Mean–variance, mean–VaR, and mean–CVaR models for portfolio selection with background risk

Abstract

This paper extends (Jiang et al. in J Bank Finance 34:3055–3060, 2010; Guo in Risk Manag 20(1):77–94, 2018) and others by investigating the impact of background risk on an investor’s portfolio choice in the mean–VaR, mean–CVaR, and mean–variance framework, and analyzes the characterization of the mean–variance, mean–VaR, and mean–CVaR boundaries and efficient frontiers in the presence of background risk. We derive the conditions that the portfolios lie on the mean–variance, mean–VaR, and mean–CVaR boundaries with and without background risk. We show that the MV, VaR, and CVaR boundaries depend on the covariance vector between the returns of the risky assets and that of the background asset and also the variance of the return of the background asset. We develop properties on MV, mean–VaR, and mean–CVaR efficient frontiers. In addition, we establish some new properties for the case with a risk-free security, extend our work to the non-normality situation, and examine the economic implication of the mean–VaR/CVaR model.

Appendix

Proof of Proposition 1

We assume that investors are mean–variance optimizers who will solve the following optimization problem for their investment decision making:

$$\min \frac{1}{2}Var(r_{\omega }) \quad {\rm s.t.} \quad \omega^\tau E(r)=\mu \ \ \text{ and } \ \ \omega^\tau I=1$$

(11)

where I is a vector of ones.

Using Lagrange multipliers to solve the two constraints in problem (11), the first-order optimality condition is

$$V\omega +{\rm Cov}(r,r_{\rm b})-\lambda_1E(r)-\lambda_2I=0.$$

Consequently, we can have

$$\omega =V^{-1}(\lambda_1E(r)+\lambda_2I-{\rm Cov}(r,r_{\rm b})).$$

(12)

Plugging Eq. (12) into the constraints in Eq. (11) obtains the following Lagrange multipliers

$$\lambda_1=\frac{C\mu -A+(EC-FA)}{D} \ \ \ \text{ and } \ \ \ \lambda_2=\frac{B-\mu A+(FB-AE)}{D},$$

where $A=I^\tau V^{-1}E(r)$, $B=E(r)^\tau V^{-1}E(r)$, $C=I^\tau V^{-1}I$, $D=BC-A^2$, $E={\rm Cov}(r,r_{\rm b})^\tau V^{-1}E(r)$, and $F={\rm Cov}(r,r_{\rm b})^\tau V^{-1}I$.

Further, for the portfolio q on the mean–variance boundary, we can have

$$\sigma^2_{\omega }= \omega^\tau V\omega +2\omega^\tau {\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \omega^\tau (\lambda_1E(r)+\lambda_2I-{\rm Cov}(r,r_{\rm b}))+2\omega^\tau {\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \lambda_1\mu +\lambda_2+\omega^\tau {\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \lambda_1(\mu +E)+\lambda_2(1+F)-{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \frac{C\mu -A+(EC-FA)}{D}(\mu +E)+\frac{B-\mu A+(FB-AE)}{D}(1+F) -{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \frac{C\mu^2-2(A-EC+FA)\mu -2AE+CE^2-2AEF+B(1+F)^2}{D} -{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \frac{C}{D}(\mu -\frac{A-EC+FA}{C})^2-\frac{C^2E^2-2CEA(1+F)+A^2(1+F)^2}{CD} +\frac{-2AE+CE^2-2AEF+B(1+F)^2}{D}-{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \frac{C}{D}(\mu -\frac{A-EC+FA}{C})^2+a ,$$

where $a=\frac{(1+F)^2}{C}-{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b})>0$.

Consequently, we can have that portfolio $\omega$ is on the mean–variance boundary with background risk if and only if

$$\frac{\sigma^2_{\omega }}{a}-\frac{(E(r_{\omega })-E(r_b)-(A-EC+FA)/C)^2}{Da/C}=1,$$

and thus, Proposition 1 follows. $\square$

Proof of Proposition 2

Over here, we only prove that the minimum-VaR portfolio is mean–CVaR efficient. The other two assertions of Proposition 2 can be obtained similarly. Assuming $\omega$ is the minimum-VaR portfolio but is not mean–CVaR efficient. From Definition 8, we notice that there is a portfolio $\nu$ such that $E[r_{\nu }]\ge E[r_{\omega }]$ and $L[t, r_{\nu }]\le L[t, r_{\omega }]$ with at least one of the inequalities holding strictly. As a result, we have

$$V(t, r_{\nu })= z_t\sigma (r_{\nu })-E(r_{\nu })=\frac{z_t}{k_t} \left[ k_t\sigma (r_{\nu })-E(r_{\nu })\right] -(1-\frac{z_t}{k_t})E(r_{\nu }) = \frac{z_t}{k_t}L(t, r_{\nu })-(1-\frac{z_t}{k_t})E(r_{\nu }) < V(t, r_{\omega }).$$

This contradicts the assumption. Hence, the minimum-VaR portfolio is mean–CVaR efficient if it exists. Thus, the assertion of Proposition 2 holds. $\square$

Proof of Proposition 3

We first prove that the condition $z_t>\sqrt{D/C}$ is a necessary and sufficient condition that the minimum-VaR portfolio exists at the $100t\%$ confidence level. For any mean–variance efficient portfolio $\omega$, applying Eq. (6), we get

$$E(r_{\omega })=E(r_{\rm b})+\frac{A-EC+FA}{C}+\sqrt{\frac{D}{C}(\sigma^2_{\omega }-a)}.$$

(13)

Now, we have to solve

$$\min_{\omega \in W} V[t; r_{\omega }].$$

Applying Eq. (6), we also know that $\sigma (r_{\omega_{\sigma }})=\sqrt{a}$. Hence, in order to obtain the VaR of the minimum-VaR portfolio, we use Proposition 2 and both Eqs. (6) and (13) to solve

$$\min_{\sigma \in [\sqrt{a},\infty ]} z_t\sigma -\left( E(r_{\rm b})+\frac{A-EC+FA}{C}+ \sqrt{\frac{D}{C}(\sigma^2-a)}\right).$$

(14)

We note that

$$\frac{\partial \left[ z_t\sigma -\left( E(r_{\rm b})+\frac{A-EC+FA}{C}+ \sqrt{\frac{D}{C}(\sigma^2-a)}\right) \right] }{\partial \sigma } =z_t-\frac{\sigma \sqrt{D/C}}{\sqrt{\sigma^2-a}}.$$

(15)

Clearly, $\bar{\sigma }=\sqrt{a}$ cannot be used to solve the minimization problem (14) because

$$\lim_{\sigma \rightarrow \sqrt{a}}z_t-\frac{\sigma \sqrt{D/C}}{\sqrt{\sigma^2-a}}=-\infty .$$

Employing Eq. (15), we obtain the following necessary condition for $\sigma_{\omega_V}$:

$$z_t=\frac{\sigma_{\omega_V}\sqrt{D/C}}{\sqrt{\sigma^2_{\omega_V}-a}}$$

to solve the minimization problem (14).

From the above equation, we can have

$$\sigma^2_{\omega_V}=\frac{az^2_t}{z^2_t-D/C}.$$

Thus, $z_t>\sqrt{D/C}$ is found to be the necessary condition that the minimum-VaR portfolio exists at the $100t\%$ confidence level.

We turn to show that $z_t>\sqrt{D/C}$ is the sufficient condition that The minimum-VaR portfolio exists at the $100t\%$ confidence level by proving that we are minimizing a convex function. Employing Eq. (15), we get

$$\frac{\partial^2\left[ z_t\sigma -\left( E(r_b)+\frac{A-EC+FA}{C}+ \sqrt{\frac{D}{C}(\sigma^2-a)}\right) \right] }{\partial \sigma^2}= \frac{\partial [z_t-\sigma \sqrt{D/C}/\sqrt{\sigma^2-a}]}{\partial \sigma }= a\sqrt{\frac{D}{C}}\frac{1}{(\sigma^2-a)^{3/2}}>0.$$

(16)

This completes the first part of our proof.

Further, from Eq. (13), we can have, when $z_t>\sqrt{D/C}$,

$$E(r_{\omega_{V(t)}})=E(r_{\rm b})+\frac{A-EC+FA}{C}+\sqrt{\frac{D}{C}(\frac{aCz^2_t}{Cz^2_t-D}-a)}.$$

$\square$

Proof of Corollary 2

For $m_{L(t)}$, we can have

$$E(r_{\omega_{L(t)}})=E(r_{\rm b})+\frac{A-EC+FA}{C}+\sqrt{\frac{D}{C}(\frac{aCk^2_t}{Ck^2_t-D}-a)}.$$

Note the following equation follows:

$$\frac{\partial E(r_{\omega_{V(t)}})}{\partial z_t}=-\sqrt{Da}\frac{D+Cz^2_t}{Cz^2_t-D}<0.$$

Since $k_t>z_t$, Corollary 2 follows. $\square$

Proof of Proposition 4

We first prove (i). To do so, we need to prove that for every mean–VaR boundary portfolio $\omega$, we have $\partial^2 V(t,r_{\omega })/\partial E(r_{\omega })^2>0$. Suppose that $\omega$ is a mean–VaR boundary portfolio being chosen arbitrarily. Then, employing Eq. (10), we get

$$\sigma_{\omega }=\sqrt{a+\frac{(E_{r_{\omega }}-E(r_{\rm b})-(A-EC+FA)/C)^2}{D/C}}.$$

As a result, we can have

$$\frac{\partial V(t,r_{\omega })}{\partial E(r_{\omega })}= \frac{\partial [z_t\sqrt{a+\frac{(E_{r_{\omega }}-E(r_{\rm b})-(A-EC+FA)/C)^2}{D/C}}-E(r_{\omega })]}{\partial E(r_{\omega })} = \frac{z_tC(E_{r_{\omega }}-E(r_{\rm b})-(A-EC+FA)/C)}{D\sqrt{a+\frac{(E_{r_{\omega }}-E(r_{\rm b})-(A-EC+FA)/C)^2}{D/C}}}-1.$$

(17)

Consequently, we can further have

$$\frac{\partial^2 V(t,r_{\omega })}{\partial E(r_{\omega })^2}= \frac{\partial [z_t\sqrt{a+\frac{(E_{r_{\omega }}-E(r_{\rm b})-(A-EC+FA)/C)^2}{D/C}}-E(r_{\omega })]}{\partial E(r_{\omega })} = \frac{z_tCa}{D\left( a+\frac{(E_{r_{\omega }}-E(r_{\rm b})-(A-EC+FA)/C)^2}{D/C}\right)^{3/2}}>0.$$

Thus, the assertion of the first part of Proposition 4 holds.

Now, we turn to prove (ii). When $z_t\le \sqrt{D/C}$, employing Eq. (17), we find that for any mean–VaR boundary portfolio, we have $\frac{\partial V(t,r_{\omega })}{\partial E(r_{\omega })}<0$$\omega$. Thus, there exists a portfolio $\nu \in W$ that has both higher expected return and smaller VaR for any portfolio $\omega \in W$. Hence, the assertion of the second part of Proposition 4 holds. $\square$

Proof of Proposition 5

We assume that the investor is a mean–variance optimizer who solves the following optimization problem for the decision in his/her investment:

$$\min \frac{1}{2}Var(r_{\omega })$$

$${\rm s.t.} \omega^\tau E(r)=\mu -\omega_{\rm f}r_{\rm f} \omega^\tau I=1-\omega_{\rm f},$$

(18)

where $r_{\omega }=\omega_{\rm f}r_{\rm f}+\omega^\tau r+r_{\rm b}$, I is a vector of ones. .

From constraint Eq. (18), we can have

$$\mu =\omega_{\rm f}r_{\rm f}+\omega^\tau E(r)=(1-\omega^\tau I)r_{\rm f}+\omega^\tau E(r)=\omega^\tau (E(r)-r_{\rm f}I)+r_{\rm f}.$$

(19)

The problem (18) can be handled by using Lagrange multipliers method

$$L=\frac{1}{2}(\omega^\tau V\omega +2\omega^\tau {\rm Cov}(r,r_{\rm b}))+\lambda_1(1-\omega_f-\omega^\tau I)+\lambda_2(\mu -\omega^\tau (E(r)-r_{\rm f}I)-r_{\rm f}).$$

Then, we obtain the following first-order optimality condition:

$$V\omega +{\rm Cov}(r,r_{\rm b})-\lambda_2(E(r)-r_{\rm f}I)=0 \omega_{\rm f}=1-\omega^\tau I.$$

Consequently, we can have

$$\omega =V^{-1}(\lambda_2(E(r)-r_{\rm f}I)-{\rm Cov}(r,r_{\rm b})).$$

(20)

Plugging Eq. (20) into the constraints in Eq. (19) obtains the following Lagrange multipliers

$$\mu -r_{\rm f}=\omega^\tau (E(r)-r_{\rm f}I)=\lambda_2(B-2r_{\rm f}A+r^2_{\rm f}C)-E+r_{\rm f}F.$$

Here $A=I^\tau V^{-1}E(r),B=E(r)^\tau V^{-1}E(r), C=I^\tau V^{-1}I,D=BC-A^2,E={\rm Cov}(r,r_{\rm b})^\tau V^{-1}E(r),F={\rm Cov}(r,r_{\rm b})^\tau V^{-1}I$. It follows that

$$\lambda_2=\frac{\mu -r_{\rm f}+E-r_{\rm f}F}{B-2r_{\rm f}A+r^2_{\rm f}C}.$$

Further, for the portfolio $\omega$ on the mean–variance boundary, we can have

$$\sigma^2_{\omega }= \omega^\tau V\omega +2\omega^\tau {\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b})= \omega^\tau (\lambda_2(E(r)-r_{\rm f}I)-{\rm Cov}(r,r_{\rm b}))+2\omega^\tau {\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \lambda_2(\mu -r_{\rm f})+\omega^\tau {\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \lambda_2(\mu -r_{\rm f})+\lambda_2E-\lambda_2r_{\rm f}F-{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b}) = \frac{(\mu -r_{\rm f}+E-r_{\rm f}F)^2}{H}+a^*.$$

Here $H=B-2r_{\rm f}A+r^2_{\rm f}C,a^*=-{\rm Cov}(r,r_{\rm b})^\tau V^{-1}{\rm Cov}(r,r_{\rm b})+{\rm Var}(r_{\rm b})>0$.

Consequently, we can have that portfolio $\omega$ is on the mean–variance boundary with background risk if and only if

$$\frac{\sigma^2_{\omega }}{a^*}-\frac{(E(r_{\omega })-E(r_{\rm b})-(r_{\rm f}+r_{\rm f}F-E))^2}{Ha^*}=1,$$

and thus, Proposition 5 follows. $\square$