Global Optimal Consumption–Portfolio Rules with Myopic Preferences and Loss Aversion

Abstract

In this paper, we construct a continuous-time market where the investor’s preferences take the form of myopic preferences used in Basak et al. (Rev Financ Stud 32:4905–4946, 2019) and the utility is assigned to the investor’s gains and losses with loss aversion. Myopic preferences make our financial model flexible and generally applicable in continuous-time markets, and loss aversion poses limitation to the investor’s consumption and investment. Then we derive the global optimal solutions to the related optimization problems through different regions, thereby obtaining global optimal consumption–portfolio rules for the investor. Candidates in different regions for global optimal consumption–portfolio rules could indicate how the investor reacts to consumption and investment with myopic preferences and loss aversion. The theoretical and numerical analysis shows that myopic preferences and loss aversion enable the investor to comply with optimal consumption–portfolio rules by balancing consumption against investment.

Appendix: Proofs

Proof of Lemma 3.1

The proof is mainly based on Karush–Kuhn–Tucker conditions. Since problem (3.2) is the same as (3.5) in mathematics and so it suffices to prove the case of the condition (a).

Firstly, we prove the unique existence of the solution to equation $h_1(n)=0$. Indeed, no matter what the term $A-r$ equals, we always have

$$\begin{aligned}&h_1'(n)=-\rho \vartheta _0 W(A-r)^2 e^{-\vartheta _0 W((A-r)n+r)}e^{\vartheta _1 W} -\vartheta _1 W\nu <0;\\&h_1(-\infty )=\lim \limits _{n\rightarrow -\infty }h_1(n)=+\infty ,\,h_1(+\infty )= \lim \limits _{n\rightarrow +\infty }h_1(n)=-\infty . \end{aligned}$$

Then by the strict monotonicity and the zero point theorem, there exists a unique $\hat{n}_1\in {\mathbb {R}}$ such that $h_1(\hat{n}_1)=0$.

Secondly, the function $-J_a^+$ is strictly convex (see, for example, Berkovitz 2002) and differentiable with respect to $(c,n)\in {\mathbb {R}}^2$, and the functions $f_i\, (i=1,2,\ldots ,4)$ are all convex and differentiable with respect to $(c,n)\in {\mathbb {R}}^2$ and satisfy the Slater constraint qualification:

$$\begin{aligned} {\text {There exists}}\, (c_0,n_0)\in {\mathbb {R}}^2 \,\,{\text {with}}\,\, f_i<0\, (i=1,2,\ldots ,4). \end{aligned}$$

(6.1)

Actually, since $f_i\, (i=1,2,\ldots ,4)$ are all linear with respect to $(c,n)\in {\mathbb {R}}^2$, it is obvious that they are convex and differentiable. Following Theorem 3.3 of Chapter III in Berkovitz (2002), the statement that $-J_a^+$ is strictly convex and differentiable could be proved by verifying that its related Hesse matrix is positive definite for each $(c,n)\in {\mathbb {R}}^2$. And this can be proved by computing the related Hesse matrix directly, i.e.,

$$\begin{aligned} \bigtriangledown ^2 (-J_a^+(c,n))=\left( \begin{aligned}&\rho \vartheta _0e^{-\vartheta _0c}&\qquad 0\\&\quad 0&\vartheta _1W^2\nu e^{-\vartheta _1W} \end{aligned} \right) . \end{aligned}$$

Now, if $A\ge r$ holds, then set $c_0=\frac{Wr}{2},n_0=\frac{\overline{n}}{2}$ in $f_i\, (i=1,2,\ldots ,4)$; and if $A<r$ holds, then set $c_0=\frac{Wr}{2},n_0=\frac{\overline{n}k_0}{2}$ with $k_0=\min \{1,\frac{r}{2(r-A)\overline{n}}\}$ in $f_i\, (i=1,2,\ldots ,4)$. So it is seen that (6.1) always holds.

Thirdly, since the function $-J_a^+$ is continuous on the compact set $R_1=\{(c,n):f_i(c,n)\le 0,i=1,2,3,4\}$, the global optimal solution $(c_1^*,n_1^*)$ to (3.2) exists. And making use of Theorem 3.1 of Chapter IV in Berkovitz (2002), strict convexity of $-J_a^+$ implies that there exists at most one global optimal solution to (3.2). Thus, the global optimal solution $(c_1^*,n_1^*)$ to (3.2) uniquely exists.

Finally, in order to derive the global optimal solution $(c_1^*,n_1^*)$, we define the associated Lagrangian function $L_1:{\mathbb {R}}^6\rightarrow [-\infty ,+\infty )$ as

$$ L_{1} (c,n,\lambda ) = \left\{ \begin{aligned} \begin{array}{*{20}l} { - J_{a}^{ + } (c,n) + \sum\limits_{{i = 1}}^{4} {\lambda _{i} } f_{i} (c,n),} \hfill & {\lambda _{i} \ge 0,i = 1,2,3,4;} \hfill \\ { - \infty ,} \hfill & {otherwise,} \hfill \\ \end{array} \end{aligned}\right. $$

where $\lambda =(\lambda _1,\lambda _2,\lambda _3,\lambda _4)$. Through convexity and the Slater constraint qualification (6.1) and Karush–Kuhn–Tucker conditions (see, for example, Proposition 3.2.3 and Theorem 3.2.8 in Borwein and Lewis (2000)), the vector $(c_1^*,n_1^*)$ is the global optimal solution to (3.2) if and only if there is a Lagrange multiplier vector $\lambda ^*$ such that $(c_1^*,n_1^*,\lambda ^*)$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll}&\frac{\partial }{\partial c}L_1(c_1^*,n_1^*,\lambda ^*)=0,\,\frac{\partial }{\partial n}L_1(c_1^*,n_1^*,\lambda ^*)=0;\\&\lambda _i^*\ge 0,\,f_i(c_1^*,n_1^*)\le 0,\,\lambda _i^*f_i(c_1^*,n_1^*)=0,\, i=1,2,3,4. \end{array}\right. \end{aligned}$$

(6.2)

In fact, whether $\lambda _1^*,\lambda _2^*,\lambda _3^*,\lambda _4^*$ are equal to zero divides the possible solutions to (6.2) into various cases, and then the necessary and sufficient conditions on which there exists a solution $(c_1^*,n_1^*,\lambda ^*)$ to (6.2) are obtained as follows:

• RG-1: $\lambda _1^*=0,\,\lambda _2^*=0,\,\lambda _3^*=0,\,\lambda _4^*=0, c_1^*=\Gamma _1,\,n_1^*=\kappa _1,\,\underline{n}\le \kappa _1\le \overline{n},\, 0\le \Gamma _1\le W((A-r)\kappa _1+r)$;

• RG-2: $\lambda _1^*=-\rho +e^{-\vartheta _1 W}>0,\,\lambda _2^*=0,\,\lambda _3^*=0,\,\lambda _4^*=0, \,c_1^*=0\le W((A-r)\kappa _1+r) ,\,n_1^*=\kappa _1,\,\underline{n}\le \kappa _1\le \overline{n};$

• RG-3: $\lambda _1^*=0,\,\lambda _2^*=e^{-\vartheta _1 W}\vartheta _1W^2\nu (\underline{n}-\kappa _1)>0, \,\lambda _3^*=0,\,\lambda _4^*=0,\,c_1^*=\Gamma _1,\,n_1^*=\underline{n},\, 0\le \Gamma _1\le W((A-r)\underline{n}+r)$;

• RG-4: $\lambda _1^*=0,\,\lambda _2^*=0, \,\lambda _3^*=e^{-\vartheta _1 W}\vartheta _1W^2\nu (\kappa _1-\overline{n})>0,\,\lambda _4^*=0, \,c_1^*=\Gamma _1,\,n_1^*=\overline{n},\, 0\le \Gamma _1\le W((A-r)\overline{n}+r)$;

• RG-5: $\lambda _1^*=0,\,\lambda _2^*=0,\,\lambda _3^*=0,\,\lambda _4^*=\rho e^{-\vartheta _0 c_1^*}-e^{-\vartheta _1 W}>0, c_1^*=W((A-r)n^*_1+r),\,h_1(n_1^*)=0,\,\underline{n}\le n_1^*\le \overline{n}$;

• RG-6: $\lambda _1^*=-\rho +e^{-\vartheta _1 W}>0, \,\lambda _2^*=e^{-\vartheta _1 W}\vartheta _1 W^2\nu (\underline{n}-\kappa _1)>0,\,\lambda _3^*=0, \,\lambda _4^*=0, \,c_1^*=0,\,n_1^*=\underline{n},\,0\le W((A-r)\underline{n}+r);$

• RG-7: $\lambda _1^*=-\rho +e^{-\vartheta _1 W}>0, \,\lambda _2^*=0,\,\lambda _3^*=e^{-\vartheta _1 W}\vartheta _1W^2\nu (\kappa _1-\overline{n})>0, \,\lambda _4^*=0, \,c_1^*=0,\,n_1^*=\overline{n},\,0\le W((A-r)\overline{n}+r);$

• RG-8: $\lambda _1^*=0,\,\lambda _2^*=-e^{-\vartheta _1 W}Wh_1(\underline{n})>0,\,\lambda _3^*=0,\,\lambda _4^* =\rho e^{-\vartheta _0c_1^*}-e^{-\vartheta _1 W}>0,\, c_1^*=W((A-r)\underline{n}+r),\,n_1^*=\underline{n},\,0\le W((A-r)\underline{n}+r);$

• RG-9: $\lambda _1^*=0,\,\lambda _2^*=0,\,\lambda _3^*=e^{-\vartheta _1W}Wh_1(\overline{n})>0,\,\lambda _4^* =\rho e^{-\vartheta _0c_1^*}-e^{-\vartheta _1W}>0,\, c_1^*=W((A-r)\overline{n}+r),\,n_1^*=\overline{n},\,0\le W((A-r)\overline{n}+r).$

If we assume the region is given (i.e., whether $\lambda _1^*,\lambda _2^*,\lambda _3^*,\lambda _4^*$ are equal to zero is determined), then necessity and sufficiency of conditions in Regions (RG-i) ($i=1,2,\ldots ,9$) are easy to verify. To complete the proof, it remains to exclude other cases of $\lambda ^*$. If $\lambda _1^*>0,\,\lambda _2^*=0,\,\lambda _3^*=0,\,\lambda _4^*>0$ hold, then (6.2) implies

$$\begin{aligned} A\ne r,\,\lambda _4^*=-e^{-\vartheta _1W}\left( 1+\frac{\vartheta _1 W\nu r}{(A-r)^2}\right) <0, \end{aligned}$$

which is a contradiction. If $\lambda _2^*>0,\,\lambda _3^*>0$ hold, then (6.2) gives $n_1^*=\underline{n},\,n_1^*=\overline{n}$, which is also a contradiction and excludes the following cases: (1) $\lambda _1^*=0,\lambda _2^*>0,\lambda _3^*>0,\lambda _4^*=0$; (2) $\lambda _1^*>0,\lambda _2^*>0,\lambda _3^*>0,\lambda _4^*=0$; (3) $\lambda _1^*=0,\lambda _2^*>0,\lambda _3^*>0,\lambda _4^*>0$, (4) $\lambda _1^*>0,\lambda _2^*>0,\lambda _3^*>0,\lambda _4^*>0$. If $\lambda _1^*>0,\,\lambda _2^*>0,\,\lambda _3^*=0,\,\lambda _4^*>0$ hold, then (6.2) gives $c_1^*=0,n_1^*=\underline{n}$ and $c_1^*=W((A-r)n_1^*+r)$. And it follows $A-r\ne 0,\,n_1^*=\underline{n}=-\frac{r}{A-r}$ and $A>r$ from $\underline{n}\le 0$. On the other hand, by using $A>r$ and $\frac{\partial }{\partial n}L_1(c_1^*,n_1^*,\lambda ^*)=0$, we have

$$\begin{aligned} \lambda _2^*+W(A-r)\lambda _4^*=-W(A-r)e^{-\vartheta _1 W}\left( 1+\frac{\vartheta _1 W\nu r}{(A-r)^2}\right) <0, \end{aligned}$$

which contradicts $\lambda _2^*>0,\lambda _4^*>0$. Thus, the case of $\lambda _1^*>0,\,\lambda _2^*>0,\,\lambda _3^*=0,\,\lambda _4^*>0$ is excluded, and similarly the case of $\lambda _1^*>0,\,\lambda _2^*=0,\,\lambda _3^*>0,\,\lambda _4^*>0$ can be also excluded. Now we have excluded all other cases. Summarizing, there exists a solution $(c_1^*,n_1^*,\lambda ^*)$ to (6.2) if and only if one of conditions in Regions $\mathrm {RG}-i$ ($i=1,2,\ldots ,9$) is satisfied, and we have completed the proof. $\square$

Proof of Lemma 3.2

Notice that the function $-J_a^-$ is coercive on $R_2=\{(c,n):g_i(c,n)\le 0,i=1,2,3,4\}$, i.e.,

$$\begin{aligned} (c,n)\in R_2, c^2+n^2\rightarrow +\infty \Rightarrow -J_a^-(c,n)\rightarrow +\infty , \end{aligned}$$

and then by Theorem 9.3-1 in Ciarlet (2013), this guarantees the existence of the solutions to (3.3). On the other hand, the function $-J_a^-$ is not convex with respect to $(c,n)\in \mathbb {R}^2$ and hence the solutions may not be unique. However, Karush–Kuhn–Tucker conditions can be still applied to obtain the optimal solution.

Firstly, by verifying the Slater constraint qualification of $g_i$ via $c_0=2Wr+\max \{W(A-r)n_0,0\},n_0=\frac{\underline{n}+\overline{n}}{2}$, we can apply Karush–Kuhn–Tucker conditions to (3.3). Define the associated Lagrangian function $L_2:{\mathbb {R}}^6\rightarrow [-\infty ,+\infty )$ as

$$\begin{aligned} L_2(c,n,\lambda )=\left\{ \begin{array}{lll}&-J_a^-(c,n)+\sum _{i=1}^4\lambda _i g_i(c,n),&\lambda _i \ge 0,i=1,2,3,4;\\&-\infty ,&otherwise. \end{array} \right. \end{aligned}$$

Then we can obtain all candidates (local optimal solutions) for the global optimal solutions to (3.2) by solving

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial }{\partial c}L_1(c_2^*,n_2^*,\lambda ^*)=0,\,\frac{\partial }{\partial n}L_1(c_1^*,n_1^*,\lambda ^*)=0;\\&\lambda _i^*\ge 0,\,g_i(c_1^*,n_1^*)\le 0,\,\lambda _i^*g_i(c_1^*,n_1^*)=0,\, i=1,2,3,4. \end{aligned}\right. \end{aligned}$$

With an analogous method in proof of Lemma 3.1, such candidates are obtained as those in Regions (RL-1)–(RL-12) of Lemma 3.2. Finally, we determine the global optimal solution (or solutions) by direct search over points $(c_{2,i}^*,n_{2,i}^*)$ ($i=1,2,\ldots ,12$) to find the point at which the function $-J_a^-$ achieves global minimum, which gives rise to (3.7). $\square$

Proof of Lemma 3.3

The existence of the solution to (3.6) can be guaranteed by making use of Theorem 9.3-1 in Ciarlet (2013) and the fact that the function $-J_b^-$ is coercive on $R_3=\{(c,n):g_i(c,n)\le 0,i=1,2,3,4\}$. Then the rest of the proof is analogous with that of Lemma 3.1, and so we omit it here. $\square$

Proof of Proposition 3.1

Under the condition (a), whether the investor expects to gain or lose wealth separates (3.1) into two cases. Then $(c_1^*,n_1^*)$ in Lemma 3.1 and $(c_2^*,n_2^*)$ in Lemma 3.2 are candidates for the global optimal solution to (3.1). Searching over above vectors, we can determine the global optimal solution (or solutions), which gives (3.8). $\square$

Proof of Proposition 3.2

Under the condition (b), whether the investor expects to gain or lose wealth separates (3.4) into two cases again. Then $(c_1^*,n_1^*)$ in Lemma 3.1 and $(c_3^*,n_3^*)$ in Lemma 3.2 are candidates for the global optimal solution to (3.4). Searching over above vectors, we can determine the global optimal solution (or solutions), which gives (3.18). $\square$

Proof of Proposition 3.3

The proof is analogous with that of Proposition 3.2, and we can prove it in a similar way. $\square$