Discrete-time mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR

Discrete-Time Mean-CVaR Portfolio Selection and Time-Consistency Induced Term Structure of the CVaR

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Discrete-Time Mean-CVaR Portfolio Selection and Time-Consistency Induced Term Structure of the CVaR Moris S. Strub, Duan Li, Xiangyu Cui, Jianjun Gao PII: DOI: Reference:

S0165-1889(19)30150-2 https://doi.org/10.1016/j.jedc.2019.103751 DYNCON 103751

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Journal of Economic Dynamics & Control

Received date: Revised date: Accepted date:

6 April 2019 7 September 2019 10 September 2019

Please cite this article as: Moris S. Strub, Duan Li, Xiangyu Cui, Jianjun Gao, Discrete-Time MeanCVaR Portfolio Selection and Time-Consistency Induced Term Structure of the CVaR, Journal of Economic Dynamics & Control (2019), doi: https://doi.org/10.1016/j.jedc.2019.103751

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Discrete-Time Mean-CVaR Portfolio Selection and Time-Consistency Induced Term Structure of the CVaR Moris S. Strub∗

Duan Li†

Xiangyu Cui‡

Jianjun Gao§

Abstract We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). A substantial challenge is the combination of a timeinconsistent objective with an incomplete and dynamic model for the financial market. We are able to solve this problem analytically by embedding the original, time-inconsistent problem into a family of time-consistent expected utility maximization problems with a piecewise linear utility function. The optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V -shaped pattern as a function of the current wealth. For the incomplete, discrete-time market considered herein, the mean-CVaR efficient frontier is a straight line in the mean-CVaR plane and thus economically meaningful. This contrasts the complete, continuous-time setting where the mean-CVaR efficient frontier is degenerate or does not exist at all. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt in order for the pre-committed optimal strategy to remain optimal at any point in time. Our result shows that a pre-committed mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises the investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.

Keywords: mean-risk portfolio choice; conditional value-at-risk; optimal investment strategies; timeinconsistency; time-consistency induced risk measure; equity premium puzzle JEL Classification: C61, G11

∗

Business School, Southern University of Science and Technology, Shenzhen, Guangdong, China. [email protected] † Corresponding author. School of Data Science, City University of Hong Kong. Email: [email protected] ‡ College of Business, Shanghai University of Finance and Economics. Email: [email protected] § Research Institute for Interdisciplinary Sciences, Shanghai University of Finance and Economics. [email protected]

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Email:

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Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

1

Introduction

The ingenious idea behind the modern portfolio theory pioneered by Markowitz (1952) is that investors aim at optimizing simultaneously two conflicting objectives: risk and return. An investor selects a portfolio minimizing the risk, originally measured by the variance, amongst all those portfolios that on average exceed a target expected return. Although measuring the risk of a portfolio by its variance is intuitively appealing and attractive because of its analytical tractability, it suffers from serious drawbacks. In particular, mean-variance preferences are not monotone and the variance punishes deviations from the mean alike, no matter whether they exceed or fall short of the mean. In response to those drawbacks, a multitude of alternative risk measures have been proposed. Amongst the most prominent are the value-at-risk (VaR) and the conditional value-at-risk (CVaR). In this paper, we study a mean-risk portfolio selection problem where the risk is measured by the CVaR. Although the VaR has been common industry practice, it has been criticized by many because it is not subadditive and one has no control over losses beyond the confidence level (e.g. Artzner et al. (1999), Dan´ıelsson et al. (2001) or Embrechts et al. (2014)). On the other hand, Acerbi and Tasche (2002) show that the CVaR is a coherent risk measure, i.e., it satisfies the axioms of translation invariance, subadditivity, positive homogeneity and monotonicity, proposed by Artzner et al. (1999). The CVaR does not only have superior mathematical properties compared with the VaR (e.g. Pflug (2000)), but will also gain in practical importance with the implementation of Basel III, as the Basel Committee has proposed to move from VaR to the CVaR in the consultative document of Basel Committee on Banking Supervision (2013). Mean-risk portfolio optimization problems where risk is measured by the CVaR have been studied in a single-period setting, see e.g. Krokhmal et al. (2002) and Bassett et al. (2004), where the problem is transformed into a linear programming problem by means of discretization, Alexander and Baptista (2004) for a model where returns follow a multivariate normal distribution, or Huang et al. (2008), Zhu and Fukushima (2009) and Zhu et al. (2014) where the worst-case CVaR is considered in a setting of robust portfolio optimization. For the context of managing portfolio depletion risk through optimal life-cycle asset allocation, Forsyth et al. (2018) use a numerical HJB approach and Monte Carlo simulations to solve a mean-CVaR optimization problem in a market where the risky asset follows a jump diffusion process. Balb´ as et al. (2010) consider mean-risk portfolio optimization in an abstract market where reachable payoffs are described by a closed subset in L2 and thus do not have to determine a dynamic investment strategy. The challenge in mean-CVaR portfolio selection when going beyond the single-period setting lies in the time-inconsistency of the CVaR (e.g. Artzner et al. (2007)), which prevents a direct application of conventional stochastic control approaches. In continuous time, complete markets, Li and Xu (2013), He et al. (2015) and Gao et al. (2017) bypass the issue of time-inconsistency by using the martingale approach, which transfers the optimization problem into an essentially single-period problem. Recently, equilibrium policies have been derived for mean-CVaR optimization problems for the continuous-time setting in He and Jiang (2018) and the discrete-time setting in Cui et al. (2019). However, Forsyth (2019) shows numerically that such strategies have undesirable properties and are outperformed even by constant strategies.

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Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Herein, we seek to determine the unconstrained, pre-committed optimal investment strategy. To that end, time-inconsistency becomes a real challenge in the generally incomplete and dynamic discretetime model for the financial market. We approach this problem by embedding the Lagrangian relaxation of the original mean-CVaR formulation into a family of expected utility maximization problems by using the dual representation for the CVaR due to Rockafellar and Uryasev (2000). Embedding a time-inconsistent mean-risk optimization problem into a family of time-consistent expected utility maximization problems resembles the breakthrough in dynamic mean-variance optimization of Li and Ng (2000) and Zhou and Li (2000). The difference between the two frameworks is that the meanvariance optimization problem can be embedded into an expected utility maximization problem with a quadratic utility function, whereas the utility function is of piecewise linear form in the case of meanCVaR optimization. When solving the mean-CVaR portfolio optimization problem, we find that the optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V -shaped pattern as a function of the current wealth. Furthermore, the mean-CVaR efficient frontier is found to be a straight line in the mean-CVaR plane. Our finding implies that, in discrete-time, there is a constant trade-off between risk and return. This is in stark contrast to the continuous-time, complete market setting, where the mean-CVaR efficient frontier is a vertical line, cf. He et al. (2015). We remark that we do not consider investment constraints in this paper. Incorporating economically relevant constraints, such as no bankruptcy, a maximum leverage ratio or position limits, remains an open problem for future research. We further consider an inverse portfolio selection problem where we seek to determine how meanCVaR preferences have to adapt such that the optimal investment strategy for an initial mean-CVaR objective remains optimal at any point in time and for all realizations of the return process. Meanrisk preferences adapting in a way that a pre-committed policy is preserved as the optimal strategy have been investigated by Cui et al. (2012) and Cui et al. (2017) for the mean-variance framework. Kov´aˇcov´a and Rudloff (2018) generalize this idea for dynamic, time-consistent convex risk measures by considering a vector optimization problem and deriving a set-valued Bellman’s principle. For general preferences and without relying on the existence of an optimal strategy, Karnam et al. (2017) introduce the notion of a dynamic utility which transfers an original time-inconsistent problem into a time-consistent one. In the context of CVaR-based hedging without mean constraint, Godin (2016) investigates a similar problem and determines a sequence of risk-measures under which the optimality of the hedging strategy is preserved by fixing the auxiliary parameter in the dual representation of the CVaR. Under some technical conditions, we find that one can remain within the class of mean-CVaR preferences in order to achieve global mean-CVaR efficiency, but needs to adjust the confidence level for the CVaR and update the target expected return along the time axis. Intriguingly, the confidence level for the CVaR is updated independently of the realized state and increasing in time. We term this the time-consistency induced term structure of the CVaR. Although conceptually different, pre-committing to a mean-CVaR optimal investment decision at time zero thus leads to the same investment behavior as naively re-optimizing the optimal strategy at every point in time with respect to mean-CVaR preferences with a time-increasing confidence level for the CVaR. The historically very large excess return of stocks over bonds has long puzzled financial economists. 2

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Mehra and Prescott (1985) found that historical returns cannot be explained by classical asset pricing models based on consumption, and named this phenomenon the equity premium puzzle. Although there has been some progress in comprehending the equity premium puzzle (see Mehra (2008) for a survey), the puzzle is still not fully understood. A final contribution of this paper is an empirical application of our mean-CVaR portfolio optimization model yielding an alternative perspective on the equity premium. Based on historical return statistics for US bills, bonds and stocks reported in De Giorgi and Post (2011), we find that a mean-CVaR investor holds a well-balanced portfolio for a range of sensible target returns. This finding does not rely on the dynamic nature of the problem studied in this paper and is robust with respect to the time horizon. Our result suggests an intuitive, risk-based view on the equity premium: when risk is perceived as the average loss in the worst states of the world, investors demand a large expected excess return of stocks in order to compensate for their large tail risk. The remainder of this paper is organized as follows. In Section 2 we introduce the discrete-time model for the financial market and the corresponding dynamic mean-CVaR portfolio optimization problem. We then embed the Lagrangian relaxation of the problem into a family of auxiliary expected utility maximization problems in Section 3. We solve the auxiliary expected utility maximization problems analytically in Section 4 and obtain the solution to the original mean-CVaR portfolio selection problem in Section 5. We discuss the time-consistency induced term structure of the CVaR in Section 6. In Section 7, we present the application of mean-CVaR portfolio selection to the equity premium puzzle. We conclude this paper in Section 8. All proofs are placed in the appendix.

2

Discrete-time mean-CVaR portfolio choice model

The financial market under consideration consists of N risky assets with random returns and one risk-free asset with a deterministic return. The deterministic total return of the risk-free asset for the period between t − 1 and t is denoted by rt > 0 and the random total returns of the risky assets for this time period by a vector Rt = [Rt1 , · · · , RtN ]0 . The total return vectors Rt , t = 1, 2, · · · , T , are assumed to be time-independent and absolutely integrable random vectors on the probability space (Ω, F, P). We define  0  0 Pt = Pt1 , Pt2 , · · · , PtN = (Rt1 − rt ), (Rt2 − rt ), · · · , (RtN − rt )

as the vector of excess returns.

Remark 2.1. Although price movements typically do not exhibit significant autocorrelation, higherorder time-dependence such as volatility clustering do exist in practice, see, e.g., Cont (2001). The assumption of independent returns is made for mathematical simplicity. Consider an investor whose wealth at time t − 1, the beginning of the t-th time period, is denoted by Xt−1 , and who chooses to invest ϑit , i = 1, 2, · · · , N , in the i-th risky asset. In order for the strategy to be self-financing the amount invested in the risk-free asset at the beginning of the t-th time period

3

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection P i T must then equal Xt−1 − N i=1 ϑt . Given an initial wealth x0 and a strategy = (ϑt )t=1 , the wealth of the investor thus evolves as Xt = rt Xt−1 + ϑ0t Pt . We denote the available information at the end of period t, t = 1, 2, . . . , T , as Ft = σ(P1 , P2 , · · · , Pt ), set F0 = {∅, Ω} and require that any strategy ϑ = (ϑt )Tt=1 is predictable with respect to this filtration and integrable. This in particular implies that ϑ0t Pt is integrable for each t = 1, 2, . . . , T since ϑt and Pt are independent. The value-at-risk at the confidence level β of a random variable representing a financial position X, VaRβ (X), represents the lowest amount which, with probability β, will not be exceeded by the loss −X,  VaRβ (X) = min z ∈ R P[−X ≤ z] ≥ β .

The conditional value-at-risk at the confidence level β is defined by 1 CVaRβ (X) = 1−β

Z

1

VaRα (X)dα

β

see, for example, Rockafellar and Uryasev (2000, 2002), Pflug (2000), or He et al. (2015). In the case where the distribution of X is absolutely continuous, the CVaR coincides with the tail conditional expectation, CVaRβ (XT ) = E [−XT | − XT ≥ VaRβ (XT )] , and represents the expected loss conditioned on the event that the loss exceeds the value-at-risk VaRβ (X). Our investor faces a mean-risk portfolio choice problem, where the objective is to minimize the risk of the portfolio among all those portfolios exceeding a given target expected return. More specifically, we consider the following dynamic mean-CVaR portfolio selection problem in discrete-time, inf ϑ CVaRβ (XT ), s.t. E [XT ] ≥ , Xt+1 = rt+1 Xt + ϑ0t+1 Pt+1 ,

(PCVaR (β, )) t = 0, 1, · · · , T − 1,

Q where  ≥ x0 Tt=1 rt = x0 ρ0 is the target expected terminal wealth. For future use we denote the Q time T value of one dollar invested in the risk-free asset at time t by ρt = T`=t+1 r` and the random terminal wealth XT which results from trading according to a strategy ϑ = (ϑt )Tt=1 by XTϑ . We suppose that E[Pti ] 6= 0 for at least one i ∈ {1, 2, . . . , N } and t ∈ {0, 1, . . . , T − 1}. This in particular 4

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

guarantees the existence a feasible solution to problem (PCV aR (β, )) for any  ≥ x0 ρ0 . Varying the target expected terminal wealth  from ρ0 x0 to infinity would trace out the mean-CVaR efficient frontier. A significant challenge behind the mean-CVaR formulation is its time-inconsistency, see for example Artzner et al. (2007), together with the incompleteness of the discrete-time financial market. This prevents us from making use of the martingale approach as in Li and Xu (2013), He et al. (2015) or Gao et al. (2017). Intuitively, time-inconsistency of the mean-CVaR optimization problem can be understood as follows: imagine a two-period setting and consider a state where the investor made a large gain in the first period. From the point of view of time zero, falling into the region of worst-case scenarios relevant for the CVaR is then very unlikely and the pre-committed investor would thus like to invest heavily in assets yielding a positive excess return. However, if the investor revisits the problem at the intermediate time the risk of falling into the region relevant for the CVaR is still significant causing the investor to be more cautious. Rudloff et al. (2014) illustrate the deviation from planned and actual behavior in a binomial tree setting and propose a time-consistent alternative for the case where the investor is not able to precommit to the investment strategy. To overcome this difficulty related to the time-inconsistency of the objective, we will lead through a series of transformations in Section 3, thereby embedding (PCVaR (β, )) into a family of time-consistent expected utility maximization problems. Note that the initial wealth level x0 does only affect the mean-CVaR optimization problem (PCVaR (β, )) through the difference between the  target expected  terminal wealth  and x0 , because PT ϑ 0 for any strategy ϑ we have CVaR(XT ) = CVaR t=1 ρt ϑt Pt − ρ0 x0 . For our analysis, we thus assume without loss of generality that x0 = 0. It will be straightforward to reintroduce a non-zero initial wealth later on. Remark 2.2. We do not consider bankruptcy prohibition when formulating (PCVaR (β, )) as is done for example in Bielecki et al. (2005) for the continuous-time mean-variance problem. As we noted above, the initial wealth only influences (PCVaR (β, )) through the difference to the target expected terminal wealth. The story would be entirely different if the wealth process were required to remain nonnegative. Solving a discrete-time mean-CVaR optimization under bankruptcy prohibition thus remains an interesting and challenging open problem. Note that the main technique of Bielecki et al. (2005), namely decomposing the problem into an optimization problem over terminal wealth levels and then a replicating the optimal terminal wealth, cannot be applied in our incomplete market setting. He et al. (2015) solved a mean-CVaR optimization problem under bankruptcy prohibition for a complete, continuous-time market. In their setting, the CVaR of the optimal solution remains independent of the expected terminal wealth target independently of whether one does or does not prohibit bankruptcy. Other economically relevant constraints, such as a no-shorting constraint (Li et al. (2002) or Cui et al. (2014)), limited leverage (Dang and Forsyth (2016)), cone constraints (Czichowsky and Schweizer (2013) or Cui et al. (2017)) or general constraints covering for example bankruptcy prohibition, noshorting, margin requirements or a combination thereof (Wang and Forsyth (2010)) have also been considered for dynamic mean-variance portfolio selection problems. Incorporating such constraints for the dynamic mean-CVaR portfolio selection problem considered herein remains a challenging open 5

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

problem for future research. The following definition weakens the notion of an arbitrage strategy. Definition 2.3. A strategy ϑ = (ϑt )Tt=1 is called a VaR-good-deal strategy at the confidence level β if h i P XTϑ > 0 > 0

and

h i P XTϑ ≥ 0 ≥ β.

If β = 1 then Definition 2.3 coincides with the definition of a classical arbitrage strategy. If an investor perceives risk by the VaR at the confidence level β, he or she ignores the worst cases happening with a probability of less than 1 − β and so a good-deal strategy at the confidence level β would look like a risk-free way to make a profit. Weakening the notion of an arbitrage to good-deals, where payoffs, which albeit not risk-free, seem to be too good to be true, was pioneered in the context of asset pricing in incomplete markets by Bernardo and Ledoit (2000) and Cochrane and Sa´a-Requejo (2000), see for example Kl¨oppel and Schweizer (2007) for an overview on this topic. For the following we assume that there are no such strategies in the market. This assumption in particular excludes arbitrage. Assumption 2.4. There is no VaR-good-deal strategy at the confidence level β. Remark 2.5. It is difficult to verify whether Assumption 2.4 is satisfied based on the dynamics of the stock price. We note that it is also often the case that it is infeasible to decide whether a general financial market admits arbitrage, cf. Karatzas and Kardaras (2007). We emphasize that mean-CVaR optimization problems are easily ill-posed if there are no constraints on trading strategies, see for example He et al. (2015). The underlying reason is that both the mean and CVaR are positive homogeneous as functionals of the terminal wealth. Therefore, if there were a strategy which starts with zero initial wealth and simultaneously achieves a positive expected terminal wealth and a negative CVaR, then one could simply scale this strategy and achieve an arbitrarily large expected terminal wealth while always satisfying the risk constraint. While Assumption 2.4 is stronger than the typical no arbitrage assumption, it is plausible if the market mainly consists of stocks and bonds. We in particular refer to the numerical examples in Example 5.2 and Section 7. Proposition 2.6. Suppose that Assumption 2.4 holds. Then problem (PCVaR (β, )) is well-posed, i.e., the objective value remains bounded. Remark 2.7. If one were solely concerned about whether the problem (PCVaR (β, )) is well-posed,
  Assumption 2.4 could be weakened to assume that CVaRβ XTϑ > 0 whenever P XTϑ > 0 > 0. We prefer the assumption of absence of a VaR-good-deal strategy at the confidence level β for two reasons. The first is economical, since we believe that a scenario where the investor starts with zero initial wealth and achieves a strictly positive terminal wealth should not be regarded as risk. To illustrate this, suppose that there would be a VaR-good deal strategy at the confidence level β, say ϑ. Then we
have by definition that VaR XTϑ < 0. Supposing that XTϑ is absolutely continuous, this would in turn imply that 6

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

  h i CVaRβ XTϑ = −E XTϑ XTϑ ≤ −VaRβ (XTϑ )    ! E XTϑ XTϑ ≤ 0 E XTϑ 0 < XTϑ ≤ −VaRβ (XTϑ )   +   = −(1 − β) . P XTϑ ≤ 0 P 0 < XTϑ ≤ −VaRβ (XTϑ ) However, we do not believe that it is reasonable that the last term influences the perceived risk of  since 0 < XTϑ ≤ −VaRβ (XTϑ ) are scenarios where a strictly positive terminal wealth is achieved from zero initial wealth. The second reason is of technical nature as Assumption 2.4 will spare us from a case distinction later on (cf. Case 3 at the beginning of Section 5). XTϑ ,

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Reformulation of the mean-CVaR portfolio selection problem

In this section, we will embed the time-inconsistent mean-CVaR optimization problem (PCVaR (β, )) into a family of expected utility maximization problems with piecewise linear utility functions. To do so, we will first consider the unconstrained minimization of the Lagrangian associated with problem (PCVaR (β, )), then use the representation of the CVaR provided by Rockafellar and Uryasev (2000) and finally carry out a change of variables. We start with showing that the mean constraint E[XT ] ≥  is binding at the optimal solution. Proposition 3.1. Under Assumption 2.4, for any strategy ϑ with E[XTϑ ] >  there is a strategy ϑˆ ˆ ˆ with E[XTϑ ] ≥  and CVaRβ (XTϑ ) < CVaRβ (XTϑ ). Introducing a Lagrangian multiplier µ ≥ 0, we consider the following Lagrangian problem, inf ϑ CVaRβ (XT ) − µE[XT ], s.t. Xt+1 = rt+1 Xt + ϑ0t+1 Pt+1 ,

t = 0, 1, · · · , T − 1,

(L(β, µ))

where µ can be viewed as a trade-off between CVaR and expected terminal wealth. Because both the CVaR and mean are positive homogeneous, there are three cases: problem (L(β, µ)) is ill-posed; not investing in risky assets is the only optimal solution; or there are infinitely many optimal solutions all achieving the objective value zero. Considering this problem is still of value since, when we are able to find a µ > 0 and a solution ϑ to (L(β, µ)) which achieves E[XTϑ ] = , then this solution is also optimal to our original problem (PCVaR (β, )). One of our main results will be an equation determining this crucial Lagrangian multiplier. The following proposition shows that it always exists and is unique. Proposition 3.2. Let  > 0. If (PCVaR (β, )) admits an optimal strategy, then there exists a unique Lagrangian multiplier µ∗ > 0 such that the optimal value of (L(β, µ∗ )) is zero and attained by a strategy satisfying E [XT ] = . On the other hand, if for some µ > 0 (L(β, µ)) admits an optimal

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Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection strategy satisfying E [XT ] = , then the objective value of (L(β, µ)) is zero and this strategy is also optimal for (PCVaR (β, )). Rockafellar and Uryasev (2000, 2002) derived the following representation of the CVaR first for continuous random variables X and later for general distributions,  CVaRβ (X) = min γ + γ∈R

   1 + E (−X − γ) . 1−β

(3.1)

Utilizing (3.1), we can convert the Lagrangian problem (L(β, µ)) into the following equivalent form: 

inf ϑ,γ

   1 (1 + µ)γ + E 1−β + µ (−XT − γ)1{XT +γ<0} + µ(−XT − γ)1{XT +γ≥0} ,

s.t. Xt+1 = rt+1 Xt + ϑ0t+1 Pt+1 ,

(3.2)

t = 0, 1, · · · , T − 1.

Our scheme to solve the mean-CVaR portfolio selection problem starts with solving, for a fixed parameter γ, the following auxiliary problem, inf E ϑ



  1 + µ (−XT − γ)1{XT +γ<0} + µ(−XT − γ)1{XT +γ≥0} , 1−β

s.t. Xt+1 = rt+1 Xt +

ϑ0t+1 Pt+1 ,

(A(γ; β, µ))

t = 0, 1, · · · , T − 1.

Note that (A(γ; β, µ)) is an expected utility maximization problem with piecewise linear utility function and in particular time-consistent. It can thus be solved by a classical dynamic programming approach.

4

The auxiliary expected utility maximization problem

In this section, we solve the auxiliary expected utility maximization problem with a piecewise linear utility function, which we reformulate below as a maximization problem, sup E ϑ



  1 + µ (XT + γ)1{XT +γ<0} + µ(XT + γ)1{XT +γ≥0} , 1−β

s.t. Xt+1 = rt+1 Xt +

ϑ0t+1 Pt+1 ,

(A(γ; β, µ))

t = 0, 1, · · · , T − 1.

The utility function in (A(γ; β, µ)) is concave and depends on γ. In order to remove this dependence, we perform the following change of variables, Yt = ρ−1 t γ + Xt . For a given strategy ϑ = (ϑt )Tt=1 , we then have
−1 0 0 Yt+1 = ρ−1 t+1 γ + Xt+1 = rt+1 ρt γ + Xt + ϑt+1 Pt+1 = rt+1 Yt + ϑt+1 Pt+1 .

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Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

With this change of variables, the auxiliary problem (A(γ; β, µ)) can be transformed to   sup E CT YT 1{YT <0} + DT YT 1{YT ≥0} , ϑ

s.t. Yt+1 = rt+1 Yt + ϑ0t+1 Pt+1 ,

where CT =

1 1−β

(AE (γ; β, µ))

t = 0, 1, · · · , T − 1,

+ µ and DT = µ. The problem still depends on γ implicitly, as we have y0 = ρ−1 0 γ.

In order to have an objective which is differentiable in the decision variable we impose the following additional assumption on the return distribution. Assumption 4.1. For any t = 1, . . . , T , Pt is an absolutely continuous distributed random vector. Assumption 4.1 is not necessary for our solution scheme, but allows us to get more explicit results. We next impose a condition on the Lagrange multiplier µ, under which we can solve (AE (γ; β, µ)). Note that this is only a condition on the parameter µ of the family of auxiliary expected utility maximization problems (A(γ; β, µ)). It is not an extra assumption needed to solve the original meanCVaR portfolio selection problem (PCVaR (β, )), which under Assumption 2.4 is well-posed according to Proposition 2.6. Let µ be such that for any t = 0, . . . , T − 1, the following condition holds,   max E Ct+1 L0 Pt+1 1{L0 Pt+1 <0} + Dt+1 L0 Pt+1 1{L0 Pt+1 ≥0} < 0,

kLk=1

(C)

where the constants (Ct )Tt=0 and (Dt )Tt=0 are recursively computed by h i h i e 0 Pt+1 < 1 + Dt+1 P K e 0 Pt+1 ≥ 1 , Ct = Ct+1 P K t t h i h i b 0 Pt+1 < −1 + Dt+1 P K b 0 Pt+1 ≥ −1 , Dt = Ct+1 P K t t

for t = T − 1, . . . , 0, with boundary conditions CT = following system of equations,

1 1−β

(4.1)

b t ∈ Rn is a solution to the + µ, DT = µ, K

i h i i 0 = E Ct+1 Pt+1 1{K + D P 1 t+1 t+1 {K b 0 Pt+1 <−1} b 0 Pt+1 ≥−1} ,

(4.2)

h i i i 0 = E Ct+1 Pt+1 1{ K e 0 Pt+1 <1} + Dt+1 Pt+1 1{K e 0 Pt+1 ≥1} ,

(4.3)

t

t

e t ∈ Rn is a solution to the following system of equations, for i = 1, . . . , n, and K t

t

for i = 1, . . . , n. The solutions to the system of equations (4.2) and (4.3) exist under Condition (C) as we will show in the proof of the theorem below. We will refer to them as the optimal allocation vectors.

Remark 4.2. The piecewise linear objective induced by the mean-CVaR problem is not strictly concave

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Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

and we are thus not able to show uniqueness of the optimal allocation vectors. However, in all our numerical experiments, the optimal allocation vectors are unique.

The following theorem yields the optimal investment strategy and value function for (AE (γ; β, µ)). b t )T −1 Theorem 4.3. Suppose that µ satisfies Condition (C) and let the optimal allocation vectors (K t=1 e t )T −1 and recursive constants (Ct )T and (Dt )T be given by the recursion outlined above. and (K t=1 t=1 t=1 Then, the optimal strategy for (AE (γ; β, µ)) is given by ϑ∗t

=

(

b t−1 , rt Yt−1 K e t−1 , −rt Yt−1 K

if Yt−1 ≥ 0, if Yt−1 < 0,

(4.4)

for t = 1, . . . , T and the corresponding sequence of value functions Jt (Yt ) is Jt (Yt ) = Ct ρt Yt 1{Yt <0} + Dt ρt Yt 1{Yt ≥0} .

(4.5)

On the other hand, if µ fails to satisfy Condition (C), then the problem (AE (γ; β, µ)) is ill-posed in the sense that the investor would like to take an infinite position in the risky assets. We emphasize again that C0 = C0 (T ; β, µ) and D0 = D0 (T ; β, µ) depend on the investment horizon T , the confidence level β and the Lagrange multiplier µ, but not on γ. The above theorem thus in particular shows that the optimal amount invested in all risky assets is V -shaped as a function of the auxiliary variable Yt−1 and the value function of problem (AE (γ; β, µ)) is piecewise linear in γ. Theorem 4.3 already foretells the structure of the optimal solution to the original problem (PCVaR (β, )) as the solution will be corresponding to a particular Lagrangian multiplier µ and a particular γ. Theorem 4.3 also states that, when µ does not satisfy condition (C) and the left-hand side thereof would be non-negative, then the auxiliary problem (A(γ; β, µ)) is ill-posed in the sense that the investor would like to take an infinite position in the risky assets. Recall that we are looking for the unique Lagrangian multiplier µ∗ under which the optimal value of (L(β, µ)) is zero. Such a Lagrangian multiplier exists according to Proposition 3.2 and satisfies condition (C) by Theorem 4.3. For the remainder of this paper, we thus only consider Lagrangian multipliers µ which satisfy condition (C).

5

Optimal solution for the mean-CVaR portfolio selection problem

Recall that we discussed three distinct cases when introducing the Lagrangian multiplier, which we repeat here for the convenience of the reader: problem (L(β, µ)) is ill-posed; not investing in risky assets is the only optimal solution; or there are infinitely many optimal solutions all achieving the objective value zero.

10

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Due to the results developed in the last section, we can now describe these three cases analytically. Indeed, Theorem 4.3 yields that the value function of problem (A(γ; β, µ)) is given by vA(β,µ) (γ) = C0 γ1{γ<0} + D0 γ1{γ≥0} . If we plug this back into the minimization of the Lagrangian, problem (L(β, µ)) becomes inf γ

(1 + µ)γ − C0 γ1{γ<0} + D0 γ1{γ≥0}





.

(5.1)

Because (5.1) is piecewise linear in γ, the three different cases are given by: - Case 1: If D0 (T ; β, µ) > 1 + µ or C0 (T ; β, µ) < 1 + µ, then the problem is ill-posed. - Case 2: If D0 (T ; β, µ) < 1 + µ < C0 (T ; β, µ), then the unique optimal solution is γ = 0. - Case 3: If D0 (T ; β, µ) = 1 + µ, then every γ ≥ 0 is optimal and if C0 (T ; β, µ) = 1 + µ, then every γ ≤ 0 is optimal. Furthermore, by Krokhmal et al. (2002), the γ optimal for (3.1) will describe the VaRβ of the corresponding terminal wealth. Because the VaRβ of any terminal wealth is nonnegative due to Assumption 2.4, the unique Lagrangian multiplier identified in Proposition 3.2 must satisfy D0 (T ; β, µ) = 1 + µ. The scheme to obtain the optimal strategy for (PCVaR (β, )) is summarized in the following theorem. b t )T −1 and Theorem 5.1. Let µ∗ be the unique µ > 0 such that D0 (T ; β, µ) = 1 + µ and let (K t=0 e t )T −1 be computed according to (4.2) and (4.3) respectively. Then there exists a unique γ ≥ 0 such (K t=0 ∗ that E[XTϑ ] = , where ϑ∗ = (ϑ∗t )Tt=1 is the optimal strategy for (AE (γ; β, µ)) determined in Theorem 4.3. Furthermore, ϑ∗ is also optimal for (PCVaR (β, )). Let us discuss the intuition behind the optimal strategy for the mean-CVaR optimization problem. Theorem 5.1 shows that the optimal investment strategy for (PCVaR (β, )) is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a V -shaped pattern as a function of the current wealth. This is sensible in the context of mean-CVaR preferences. When the investor has been doing very well up to the current time, the wealth is large and the chance of falling into the region of worst cases (as seen from time zero) is small. Hence, the investor invests heavily into the assets providing a large excess return as he or she is close to risk neutral at this point. On the other hand, if the investor was unfortunate up to the current time the terminal wealth is already in the region of worst cases (again as viewed from time zero). Within this region, a position achieving a positive expected return increases the expected terminal wealth and at the same time reduces the expected tail loss. By investing heavily into the assets providing a large excess return, the investor can thus optimize return and risk simultaneously. The V -shaped pattern then comes naturally from the fact that both expected return and the CVaR are positive homogeneous functionals of the terminal wealth. A similar V -shaped pattern was also found to be optimal for the continuous-time setting studied in Gao et al. (2017).

11

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection When we denote by µ∗ the solution to D0 (T ; β, µ) = 1 + µ, then, under the optimal strategy, CVaRβ (XT∗ ) = µ∗ . Reintroducing a possibly nonzero initial wealth x0 , the mean-CVaR efficient frontier is thus given by CVaRβ (XT∗ ) = µ∗ ( − ρ0 x0 ) − ρ0 x0

(5.2)

for  ≥ ρ0 x0 . It is clear that µ∗ is in particular the unique trade-off between risk and return. Having a constant trade-off between risk and return is a natural consequence of using a coherent risk measure when there are no investment constraints as both the expectation and risk measure are positive homogeneous. Obtaining a unique and meaningful mean-CVaR efficient frontier stands in stark contrast to the continuous and complete market setting studied by Li and Xu (2013), He et al. (2015) and Gao et al. (2017). He et al. (2015) find that the mean-CVaR optimization problem in a continuous-time complete market is mostly ill-posed, see also Jin et al. (2005), who show that problems with a meandownside risk measure objective are generally ill-posed in complete, continuous-time markets. If it is well-posed, the mean-CVaR efficient frontier is a vertical line in the mean-CVaR plane and there is thus no connection between target expected terminal wealth and achieved risk level. Li and Xu (2013) and Gao et al. (2017) impose an upper bound on the terminal wealth, which leads to a well-posed portfolio selection problem and a non-linear tradeoff between risk and target expected return. It is however not clear why an investor would impose an upper bound on his or her terminal wealth level and how to select this upper bound. Moreover, the upper bound crucially determines the mean-CVaR efficient frontier and changing this somewhat arbitrary constraint on the terminal wealth thus leads to a different tradeoff between CVaR and target expected return. Analogously to the minimum variance set, which denotes the set of strategies such that there is no other strategy achieving the same expected return but a lower variance, one could also be interested in deriving a minimum CVaR set. To obtain the lower branch of the minimum CVaR set one would need to solve inf ϑ CVaRβ (XT ), s.t. E [XT ] ≤ , Xt+1 = rt+1 Xt + ϑ0t+1 Pt+1 ,

(PL−CVaR (β, )) t = 0, 1, · · · , T − 1,

for a target expected terminal wealth  ≤ x0 ρ0 . The solution scheme to solve (PL−CVaR (β, )) is very similar to that employed for solving (PCVaR (β, )). The differences are that the Lagrange multiplier would have to be negative and the auxiliary expected utility maximization problem has a decreasing, or inverse-V -shaped utility function, which however is still concave. So in order to obtain the lower branch of the minimum CVaR set one would need to find a µL ≤ 0 such that D0L (T ; β, µL ) = 1 + µL and would then have CVaRβ (XTL ) = µL ( − ρ0 x0 ) − ρ0 x0

(5.3)

12

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

for  < ρ0 x0 . Example 5.2. Consider a market with one risk-free asset paying an interest rate of r = 1.05 and three risky assets with i.i.d. lognormal returns having a mean and covariance structure given by 

 1.10   E[Rt ] =  1.11  , 1.12



 0.030 0.025 0.022   Cov(Rt ) =  0.025 0.050 0.055  . 0.022 0.055 0.075

(5.4)

The investment horizon is T = 4 and the confidence level β = 0.95. Our first task is to find the Lagrangian multiplier µ∗ satisfying D0 (T ; β, µ) = 1 + µ. For 10 different µ between an initial lower bound µ(0) and upper bound µ(0), we simulate the returns using a random number generator, test whether µ satisfies Condition (C) and, if it does, compute D0 (T ; β, µ). We then find the µ0 (0) where |D0 (T ; β, µ) − (1 + µ)| is smallest and update µ(1) = µ0 (0) − 1.5(µ(0) − µ(0))/10 and µ(1) = µ0 (0) + 1.5(µ(0) − µ(0))/10. Following this procedure of narrowing down the search interval eight times leads us to the Lagrange multiplier µ∗ = 1.0248 achieving |D0 (T ; β, µ) − (1 + µ)| = 1.09 × 10−4 in our simulation. We remark that this µ∗ in particular satisfies Condition (C). The corresponding optimal allocation vectors are calculated as         3.02 2.66 2.45 2.28  b   b   b   b0 =  K  −1.27  , K 1 =  −1.17  , K2 =  −1.13  , K3 =  −1.08  , 

2.45





2.25





2.12





2.03

 4.12 3.89 3.77 3.68  e   e   e   e0 =  K  −1.07  , K 1 =  −1.00  , K2 =  −0.97  , K3 =  −0.95  . 2.23 2.09 2.02 1.97

With a completely analogous scheme of narrowing down the search interval we find the negative Lagrange multiplier for the lower branch of the minimum CVaR set µL = −2.9617 achieving |D0 (T ; β, µ)− (1 + µ)| = 6.63 × 10−5 and the corresponding optimal allocation vectors 

 −1.68  bL =  K  0.34  , 0 −0.78   3.04  eL =  K  −0.74  , 0 1.57



 −1.66  bL =  K  0.34  , 1 −0.77   3.06  eL =  K  −0.75  , 1 1.58



 −1.65  bL =  K  0.34  , 2 −0.76   3.08  eL =  K  −0.75  , 2 1.60



 −1.63  bL =  K  0.34  , 3 −0.76   3.10  eL =  K  −0.76  . 3 1.61

We then compute the achieved mean-CVaR pair of the optimal strategy by simulating 10, 000 scenarios for the risky assets. We assume that the initial wealth is x0 = 1 and vary γ to trace out the minimum CVaR set. For the sake of comparison, we also compute the achieved mean-CVaR tradeoff for two alternative strategies. The first is a 1/N -type strategy which invests the same amount in each risky 1/N asset at any time. Specifically, this strategy is given by ϑt = (c, . . . , c) ∈ RN , t = 1, . . . , T for some 13

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection c ∈ R. We vary the amount c invested in the risky assets in order to achieve the corresponding target expected terminal wealth. The second alternative strategy keeps the insight of a V -shaped strategy being optimal, but invests equally in each risky asset. So this V -shaped alternative invests as the optimal b t and K e t replaced by all-one vectors (1, . . . , 1) ∈ RN normalized such that they strategy, but with K b t and K e t respectively. Figure 1 shows that the optimal strategy achieve the same expected return as K Figure 1: Mean-CVaR efficient frontier and tradeoff for alternative strategies

1.5 1.45 1.4

Mean

1.35 1.3 1.25 Optimal strategy V-shaped alternative 1/N-type alternative

1.2 1.15 1.1 -1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

CVaR Notes. This figure shows the mean-CVaR frontier generated by the optimal strategy, a V -shaped alternative strategy and a 1/N -shaped alternative strategy in a lognormal market with parameters given in (5.4).

dominates the two alternative strategies as expected. Interestingly, the V -shaped alternative performs almost as well as the optimal strategy, whereas the 1/N -type alternative is clearly doing much worse. We find this remarkable because it is optimal to short one of the risky assets and long the other two. The phenomena that a V -shaped alternative is performing almost as well as the optimal strategy can be observed for a range of parameter values for the risky assets and, not surprisingly, is even stronger when it is optimal to long all risky assets and the optimal strategy prescribes to hold a more balanced portfolio of risky assets. This example shows that, even if one does not know the exact distribution b t and K e t , the loss of the assets and/or is not able to exactly compute the optimal allocation vectors K 14

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

in efficiency is minimal as long as one keeps the crucial V -shaped pattern of the cumulated amount invested in the risky asset. The sampling distribution of the terminal wealth with mean  = 1.33 is reported in Figure 2. The column on the left shows the complete distribution, whereas the column on the right focuses on the tail distribution relevant for the CVaR. The distributions obtained by the optimal strategy and the V -shaped alternative are very similar: strongly positive skewed. This is intuitive; an attractive distribution for a mean-CVaR investor has a heavy right tail to contribute for the mean objective, but a light left tail to limit the risk measured by the CVaR. The superiority of the distribution generated by the optimal strategy over the distribution generated by the V -shaped alternative can be seen by looking at the taildistribution relevant for the CVaR. The optimal policy leads to a terminal wealth distribution with even lighter right tail than the one obtained by following the V -shaped alternative. The difference between the distributions is however small, consistent with the result of Figure 1. The distribution obtained by the 1/N -type alternative on the other hand is very different. It resembles a normal distribution and is symmetric around its mean. It is obvious from the picture showing the tail distribution that this results in a larger CVaR. Example 5.3. This examples serves to robusitfy the findings of Example 5.2 with regards to the return distribution. In particular, we consider a market setting exactly as in Example 5.2, but now assume that the returns follow a modified Log Student’s t-distribution and thus exhibit heavier tails. We normalize the returns such that the mean and correlation among the risky assets are as in (5.4). We then proceed according to exactly the same numerical procedure as in Example 5.2 to determine the optimal allocation vectors and then compute the achieved mean-CVaR frontiers of the optimal, V -shaped and 1/N -type strategies by again simulating 10, 000 scenarios for the risky assets. Figure 3 shows the mean-CVaR tradeoffs generated by the optimal, V -shaped and 1/N -type alternative strategies respectively, while Figure 4 reports the corresponding terminal wealth distributions. These figures confirm the main observations we discussed in Example 5.2: the optimal strategy dominates the two alternative strategies in terms of mean-CVaR performance; the V -shaped alternative strategy performs considerably better than the 1/N -type alternative and almost as well as the optimal strategy; the terminal wealth distributions under the optimal and V -shaped alternative strategy are strongly positive skewed and intuitive from a mean-CVaR perspective. On the other hand, the terminal wealth distribution under the 1/N -type alternative strategy is more symmetric, resembles a normal distribution and is thus not attractive for a mean-CVaR investor.

6

Time-consistency induced term structure of the CVaR

Forsyth (2019) shows that imposing a time-consistency constraint leads to poor performance compared to the pre-committed solution, and even compared to constant strategies. It is thus important to understand conditions under which the pre-committed solution remains optimal over time. To this end, we investigate an inverse portfolio selection problem, where we seek to determine how mean-CVaR preferences have to adapt in order for the pre-committed optimal strategy to remain optimal at any

15

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Figure 2: Simulated distribution of terminal wealth resulting from optimal mean-CVaR and alternative strategies Distribution of terminal wealth

Tail distribution relevant for the CVaR

500

140 Optimal strategy

450

Optimal strategy

120

350

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400

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140 V-shaped alternative

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1.1

Terminal wealth

Notes. This figure shows the simulated distribution of the terminal wealth generated by the optimal strategy, a V -shaped alternative strategy and a 1/N -shaped alternative strategy in a lognormal market with parameters given in (5.4). The left column shows the full distribution of the corresponding terminal wealth whereas the right column only shows the tail distribution relevant for the CVaR.

16

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Figure 3: Mean-CVaR efficient frontier and tradeoff for alternative strategies when the returns follow a log Student t-distribution

1.45

1.4

Mean

1.35

1.3

1.25 Optimal strategy V-shaped alternative 1/N-type alternative

1.2

1.15 -1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

CVaR Notes. This figure shows the mean-CVaR frontier generated by the optimal strategy, a V -shaped alternative strategy and a 1/N -shaped alternative strategy in a market where the returns follow a log-Student-t distribution.

point in time and for all realizations of the return process. In the previous section, we derived the pre-committed optimal strategy for minimizing the β0 -CVaR constrained by a requirement of achieving a specified target expected terminal wealth 0 , inf CVaRβ0 (XT |x0 ), ϑ

(PCV aR−0 (β0 , 0 ))

s.t. E [XT |x0 ] ≥ 0 ,

Xt+1 = rt+1 Xt + ϑ0t+1 Pt+1 ,

t = 0, 1, · · · , T − 1,

where we write β0 and 0 to emphasize that these two parameters correspond to the risk-level and target expected terminal wealth at the initial time zero. The optimal investment strategy is then given

17

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Figure 4: Simulated distribution of terminal wealth resulting from optimal mean-CVaR and alternative strategies when the returns follow a log Student t-distribution Tail distribution relevant for the CVaR

Distribution of terminal wealth 500

140 Optimal strategy

450

Optimal strategy

120

350

Number of simulations

Number of simulations

400

300 250 200 150

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140 V-shaped alternative

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1

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Notes. This figure shows the simulated distribution of the terminal wealth generated by the optimal strategy, a V -shaped alternative strategy and a 1/N -shaped alternative strategy in a market where the returns follow a log-Student-t distribution. The left column shows the full distribution of the corresponding terminal wealth whereas the right column only shows the tail distribution relevant for the CVaR.

18

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

by ϑ∗t

=

(


b rt Xt−1 + ρ−1 t−1 γ0 Kt−1 ,
e −rt Xt−1 + ρ−1 t−1 γ0 Kt−1 ,

if if


Xt−1 + ρ−1 t−1 γ0 ≥ 0,
Xt−1 + ρ−1 t−1 γ0 < 0,

b t−1 satisfies where γ0 is chosen such that the mean constraint is satisfied, K

h i i i 0 = E Ct+1 Pt+1 1{K b 0 Pt+1 <−1} + Dt+1 Pt+1 1{K b 0 Pt+1 ≥−1} , t

t

e t satisfies for i = 1, . . . , n, and K

h i i i 0 = E Ct+1 Pt+1 1{ K + D P 1 0 0 t+1 e Pt+1 <1} e Pt+1 ≥1} , t+1 {K t

t

for i = 1, . . . , n. The constants (Ct )Tt=0 and (Dt )Tt=1 are recursively given by CT = and h i h i e 0 Pt+1 < 1 + Dt+1 P K e 0 Pt+1 ≥ 1 , Ct = Ct+1 P K t t h i h i b 0 Pt+1 ≥ −1 , b 0 Pt+1 < −1 + Dt+1 P K Dt = Ct+1 P K t t

1 1−β0

+ µ0 , DT = µ0

for t = T − 1, . . . , 0, and, crucially, (1 + µ0 ) = D0 = D0 (T ; β0 , µ0 ). It is well known that the CVaR is not a time-consistent risk measure, i.e., if the investor revisits the following problem, inf CVaRβ0 (XT |Xτ ), ϑ

(PCV aR−τ (β0 , 0 ))

s.t. E [XT |Xτ ] ≥ 0 ,

Xt+1 = rt+1 Xt + ϑ0t+1 Pt+1 ,

t = τ, τ + 1, · · · , T − 1,

at an intermediate time 0 < τ < T , then the investors’ optimal investment policy derived from solving (PCV aR−τ (β0 , 0 )) would deviate from the optimal strategy determined at time zero. Indeed, because (Ct )Tt=0 and (Dt )Tt=0 are updated according to (4.1) and since Dt < Ct for any t = 1, . . . , T due to Condition (C), it holds that Dt−1 > Dt for any t = 1, . . . T and in particular 1 + µ0 = D0 > Dτ . Hence, the investor would have to take a larger Lagrangian multiplier, leading to different recursively  T −1 bt defined constants (Ctτ )Tt=τ and (Dtτ )Tt=τ , and in turn to different optimal allocation vectors K t=τ  T −1 e and Kt . t=τ Our goal in this section is to investigate how preferences have to change, such that the precommitted optimal strategy determined at time zero remains optimal at any intermediate time τ . Karnam et al. (2017) introduce the idea of a dynamic utility, under which the original time-inconsistent problem with fixed preferences becomes time-consistent. The following theorem shows that, in our setting, this dynamic utility remains within the class of mean-CVaR preferences. However, the confidence level β of the CVaR has to be updated as time evolves. We call this the time-consistency

19

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

induced term structure of the CVaR. In essence, one needs to adjust the local mean-CVaR preferences along the time axis in order to achieve global mean-CVaR efficiency. We remark that there are several alternative definitions of time-consistency in the literature on dynamic risk measures and refer to Acciaio and Penner (2011) or Bielecki et al. (2017) for an overview. Our notion of time-consistency is based on the preservation of optimality of an investment strategy across time. Theorem 6.1. The optimal strategy determined at time zero ϑ∗ remains optimal for the mean constrained CVaR minimization problem PCVaR−τ (βτ , τ ) at an intermediate time 0 < τ < T , if there is a Lagrange multiplier µτ such that (βτ , µτ ) satisfy 1 + µτ = D0 (T − τ ; βτ , µτ ) , 1 − β0 µτ = , µ0 1 − βτ

(6.1) (6.2)

 ∗  ∗ and the target expected terminal wealth is updated as τ = E XTϑ Xτ , where XTϑ is the terminal wealth when following the strategy ϑ∗ . A similar result was obtained in the context of dynamic, CVaR-based hedging in Godin (2016). Different from our setting, there is no mean-constraint therein and the constructed sequence of risk measures is obtained by fixing the auxiliary parameter in the dual representation of the CVaR. Our result in Theorem 6.1 on the other hand allows us to remain within the original mean-CVaR preference structure and leads to a more transparent economic interpretation in terms of an induced dynamic confidence level as we will further discuss below. Theorem 6.1 shows us how we can compute the time-consistency induced dynamic confidence level βτ and update the target expected terminal wealth τ . Whereas the updated target expected terminal wealth depends on the realized state, the confidence level does not. The following theorem shows that the time-consistency induced confidence level of the CVaR is increasing in time and that the target expected terminal wealth is increasing in the realized wealth at an intermediate time. Theorem 6.2. Suppose that there exist solutions (βτ , µτ ) to (6.1) and (6.2) for τ = 0, . . . , T − 1. Then the following hold: i) The time-consistency induced dynamic confidence level βτ is increasing in τ = 0, . . . , T − 1. ii) The updated target expected terminal wealth τ is increasing in the wealth Xτ realized at time τ . From the proof of Theorem 6.2 one can observe that also µτ is increasing in time. If one would simply keep the confidence level fixed at β0 at an intermediate time τ and denote by µ ˜τ the solution to D0 (T − τ ; β0 , µ ˜τ ) = 1 + µ ˜τ , then obviously µ ˜τ ≥ µ0 , because there are fewer remaining trading dates and µ represents the tradeoff between return and risk measured by the CVaR at the same confidence level. Because the confidence level is increasing, we have that µτ ≥ µ ˜τ ≥ µ0 , so the increasing tradeoff between return and risk is on one hand due to fewer remaining trading opportunities, on the other hand because of a change in the risk measure. Theorem 6.2 shows that a pre-committed mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR who re-optimizes at every point in time. 20

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

This means that, besides updating the target expected terminal wealth, the mean-CVaR investor would also perceive risk as the expected loss in an ever smaller range of the terminal distribution. Note however that updating the target expected terminal wealth is crucial because it is the updated target expected terminal wealth which ultimately forces the optimal strategy to remain within the V shaped structure as determined at time zero, whereas the confidence level of the CVaR only determines the distribution among the risky assets. We have seen in Example 5.2 that even when the distribution among the risky assets is anything but optimal, the loss in efficiency is small as long as the V -shaped structure is preserved. So one could expect that if one would only update the target expected terminal wealth, but not the confidence level for the CVaR, then the precommited strategy would be inefficient at an intermediate time, but the loss of inefficiency would likely to be very small. Although providing general conditions under which a solution (βτ , µτ ) to (6.1) and (6.2) exist remains a challenging open problem, the following example shows the existence in the setting of Example 5.2. Example 6.3. This example shows the time-consistency induced term structure of the CVaR in the market setting of Example 5.2. To determine the term structure, we need to solve equations (6.1) and (6.2). We do this numerically using a search approach similar to the one utilized to find the Lagrangian multiplier in Example 5.2. We consider a number of β equidistantly distributed between a 0 lower bound β L and β U . For each β in this interval we define µ(β) = µ0 1−β 1−β such that equation (6.2) is satisfied exactly. We next compute |D0 (T − τ ; β, µ(β)) − (1 + µ(β))| for each β and update our β τ to be the one where this value is smallest. For the next iteration we consider a smaller interval around the current optimal β τ . Using this methodology we numerically compute the following time-consistency induced term structure of the CVaR β0 = 0.9500,

β1 = 0.9673,

β2 = 0.9805,

β3 = 0.9900.

µ2 = 2.6303,

µ3 = 5.1315.

The corresponding Lagrange multipliers are given by µ0 = 1.0248,

7

µ1 = 1.5690,

Mean-CVaR optimization and the equity premium puzzle

In this section, we apply our results on mean-CVaR portfolio selection to one of the long-standing puzzles in financial economics: the equity premium puzzle. This puzzle, first described by Mehra and Prescott (1985), refers to the empirical fact that historical average returns of stocks dramatically outperform those of bonds. The outperformance is to such an extent that it is difficult for classical theories to explain why people in the market still diversify their investments between bonds and stocks, cf. also Mehra and Prescott (2003). Some success in explaining the equity premium puzzle has been made in the field of behavioral finance. Benartzi and Thaler (1995) show that, under a model with loss-aversion and mental accounting, the evaluation period where bonds and stocks result in the same prospect theory value is approximately one year, and that with this time horizon an optimal portfolio 21

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

for prospect theory preferences would be roughly balanced between stocks and bonds. Amongst many, some notable works using behavioral preferences to explain the equity premium puzzle are Campbell and Cochrane (1999), who use a model with habit formation, and Barberis et al. (2001), where the investor obtains utility from consumption as well as a gain-loss utility from the holdings in risky assets. More recently, De Giorgi and Post (2011) study a behavioral portfolio selection problem with an endogenously determined, state-dependent reference point. They apply their model to historical data in order to investigate whether it can explain the equity premium puzzle and conclude that the reference point must contain an important exogenous component. Our objective is to follow the approach of Benartzi and Thaler (1995) and De Giorgi and Post (2011) and examine whether diversification across bonds and stocks is optimal for a mean-CVaR investor. For this purpose, we consider the same setting and historical return statistics as in De Giorgi and Post (2011). The market under consideration contains three assets. One-month U.S. Treasury bills, termed Bills, serve as the risk-free asset. As risky assets we consider 10-year constant maturity U.S. Treasury bonds (Bonds) and the S&P 500 stock market index (Stocks). We suppose the investor rebalances his or her portfolio every month and, as in Benartzi and Thaler (1995), consider a time horizon of one year, i.e., T = 12. The confidence level for the CVaR is β = 0.95. Between July 1926 and June 2010, Bills yielded an average monthly return of r = 1.003, which we take as the risk-free return. In the same time period Bonds and Stocks had a mean return and covariance structure of returns given by E[Rt ] =

1.0044 1.0088

!

,

Cov(Rt ) =

0.0005 0.0001 0.0001 0.0021

!

.

(7.1)

We refer to De Giorgi and Post (2011) for more information on the historical data. Using exactly the same methodology of narrowing down the search interval for the Lagrangian multiplier µ∗ as in Example 5.2 we find that µ∗ = 1.1556 and the optimal allocation vectors are given by b0 = K

b4 = K

b8 = K

! 7.68 , 10.09 ! 6.76 , 9.05 ! 6.15 , 8.36

b1 = K

b5 = K

b9 = K

7.37 9.78 6.61 8.83 6.05 8.21

!

!

!

b2 = , K

b6 = , K

b 10 = , K

7.18 9.49 6.46 8.65 5.93 8.07

!

!

b3 = , K

b7 = , K

!

b 11 = , K

6.96 9.25 6.28 8.49

!

!

5.77 7.97

,

,

!

22

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

and e0 = K

e4 = K

e8 = K

11.54 12.59 11.08 12.03 10.86 11.76

!

!

!

e1 = , K

e5 = , K

e9 = , K

11.39 12.41 11.01 11.94 10.82 11.71

!

11.26 12.26

e2 = , K

!

!

!

10.96 11.87

e6 = , K

!

10.79 11.68

e 10 = , K

e3 = , K

e7 = , K

!

11.16 12.14 10.90 11.81

e 11 = , K

!

,

!

10.75 11.64

,

!

.

We can already observe that the amount invested in Bonds and Stocks is roughly equal. To confirm this observation, we simulate 10, 000 scenarios and consider the average percentage allocation to each asset class for different target expected terminal wealth levels. The target returns chosen below approximately correspond to a (2/3 Bills and 1/3 Stocks), (1/2 Bills and 1/2 Stocks) and (1/3 Bills and 2/3 Stocks) allocation respectively. Table 1 shows that a mean-CVaR investor holds a balanced Table 1: Average asset class allocation for different target expected terminal wealth levels of a meanCVaR investor Average allocation to Bills Bonds Stocks

 = 1.065

 = 1.075

 = 1.085

43.07% 24.30% 32.63%

25.26% 31.90% 42.84%

08.59% 39.01% 52.40%

Notes. The table shows the average allocation to the asset classes Bills, Bonds and Stocks of the optimal mean-CVaR strategy for different target returns in a lognormal market with parameters given in (7.1).

portfolio on average which could very well be observed in practice. To provide further information on typical trading behavior of a mean-CVaR investor we show the 10th percentile, median and 90th percentile of the investment allocation to each asset class over the whole investment period in Figure 5. The solid line representing the median allocation shows that, typically, a mean-CVaR investor holds a balanced portfolio throughout the period and increases the allocation to the risk-free asset as the time-horizon approaches conforming with common investment advice. We also observe that, at all percentiles shown in Figure 5, the allocation to Bonds and Stocks is roughly equal. The mean-CVaR tradeoff of a non-constant return R is defined to be E [R] − r . CVaRβ (R) + r Note that the denominator corresponds to CVaRβ (R) − CVaRβ (r). In analogy to the Sharpe ratio, the mean-CVaR tradeoff as defined above describes the slope of the line connecting the points in the mean-CVaR plane generated by the risky return R and the risk-free return r. De Giorgi and Post 23

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Figure 5: Optimal allocation to different asset classes for a mean-CVaR investor

150%

Percentage Allocation

100%

50%

0%

10th Percentile Bills 10th Percentile Bonds 10th Percentile Stocks Median Bills Median Bonds Median Stocks 90th Percentile Bills 90th Percentile Bonds 90th Percentile Stocks

-50%

-100%

0

1

2

3

4

5

6

7

8

9

10

11

time t Notes. This figure shows the 10th percentile, median and 90th percentile of the allocations to Bills, Bonds and Stocks throughout the investment horizon in a lognormal market with parameters given in (7.1) The target expected terminal wealth level is  = 1.075.

(2011) only consider four different strategies: pure Bills, Bonds and Stocks strategies where everything is invested in the respective asset class and a 50/50 mixture of Bonds and Stocks termed Mixtures. Figure 6 shows that all those strategies are clearly dominated by the optimal strategy in terms of mean-CVaR tradeoff with the exception of the Bills strategy which trivially is mean-CVaR efficient. However, among the strategies investing in risky assets, Mixtures achieves a mean-CVaR tradeoff of 0.3355 and is thus superior to Bonds, which achieve a mean-CVaR tradeoff of 0.1327 and in particular also to Stocks with a mean-CVaR tradeoff of 0.3214. A mean-CVaR investor only considering those strategies would thus hold the balanced Mixture portfolio if wanting to invest in risky assets at all. The better performance of the Mixtures strategy in terms of mean-CVaR tradeoff is not due to the dynamic nature of the portfolio selection problem and robust with respect to the length of the time interval of the historical returns. Table 2 shows the single-period mean-CVaR tradeoff at a β = 0.95

24

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Figure 6: Mean-CVaR frontier and tradeoff for bills, bonds, stocks and mixtures

1.25 Optimal strategy Bills Bonds Stocks Mixtures

1.2

Mean

1.15

1.1

1.05

1 -1.05

-1

-0.95

-0.9

-0.85

-0.8

-0.75

CVaR Notes. This figure shows the mean-CVaR frontier generated by the optimal strategy as well as the mean-CVaR trade-off obtained by the strategies Bills, Bonds, Stocks and Mixtures in a lognormal market with parameters given in (7.1).

confidence level achieved by the returns of Bonds, Stocks and Mixtures and contrasts this tradeoff with the expected log-utility value achieved by the corresponding strategies. We can observe that, for any time horizon, Mixtures outperform both Bonds and Stocks in terms of mean-CVaR performance whereas Stocks achieve a better expected log-utility value than Bonds or Mixtures. The latter observation simply describes the equity premium puzzle. The dominance of Stocks over Mixtures in the expected utility framework is robust with respect to the utility function and would be even stronger for a power utility of the form u(x) = X α /α with α ∈ (0, 1). The superior performance of Mixtures over Bonds and Stocks in terms of mean-CVaR tradeoff on the other hand offers a potential explanation for diversification. To summarize, we have found that it is optimal for a mean-CVaR investor to hold a well diversified portfolio for historical Bill, Bond and Stock returns, both under the dynamic model which is the focus of this paper as well as a single-period analysis of the mean-CVaR tradeoff achieved by the Bonds,

25

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

Table 2: Performance of Bonds, Stocks and Mixtures from a mean-CVaR and expected log-utility perspective

Time horizon

Mean-CVaR tradeoff Bonds Stocks Mixtures

Expected utility value Bonds Stocks Mixtures

1 month 3 months 6 months 12 months 18 months

0.0338 0.0547 0.0791 0.1125 0.1420

0.0042 0.0125 0.0250 0.0495 0.0742

0.0677 0.1163 0.1786 0.2749 0.3663

0.0735 0.1256 0.1904 0.2853 0.3750

0.0077 0.0233 0.0466 0.0932 0.1385

0.0062 0.0191 0.0382 0.0767 0.1140

Notes. The table shows the performance of the strategies Bonds, Stocks and Mixtures for the mean-CVaR and expected log utility objective for different time horizons in a lognormal market with parameters given in (7.1).

Stocks and Mixtures strategies suggested by De Giorgi and Post (2011). The intuitive explanation for this is that, when risk is measured by the conditional value-at-risk, the investor is particularly concerned about the average wealth in the worst scenarios. Because of the higher likelihood of a major drop, stocks are required to have a considerable excess return over bonds in order to compensate for their inherent risk as measured by the CVaR and remain attractive for investors. We remark that this conclusion is only suggestive for the role risk measured by CVaR might play to help explaining the equity premium puzzle in the same sense as Benartzi and Thaler (1995) or De Giorgi and Post (2011). We do not derive the historical equity premium in the sense of an asset pricing model, but only show that optimal strategies of a mean-CVaR investor are consistent with real world behavior for historical returns. It remains an interesting open question whether one is able to derive historically observed returns in a CVaR-based asset pricing model. We leave this question as a future research direction.

8

Conclusions

We solved the mean-CVaR portfolio optimization problem in an incomplete, discrete-time setting by invoking the Lagrangian relaxation of the problem and using the dual representation of the CVaR of Rockafellar and Uryasev (2000). The cumulated amount invested in risky assets is a V -shaped function of the current wealth and the mean-CVaR efficient frontier is a straight line in the meanCVaR plane. Whereas He et al. (2015) find that mean-CVaR portfolio selection in a continuous and complete market setting is often ill-posed and, even if not, displays some pathological features, our findings suggest that a mean-CVaR objective leads to intuitive and sensible investment behavior where there is a real trade-off between risk and return. So while He et al. (2015) conclude that their paper economically is a critique of using the CVaR on terminal wealth only to model risk for portfolio choice, our findings show that the CVaR can be a reasonable risk measure for mean-risk portfolio optimization in incomplete markets. Contingent to technical conditions, we solved the inverse investment problem of how mean-CVaR preferences have to adapt such that the pre-committed optimal strategy remains optimal as time

26

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

evolves, thus achieving global mean-CVaR optimality. The time-consistency induced term structure of the CVaR was found to have an increasing confidence level with respect to time. Finally, we applied our mean-CVaR optimization model to the equity premium puzzle, one of the long-standing puzzles in financial economics. We showed that a mean-CVaR investor typically holds a balanced portfolio of bills, bonds and stocks when faced with historical return statistics. This finding suggests that the large equity premium demanded by investors might be due to the larger tail risk inherent to stocks. We close this paper with suggesting two main directions for future research. First, it remains an open and challenging problem to incorporate economically relevant investment constraints for discretetime mean-CVaR portfolio selection. The second direction is to develop a CVaR-based asset pricing model and investigate whether such a model is able to confirm the risk-based explanation of the equity premium puzzle suggested herein.

Acknowledgments The authors thank Tomasz Bielecki, Enrico De Giorgi, Xuedong He, Martin Herdegen, Birgit Rudloff, Ke Zhou, Xunyu Zhou, and two anonymous referees for their very useful comments and suggestions. The research of the second author is partially supported by Hong Kong Research Grants Council under project number 11200219.

Appendix. Proofs Proof of Proposition 2.6
Let ϑ = (ϑt )Tt=1 be any strategy with P[XTϑ > 0] > 0. Assumption 2.4 implies that CVaRβ XTϑ ≥
VaRβ XTϑ > 0.

Proof of Proposition 3.1

Let ϑ be such that E[XTϑ ] >  and set c = /E[XTϑ ]. Then 0 < c < 1 and with ϑˆt = cϑt+1 , ˆ t = 0, . . . , T − 1, one has E[XTϑ ] = . The result follows immediately by Assumption 2.4.

Proof of Proposition 3.2 Let v be the optimal value of (PCVaR (β, )) and V (µ) be the optimal value of (L(β, µ)). Then v > 0 by Assumption 2.4. Set µ∗ := v . Clearly V (µ∗ ) ≤ 0. If V (µ∗ ) < 0 then there is a strategy ϑ such that CVaRβ (XTϑ ) − µ∗ E[XTϑ ] < 0. This implies that E[XTϑ ] > 0 again by Assumption 2.4. We define the ˆ ˆ strategy (ϑˆt )Tt=1 by ϑˆt := E[X ϑ ] ϑt , t = 1, . . . , T . Then E[XTϑ ] =  and CVaRβ (XTϑ ) < v contradicting T the definition of v. Uniqueness of µ∗ is obvious. For the other direction, let µ > 0 and suppose that (L(β, µ)) admits an optimal strategy ϑ∗  ∗ satisfying E XTϑ = . The objective value of (L(β, µ)) is nonpositive since the strategy which never invests in risky assets achieves the objective value zero. If the objective value were negative, then cϑ∗ 27

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection would achieve a smaller objective value for any c > 1 in contradiction to the optimality of ϑ∗ . Hence the objective value of (L(β, µ) is zero. Clearly, ϑ∗ is also optimal for (PCVaR (β, )).

Proof of Theorem 4.3 At terminal time T , the value function JT coincides with the objective function. When we show that b t )T −1 and (K e t )T −1 exist as solutions to the equations (4.2) and (4.3) respectively the constants (K t=0 t=0 and, given Jt (Yt ) as above, the optimal strategy ϑ∗t and subsequent value function Jt−1 (Yt−1 ) at time t − 1 are as claimed, then the theorem follows by Bellman’s principle of optimality and backward induction. We fix a t ∈ {1, . . . , T } and distinguish the three cases Yt−1 > 0, Yt−1 < 0 and Yt−1 = 0. Let us start with Yt−1 > 0. We want to choose ϑt in order to maximize     E Jt (Yt ) Ft−1 = E Ct ρt Yt 1{Yt <0} + Dt ρt Yt 1{Yt ≥0} Ft−1 h
= E Ct ρt rt Yt−1 + ϑ0t Pt 1{rt Yt−1 +ϑ0t Pt <0} i
+Dt ρt rt Yt−1 + ϑ0t Pt 1{rt Yt−1 +ϑ0t Pt ≥0} Ft−1 h i

= ρt−1 Yt−1 E Ct 1 + K0t−1 Pt 1{K0 Pt <−1} + Dt 1 + K0t−1 Pt 1{K0 Pt ≥−1} , t−1 t−1

where we set ϑt = rt Yt−1 Kt−1 for an arbitrary Kt−1 and drop the conditioning on the filtration due to the independence of the returns in the last step. We show that Condition (C) implies that an optimal b t−1 to the last expectation, which is independent of Yt−1 , exists by conditioning on the length k of K Kt−1 . h
sup E Ct 1 + K0t−1 Pt 1{K0

Kt−1

i
0 + D 1 + K P 1 t t−1 t {K0t−1 Pt ≥−1} t−1 Pt <−1} i h ≤ E Ct 1{K0 Pt <−1} + Dt 1{K0 Pt ≥−1} t−1 t−1    + sup k max E Ct L0 Pt 1{L0 Pt <−1/k} + Dt L0 Pt 1{L0 Pt ≥−1/k} kLk=1 k>0     Ct 0 0 , ≤ Ct + sup k max E Ct L Pt 1{L0 Pt <0} + Dt L Pt 1{L0 Pt ≥0} + k kLk=1 k>0

where we use the fact of Ct > Dt for the second inequality, which holds by virtue of Condition (C). Again due to Condition (C), the term within the parentheses becomes negative for sufficiently large k. Hence we can limit ourself to a compact set and the optimal solution thus exists due to the continuity of the objective. This implies that x 7→ Ct x1{x<0} + Dt x1{x≥0} is concave. Therefore, the objective is concave as a composition between a concave and an affine function and hence that first b t−1 to be optimal. K b t−1 thus satisfies and can be order conditions are necessary and sufficient for K determined via the system of equations (4.2). On the other hand, if Condition (C) would not hold, then there exists an L0 ∈ RN with kL0 k = 1

28

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

and h i E Ct+1 L00 Pt+1 1{L0 Pt+1 <0} + Dt+1 L00 Pt+1 1{L0 Pt+1 ≥0} ≥ 0. 0

0

Consider the strategy Kt−1 = kL0 for k > 0 and let 0

h

kL00 Pt




kL00 Pt




i

g (k) : = E Ct 1 + 1{kL0 Pt <−1} + Dt 1 + 1{kL0 Pt ≥−1} 0 0     h i 1 1 = Ct P L00 Pt < − + Dt P L00 Pt ≥ − + kE Ct L00 Pt 1{L0 Pt <− 1 } + Dt L00 Pt 1{L0 Pt ≥− 1 } . 0 0 k k k k Let f 0 denote the pdf of L00 Pt . Differentiating with respect to k yields   h i 1 1 d 0 0 0 0 g (k) = (Ct − Dt )f − + E C L P 1 + D L P 1 1 1 0 0 t t t t 0 0 {L0 Pt <− k } {L0 Pt ≥− k } dk k k2       −1 0 1 1 −1 0 1 1 + k Ct − Dt f − f − k k k2 k k k2 i h = E Ct L00 Pt 1{L0 Pt <− 1 } + Dt L00 Pt 1{L0 Pt ≥− 1 } 0 0 k h ik 0 0 ≥ E Ct L0 Pt 1{L0 Pt <0} + Dt L0 Pt 1{L0 Pt ≥0} 0

0

≥ 0.

Therefore, the investor would like to hold an infinite number of the risky assets represented by L0 .

For the remainder of this proof we assume that µ satisfies Condition (C), plug the optimal strategy back into the objective and compute h   b 0 Pt 1 b 0 Jt−1 (Yt−1 ) = ρt−1 Yt−1 E Ct 1 + K t−1 {K

t−1

= Dt−1 ρt−1 Yt−1 ,

  b 0 Pt 1 b 0 + D 1 + K t t−1 Pt <−1} {K

}

t−1 Pt ≥−1

i

b t−1 and the updating rule for the constant where we use the equations in (4.2) which determine K Dt−1 in equation (4.1). The case of Yt−1 < 0 is very similar and is thus omitted. It remains to argue that it is optimal to invest everything into the risk-free account when Yt−1 = 0. In this case ϑt is chosen such that h i E [Jt (Yt )] = E Ct ρt ϑ0t Pt 1{ϑ0t Pt <0} + Dt ρt ϑ0t Pt 1{ϑ0t Pt ≥0} |Ft−1 is maximized. If the investor does not invest in risky assets, the objective value is zero. On the other

29

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection hand, if ϑt ∈ Rn \ {0}, then

h i ρt E Ct ϑ0t Pt 1{ϑ0t Pt <0} + Dt ϑ0t Pt 1{ϑ0t Pt ≥0}   ϑt 0 ϑt 0 o o n n = ρt kϑt kE Ct Pt 1 ϑt 0 P <0 + Dt Pt 1 ϑt 0 P ≥0 t t kϑt k kϑt k kϑt k kϑt k <0

due to Condition (C).

Proof of Theorem 5.1 This follows by the arguments given above. The µ∗ solving D0 (T ; β, µ) = 1 + µ is unique due to Proposition 3.2. Combining the definition of Yt with (4.4) yields that the optimal terminal wealth is ∗ linear in γ. Therefore, there exists a unique γ ≥ 0 such that E[XTϑ ] = . The terminal wealth optimal for (AE (γ; β, µ)) will always achieve a positive expected return as a solution to an expected utility maximization problem with an increasing and concave utility function.

Proof of Theorem 6.1 Let (βτ , µτ ) satisfy equations (6.1) and (6.2). We define the constants (Ctτ )Tt=τ and (Dtτ )Tt=τ exactly as in (4.1). Because of (6.2), we have CTτ 1/(1 − βτ ) + µτ 1/(1 − β0 ) + µ0 CT . = = = τ DT µτ µ0 DT bτ b eτ e We thus find that K T −1 = KT −1 and KT −1 = KT −1 , and thus also Csτ Dsτ

CTτ −1 τ DT −1

=

CT −1 DT −1 .

By backwards

Cs bτ = K b s, K eτ = K e s and thus also induction, we have K =D for any s = τ, . . . , T − 1. By equation s s s (6.1) µτ satisfies the equation for the Lagrange multiplier. γτ is chosen such that the mean constraint  τ  E XTϑ Xτ = τ is satisfied. By the choice of τ , the optimal strategy determined at time zero ϑ∗ , and thus also the original γ0 , satisfies this constraint.

Proof of Theorem 6.2 We first show that βτ is increasing in τ . Let τ ∈ {1, . . . , T − 1} and note that it is sufficient to show  T −1  T −1 et bt do not depend on whether they are computed at that βτ > β0 . Recall that K and K t=τ t=τ time zero with respect to the time horizon T and parameters β0 and µ0 or at time τ with time horizon Cτ CT T − τ and parameters βτ and µτ by Theorem 6.1. Since D = DTτ we have T T

DTτ −1 =

h i h i τ DTτ b 0 PT < −1 + DT DT P K b 0 PT ≥ −1 = Dτ DT −1 . CT P K T −1 T −1 T DT DT DT

30

Strub, Li, Cui and Gao: Discrete-Time Mean-CVaR Portfolio Selection

By induction one obtains Dττ = DTτ

Dτ DT

and therefore 1 + µ0 = µ0

Dτ 1 + µτ D0 > µ0 ττ = µ0 . DT DT µτ

It follows that µτ > µ0 and thus also βτ > β0 by virtue of (6.2).  ∗  In order to show that τ is increasing in the realized wealth Xτ , we need to prove that E XTϑ X τ <  ∗  E XTϑ X τ whenever X τ < X τ . Fix an initial wealth x0 and denote the optimal strategy for (PCVaR (β, )) by ϑ∗ = (ϑ∗t )Tt=1 with corresponding γ. Let τ ∈ {1, . . . , T − 1} and X τ < X τ . Set −1 Y τ = ρ−1 τ γ + X τ and Y τ = ρτ γ + X τ , consider sup E [JT (YT )] s.t. Yt+1 = rt+1 Yt + ϑ0t+1 Pt+1 ,

t = τ, · · · , T − 1,

for Yτ = Y τ and Yτ = Y τ and denote the corresponding optimal strategies by ϑ = (ϑt )Tt=τ +1 and ϑ = (ϑt )Tt=τ +1 and the corresponding optimal solution processes by (Y t )Tt=τ and (Y t )Tt=τ respectively. Write ZT = Y T − Y T . Because JT is subadditive and by Jensen’s inequality we have 
 E JT Y T = E [JT (Y T + ZT )] ≤ E [JT (Y T )] + E [JT (ZT )] ≤ E [JT (Y T )] + JT (E [ZT ]) .


 Because E JT Y T > E [JT (Y T )], this implies that E[ZT ] > 0.

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