and standard deviations are (1.1989, 8.1898), (0.0878, 11.5249), and (0.1231, 14.2933), respectively.(2)
HSNU: the same numbers of samples in three periods with HSNF. Their mean returns and standard deviations are (0.4854, 6.1943), (0.0898, 7.7243), and (0.06677, 12.2633), respectively.(3)
The other information is the same as that in Section5.1.1.Based on these messages, the allocation solved by robust Mean-CVaR ratio model (57) is (0.676, 0.323) and Sharpe ratio model (58) and our model (47) give the same allocation (0.416, 0.584). We consider returns on HSNF and HSNU in the next 45 days as shown in Figure 4, compute the corresponding expected returns with respective allocations and plot them in Figure 5.Figure 4
Returns of future 45 days on two financial assets.Figure 5
The expected returns of three models in future 45 days.As shown, both of the mean returns on the two allocations fluctuate around 0.7. However, the volatility of returns corresponding to model (57) which only involves CVaR is apparently larger. On the other hand, the worst expected return with our portfolio is lightly higher than that with the allocation solved by model (57). It should be noted that samples in this test have similar mean returns and standard deviations. With the known parameters in Section 5.1.1, the new model (47) obtains the same results with robust Sharpe ratio model (58). In all, for conservative investors, the portfolio solved by our model would be a better choice resulting in an acceptable reward with a smaller volatility because of the advantage of involving standard deviation.


## 5.2.1. Test on Trade Data in the Power Market

The three models are applied to two types history trade data in power market in Section5.1.1. The allocations of assets obtained by solving robust Mean-CVaR ratio model (57), Robust Sharpe ratio model (58), and our model (47) are (0.286, 0.714), (0.611, 0.389), and (0.370, 0.630), respectively.Now we focus on some rewards on spot and future markets in the future (45 transactions), respectively, as shown in Figure2. In terms of the three portfolios above, we compute the corresponding expected rewards in the future 45 transactions. Figure 3 plots the expected rewards based on model (57), (58), and our model (47).Figure 2
Rewards of future 45 transactions on two markets.Figure 3
The expected rewards obtained by three models (57), (58), and (47).We see from Figure3 that rewards under the optimal portfolio in our model (47) are better than ones in the reward-risk ratio model (57), but less than Sharpe ratio model (58). In fact, robust Sharpe ratio model gives higher rewards due to the data of spot contract have more lager mean return and Sharpe ratio. However, the standard deviation of (57), (58), and (47) is 0.8497, 2.8182, and 1.2231, respectively. From Figure 3, it is obvious that the results of Sharpe ratio model have more larger volatility. In other words, our portfolio can result in relatively higher rewards with acceptable volatility, which is more attractive to investors. So our model (47) is a preferable choice in certain cases when the investors would pursue better rewards within a reasonable volatility level.In addition, we note that the historical rewards in two markets in Section5.1.1 differ apparently in magnitude. In other words, the rewards in future market are much lower than the ones in spot market. In this case, even though spot contract possess highly volatile rewards, investors would most likely invest heavily in this market. This is the other explanation of Figure 3.


## 5.2.2. Test on Portfolios of the Two Hang Seng Index

In contrary, we consider the following case without too large difference of returns in magnitude. Referring to [28,31], we test three models (57), (58), and (47) on portfolios of the two Hang Seng Index. We consider the day returns of these assets. 1000 samples of returns of the two assets are collected from the time period (2017.01.03–2021.01.22).(1)
HSNF: the numbers of samples in three periods are 200, 500 and 300, respectively. Their mean returns and standard deviations are (1.1989, 8.1898), (0.0878, 11.5249), and (0.1231, 14.2933), respectively.(2)
HSNU: the same numbers of samples in three periods with HSNF. Their mean returns and standard deviations are (0.4854, 6.1943), (0.0898, 7.7243), and (0.06677, 12.2633), respectively.(3)
The other information is the same as that in Section5.1.1.Based on these messages, the allocation solved by robust Mean-CVaR ratio model (57) is (0.676, 0.323) and Sharpe ratio model (58) and our model (47) give the same allocation (0.416, 0.584). We consider returns on HSNF and HSNU in the next 45 days as shown in Figure 4, compute the corresponding expected returns with respective allocations and plot them in Figure 5.Figure 4
Returns of future 45 days on two financial assets.Figure 5
The expected returns of three models in future 45 days.As shown, both of the mean returns on the two allocations fluctuate around 0.7. However, the volatility of returns corresponding to model (57) which only involves CVaR is apparently larger. On the other hand, the worst expected return with our portfolio is lightly higher than that with the allocation solved by model (57). It should be noted that samples in this test have similar mean returns and standard deviations. With the known parameters in Section 5.1.1, the new model (47) obtains the same results with robust Sharpe ratio model (58). In all, for conservative investors, the portfolio solved by our model would be a better choice resulting in an acceptable reward with a smaller volatility because of the advantage of involving standard deviation.


## 6. Conclusion

The main work in this paper is that robust reward-risk ratio optimization models with both CVaR and standard deviation are proposed. In terms of assumptions on measures of the general reward, CVaR, and standard deviation and under an ellipsoid uncertainty set, the ratio models can be reformulated to second-order conic optimization models using conic duality theorem. Compared with existed reward-risk ratio optimization models in [30,31,38], which only consider CVaR as risk measure, the proposed ratio models involve standard deviation as well as CVaR. That is, we further constrain standard deviation over corresponding models. Thus, the proposed ratio models can make a new balance between reward and risk since involving standard deviation in the risk measure. The investors would obtain reasonable returns without very high volatility. Namely, new model can give a better but a little bit conservative decision to some extent. However, like the other reward-risk ratio optimization models, our models are sensitive to the scaling of matrices in ellipsoid uncertainty sets as well. In other words, it indicates the important of choosing appropriate parameters to represent incomplete distribution information when utilizing any reward-risk ratio models. While there are some useful results like the model convexity and the equivalence of solutions among various simplified models. The models proposed in this paper can be extended to other different useful uncertainty sets in practice.



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*Source: 8304411-2022-04-16.xml*