Investment decisions under incomplete markets in the presence of wealth effects

Abstract

This paper investigates the impact of market incompleteness on investment decisions. In contrast to previous studies, we assume an entrepreneur has power utility and investment causes an additive increase in her wealth. By using the binomial theorem, we derive the analytical solutions for the option value and investment threshold. First, we show that an increase in wealth alleviates the over-investment problem caused by the market incompleteness. In addition, the marginal value of wealth is always more than one and decreases in the wealth level. Finally, the increase in wealth substantially enhances the value of the investment option and lowers the idiosyncratic risk premium.

Appendices

Appendix A: Proof of Proposition 1

First, we find that it is difficult to directly derive the solution of $P\left( W,X\right)$ from the PDE (16). However, we can rewrite (5) as

$$\begin{aligned} J\left( W_{t},X_{t}\right) =\underset{\tau \ge t}{\max }\text { }\mathbb {E} _{t}\left\{ \frac{\sum \limits _{n=0}^{\infty }\left( \begin{array}{c} 1-\gamma \\ n \end{array} \right) W_{t}^{1-\gamma -n}e^{-nr\left( \tau -t\right) }\left( X_{\tau }-I\right) ^{n}}{1-\gamma }\right\} . \end{aligned}$$

(A.1)

Here, we use the binomial theorem. Since $W_{t}$ is known at time t, we have

$$\begin{aligned} J\left( W_{t},X_{t}\right) =\underset{\tau \ge t}{\max }\text { }\frac{ \sum \limits _{n=0}^{\infty }\left( \begin{array}{c} 1-\gamma \\ n \end{array} \right) W_{t}^{1-\gamma -n}\mathbb {E}_{t}\left\{ e^{-nr\left( \tau -t\right) }\left( X_{\tau }-I\right) ^{n}\right\} }{1-\gamma }. \end{aligned}$$

(A.2)

Then, we treat $V_{n,t}=$ $\mathbb {E}_{t}\left\{ e^{-nr\left( \tau -t\right) }\left( X_{\tau }-I\right) ^{n}\right\}$ as the value of a conjectured real option with discount rate nr and lump-sum payoff $\left( X_{\tau }-I\right) ^{n}$. Based on the standard arguments, $V_{n}$ satisfies the following ODE:

$$\begin{aligned} nrV_{n}\left( X\right) =\mu XV_{n}^{\prime }\left( X\right) +\frac{1}{2} \sigma ^{2}X^{2}V_{n}^{\prime \prime }\left( X\right) . \end{aligned}$$

(A.3)

Combining this with boundary condition $V_{n}\left( \overline{X}\right) =\left( \overline{X}-I\right) ^{n}$, we have

$$\begin{aligned} V_{n}\left( X\right) =\left( \frac{X}{\overline{X}}\right) ^{\beta _{n}}\left( \overline{X}-I\right) ^{n}, \end{aligned}$$

(A.4)

where

$$\begin{aligned} \beta _{n}=\frac{1}{\sigma ^{2}}\left[ -\left( \mu -\frac{1}{2}\sigma ^{2}\right) +\sqrt{\left( \mu -\frac{1}{2}\sigma ^{2}\right) ^{2}+2nr\sigma ^{2}}\right] . \end{aligned}$$

(A.5)

Therefore, we obtain the closed-form solution for $P\left( W,X\right)$ as

$$\begin{aligned} P\left( W,X\right) =\left[ \left( 1-\gamma \right) J\left( W,X\right) \right] ^{\frac{1}{1-\gamma }}=W\left[ \sum \limits _{n=0}^{\infty }\left( \begin{array}{c} 1-\gamma \\ n \end{array} \right) W^{-n}V_{n}\left( X\right) \right] ^{\frac{1}{1-\gamma }}. \end{aligned}$$

(A.6)

First, we can easily verify that the above solution satisfies boundary conditions (17), (18) and (20). From boundary condition ( 19), we obtain the optimal investment threshold as

$$\begin{aligned} \frac{\left( W+\overline{X}-I\right) ^{\gamma }}{1-\gamma } \sum \limits _{n=1}^{\infty }\left( \begin{array}{c} 1-\gamma \\ n \end{array} \right) \beta _{n}W^{1-\gamma -n}\left( \overline{X}-I\right) ^{n}=\overline{ X}. \end{aligned}$$

(A.7)

To guarantee convergence for the left-hand side of the above equation, we need $W>\overline{X}-I$.

For the special cases of $\gamma =1$ and $\gamma =2$, the solutions for the entrepreneur’s certainty equivalent wealth and the investment threshold can be further simplified.

Special case I: $\gamma =1$

When the entrepreneur has log utility, her certainty equivalent wealth $P\left( W,X\right)$ becomes

$$\begin{aligned} P\left( W,X\right) =W\exp \left[ \sum _{n=1}^{\infty }\frac{\left( -1\right) ^{n+1}}{n}W^{-n}V_{n}\left( X\right) \right] , \end{aligned}$$

(A.8)

where $V_{n}\left( X\right)$ is given by (22). The optimal investment threshold $\overline{X}\left( W\right)$ becomes the solution of the following equation

$$\begin{aligned} \left( W+\overline{X}-I\right) \sum \limits _{n=1}^{\infty }\frac{\left( -1\right) ^{n+1}}{n}\beta _{n}W^{-n}\left( \overline{X}-I\right) ^{n}= \overline{X}. \end{aligned}$$

(A.9)

Special case II: $\gamma =2$

When the entrepreneur’s risk aversion coefficient equals 2, her certainty equivalent wealth $P\left( W,X\right)$ becomes

$$\begin{aligned} P\left( W,X\right) =\frac{W}{\sum \limits _{n=0}^{\infty }\left( -1\right) ^{n}W^{-n}V_{n}\left( X\right) }, \end{aligned}$$

(A.10)

and the optimal investment threshold $\overline{X}\left( W\right)$ becomes the solution of the following equation:

$$\begin{aligned} \left( W+\overline{X}-I\right) ^{2}\sum \limits _{n=1}^{\infty }\left( -1\right) ^{n+1}\beta _{n}W^{-1-n}\left( \overline{X}-I\right) ^{n}= \overline{X}. \end{aligned}$$

(A.11)

B Proof of Proposition 2

Following Wang, Wang and Yang (2016), we complete markets by introducing a tradable asset that is perfectly correlated with the project value. Since the project risk is idiosyncratic, it can be diversified away at no premium. Hence, the dynamics of this new financial asset are given by

$$\begin{aligned} \frac{dS_{t}}{S_{t}}=rdt+\sigma _{S}dZ_{t}, \end{aligned}$$

(B.1)

where $\sigma _{S}$ is the volatility parameter and Z is the same Brownian motion driving the dynamics of project value. When the market is incomplete, there is no such a tradable asset that is perfectly correlated with the project value. Hence this variable S disappears in the incomplete market case. Denote $\phi _{t}$ as the fraction of the agent’s wealth in this asset; then, the entrepreneur’s wealth W evolves as follows:

$$\begin{aligned} dW_{t}=rW_{t}dt+\sigma _{S}\phi _{t}W_{t}dZ_{t}. \end{aligned}$$

(B.2)

Using the standard principle of optimality, the value function satisfies the following PDE:

$$\begin{aligned} r\left( 1-\gamma \right) J\left( W,X\right)= & {} \underset{\phi }{\max }\text { }rWJ_{W}\left( W,X\right) +\mu XJ_{X}\left( W,X\right) +\frac{1}{2}\phi ^{2}\sigma _{S}^{2}W^{2}J_{WW}\left( W,X\right) \nonumber \\&+\phi \sigma _{S}\sigma WXJ_{WX}\left( W,X\right) +\frac{1}{2}\sigma ^{2}X^{2}J_{XX}\left( W,X\right) . \end{aligned}$$

(B.3)

Compared with (12), there exist two additional terms on the right-hand side of the PDE: $\frac{1}{2}\phi ^{2}\sigma _{S}^{2}W^{2}J_{WW}$ and $\phi _{t}\sigma _{S}\sigma WXJ_{WX}$. We have $J_{WW}$ and $J_{WX}$ due to the stochastic returns for the new risky asset and dynamic hedging, respectively.

Similarly, we write the value function as

$$\begin{aligned} J\left( W,X\right) =\frac{\left[ P\left( W,X\right) \right] ^{1-\gamma }}{ 1-\gamma }. \end{aligned}$$

(B.4)

Then, we have

$$\begin{aligned} rP\left( W,X\right)= & {} \underset{\phi }{\max }\text { }rWP_{W}\left( W,X\right) +\mu XP_{X}\left( W,X\right) +\frac{1}{2}\phi ^{2}\sigma _{S}^{2}W^{2}P_{WW}\left( W,X\right) \nonumber \\&+\phi \sigma _{S}\sigma WXP_{WX}\left( W,X\right) +\frac{1}{2}\sigma ^{2}X^{2}P_{XX}\left( W,X\right) -\frac{1}{2}\gamma \phi ^{2}\sigma _{S}^{2}W^{2}\frac{P_{W}\left( W,X\right) ^{2}}{P\left( W,X\right) } \nonumber \\&-\gamma \phi \sigma _{S}\sigma WX\frac{P_{W}\left( W,X\right) P_{X}\left( W,X\right) }{P\left( W,X\right) }-\frac{1}{2}\gamma \sigma ^{2}X^{2}\frac{ P_{X}\left( W,X\right) ^{2}}{P\left( W,X\right) }. \end{aligned}$$

(B.5)

The FOC for hedging demand $\phi$ implies

$$\begin{aligned} \phi =-\frac{\sigma XP_{WX}\left( W,X\right) -\gamma \sigma X\frac{ P_{W}\left( W,X\right) P_{X}\left( W,X\right) }{P\left( W,X\right) }}{\sigma _{S}WP_{WW}\left( W,X\right) -\gamma \sigma _{S}W\frac{P_{W}\left( W,X\right) ^{2}}{P\left( W,X\right) }}. \end{aligned}$$

(B.6)

Substituting the optimal hedging demand into the PDE (B.5), we obtain

$$\begin{aligned} rP\left( W,X\right)= & {} rWP_{W}\left( W,X\right) +\mu XP_{X}\left( W,X\right) +\frac{1}{2}\sigma ^{2}X^{2}P_{XX}\left( W,X\right) -\frac{1}{2}\gamma \sigma ^{2}X^{2}\frac{P_{X}\left( W,X\right) ^{2}}{P\left( W,X\right) } \nonumber \\&-\frac{\sigma ^{2}X^{2}\left( P_{WX}\left( W,X\right) -\gamma \frac{ P_{W}\left( W,X\right) P_{X}\left( W,X\right) }{P\left( W,X\right) }\right) ^{2}}{2\left( P_{WW}\left( W,X\right) -\gamma \frac{P_{W}\left( W,X\right) ^{2}}{P\left( W,X\right) }\right) }. \end{aligned}$$

(B.7)

By using the insights for the CM case, we conjecture

$$\begin{aligned} P\left( W,X\right) =W+V\left( X\right) . \end{aligned}$$

(B.8)

Substituting the above-conjectured solution into (B.7) and the four boundary conditions (17) to (20), we have

$$\begin{aligned}&rV\left( X\right) =\mu XV^{\prime }\left( X\right) +\frac{1}{2}\sigma ^{2}X^{2}V^{\prime \prime }\left( X\right) , \end{aligned}$$

(B.9)

$$\begin{aligned}&V\left( X^{*}\right) =X^{*}-I, \end{aligned}$$

(B.10)

$$\begin{aligned}&V^{\prime }\left( X^{*}\right) =1. \end{aligned}$$

(B.11)

Therefore, (B.8) is indeed the solution for the CM model. Substituting the solution into (B.6) yields the following optimal hedging portfolio:

$$\begin{aligned} \phi =-\frac{\sigma XV^{\prime }\left( X\right) }{\sigma _{S}W}. \end{aligned}$$

(B.12)

Then, we obtain the wealth dynamics $W_{t}$:

$$\begin{aligned} dW_{t}=rW_{t}dt-\sigma X_{t}V^{\prime }\left( X_{t}\right) dZ_{t}. \end{aligned}$$

(B.13)

Thus, the entrepreneur’s wealth is negatively correlated with the project value. For the total wealth $P^{*}\left( W,X\right)$, we have

$$\begin{aligned} dP_{t}^{*}= & {} dW_{t}+\left( \mu X_{t}V^{\prime }\left( X_{t}\right) + \frac{1}{2}\sigma ^{2}X_{t}^{2}V^{\prime \prime }\left( X_{t}\right) \right) dt+\sigma X_{t}V^{\prime }\left( X_{t}\right) dZ_{t} \nonumber \\= & {} dW_{t}+rV\left( X_{t}\right) dt+\sigma X_{t}V^{\prime }\left( X_{t}\right) dZ_{t}=rP_{t}^{*}dt, \end{aligned}$$

(B.14)

which means that the total wealth is deterministic and increases at rate r.

Appendix C: Proof of Proposition 3

When the project has value $X_{t}$ follows a arithmetic Brownian motion, the ODE for $V_{n}$ is replaced by

$$\begin{aligned} nrV_{n}\left( X\right) =\mu V_{n}^{\prime }\left( X\right) +\frac{1}{2} \sigma ^{2}V_{n}^{\prime \prime }\left( X\right) . \end{aligned}$$

(C.1)

Combining this with boundary condition $V_{n}\left( \overline{X}\right) =\left( \overline{X}-I\right) ^{n}$, we have

$$\begin{aligned} V_{n}\left( X\right) =e^{\theta _{n}\left( X-\overline{X}\right) }\left( \overline{X}-I\right) ^{n}, \end{aligned}$$

(C.2)

where

$$\begin{aligned} \theta _{n}=\frac{\sqrt{\mu ^{2}+2nr\sigma ^{2}}-\mu }{\sigma ^{2}}. \end{aligned}$$

(C.3)

Therefore, we obtain the closed-form solution for $P\left( W,X\right)$ as

$$\begin{aligned} P\left( W,X\right) =\left[ \left( 1-\gamma \right) J\left( W,X\right) \right] ^{\frac{1}{1-\gamma }}=W\left[ \sum \limits _{n=0}^{\infty }\left( \begin{array}{c} 1-\gamma \\ n \end{array} \right) W^{-n}V_{n}\left( X\right) \right] ^{\frac{1}{1-\gamma }}. \end{aligned}$$

(C.4)

Then we obtain the optimal investment threshold as

$$\begin{aligned} \frac{\left( W+\overline{X}-I\right) ^{\gamma }}{1-\gamma } \sum \limits _{n=1}^{\infty }\left( \begin{array}{c} 1-\gamma \\ n \end{array} \right) \theta _{n}W^{1-\gamma -n}\left( \overline{X}-I\right) ^{n}=1. \end{aligned}$$

(C.5)

To guarantee convergence for the left-hand side of the above equation, we need $W>\overline{X}-I$ as well.

When the market is complete, the optimal investment threshold $X^{*}$ is standard as

$$\begin{aligned} X^{*}=I+\frac{1}{\theta }, \end{aligned}$$

(C.6)

where

$$\begin{aligned} \theta =\frac{\sqrt{\mu ^{2}+2r\sigma ^{2}}-\mu }{\sigma ^{2}}. \end{aligned}$$

(C.7)