Multi-period portfolio selection with interval-based conditional value-at-risk

Abstract

We propose a multi-period mean-risk portfolio model based as a risk measure on the interval conditional value at risk (ICVaR). The ICVaR was introduced in Liu et al. (Ann Op Res 307:329–361, 2021) in a strict relationship with second-order stochastic dominance and adopted as risk measure in the formulation of a static portfolio optimization problem: in this article we reconsider its key properties and specify a multistage portfolio model based on the trade-off between expected wealth and terminal ICVaR. The definition of this risk measure depends on a reference point, that by discriminating between contiguous stochastic dominance orders motivated in Liu et al. (2021) the introduction of interval stochastic dominance (ISD) of the first and second-order specifically in a financial context. We develop from there in this article and present a set of results that help characterizing rigorously the relationship between the solution of the multistage stochastic programming portfolio problem and the underlying ISD principles. An extended set of computational results is presented to validate in- and out-of-sample a set of mathematical results and the modeling framework over the 2021–2022 period.

Proof of Propositions 3, 5 and 6

We present the proof of Proposition 3.

Proof of Proposition 3

• The monotonicity follows naturally by the monotonicity of the positive part operator, expected value operator as well as the supremum operator.

• To show the concavity, for any $\lambda \in [0,1]$, we have

  $$\begin{aligned}&\rho _{\alpha ,\beta }(\lambda W + (1-\lambda ) Y) \\&\quad = \sup _{\eta \le \beta } \{ \eta -\frac{1}{1-\alpha } \mathbb {E}[ \eta - \lambda W - (1-\lambda ) Y ]_+ \} \\&\quad \ge \sup \limits _{\eta =\lambda \eta _1+(1-\lambda )\eta _2, \eta _1\le \beta ,\eta _2 \le \beta } \{ \eta -\frac{1}{1-\alpha } \mathbb {E}[ \eta - \lambda W - (1-\lambda ) Y ]_+ \} \\&\quad = \sup _{\eta _1\le \beta ,\eta _2 \le \beta } \{ \lambda \eta _1+(1-\lambda )\eta _2 -\frac{1}{1-\alpha } \mathbb {E}[ \lambda (\eta _1- W) - (1-\lambda ) Y ]_+ \} \\&\quad \ge \sup _{\eta _1\le \beta ,\eta _2 \le \beta } \{ \lambda \eta _1+(1-\lambda )\eta _2 -\frac{\lambda }{1-\alpha } \mathbb {E}[ \eta _1- W ]_+ - \frac{1-\lambda }{1-\alpha } \mathbb {E}[ \eta _2 - Y ]_+ \} \\&\quad = \sup _{\eta _1\le \beta } \{ \lambda \eta _1 -\frac{\lambda }{1-\alpha } \mathbb {E}[ \eta _1- W ]_+ \} + \sup _{\eta _2\le \beta } \{ (1-\lambda )\eta _2 - \frac{1-\lambda }{1-\alpha } \mathbb {E}[ \eta _2 - Y ]_+ \} \\&\quad = \lambda \rho _{\alpha ,\beta }( W ) + (1-\lambda ) \rho _{\alpha ,\beta }(Y). \end{aligned}$$

• The positive homogeneity comes from the fact for any $k\in \mathbb {R}_{++}$ that

  $$\begin{aligned} \rho _{\alpha ,\beta }( kW)&= \sup _{\eta \le \beta } \{ \eta -\frac{1}{1-\alpha } \mathbb {E}[\eta -kW]_+ \} {\mathop {=}\limits ^{{\eta =k\tilde{\eta }}}} \sup _{ k\tilde{\eta }\le \beta } \{ k\tilde{\eta }-\frac{1}{1-\alpha } \mathbb {E}[k\tilde{\eta }-kW]_+ \} \\&=k \sup _{ \tilde{\eta }\le \beta /k} \{ \tilde{\eta }-\frac{1}{1-\alpha } \mathbb {E}[\tilde{\eta }-W]_+ \} = k\rho _{\alpha ,\beta /k}( W ) . \end{aligned}$$

• The cash additivity can be derived from the fact for any $c\in \mathbb {R}$ that

  $$\begin{aligned} \rho _{\alpha ,\beta }( W + c)&= \sup _{\eta \le \beta } \{ \eta -\frac{1}{1-\alpha } \mathbb {E}[\eta -W-c]_+ \\&{\mathop {=}\limits ^{{\eta =\tilde{\eta }+c}}} \sup _{\tilde{\eta }+c\le \beta } \{ \tilde{\eta }+c-\frac{1}{1-\alpha } \mathbb {E}[\tilde{\eta }+c-W-c]_+ \\&= c+ \sup _{\tilde{\eta }\le \beta -c} \{ \tilde{\eta }-\frac{1}{1-\alpha } \mathbb {E}[\tilde{\eta }-W]_+ = \rho _{\alpha ,\beta -c}( W ) + c. \end{aligned}$$

$\square$

We show now the technical details of the proof of Propositions 5 and 6. First, we prove the following auxiliary result. Consider the random variable $W_{Y}=(W-Y)$

Proposition 7

For $0 \le \alpha < 1$, $\zeta \le \beta$, and $\eta \in \mathbb {R}$, we have

$$\begin{aligned} \frac{1}{1-\alpha } \mathbb {E}[\zeta -W_Y]_+ \ge \frac{F_2(W,\eta ) - F_2(Y+\zeta ,\eta )}{(1-\alpha )} \end{aligned}$$

Proof of Proposition 7

Since for all $z \in \mathbb {R}$, the positive part is given by $|z|_{+}=\frac{z+|z|}{2}$, then for any $\zeta \le \beta$

$$\begin{aligned} \frac{1}{1-\alpha } \mathbb {E}[\zeta -W_Y]_+&=\frac{\mathbb {E}[\zeta - (W-Y) + |\zeta - (W-Y) |]}{2(1-\alpha )} \nonumber \\&=\frac{\mathbb {E}[(\eta -W)-(\eta -(Y+\zeta )) + |(\eta -W)-(\eta -(Y+\zeta )) |]}{2(1-\alpha )} \nonumber \\&\ge \frac{ \mathbb {E}[ (\eta -W) + |\eta -W|]-\mathbb {E}[ (\eta -(Y+\zeta )) + |\eta -(Y+\zeta )|] }{2(1-\alpha )} \nonumber \\&= \frac{F_2(W,\zeta ) - F_2(Y+\zeta ,\eta )}{(1-\alpha )} \end{aligned}$$

(17)

$\square$

1.1 Lower bound for the second-order gap function

A relevant implication of Proposition 7 is that the ICVaR function $\rho _{\alpha ,\beta }$ provides a lower bound for the second-order gap function $\mathscr {H}_{2}$ as established in Proposition 5. Now, we show the technical details of the proof.

Proof of Proposition 5

Using the closed form for the ICVaR risk measure as in Proposition 1 and the inequality in Proposition 7, we have that for $\eta \in \mathbb {R}$

$$\begin{aligned} \rho _{\alpha ,\beta }(W_Y) -\beta&= -\frac{1}{1-\alpha } \mathbb {E}[\beta -(W_Y)]_+ \\&\le \frac{F_2(Y+\beta ,\eta )-F_2(W,\eta ) }{(1-\alpha )} \end{aligned}$$

The result follows taking the infimum.$\square$

In addition, the result of Proposition 7 gives a lower bound for the first-order gap function as stated in Proposition 6. Using the results of this section, we prove the Proposition 6. The proof involves Chebyshev’s inequality which allows dominating the cumulative distribution using the $L_1$ norm.

Proof of Proposition 6

By Chebyshev’s inequality, we have that for $\eta \le 0$ and $\beta$ a non-positive number.

$$\begin{aligned} F_2(W,\eta )&\ge |\beta | \times \mathbb {P}(\eta -W \ge \beta |) \nonumber \\&= |\beta | \times F_1(W,\eta +\beta ) \end{aligned}$$

(18)

Combining Proposition 7 and Eq.(18), we obtain that for $\eta \le 0$

$$\begin{aligned} \rho _{\alpha ,\beta }(W_Y) -\beta \le \frac{F_2(Y+\beta ,\eta )-|\beta | F_1(W,\eta +\beta )}{(1-\alpha )}. \end{aligned}$$

Thus,

$$\begin{aligned} \rho _{\alpha ,\beta }(W_Y) -\beta \le \frac{F_2(Y+\beta ,\eta )-|\beta | F_1(Y+\beta ,\eta )}{(1-\alpha )}+\frac{|\beta |}{(1-\alpha )} \left( F_1(Y+\beta ,\eta )-F_1(W,\eta +\beta )\right) . \end{aligned}$$

Since $F_1(Y+\beta ,\eta )=F_1(Y+2\beta ,\eta +\beta )$, the result follows taking supremum and infimum on the respective terms. $\square$