• Introduction

• Warm-Up: Markowitz’s Portfolio

  • Signal model
  • Markowitz’s formulation
  • Drawbacks of Markowitz portfolio

• Alternative Measures of Risk: DR, VaR, CVaR, and DD

• Mean-DR portfolio

• Mean-CVaR portfolio

• Mean-DD portfolio

• Conclusions

Markowitz’s portfolio has never been fully embraced by practitioners, among other reasons because

1. variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses: the solution is to use alternative measures for risk, e.g., VaR and CVaR,

2. it is highly sensitive to parameter estimation errors (i.e., to the covariance matrix \(\boldsymbol{\Sigma}\) and especially to the mean vector \(\boldsymbol{\mu}\)): solution is robust optimization and improved parameter estimation,

3. it only considers the risk of the portfolio as a whole and ignores the risk diversification (i.e., concentrates risk too much in few assets, this was observed in the 2008 financial crisis): solution is the risk parity portfolio.

👉 We will here consider more meaningful measures for risk than the variance, like the downside risk (DR), Value-at-Risk (VaR), Conditional VaR (CVaR) or Expected Shortfall (ES), and drawdown (DD).

• Let us denote the log-returns of \(N\) assets at time \(t\) with the vector \(\mathbf{r}_{t}\in\mathbb{R}^{N}\) (i.e., \(r_{it}=\log{p_{i,t}}-\log{p_{i,t-1}}\)).

• Note that the log-returns are almost the same as the linear returns \(R_{it}=\frac{p_{i,t}-p_{i,t-1}}{p_{i,t-1}}\), i.e., \(r_{it}\approx R_{it}\).

• The time index \(t\) can denote any arbitrary period such as days, weeks, months, 5-min intervals, etc.

• \(\mathcal{F}_{t-1}\) denotes the previous historical data.

• Econometrics aims at modeling \(\mathbf{r}_{t}\) conditional on \(\mathcal{F}_{t-1}\).

• \(\mathbf{r}_{t}\) is a multivariate stochastic process with conditional mean and covariance matrix denoted as (Feng and Palomar 2016) \[\begin{aligned} \boldsymbol{\mu}_{t} &\triangleq\textsf{E}\left[\mathbf{r}_{t}\mid\mathcal{F}_{t-1}\right]\\ \boldsymbol{\Sigma}_{t} &\triangleq\textsf{Cov}\left[\mathbf{r}_{t}\mid\mathcal{F}_{t-1}\right]=\textsf{E}\left[(\mathbf{r}_{t}-\boldsymbol{\mu}_{t})(\mathbf{r}_{t}-\boldsymbol{\mu}_{t})^{T}\mid\mathcal{F}_{t-1}\right]. \end{aligned}\]

• For simplicity we will assume that \(\mathbf{r}_{t}\) follows an i.i.d. distribution (which is not very innacurate in general).

• That is, both the conditional mean and conditional covariance are constant: \[\begin{aligned} \boldsymbol{\mu}_{t} &= \boldsymbol{\mu},\\ \boldsymbol{\Sigma}_{t} &= \boldsymbol{\Sigma}. \end{aligned}\]

• Very simple model, however, it is one of the most fundamental assumptions for many important works, e.g., the Nobel prize-winning Markowitz portfolio theory (Markowitz 1952).

• Consider the i.i.d. model: \[\mathbf{r}_{t}=\boldsymbol{\mu}+\mathbf{w}_{t},\] where \(\boldsymbol{\mu}\in\mathbb{R}^{N}\) is the mean and \(\mathbf{w}_{t}\in\mathbb{R}^{N}\) is an i.i.d. process with zero mean and constant covariance matrix \(\boldsymbol{\Sigma}\).

• The mean vector \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\) have to be estimated in practice based on \(T\) observations.

• The simplest estimators are the sample estimators:

  • sample mean: \(\quad\hat{\boldsymbol{\mu}} =\frac{1}{T}\sum_{t=1}^{T}\mathbf{r}_{t}\)
  • sample covariance matrix: \(\quad\hat{\boldsymbol{\Sigma}} =\frac{1}{T-1}\sum_{t=1}^{T}(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}})(\mathbf{r}_{t}-\hat{\boldsymbol{\mu}})^{T}.\)

• Many more sophisticated estimators exist, namely: shrinkage estimators, Black-Litterman estimators, etc.

• The parameter estimates \(\hat{\boldsymbol{\mu}}\) and \(\hat{\boldsymbol{\Sigma}}\) are only good for large \(T\), otherwise the estimation error is unacceptable.

• For instance, the sample mean is particularly a very inefficient estimator, with very noisy estimates (Meucci 2005).

• In practice, \(T\) cannot be large enough due to either:

  • unavailability of data or
  • lack of stationarity of data.

• As a consequence, the estimates contain too much estimation error and a portfolio design (e.g., Markowitz mean-variance) based on those estimates can be severely affected (Chopra and Ziemba 1993).

• Indeed, this is why Markowitz portfolio and other extensions are rarely used by practitioners.

• Suppose the capital budget is \(B\) dollars.

• The portfolio \(\mathbf{w}\in\mathbb{R}^{N}\) denotes the normalized dollar weights of the \(N\) assets such that \(\mathbf{1}^{T}\mathbf{w}=1\) (so \(B\mathbf{w}\) denotes dollars invested in the assets).

• For each asset \(i\), the initial wealth is \(Bw_{i}\) and the end wealth is \[Bw_{i}\left(p_{i,t}/p_{i,t-1}\right)=Bw_{i}\left(R_{it}+1\right).\]

• Then the portfolio return is \[R_{t}^{p}= \frac{\sum_{i=1}^{N}Bw_{i}\left(R_{it}+1\right)-B}{B}=\sum_{i=1}^{N}w_{i}R_{it}\approx\sum_{i=1}^{N}w_{i}r_{it}=\mathbf{w}^{T}\mathbf{r}_{t}\]

• The portfolio expected return and variance are \(\mathbf{w}^{T}\boldsymbol{\mu}\) and \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\), respectively.

• Capital budget constraint: \[\mathbf{1}^T\mathbf{w} = 1.\]

• Long-only constraint: \[\mathbf{w} \geq 0.\]

• Dollar-neutral or self-financing constraint: \[\mathbf{1}^T\mathbf{w} = 0.\]

• Holding constraint: \[\mathbf{l}\leq\mathbf{w}\leq \mathbf{u}\] where \(\mathbf{l}\in\mathbb{R}^{N}\) and \(\mathbf{u}\in\mathbb{R}^{N}\) are lower and upper bounds of the asset positions, respectively.

• Leverage constraint: \[\left\Vert \mathbf{w}\right\Vert _{1}\leq L.\]

• Cardinality constraint: \[\left\Vert \mathbf{w}\right\Vert _{0} \leq K.\]

• Turnover constraint: \[\left\Vert \mathbf{w}-\mathbf{w}_{0}\right\Vert _{1} \leq u\] where \(\mathbf{w}_{0}\) is the currently held portfolio.

• Market-neutral constraint: \[\boldsymbol{\beta}^T\mathbf{w} = 0.\]

• In finance, the expected return \(\mathbf{w}^{T}\boldsymbol{\mu}\) is very relevant as it quantifies the average benefit.

• However, in practice, the average performance is not enough to characterize an investment and one needs to control the probability of going bankrupt.

• Risk measures control how risky an investment strategy is.

• The most basic measure of risk is given by the variance (Markowitz 1952): a higher variance means that there are large peaks in the distribution which may cause a big loss.

• There are more sophisticated risk measures such as downside risk, VaR, ES, etc.

• The mean return \(\mathbf{w}^{T}\boldsymbol{\mu}\) and the variance (risk) \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\) (equivalently, the standard deviation or volatility \(\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}\)) constitute two important performance measures.

• Usually, the higher the mean return the higher the variance and vice-versa.

• Thus, we are faced with two objectives to be optimized: it is a multi-objective optimization problem.

• They define a fundamental mean-variance tradeoff curve (Pareto curve).

• The choice of a specific point in this tradeoff curve depends on how agressive or risk-averse the investor is.

• The idea of the Markowitz mean-variance portfolio (MVP) (Markowitz 1952) is to find a trade-off between the expected return \(\mathbf{w}^{T}\boldsymbol{\mu}\) and the risk of the portfolio measured by the variance \(\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\): \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\mu}-\lambda\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1 \end{array}\] where \(\mathbf{w}^{T}\mathbf{1}=1\) is the capital budget constraint and \(\lambda\) is a parameter that controls how risk-averse the investor is.

• This is a convex quadratic problem (QP) with only one linear constraint which admits a closed-form solution: \[\mathbf{w}_{\sf MVP} = \frac{1}{2\lambda}\boldsymbol{\Sigma}^{-1}\left(\boldsymbol{\mu}+\nu\mathbf{1}\right),\] where \(\nu\) is the optimal dual variable \(\nu=\frac{2\lambda-\mathbf{1}^{T}\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}{\mathbf{1}^{T}\boldsymbol{\Sigma}^{-1}\mathbf{1}}\).

• The global minimum variance portfolio (GMVP) ignores the expected return and focuses on the risk only: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1. \end{array}\]

• It is a simple convex QP with solution \[\mathbf{w}_{\sf GMVP}=\frac{1}{\mathbf{1}^{T}\boldsymbol{\Sigma}^{-1}\mathbf{1}}\boldsymbol{\Sigma}^{-1}\mathbf{1}.\]

• It is widely used in academic papers for simplicity of evaluation and comparison of different estimators of the covariance matrix \(\boldsymbol{\Sigma}\) (while ignoring the estimation of \(\boldsymbol{\mu}\)).

Markowitz’s portfolio has never been fully embraced by practitioners, among other reasons because

1. variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses: the solution is to use alternative measures for risk, e.g., VaR and CVaR,

2. it is highly sensitive to parameter estimation errors (i.e., to the covariance matrix \(\boldsymbol{\Sigma}\) and especially to the mean vector \(\boldsymbol{\mu}\)): solution is robust optimization and improved parameter estimation,

3. it only considers the risk of the portfolio as a whole and ignores the risk diversification (i.e., concentrates risk too much in few assets, this was observed in the 2008 financial crisis): solution is the risk parity portfolio.

👉 We will here consider more meaningful measures for risk than the variance, like the downside risk (DR), Value-at-Risk (VaR), Conditional VaR (CVaR) or Expected Shortfall (ES), and drawdown (DD).

• In finance, the mean return is very relevant as it quantifies the average benefit of the investment.

• However, in practice, the average performance is not good enough and one needs to control the probability of going bankrupt.

• Risk measures control how risky an investment strategy is.

• The most basic measure of risk is the variance as considered by Markowitz (1952): a higher variance means that there are large peaks in the risk distribution which may cause a big loss.

• However, Markowitz himself already recognized and stressed the limitations of the mean-variance analysis (Markowitz 1959).

• Variance is not a good measure of risk in practice since it penalizes both the unwanted high losses and the desired low losses (or gains) (McNeil et al. 2005).

• Indeed, the mean-variance portfolio framework penalizes up-side and down-side risk equally, whereas most investors don’t mind up-side risk.

• To overcome the limitations of the variance as risk measure, a number of alternative risk measures have been proposed, for example:

  • Downside Risk (DR)
  • Value-at-Risk (VaR)
  • Conditional Value-at-Risk (CVaR)
  • Drawdown (DD):
    • maximum DD
    • average DD
      
    • Conditional Drawdown at Risk (CDaR)

• Let \(R\) be a random variable representing the return of an asset or portfolio (e.g., \(R=\mathbf{w}^T\mathbf{r}\) where \(\mathbf{r}\) denotes the vector of random returns of the assets).

• We are familiar with the mean return \(\mu=\mathsf{E}[R]\) and with the variance \(\sigma^2=\mathsf{E}[(R-\mu)^2]\).

• The idea of downside risk is that the left-handside of the return distribution involves risk while the right-handside contains the better investment opportunities.

• Interest in downside risk arose in the early 1950s.

• One example is the semi-variance, already considered by Markowitz (1959).

• The semi-variance measures the variability of the returns below the mean.

• The semi-variance is a special case of the more general lower partial moments (LPM): \[\textsf{LPM} = \mathsf{E}\left[\left((\tau - R)^+\right)^\alpha\right],\] where \((\cdot)^+=\max(0, \cdot)\).
• The parameter \(\tau\) is termed the disaster level.
• The parameter \(\alpha\) reflects the investor’s feeling about the relative consequences of falling short of \(\tau\) by various amounts:
  • the value \(\alpha=1\) (which suits a neutral investor) separates risk-seeking (\(0<\alpha<1\)) from risk-averse (\(\alpha>1\)) behavior with regard to returns below the target \(\tau\).
• By changing the parameters \(\alpha\) and \(\tau\) most downside measures used in practice can be formed.
• In particular, setting \(\alpha=2\) and \(\tau=\mathsf{E}[R]\) yields the semi-variance (or lower partial variance): \[\textsf{SV} = \mathsf{E}\left[\left((E[R] - R)^+\right)^2\right].\]

• To overcome the drawback of variance, another popular single side risk measurement is the Value-at-Risk (VaR) initially proposed by J.P. Morgan.

• VaR denotes the maximum loss with a specified confidence level (e.g., confidence level = 95%, period = 1 day).

• Let \(\xi\) be a random variable representing the loss from a portfolio over some period of time (e.g., \(\xi=-\mathbf{w}^T\mathbf{r}\) where \(\mathbf{r}\) denotes the vector of random returns of the assets).

• The VaR is defined as \[\mathsf{VaR}_{\alpha} = \inf\left\{\xi_0:\mathsf{Pr}\left(\xi\leq\xi_0\right)\geq\alpha\right\}\] with \(\alpha\) the confidence level, say, \(\alpha=0.95\).

• However, this measure does not take into account losses exceeding VaR, is nonconvex, and is not subadditive.

• The drawdown (DD) at time \(t\) is defined as the decline from a historical peak of the cumulative profit \(X(t)\).

• The unnormalized version is \[D(t)^{\sf unnorm} = \max_{1\le\tau\le t}X(\tau) - X(t).\]

• But in practice, the normalized version is used: \[D(t) = \frac{{\sf HWM}(t) - X(t)}{{\sf HWM}(t)}\] where \({\sf HWM}(t)\) is the high water mark of \(X(t)\) defined as \[{\sf HWM}(t) = \max_{1\le\tau\le t}X(\tau).\]

• Then one can define the maximum DD (Max-DD) over a period \(t=1,\ldots,T\) as \[M(T)=\max_{1\le t\le T}D(t)\]

• Also the average DD (Ave-DD) over a period \(t=1,\ldots,T\) as \[A(T)=\frac{1}{T}\sum_{1\le t\le T}D(t)\]

• Similarly to the CVaR, we can define the Conditional Drawdown at Risk (CDaR) as the mean of the worst \(100(1-\alpha)\%\) drawdowns.

• For less risk-averse investors, we can consider the mean-LPM portfolio formulation with \(\alpha=1\), which is an LP: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^T\boldsymbol{\mu}-\lambda \frac{1}{T}\sum_{t=1}^T\left(\mathbf{w}^T\boldsymbol{\mu} - \mathbf{w}^T\mathbf{r}_t\right)^+\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0}. \end{array}\]

• For more risk-averse investors, we can consider the mean-LPM convex portfolio formulation with \(\alpha=3\): \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^T\boldsymbol{\mu}-\lambda \frac{1}{T}\sum_{t=1}^T\left(\left(\mathbf{w}^T\boldsymbol{\mu} - \mathbf{w}^T\mathbf{r}_t\right)^+\right)^3\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0}. \end{array}\]

• A portfolio formulation dealing directly with VaR and CVaR quantities is not tractable.

• Let \(f\left(\mathbf{w},\mathbf{r}\right)\) be an arbitrary cost function, where \(\mathbf{w}\) is the optimization variable (portfolio) and \(\mathbf{r}\) denotes the random asset returns.

  • Example: \(f\left(\mathbf{w},\mathbf{r}\right)=-\mathbf{w}^{T}\boldsymbol{r}\).

• Consider, for example, the maximization of the mean return subject to a CVaR risk constraint on the loss: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{maximize}} & \mathbf{w}^{T}\boldsymbol{\mu}\\ \textsf{subject to} & \mathsf{CVaR}_{\alpha}\left(f\left(\mathbf{w},\mathbf{r}\right)\right)\leq c\\ & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0} \end{array}\] where \[\mathsf{CVaR}_{\alpha}\left(f\left(\mathbf{w},\mathbf{r}\right)\right) = \textsf{E}\left[f\left(\mathbf{w},\mathbf{r}\right)\mid f\left(\mathbf{w},\mathbf{r}\right)\geq\mathsf{VaR}_{\alpha}\left(f\left(\mathbf{w},\mathbf{r}\right)\right)\right].\]

• Rockafellar and Uryasev (2000) first proposed to minimize the CVaR of the portfolio loss as follows: \[\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \mathsf{CVaR}_{\alpha}\left(\mathbf{w}^T\mathbf{r}\right)\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0} \end{array}\] where \[\mathsf{CVaR}_{\alpha}\left(\mathbf{w}^T\mathbf{r}\right) = \textsf{E}\left[\mathbf{w}^T\mathbf{r}\mid \mathbf{w}^T\mathbf{r}\geq\mathsf{VaR}_{\alpha}\left(\mathbf{w}^T\mathbf{r}\right)\right].\]

• Define the auxiliary convex function \[F_{\alpha}(\mathbf{w},\zeta)=\zeta+\frac{1}{1-\alpha}\mathsf{E}\left[-\mathbf{w}^{T}\mathbf{r}-\zeta\right]^{+},\] where \(\left[x\right]^{+}=\max\left(x,0\right)\).

• Rockafellar and Uryasev show that

  1. \(\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})\) is a minimizer of \(F_{\alpha}(\mathbf{w},\zeta)\) w.r.t. \(\zeta\): \[\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})\in \arg\min_{\zeta}F_{\alpha}(\mathbf{w},\zeta).\]
  2. \(\mathsf{CVaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})\) equals minimum \(F_{\alpha}(\mathbf{w},\zeta)\) w.r.t. \(\zeta\): \[\mathsf{CVaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})= \min_{\zeta}F_{\alpha}(\mathbf{w},\zeta).\]

• The minimizer of \(F_{\alpha}(\mathbf{w},\zeta)\) w.r.t. \(\zeta\) satisfies: \(0\in\partial_{\zeta}F_{\alpha}(\mathbf{w},\zeta^{\star})\). For example, we choose the following subgradient: \[\begin{aligned} 0=s_{\zeta}F_{\alpha}(\mathbf{w},\zeta^{\star}) & =1-\frac{1}{1-\alpha}\int\mathbf{1}_{\left\{-\mathbf{w}^{T}\mathbf{r}>\zeta^{\star}\right\}}p(\mathbf{r})d\mathbf{r}\\ & =1-\frac{1}{1-\alpha}P\left(-\mathbf{w}^{T}\mathbf{r}>\zeta^{\star}\right), \end{aligned}\] where \(\mathbf{1}_{\{\cdot\}}\) is the indicator function. Solving the above equation, we have \[P\left(-\mathbf{w}^{T}\mathbf{r}>\zeta^{\star}\right)=1-\alpha \Longrightarrow \zeta^{\star}=\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r}).\]

• First, we have \[\min_{\zeta}F_{\alpha}(\mathbf{w},\zeta)=F_{\alpha}(\mathbf{w},\zeta^{\star}) =\zeta^{\star}+\frac{1}{1-\alpha}\mathsf{E}[-\mathbf{w}^{T}\mathbf{r}-\zeta^{\star}]^{+}.\] Then, recall that
  \[\begin{aligned} \mathsf{CVaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r}) & = \mathsf{E}\left[-\mathbf{w}^{T}\mathbf{r}\big|-\mathbf{w}^{T}\mathbf{r}>\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})\right]\\ & = \frac{1}{1-\alpha}\int_{-\mathbf{w}^{T}\mathbf{r}>\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})}\left(-\mathbf{w}^{T}\mathbf{r}\right)p(\mathbf{r})d\mathbf{r}\\ & = \frac{1}{1-\alpha}\int\left[-\mathbf{w}^{T}\mathbf{r}-\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r})\right]^{+}p(\mathbf{r})d\mathbf{r}\\ & \qquad +\mathsf{VaR}_{\alpha}(-\mathbf{w}^{T}\mathbf{r}). \end{aligned}\]

• In words, minimizing \(F_{\alpha}\left(\mathbf{w},\zeta\right)\) simultaneously calculates the optimal CVaR and VaR.

Proof: \[F_{\alpha}\left(\mathbf{w},\zeta\right)=\zeta+\frac{1}{1-\alpha}\int\left[-\mathbf{w}^{T}\mathbf{r}-\zeta\right]^{+}p\left(\mathbf{r}\right)d\mathbf{r}.\]

• Sample average approximation of \(F_{\alpha}(\mathbf{w},\zeta)\): \[\begin{aligned} F_{\alpha}(\mathbf{w},\zeta) & =\zeta+\frac{1}{1-\alpha}\mathsf{E}\left[-\mathbf{w}^T\mathbf{r}-\zeta\right]^{+}\\ & \approx\zeta+\frac{1}{1-\alpha}\frac{1}{T}\sum_{t=1}^{T}\left[-\mathbf{w}^T\mathbf{r}_t-\zeta\right]^{+}. \end{aligned}\]

• We first include the dummy variables \(z_t\): \[z_t\geq\left[-\mathbf{w}^T\mathbf{r}_t-\zeta\right]^{+} \Longrightarrow z_t\geq -\mathbf{w}^T\mathbf{r}_t-\zeta,\,z_t\geq0\]

• CVaR portfolio problem can be approximated by an LP: \[\begin{array}{ll} \underset{\mathbf{w}, \{z_t\}, \zeta}{\textsf{minimize}} & \zeta+\frac{1}{1-\alpha}\frac{1}{T}\sum_{t=1}^{T}z_{t}\\ \textsf{subject to} & 0\leq z_{t}\geq-\mathbf{w}^{T}\mathbf{r}_{t}-\zeta,\quad t=1,\dots,T\\ & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0}. \end{array}\]

• We can also consider the maximization of the mean return subject to a CVaR constraint: \[\begin{array}{ll} \underset{\mathbf{w}, \{z_t\}, \zeta}{\textsf{maximize}} & \mathbf{w}^T\boldsymbol{\mu}\\ \textsf{subject to} & \zeta+\frac{1}{1-\alpha}\frac{1}{T}\sum_{t=1}^{T}z_{t} \le c\\ & 0\leq z_{t}\geq-\mathbf{w}^{T}\mathbf{r}_{t}-\zeta,\quad t=1,\dots,T\\ & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0}. \end{array}\]

• Or a mean-CVaR objective: \[\begin{array}{ll} \underset{\mathbf{w}, \{z_t\}, \zeta}{\textsf{maximize}} & \mathbf{w}^T\boldsymbol{\mu} - \lambda\left(\zeta+\frac{1}{1-\alpha}\frac{1}{T}\sum_{t=1}^{T}z_{t}\right)\\ \textsf{subject to} & 0\leq z_{t}\geq-\mathbf{w}^{T}\mathbf{r}_{t}-\zeta,\quad t=1,\dots,T\\ & \mathbf{1}^T\mathbf{w}=1,\quad \mathbf{w}\ge\mathbf{0}. \end{array}\]