Portfolio Optimization with Relative Tail Risk

Harry Markowitz’s mean-variance portfolio optimization technique (Markowitz (1952)) has made a remarkable impact on portfolio theory in finance. This innovative method has been extensively utilized successfully in portfolio selection, allocation, and risk management practices. The adoption of Markowitz’s mean-variance model has resulted in significant improvements in investment outcomes and become a cornerstone of modern portfolio theory. Many investors have utilized Markowitz’s method to determine an efficient portfolio weight vector. Still, constructing a portfolio that surpasses the market index (e.g., S&P 500 index or DJIA) is not easy to be achieved. For that reason, some investors use relative portfolio optimization, that is, relaxing the assumption of the portfolio variance as the tracking error, which is a variance of the relative return. Here, the relative return is the difference between the portfolio return and a benchmark return, such as the market index return

This paper presents how to improve relative portfolio optimization based on two aspects. First, we loosen the Gaussian assumption of Markowitz’s model, which was empirically rejected in literature including Fama (1963), Mandelbrot (1963a, b), and Cont and Tankov (2004). Non-Gaussian multivariate distributions have been introduced to capture stylized facts, such as fat-tails and asymmetric dependence, that are not accounted for by the Gaussian model. This paper proposes the normal tempered stable (NTS) distribution as an alternative distribution to the Gaussian distribution. The NTS is defined by the tempered stable subordinated Gaussian distribution (Barndorff-Nielsen (1978), Barndorff-Nielsen and Levendorskii (2001) and Barndorff-Nielsen and Shephard (2001)). The NTS distribution has been popularly applied in finance by capturing the fat-tails of the asset return distribution and describing asymmetric dependence (See Kim and Volkmann (2013)). For example, it is applied to financial risk management in Anand et al. (2017) and Kurosaki and Kim (2018), and portfolio management in Eberlein and Madan (2010), Anand et al. (2016), and Kim (2022). In addition, Kim et al. (2015) applies a Lévy  process generated by the two-dimensional NTS distribution to Quanto option pricing, and its extension to capture stochastic dependence is further discussed in Kim et al. (2023). Recently, the NTS distribution was applied to cryptocurrency portfolio optimization in Kurosaki and Kim (2022).

Second, we enhance the portfolio theory by replacing CoVaR and CoCVaR instead of the variance of the relative return as the risk measure. Traditionally, portfolio variance, value at risk (VaR), and conditional value at risk (CVaR) have been employed as the portfolio risk measure, but those are absolute risk measures. Relative traders prefer to use tracking errors such as the variance of relative return with respect to the benchmark return instead of the absolute risk measures. In this paper, we take CoVaR (Adrian and Brunnermeier (2016)) and CoCVaR (Huang and Uryasev (2018)) as the risk measures of relative portfolio optimization. CoVaR is proposed as a measure of systemic risk, which is the VaR of the financial system under the condition of a distressed market. Girardi and Tolga Ergün (2013) calculate CoVaR on the multivariate GARCH model and Reboredo and Ugolini (2015) use the copula method. CoCVaR (or CoES by Adrian and Brunnermeier (2016)) is an expected downturn of the financial system under the condition of a distressed market. Huang and Uryasev (2018) define the mathematical formula of CoCVaR and apply it to measure the risk of the ten largest publicly traded banks in the United States under a distressed condition of market factors²²2VIX, liquidity spread, three-month treasury change, term spread change, credit spread change, equity market return, and real estate sector excess return.. Recently, Liu et al. (2021) discuss CoVaR and CoCVaR on the GARCH model with two-dimensional NTS innovations and do the back-testing.

The paper discusses portfolio optimization minimizing the portfolio CoVaR and CoCVaR concerning the market index and derives the marginal contributions to risk for the portfolio CoVaR and CoCVaR. The marginal contribution to risk is the rate of change in risk with respect to a small percentage change in proportion to a member’s asset. Mathematically it is defined by the first derivative of CoVaR or CoCVaR with respect to the marginal weight. In the context of absolute optimization, the marginal contributions to VaR and CVaR are employed to identify assets with high and low levels of risk. The general form of marginal risk contributions for the VaR and CVaR are provided in Gourieroux et al. (2000). The analytic forms of the marginal contributions for VaR and CVaR are discussed under the skewed-$t$ distribution in Stoyanov et al. (2013), and under the NTS distribution in Kim et al. (2012) and Kim (2022). Since we focus on relative optimization in this paper, we provide analytic formulas for the marginal contributions to CoVaR and CoCVaR under the NTS market model instead of VaR and CVaR. Those marginal contributions to CoVaR and CoCVaR help relative traders to make decisions in portfolio rebalancing. However, the multiple integrals involved in these formulas can present technical difficulties for numerical calculations. To solve this problem, we use Monte-Carlo simulation (MCS) methods. Furthermore, we perform the risk budgeting based on the marginal contributions to CoVaR and CVaR empirically.

The remainder of this paper is organized as follows. The definition of portfolio CoVaR and portfolio CoCVaR are presented in Section 2. Section 3 reviews the NTS market model. Section 4 presents the portfolio CoVaR and CoCVaR on the NTS market model, along with a detailed analysis of the marginal contributions to CoVaR and CoCVaR. Empirical illustrations are given in section 5. In this section, we exhibit MCS method to calculate CoCVaR on the NTS market model and do the portfolio CoCVaR minimizing portfolio optimization with the estimated parameters. We also discuss portfolio budgeting using the marginal CoVaR and the marginal CoCVaR. Finally, Section 6 concludes. Proofs and mathematical details are presented in Appendix.

Let $X$ be a random variable for a market factor, for instance, the market index returns, and $Y$ be a random variable for asset or portfolio returns. We consider a random vector $(X,Y)$ following a joint distribution. The CoVaR and CoCVaR of $Y$ at a significant level $\eta$ under a condition of the event $X\leq-\textup{\rm VaR}_{\zeta}(X)$ are defined by

$$\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{Y}|{X}}\right)=\textup{\rm VaR}_{\eta}(Y|X\leq-\textup{\rm VaR}_{\zeta}(X))=-\inf_{x}\{x|P(Y\leq x|X\leq-\textup{\rm VaR}_{\zeta}(X))\geq\eta\},$$

and

$$\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{Y}|{X}}\right)=-E\left[Y|Y<-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{Y}|{X}}\right),X\leq-\textup{\rm VaR}_{\zeta}(X)\right],$$

respectively³³3See Adrian and Brunnermeier (2016),Huang and Uryasev (2018), and Liu et al. (2021) for more details, where $\textup{\rm VaR}_{\zeta}(X)$ is the Value-at-Risk (VaR) of $X$ for a significant level $\zeta$ given as

$$\textup{\rm VaR}_{\zeta}(X)=-\inf_{x}\{x|P\left(X\leq x\right)\geq\zeta\}.$$

If the joint distribution of $(X,Y)$ is continuous, then $\textup{\rm VaR}_{\zeta}(X)=-F_{X}^{-1}(\zeta)$ where $F_{X}$ is the cumulative distribution function (cdf) for the marginal distribution of $X$ and $F_{X}^{-1}$ is the inverse function of $F_{X}$. Moreover, we have

$$F_{(X,Y)}(-\textup{\rm VaR}_{\zeta}(X),-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{Y}|{X}}\right))=\eta\zeta$$

for the joint cdf of $(X,Y)$.

Consider $N$ number of stocks and a market index and an $(N+1)$-dimensional random vector $R=(R_{0}$, $R_{1}$, $\cdots$, $R_{N})^{{\texttt{T}}}$, where $R_{0}$ is the index return and $R_{n}$ is the return of the $n$th stock for $n\in\{1$, $2$, $\cdots$, $N\}$. Let $w=(w_{1}$, $w_{2}$, $\cdots$, $w_{N})^{\texttt{T}}\in I^{N}$ be an $N$-dimensional vector satisfying $\sum_{n=1}^{N}w_{n}=1$ for $I=[0,1]$. The $n$th element $w_{n}$ of $w$ is the proportion of capital invested in the $n$th stock for $n\in\{1,2,\cdots,N\}$. In this case, $w$ be referred to as the capital allocation weight vector for the $N$ stocks in the market⁴⁴4In this paper, we consider the long-only portfolio.. Then we have the random variable of the portfolio return as $R_{p}(w)=\sum_{n=1}^{N}w_{n},R_{n}$.

Suppose that $R$ follows $(N+1)$-dimensional normal distribution with a mean vector $\mu=(\mu_{0}$, $\mu_{1}$, $\cdots$, $\mu_{N})^{\texttt{T}}$ and a covariance matrix $\Sigma$, that is $R\sim\Phi(\mu,\Sigma)$. Let $\sigma_{n,m}$ be the $(n+1,m+1)$-th element of $\Sigma$ for $n,m\in\{0,1,\cdots,N\}$. Then the bivariate random vector $(R_{0},R_{p}(w))^{\texttt{T}}$ follows the bivariated normal distribution, $(R_{0},R_{p}(w))^{\texttt{T}}\sim\Phi(\bar{\mu},\bar{\Sigma})$ with

$$\bar{\mu}=(\mu_{0},\mu_{p}(w))^{\texttt{T}},\text{ and }\bar{\Sigma}=\left(\begin{matrix}\sigma_{0,0}&\sigma_{p,0}(w)\\ \sigma_{p,0}(w)&\sigma_{p}^{2}(w)\end{matrix}\right),$$

where $\mu_{p}(w)=\sum_{n=1}^{N}w_{n}\mu_{n}$, $\sigma_{p,0}(w)=\sum_{n-1}^{N}w_{m}\sigma_{0,n}$, $\sigma_{p}^{2}(w)=\sum_{n=1}^{N}\sum_{m=1}^{N}w_{m}w_{m}\sigma_{n,m}$. We can calculate VaR of $R_{0}$ as $\textup{\rm VaR}_{\zeta}(R_{0})=q_{1-\zeta}\sqrt{\sigma_{0,0}}-\mu_{0}$, where $q_{1-\zeta}$ is $(1-\zeta)$-quantile value of the standard normal distribution. Suppose the value $x$ satisfies

$$\int_{-\infty}^{x}\int_{-\infty}^{-\textup{\rm VaR}_{\zeta}(R_{0})}f_{\Phi(\bar{\mu},\bar{\Sigma})}(x_{1},x_{2})dx_{1}dx_{2}=\eta\zeta,$$

where $f_{\Phi(\bar{\mu},\bar{\Sigma})}$ is the probability density function of $\Phi(\bar{\mu},\bar{\Sigma})$. Then $\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{R_{p}(w)}|{R_{0}}}\right)=-x$. Moreover,

$$\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{R_{p}(w)}|{R_{0}}}\right)=-\frac{1}{\eta\zeta}\int_{-\infty}^{-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{R_{p}(w)}|{R_{0}}}\right)}\int_{-\infty}^{-\textup{\rm VaR}_{\zeta}(R_{0})}x_{1}f_{\Phi(\bar{\mu},\bar{\Sigma})}(x_{1},x_{2})dx_{1}dx_{2}.$$

In this section, we define the multivariate standard NTS distribution and construct the market model based on the distribution.

Consider $N$ number of stocks and a market index for a given market as follows:

• •

  $R_{0}$ is the market index return.

• •

  $R_{n}$ for $n\in\{1,2,\cdots,N\}$ is the individual stock return of a stock portfolio consisting of $N$ stocks.

Suppose $R=(R_{0},R_{1},\cdots,R_{N})^{{\texttt{T}}}$ follows a ($N+1$)-dimensional NTS market model, that is, we have

$$R=\mu+\textup{diag}(\sigma)\Xi\text{ with }\Xi\sim\textup{stdNTS}_{N+1}(\alpha,\theta,\beta,\Sigma)$$

(7)

where

• •

  $\mu=(\mu_{0},\mu_{1},\cdots,\mu_{N})^{{\texttt{T}}}\in\mathbb{R}^{N+1}$, and $\sigma=(\sigma_{0},\sigma_{1},\cdots,\sigma_{N})^{{\texttt{T}}}\in\mathbb{R}_{+}^{N+1}$,

• •

  $\alpha\in(0,2)$, $\theta>0$, $\beta=(\beta_{0},\beta_{1},\cdots,\beta_{N})^{{\texttt{T}}}\in\mathbb{R}^{N+1}$ with $|\beta_{n}|<\sqrt{\frac{2\theta}{2-\alpha}}$ for $n\in\{0,1,\cdots,N\}$,

• •

  $\Sigma$ is the $(N+1)\times(N+1)$ covariance matrix and $\rho_{n,m}$ is the $(n+1,m+1)$-th element of $\Sigma$ for $n,m\in\{0,1,\cdots,N\}$.

Based on the assumption, we obtain the following proposition whose proof is in Appendix.

Remark Let $\sigma_{*}=(\sigma_{1},\sigma_{2},\cdots,\sigma_{N})^{\texttt{T}}$, $\rho^{*}=(\rho_{0,1},\rho_{0,2},\cdots,\rho_{0,N})^{\texttt{T}}$, $\Sigma^{*}=\left(\rho_{n,m}\right)_{n,m\in\{1,2,\cdots,N\}}$ and $\gamma^{*}=(\gamma_{1},\gamma_{2},\cdots,\gamma_{N})^{\texttt{T}}$. Then

$\displaystyle\rho_{p}(w)$

$\displaystyle:=\frac{w^{\texttt{T}}V_{\gamma,\sigma,\rho}^{*}}{\sqrt{w^{\texttt{T}}\Sigma^{*}_{\gamma,\sigma}w}}$

for $V_{\gamma,\sigma,\rho}^{*}=\textrm{\rm diag}(\gamma^{*})\textrm{\rm diag}(\sigma^{*})\rho^{*}$ and $\Sigma_{\gamma,\sigma}^{*}=\textrm{\rm diag}(\gamma^{*})\textrm{\rm diag}(\sigma^{*})\Sigma^{*}\textrm{\rm diag}(\sigma^{*})\textrm{\rm diag}(\gamma^{*})$.

With the positive homogeneity and translation invariance properties of VaR and CoVaR, we have

$$\textup{\rm VaR}_{\zeta}(R_{0})=\sigma_{0}(w)\textup{\rm VaR}_{\zeta}(\Xi_{0})-\mu_{0}$$

and

$$\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{R_{p}(w)}|{R_{0}}}\right)=\sigma_{p}(w)\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)-\mu_{p}(w).$$

Since marginal distributions of $\Xi_{0}$, and $\Xi_{p}(w)$ are continuous, we have $\textup{\rm VaR}_{\zeta}(\Xi_{0})=-F_{\Xi_{0}}^{-1}(\zeta)$, where $F_{\Xi_{0}}$ and $F_{\Xi_{0}}^{-1}$ are the cdf and inverse cdf of $\Xi_{0}$, respectively. Moreover, $\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)$ is the value satisfying

$$\mathbb{P}\left(\Xi_{p}(w)<-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right),\Xi_{0}<-\textup{\rm VaR}_{\zeta}(\Xi_{0})\right)=\zeta\eta,$$

i.e.

$$F_{(\Xi_{0},\Xi_{p}(w))}\left(-\textup{\rm VaR}_{\zeta}(\Xi_{0}),-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)\right)=\zeta\eta,$$

(9)

where $F_{(\Xi_{0},\Xi_{p}(w))}$ is the cdf of $(\Xi_{0},\Xi_{p}(w))$. By (1), $F_{(\Xi_{0},\Xi_{p}(w))}$ is given as

	$\displaystyle F_{(\Xi_{0},\Xi_{p}(w))}(\xi_{0},\xi_{p})$
	$\displaystyle=\int_{0}^{\infty}\int_{-\infty}^{\frac{\xi_{p}-\beta_{p}(w)(t-1)}{\gamma_{p}(w)\sqrt{t}}}\int_{-\infty}^{\frac{\xi_{0}-\beta_{0}(t-1)}{\gamma_{0}\sqrt{t}}}f^{\Phi_{2}}_{\rho_{p}(w)}(x_{1},x_{2})dx_{1}dx_{2}f_{\mathcal{T}}(t)dt,$

where $\gamma_{0}=\sqrt{1-\beta_{0}^{2}\left(\frac{2\theta}{2-\alpha}\right)}$, $\gamma_{p}(w)=\sqrt{1-(\beta_{p}(w))^{2}\left(\frac{2\theta}{2-\alpha}\right)}$ and $f^{\Phi_{2}}_{\rho}$ is the pdf of the bivariate standard normal distribution with covariance $\rho$.

By the same arguments, we have CoCVaR as

$$\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{R_{p}(w)}|{R_{0}}}\right)=\sigma_{p}(w)\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)-\mu_{p}(w)$$

(10)

and

$\displaystyle\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)=-\frac{1}{\zeta\eta}\int_{-\infty}^{-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)}\int_{-\infty}^{-\textup{\rm VaR}_{\zeta}(\Xi_{0})}x_{2}f_{(\Xi_{0},\Xi_{p}(w))}(x_{1},x_{2})dx_{1}\,dx_{2}$

where

$\displaystyle f_{(\Xi_{0},\Xi_{p}(w))}(x_{1},x_{2})=\int_{0}^{\infty}g\left(x_{1}-\beta_{0}(t-1),x_{2}-\beta_{p}(w)(t-1),\gamma_{0}\sqrt{t},\gamma_{p}(w)\sqrt{t},\rho_{p}(w)\right)f_{\mathcal{T}}(t)dt,$

and

$$g(x,y,u,v,\rho)=\frac{1}{2\pi uv\sqrt{1-\rho}}\exp\left(-\frac{1}{2(1-\rho^{2})}\left(\left(\frac{x}{u}\right)^{2}-\frac{2\rho xy}{uv}+\left(\frac{y}{v}\right)^{2}\right)\right),$$

by (2). By the change of variables, we have

		$\displaystyle\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)$
		$\displaystyle=-\frac{1}{\zeta\eta}\int_{0}^{\infty}\int_{-\infty}^{C(w,t)}\int_{-\infty}^{v(t)}\left(\beta_{p}(w)(t-1)+x_{2}\gamma_{p}(w)\sqrt{t}\right)f^{\Phi_{2}}_{\rho_{p}(w)}(x_{1},x_{2})dx_{1}\,dx_{2}f_{\mathcal{T}}(t)dt,$		(11)

or

$\displaystyle\textup{\rm CoCVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)=-\frac{1}{\zeta\eta}E\left[\left(\beta_{p}(w)({\mathcal{T}}-1)+\epsilon_{p}\gamma_{p}(w)\sqrt{{\mathcal{T}}}\right)1_{\epsilon_{p}<C(w,{\mathcal{T}})}1_{\epsilon_{0}<v({\mathcal{T}})}\right]$

(12)

where $C(w,t)=\frac{-\textup{\rm CoVaR}_{{\eta},{\zeta}}\left({{\Xi_{p}(w)}|{\Xi_{0}}}\right)-\beta_{p}(w)(t-1)}{\gamma_{p}(w)\sqrt{t}}$, $v(t)=\frac{\xi_{0}-\beta_{0}(t-1)}{\gamma_{0}\sqrt{t}}$, $\xi_{0}=-\textup{\rm VaR}_{\zeta}(\Xi_{0})$, $(\epsilon_{0},\epsilon_{p})$ is the bivariate standard normal random vector with covariance $\rho_{p}(w)$, and ${\mathcal{T}}$ is the tempered stable subordinator with parameters $(\alpha,\theta)$ independent of $(\epsilon_{0},\epsilon_{p})$.