Optimal Dynamic Portfolio with Mean-CVaR Criterion

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are popular risk measures from academic, industrial and regulatory perspectives. The problem of minimizing CVaR is theoretically known to be of Neyman-Pearson type binary solution. We add a constraint on expected return to investigate the Mean-CVaR portfolio selection problem in a dynamic setting: the investor is faced with a Markowitz type of risk reward problem at final horizon where variance as a measure of risk is replaced by CVaR. Based on the complete market assumption, we give an analytical solution in general. The novelty of our solution is that it is no longer Neyman-Pearson type where the final optimal portfolio takes only two values. Instead, in the case where the portfolio value is required to be bounded from above, the optimal solution takes three values; while in the case where there is no upper bound, the optimal investment portfolio does not exist, though a three-level portfolio still provides a sub-optimal solution.


Introduction
The portfolio selection problem published by Markowitz [23] in 1952 is formulated as an optimization problem in a one-period static setting with the objective of maximizing expected return, subject to the constraint of variance being bounded from above. In 2005, Bielecki et al. [8] published the solution to this problem in a dynamic complete market setting. In both cases, the measure of risk of the portfolio is chosen as variance, and the risk-reward problem is understood as the "Mean-Variance" problem.
is the Fenchel-Legendre dual of the Expected Shortfall) given by Rockafellar and Uryasev [26]. Föllmer and Leukert [14] characterized the solution to the latter problem in a general semimartingale complete market model to be binary, where they have demonstrated its close relationship to the Neyman-Pearson problem of hypothesis testing between the risk neutral probability measureP and the physical probability measure P .
Adding the expected return constraint to WVaR minimization (CVaR is a particular case of WVaR), Cherny [10] found conditions under which the solution to the Mean-WVaR problem was still binary or nonexistent. In this paper, we discuss all cases for solving the Mean-CVaR problem depending on a combination of two criteria: the level of the Radon-Nikodým derivative dP dP relative to the confidence level of the risk measure; and the level of the return requirement. More specifically, when the portfolio is uniformed bounded from above and below, we find the optimal solution to be nonexistent or binary in some cases, and more interestingly, take three values in the most important case (see Case 4 of Theorem 3.15). When the portfolio is unbounded from above, in most cases (see Case 2 and 4 in Theorem 3.17), the solution is nonexistent, while portfolios of three levels still give sub-optimal solutions. Since the new solution we find can take not only the upper or the lower bound, but also a level in between, it can be viewed in part as a generalization of the binary solution for the Neyman-Pearson problem with an additional constraint on expectation.
This paper is organized as follows. Section 2 formulates the dynamic portfolio selection problem, and compares the structure of the binary solution and the 'three-level' solution, with an application of exact calculation in the Black-Scholes model. Section 3 details the analytic solution in general where the proofs are delayed to Appendix 5; Section 4 lists possible future work. 2 The Structure of the Optimal Portfolio

Main Problem
Let (Ω, F, (F) 0≤t≤T , P ) be a filtered probability space that satisfies the usual conditions where F 0 is trivial and F T = F. The market model consists of d + 1 tradable assets: one riskless asset (money market account) and d risky asset (stock). Suppose the risk-free interest rate r is a constant and the stock S t is a d-dimensional real-valued locally bounded semimartingale process. Let the number of shares invested in the risky asset ξ t be a d-dimensional predictable process such that the stochastic integral with respect to S t is well-defined. Then the value of a self-financing portfolio X t evolves according to the dynamics dX t = ξ t dS t + r(X t − ξ t S t )dt, X 0 = x 0 .
Here ξ t dS t and ξ t S t are interpreted as inner products if the risky asset is multidimensional d > 1. The portfolio selection problem is to find the best strategy (ξ t ) 0≤t≤T to minimize the Conditional Value-at-Risk (CVaR) of the final portfolio value X T at confidence level 0 < λ < 1 , while requiring the expected value to remain above a constant z. † In addition, we require uniform lower bound x d and upper bound x u on the value of the portfolio over time such that −∞ < x d < x 0 < x u ≤ ∞. Therefore, our Main Dynamic Problem is inf ξt CV aR λ (X T ) (1) subject to E[X T ] ≥ z, x d ≤ X t ≤ x u a.s. ∀t ∈ [0, T ].
Note that the no-bankruptcy condition can be imposed by setting the lower bound to be x d = 0, and the portfolio value can be unbounded from above by taking the upper bound as x u = ∞. Our solution will be based on the following complete market assumption.
Assumption 2.1 There is No Free Lunch with Vanishing Risk (as defined in Delbaen and Schachermayer [11]) and the market model is complete with a unique equivalent local martingale measureP such that the Radon-Nikodým derivative dP dP has a continuous distribution.
Under the above assumption any F-measurable random variable can be replicated by a dynamic portfolio.
Thus the dynamic optimization problem (1) can be reduced to: first find the optimal solution X * * to the if it exists, and then find the dynamic strategy that replicates the F-measurable random variable X * * . Here the expectations E andẼ are taken under the physical probability measure P and the risk neutral probability measureP respectively. Constant x r = x 0 e rT is assumed to satisfy −∞ < x d < x 0 ≤ x r < x u ≤ ∞ and the additional capital constraintẼ[X] = x r is the key to make sure that the optimal solution can be replicated by a dynamic self-financing strategy with initial capital x 0 . † Krokhmal et al. [21] showed conditions under which the problem of maximizing expected return with CVaR constraint is equivalent to the problem of minimizing CVaR with expected return constraint. In this paper, we use the term Mean-CVaR problem for both cases.
Using the equivalence between Conditional Value-at-Risk and the Fenchel-Legendre dual of the Expected Shortfall derived in Rockafellar and Uryasev [26], the CVaR optimization problem (2) can be reduced to an Expected Shortfall optimization problem which we name as the Two-Constraint Problem: Step 1: Step 2: Minimization of Conditional Value-at-Risk To compare our solution to existing ones in literature, we also name an auxiliary problem which simply minimizes Conditional Value-at-Risk without the return constraint as the One-Constraint Problem: Step 1 in (4) is replaced by Step 1: Step 2 in (5) remains the same.

Main Result
This subsection is devoted to a conceptual comparison between the solutions to the One-Constraint Problem and the Two-Constraint Problem. The solution to the Expected Shortfall Minimization problem in Step 1 of the One-Constraint Problem is found by Föllmer and Leukert [14] under Assumption 2.1 to be binary in nature: where I · (ω) is the indicator function and set A is defined as the collection of states where the Radon-Nikodým derivative is above a threshold ω ∈ Ω : dP dP (ω) > a . This particular structure where the optimal solution X(x) takes only two values, namely the lower bound x d and x, is intuitively clear once the problems of minimizing Expected Shortfall and hypothesis testing between P andP are connected in Föllmer and Leukert [14], the later being well-known to possess a binary solution by Neyman-Pearson Lemma. There are various ways to prove the optimality. Other than the Neyman-Pearson approach, it can be viewed as the solution from a convex duality perspective, see Theorem 1.19 in Xu [32]. In addition, a simplified version to the proof of Proposition 3.14 gives a direct method using Lagrange multiplier for convex optimization.

The solution to
Step 2 of One-Constraint Problem, and thus to the Main Problems in (1) and (2) as a pure risk minimization problem without the return constraint is given in Schied [30], Sekine [31], and Li and Xu [22]. Since Step 2 only involves minimization over a real-valued number x, the binary structure is preserved through this step. Under some technical conditions, the solution to Step 2 of the One-Constraint Problem is shown by Li and Xu [22] (Theorem 2.10 and Remark 2.11) to be where (a * , x * ) is the solution to the capital constraint (Ẽ[X(x)] = x r ) in Step 1 and the first order Euler condition (v (x) = 0) in Step 2: A static portfolio holding only the riskless asset will yield a constant portfolio value X ≡ x r with CV aR(X) = −x r . The diversification by managing dynamically the exposure to risky assets decreases the risk of the overall portfolio by an amount shown in (9). One interesting observation is that the optimal portfolio exists regardless whether the upper bound on the portfolio is finite x u < ∞ or not x u = ∞. This conclusion will change drastically as we add the return constraint to the optimization problem.
The main result of this paper is to show that the optimal solution to the Two-Constraint Problem, and thus the Main Problem (1) and (2), does not have a Neyman-Pearson type of binary solution, which we call Two-Line Configuration in (8); instead, it has a Three-Line Configuration. Proposition 3.14 and Theorem 3.15 prove that, when the upper bound is finite x u < ∞ and under some technical conditions, the solution to Step 2 of the Two-Constraint Problem turns out to be where (a * * , b * * , x * * ) is the solution to the capital constraint and the first order Euler condition, plus the additional return constraint (E[X(x)] = z): x d P (A) + xP (B) + x u P (D) = z, (return constraint) (14) x dP (A) + xP (B) + x uP (D) = x r , (capital constraint) The sets in equation (14)- (16) are defined by different levels of the Radon-Nikodým derivative: When the upper bound is infinite x u = ∞, Theorem 3.17 shows that the solution for the optimal portfolio is no longer a Three-Line Configuration. It can be pure money market account investment (One-Line), binary (Two-Line), or very likely nonexist. In the last case, the infimum of the CVaR can still be computed, and a sequence of Three-Line Configuration portfolios can be found with their CVaR converging to the infimum.

Example: Exact Calculation in the Black-Scholes Model
We show the closed-form calculation of the Three-Line Configuration (12)- (16), as well as the corresponding optimal dynamic strategy in the benchmark Black-Scholes Model. Suppose an agent is trading between a money market account with interest rate r and one stock ‡ that follows geometric Brownian motion dS t = µS t dt + σS t dW t with instantaneous rate of return µ, volatility σ, and initial stock price S 0 . The endowment starts at x 0 and bankruptcy is prohibited at any time, x d = 0, before the final horizon T . The expected terminal value E[X T ] is required to be above a fixed level 'z' to satisfy the return constraint. When 'z' where X * is the optimal portfolio (8) for the One Constraint Problem, the return constraint is non-binding and obviously the Two-Line Configuration X * is optimal. Letz be the highest expected value achievable by any self-financing portfolio starting with initial capital x 0 (see Definition 3.2 and Lemma 3.3). When the return requirement becomes meaningful, i.e., z ∈ (z * ,z], the Three-Line Configuration X * * in (12) becomes optimal.
Since the Radon-Nikodým derivative dP dP is a scaled power function of the final stock price which has a log-normal distribution, the probabilities in equations (15)- (16) can be computed in closed-form: Numerical results comparing the minimal risk for various levels of upper-bound x u and return constraint z are summarized in Table 1. As expected the upper bound on the portfolio value x u has no impact on the One-Constraint Problem, as (x * , a * ) and CV aR λ (X * T ) are optimal whenever x u ≥ x * . On contrary in the Two-Constraint Problem, the stricter the return requirement z, the more the Three-Line Configuration X * * deviates from the Two-Line Configuration X * . Stricter return requirement (higher z) implies higher minimal risk CV aR λ (X * * T ); while less strict upper bound (higher x u ) decreases minimal risk CV aR λ (X * * T ). Notably, under certain conditions in Theorem 3.17, for all levels of return z ∈ (z * ,z], when x u → ∞, CV aR λ (X * * T ) approaches CV aR λ (X * T ), as the optimal solution cease to exist in the limiting case.
1. Any Three-Line Configuration has the structure X = x d I A + xI B + x u I D .
2. The Two-Line Configuration X = xI B + x u I D is associated to the above definition in the case Moreover, 1. General Constraints are the capital constraint and the equality part of the expected return constraint for a Three-Line Configuration X = x d I A + xI B + x u I D : 2. Degenerated Constraints 1 are the capital constraint and the equality part of the expected return constraint for a Two-Line Configuration X = xI B + x u I D : Degenerated Constraints 2 are the capital constraint and the equality part of the expected return constraint for a Two-Line Configuration X = x d I A + xI B : Degenerated Constraints 3 are the capital constraint and the equality part of the expected return constraint for a Two-Line Configuration X = x d I A + x u I D : Note that Degenerated Constraints 1 correspond to the General Constraints when a = ∞; Degenerated Constraints 2 correspond to the General Constraints when b = 0; and Degenerated Constraints 3 correspond to the General Constraints when a = b.
We use the Two-Line Configuration where the value of the random variable X takes either the upper or the lower bound, as well as its capital constraint to define the 'Bar-System' from which we calculate the highest achievable return.
In the 'Bar-System',Ā,D andX are associated to the constantā in Lemma 3.3z is the highest expected return that can be obtained by a self-financing portfolio with initial capital x 0 whose value is bounded between x d and x u : In the following lemma, we vary the 'x' value in the Two-Line Configurations X = xI B + x u I D and X = x d I A + xI B , while maintaining the capital constraints respectively. We observe their expected returns to vary between values x r andz in a monotone and continuous fashion. From now on, we will concern ourselves with requirements on the expected return in the interval z ∈ [x r ,z] because on one side Lemma 3.3 ensures that there are no feasible solutions to the Main Problem (2) if we require an expected return higher thanz. On the other side, Lemma 3.3, Lemma 3.4 and Theorem 3.11 lead to the conclusion that a return constraint where z ∈ (−∞, x r ) is too weak to differentiate the Two-Constraint Problem from the One-Constraint Problem as their optimal solutions concur.
, define x z1 and x z2 to be the corresponding x value for Two-Line Configurations X = xI B + x u I D and X = x d I A + xI B that satisfy Degenerated Constraints 1 and Degenerated Constraints 2 respectively. § Threshold 'b' and consequently sets 'B' and 'D' are all dependent on 'x' through the capital constraint, therefore z(x) is not a linear function of x. Definition 3.5 implies when we fix the level of expected return z, we can find two particular feasible The values x z1 and x z2 are well-defined because Lemma 3.4 guarantees z(x) to be an invertible function in both cases. We summarize in the following lemma whether the Two-Line Configurations satisfying the capital constraints meet or fail the return constraint as x ranges over its domain [x d , x u ] for the Two-Line and Three-Line Configurations in Definition 3.1.
We turn our attention to solving Step 1 of the Two-Constraint Problem (4): Step 1: Notice that a solution is called for any given real number x, independent of the return level z or capital level x r . From Lemma 3.6 and the fact that the Two-Line Configurations are optimal solutions to Step 1 of the One-Constraint Problem (see Theorem 3.11), we can immediately draw the following conclusion.
Proposition 3.7 For fixed −∞ < x d < x r < x u < ∞, and a fixed level z ∈ [x r ,z].
1. If we fix x ∈ [x d , x z1 ], then there exists a Two-Line Configuration X = xI B + x u I D which is the optimal solution to Step 1 of the Two-Constraint Problem; 2. If we fix x ∈ [x z2 , x u ], then there exists a Two-Line Configuration X = x d I A + xI B which is the optimal solution to Step 1 of the Two-Constraint Problem.  3. In the extreme case when a = ∞, the Three-Line configuration becomes the Two-Line Configuration X = xI B + x u I D ; in the extreme case when b = 0, the Three-Line configuration becomes the Two-Line Configuration X = x d I A + xI B . In either case, the expected value is below z by Lemma 3.6. Proposition 3.9 For fixed −∞ < x d < x r < x u < ∞, and a fixed level z ∈ [x r ,z]. If we fix x ∈ (x z1 , x z2 ), then there exists a Three-Line Configuration X = x d I A + xI B + x u I D that satisfies the General Constraints which is the optimal solution to Step 1 of the Two-Constraint Problem.
Combining Proposition 3.7 and Proposition 3.9, we arrive to the following result on the optimality of the Three-Line Configuration.   (17) satisfying the General Constraints: To solve Step 2 of the Two-Constraint Problem (5), and thus the Main Problem (2), we need to where v(x) has been computed in Theorem 3.10. Depending on the z level in the return constraint being lenient or strict, the solution is sometimes obtained by the Two-Line Configuration which is optimal to the One-Constraint Problem, and other times obtained by a true Three-Line configuration. To proceed in this direction, we recall the solution to the One-Constraint Problem from Li and Xu [22]. 1. Suppose ess sup dP dP ≤ 1 λ . X = x r is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the One-Constraint Problem and the associated minimal risk is CV aR(X) = −x r .
• If 1 a ≤ λ−P (Ā) 1−P (Ā) (see Definition 3.2 for the 'Bar-System'), thenX = x d IĀ + x u ID is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the One-Constraint Problem and the associated minimal risk is • Otherwise, let a * be the solution to the equation 1 a = λ−P (A) 1−P (A) . Associate sets A * = ω ∈ Ω : dP dP (ω) > a * and B * = ω ∈ Ω : dP dP (ω) ≤ a * to level a * . Define x * = xr−x dP (A * ) so that configuration satisfies the capital constraintẼ[X * ] = x dP (A * ) + x * P (B * ) = x r . ¶ Then X * (we call the 'Star-System') is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the One-Constraint Problem and the associated minimal risk is We see that when z is smaller than z * , the binary solutions X * andX provided in Theorem 3.11 are indeed the optimal solutions to Step 2 of the Two-Constraint Problem. However, when z is greater than z * , these Two-Line Configurations are no longer feasible in the Two-Constraint Problem. We now show that the Three-Line Configuration is not only feasible but also optimal. First we establish the convexity of the objective function and its continuity in a Lemma.
Lemma 3.13 v(x) is a convex function for x ∈ R, and thus continuous.
Proposition 3.14 For fixed −∞ < x d < x r < x u < ∞, and a fixed level z ∈ (z * ,z]. ¶ Equivalently, (a * , x * ) can be viewed as the solution to the capital constraint and the first order Euler condition in equations (10) and (11).
Suppose ess sup dP dP > 1 λ . The solution (a * * , b * * , x * * ) (and consequently, A * * , B * * and D * * ) to the equations exists. X * * = x d I A * * +x * * I B * * +x u I D * * (we call the 'Double-Star System') is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem and the associated minimal risk is Putting together Proposition 3.14 with Theorem 3.11, we arrive to the Main Theorem of this paper.

Theorem 3.15 (Minimization of Conditional Value-at-Risk When
1. Suppose ess sup dP dP ≤ 1 λ and z = x r . The pure money market account investment X = x r is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem and the associated minimal risk is CV aR(X) = −x r .
2. Suppose ess sup dP dP ≤ 1 λ and z ∈ (x r ,z]. The optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem does not exist and the minimal risk is 3. Suppose ess sup dP dP > 1 λ and z ∈ [x r , z * ] (see Definition 3.12 for z * ).
• If 1 a ≤ λ−P (Ā) 1−P (Ā) (see Definition 3.2), then the 'Bar-System'X = x d IĀ + x u ID is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem and the associated minimal risk is • Otherwise, the 'Star-System' X * = x d I A * +x * I B * defined in Theorem 3.11 is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem and the associated minimal risk is 4. Suppose ess sup dP dP > 1 λ and z ∈ (z * ,z]. the 'Double-Star-Sytem' X * * = x d I A * * + x * * I B * * + x u I D * * defined in Proposition 3.14 is the optimal solution to Step 2: Minimization of Conditional Valueat-Risk of the Two-Constraint Problem and the associated minimal risk is We observe that the pure money market account investment is rarely optimal. When the Radon-Nikodým derivative is bounded above by the reciprocal of the confidence level of the risk measure (ess sup dP dP ≤ 1 λ ), a condition not satisfied in the Black-Scholes model, the solution does not exist unless the return requirement coincide with the risk-free rate. When the Radon-Nikodým derivative exceeds 1 λ with positive probability, and the return constraint is low z ∈ [x r , z * ], the Two-Line Configuration which is optimal to the CV aR minimization problem without the return constraint is also the optimal to the Mean-CVaR problem. However, in the more interesting case where the return constraint is materially high z ∈ (z * ,z], the optimal Three-Line-Configuration sometimes takes the value of the upper bound x u to raise the expected return at the cost the minimal risk will be at a higher level. This analysis complies with the numerical example shown in Section 2.3.

Case x u = ∞: No Upper Bound
We first restate the solution to the One-Constraint Problem from Li and Xu [22] in the current context: 2. Suppose ess sup dP dP > 1 λ . The 'Star-System' X * = x d I A * + x * I B * defined in Theorem 3.11 is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the One-Constraint Problem and the associated minimal risk is We observe that although there is no upper bound for the portfolio value, the optimal solution remains bounded from above, and the minimal CV aR is bounded from below. The problem of purely minimizing CV aR risk of a self-financing portfolio (bounded below by x d to exclude arbitrage) from initial capital x 0 is feasible in the sense that the risk will not approach −∞ and the minimal risk is achieved by an optimal portfolio. When we add substantial return constraint to the CV aR minimization problem, although the minimal risk can still be calculated in the most important case (Case 4 in Theorem 3.17), it is truly an infimum and not a minimum, thus it can be approximated closely by a sub-optimal portfolio, but not achieved by an optimal portfolio.
1. Suppose ess sup dP dP ≤ 1 λ and z = x r . The pure money market account investment X = x r is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem and the associated minimal risk is CV aR(X) = −x r .
2. Suppose ess sup dP dP ≤ 1 λ and z ∈ (x r , ∞). The optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem does not exist and the minimal risk is 3. Suppose ess sup dP dP > 1 λ and z ∈ [x r , z * ]. The 'Star-System' X * = x d I A * + x * I B * defined in Theorem 3.11 is the optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem and the associated minimal risk is 4. Suppose ess sup dP dP > 1 λ and z ∈ (z * , ∞). The optimal solution to Step 2: Minimization of Conditional Value-at-Risk of the Two-Constraint Problem does not exist and the minimal risk is Remark 3.18 From the proof of the above theorem in Appendix 5, we note that in case 4, we can always find a Three-Line Configuration as a sub-optimal solution, i.e., there exists for every > 0, a corresponding portfolio X = x d I A + x I B + α I D which satisfies the General Constraints and produces a CV aR level close to the lower bound: CV aR(X ) ≤ CV aR(X * ) + .

Future Work
The second part of Assumption 2.1, namely the Radon-Nikodým derivative dP dP having a continuous distribution, is imposed for the simplification it brings to the presentation in the main theorems. Further work can be done when this assumption is weakened. We expect that the main results should still hold, albeit in a more complicated form. It will also be interesting to extend the closed-form solution for Mean-CVaR minimization by replacing CVaR with Law-Invariant Convex Risk Measures in general. Another direction will be to employ dynamic risk measures into the current setting.
Although in this paper we focus on the complete market solution, to solve the problem in an incomplete market setting, the exact hedging argument via Martingale Representation Theorem that translates the dynamic problem (1) into the static problem (2) has to be replaced by a super-hedging argument via Optional Decomposition developed by Kramkov [20], and Föllmer and Kabanov [13]. The detail is similar to the process carried out for Shortfall Risk Minimization in Föllmer and Leukert [14], Convex Risk Minimization in Rudloff [28], and law-invariant risk preference in He and Zhou [17]. The curious question is: Will the Third-Line

Configuration remain optimal?
The outcome in its format resembles techniques employed in Föllmer and Leukert [14] and Li and Xu [22] where the point masses on the thresholds for the Radon-Nikodým derivative in (17) have to be dealt with carefully.

Appendix
Proof of Lemma 3.3. The problem of is equivalent to the Expected Shortfall Problem Therefore, the answer is immediate.
For any given > 0, choose x 2 − x 1 ≤ , then Therefore, z decreases continuously as x increases when x ∈ [x d , x r ]. When x = x d , z =z from Definition 3.2. When x = x r , X ≡ x r and z = E[X] = x r . Similarly, we can show that z increases continuously from x r toz as x increases from x r to x u . Lemma 3.6 is a logical consequence of Lemma 3.4 and Definition 3.5; Proposition 3.7 follows from Lemma 3.6; so their proofs will be skipped.
Similarly, let configuration X 2 = x d I A2 + xI B2 + x u I D2 correspond to the pair (a 2 , b 2 ). Define z 1 = E[X 1 ] and z 2 = E[X 2 ]. Since both X 1 and X 2 satisfy the capital constraint, we have This simplifies to the equation Then Suppose the pair (a 1 , b 1 ) is chosen so that X 1 satisfies the budget constraintẼ[X 1 ] = x r . For any given Now choose a 2 such that a 2 < a 1 and equation (18) is satisfied. Then X 2 also satisfies the budget constraintẼ[X 2 ] = x r , and We conclude that the expected value of the Three-Line configuration decreases continuously as b decreases and a increases.
In the following we provide the main proof of the paper: the optimality of the Three-Line configuration.
As standard for convex optimization problems, if we can find a pair of Lagrange multipliers λ ≥ 0 and µ ∈ R such thatX is the solution to the minimization problem (19) inf thenX is the solution to the constrained problem Choose any X ∈ F where x d ≤ X ≤ x u , and denote G = {ω ∈ Ω : X(ω) ≥ x} and L = {ω ∈ Ω : X(ω) < x}.
The last inequality holds because each term inside the expectation is greater than or equal to zero.
Theorem 3.10 is a direct consequence of Lemma 3.6, Proposition 3.7, and Proposition 3.9.
where a xz1 = ∞. Therefore, Case 3 dominates Case 2. In Case 5, Therefore, Case 4 dominates Case 5. When x ∈ [x z2 , x u ] and ess sup dP dP > 1 λ , Theorem 3.10 and Theorem 3.11 imply that the infimum in Case 4 is achieved either byX or X * . Since we restrict z ∈ (z * ,z] where z * =z by Definition 3.12 in the first case, we need not consider this case in the current proposition. In the second case, Lemma 3.4 implies that x * < x z2 (because z > z * ). By the convexity of v(x), and then the continuity of v(x), Therefore, Case 3 dominates Case 4. We have shown that Case 3 actually provides the globally infimum: Now we focus on x ∈ (x z1 , x z2 ), where X(x) = x d I Ax + xI Bx + x u I Dx satisfies the general constraints: and the definition for sets A x , B x and D x are Note that v(x) = (x − x d )P (A x ) (see Theorem 3.10). Since P (A x ) + P (B x ) + P (D x ) = 1 andP (A x ) + P (B x ) +P (D x ) = 1, we rewrite the capital and return constraints as Differentiating both sides with respect to x, we get Therefore, When the above derivative is zero, we arrive to the first order Euler condition To be precise, the above differentiation should be replaced by left-hand and right-hand derivatives as detailed in the Proof for Corollary 2.8 in Li and Xu [22]. But the first order Euler condition will turn out to be the same because we have assumed that the Radon-Nikodým derivative dP dP has continuous distribution. To finish this proof, we need to show that there exists an x ∈ (x z1 , x z2 ) where the first order Euler condition is satisfied. From Lemma 3.8, we know that as x x z1 , a x ∞, and P (A x ) 0. Therefore, As x x z2 , b x 0, and P (D x ) 0. Therefore, This derivative coincides with the derivative of the value function of the Two-Line configuration that is optimal on the interval x ∈ [x z2 , x u ] provided in Theorem 3.10 (see Proof for Corollary 2.8 in Li and Xu [22]).
Again when x ∈ [x z2 , x u ] and ess sup dP dP > 1 λ , Theorem 3.10 and Theorem 3.11 imply that the infimum of v(x) − λx is achieved either byX or X * . Since we restrict z ∈ (z * ,z] where z * =z by Definition 3.12 in the first case, we need not consider this case in the current proposition. In the second case, Lemma 3.4 implies that x * < x z2 (because z > z * ). This in turn implies We have just shown that there exist some x * * ∈ (x z1 , x z2 ) such that (v(x) − λx) | x=x * * = 0. By the convexity of v(x) − λx, this is the point where it obtains the minimum value. Now Proof of Theorem 3.15. Case 3 and 4 are already proved in Theorem 3.11 and Proposition 3.14. In Case 1 where ess sup dP dP ≤ 1 λ and z = x r , X = x r is both feasible and optimal by Theorem 3.11. In Case 2, fix arbitrary > 0. We will look for a Two-Line solution X = x I A + α I B with the right parameters a , x , α which satisfies both the capital constraint and return constraint: where A = ω ∈ Ω : dP dP (ω) > a , B = ω ∈ Ω : dP dP (ω) ≤ a , and produces a CVaR level close to the lower bound: CV aR(X ) ≤ CV aR(x r ) + = −x r + .
First, we choose x = x r − . To find the remaining two parameters a and α so that equations (20) and (21) are satisfies, we note x r P (A ) + x r P (B ) = x r , x rP (A ) + x rP (B ) = x r , and conclude that it is equivalent to find a pair of a and α such that the following two equalities are  (20) and (21). It is not difficult to prove that the fractionP (B)
Under Assumption 2.1, the solution in Case 2 is almost surely unique, the result is proved.
Proof of Theorem 3.17. Case 1 and 3 are obviously true in light of Theorem 3.16. The proof for Case 2 is similar to that in the Proof of Theorem 3.15, so we will not repeat it here. Since E[X * ] = z * < z in case 4, CV aR(X * ) is only a lower bound in this case. We first show that it is the true infimum obtained in Case 4. Fix arbitrary > 0. We will look for a Three-Line solution X = x d I A + x I B + α I D with the right parameters a , b , x , α which satisfies the general constraints: where A = ω ∈ Ω : dP dP (ω) > a , B = ω ∈ Ω : b ≤ dP dP (ω) ≤ a , D = ω ∈ Ω : dP dP (ω) < b , and produces a CVaR level close to the lower bound: CV aR(X ) ≤ CV aR(X * ) + .
First, we choose a = a * , A = A * , x = x * − δ, where we define δ = λ λ−P (A * ) . To find the remaining two parameters b and α so that equations (23) and (24) are satisfies, we note E[X * ] = x d P (A * ) + x * P (B * ) = z * , E[X * ] = x dP (A * ) + x * P (B * ) = x r , and conclude that it is equivalent to find a pair of b and α such that the following two equalities are satisfied:  (23) and (24). It is not difficult to prove that the fractionP (D) increases continuously from 0 toP (B * ) P (B * ) as b increases from 0 to a * . Therefore, we can find a solution b ∈ (0, a * ) where (25) is satisfied. By definition (3), The difference Under Assumption 2.1, the solution in Case 4 is almost surely unique, the result is proved.