# Investment Decisions with Two-Factor Uncertainty

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^{*}

## Abstract

**:**

## 1. Introduction

_{2}prices, all following (correlated) Geometric Brownian motions.

## 2. Investment Decision Given Two Uncertain Revenue Flows

**Theorem**

**1.**

- 1.
- $b\left(x\right)=sup\left\{\phantom{\rule{0.222222em}{0ex}}y\in {\Re}_{+}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}V(x,y)>F(x,y)\phantom{\rule{0.222222em}{0ex}}\right\}$ for all $x\in (0,{x}^{*})$;
- 2.
- b is non-increasing on $(0,{x}^{*})$;
- 3.
- b is convex on $(0,{x}^{*})$;
- 4.
- b is continuous;
- 5.
- $b\left(x\right)<{y}^{*}$ on $(0,{x}^{*})$, and $b\left(x\right)=0$ on $[{x}^{*},\infty )$.

- 1.
- closed;
- 2.
- convex.

- 1.
- $V>0$ on ${\Re}_{++}^{2}$;
- 2.
- V is convex;
- 3.
- V is continuous;
- 4.
- V is increasing in x and y.

**Remark**

**1.**

- 1.
- We can write$$D=\left\{\phantom{\rule{0.222222em}{0ex}}(x,y)\in {\Re}_{+}^{2}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}y<b\left(x\right)\phantom{\rule{0.222222em}{0ex}}\right\},\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}S=\left\{\phantom{\rule{0.222222em}{0ex}}(x,y)\in {\Re}_{+}^{2}\phantom{\rule{0.277778em}{0ex}}|\phantom{\rule{0.277778em}{0ex}}y\ge b\left(x\right)\phantom{\rule{0.222222em}{0ex}}\right\}.$$
- 2.
- The optimal stopping boundary can never lie below the Net Present Value boundary $\overline{b}$, i.e.,$$b\left(x\right)>\overline{b}\left(x\right):={\delta}_{2}(I-x/{\delta}_{1}),\phantom{\rule{1.em}{0ex}}all\phantom{\rule{4.pt}{0ex}}x\in (0,{\delta}_{1}I).$$

## 3. The Quasi-Analytical Approach

#### Results of the Quasi-Analytical Approach

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 4. Numerical Solution

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem 1.**

- ($V>0$ on ${\Re}_{++}^{2}$) On S, the result is trivial. Let $(x,y)\in D\cap {\Re}_{++}^{2}$. Consider the stopping time:$$\tau =inf\{t\ge 0|F({X}_{\tau},{Y}_{\tau})>0\}.$$Since ${e}^{-r\tau}F({X}_{\tau},{Y}_{\tau})=0$ on $\{\tau =\infty \}$ (since $r>max\{{\alpha}_{1},{\alpha}_{2}\}$) and $\mathrm{P}(\tau <\infty )>0$, it holds that:$$V(x,y)\ge \mathbb{E}\left[{e}^{-r\tau}F({X}_{\tau},{Y}_{\tau})\right]>0.$$
- (Convexity of V) On S, the result is trivial. Take $({x}^{\prime},{y}^{\prime}),({x}^{\u2033},{y}^{\u2033})\in D$ and $\lambda \in (0,1)$. Define $(x,y):=\lambda ({x}^{\prime},{y}^{\prime})+(1-\lambda )({x}^{\u2033},{y}^{\u2033})$. It then holds that:$$\begin{array}{cc}\hfill V(x,y)=& \underset{\tau}{sup}\mathbb{E}\left[{e}^{-r\tau}F({x}^{\prime},{y}^{\prime})\right]\hfill \\ \hfill =& \underset{\tau}{sup}\mathbb{E}\left[{e}^{-r\tau}\left(\frac{x{X}_{\tau}^{1}}{{\delta}_{1}}+\frac{y{Y}_{\tau}^{1}}{{\delta}_{2}}-I\right)\right]\hfill \\ \hfill =& \underset{\tau}{sup}\mathbb{E}\left[{e}^{-r\tau}\left(\frac{(\lambda {x}^{\prime}+(1-\lambda ){x}^{\u2033}){X}_{\tau}^{1}}{{\delta}_{1}}+\frac{(\lambda {y}^{\prime}+(1-\lambda ){y}^{\u2033}){Y}_{\tau}^{1}}{{\delta}_{2}}-I\right)\right]\hfill \\ \hfill =& \underset{\tau}{sup}\mathbb{E}\left[\lambda {e}^{-r\tau}\left(\frac{{x}^{\prime}{X}_{\tau}^{1}}{{\delta}_{1}}+\frac{{y}^{\prime}{Y}_{\tau}^{1}}{{\delta}_{2}}-I\right)+(1-\lambda ){e}^{-r\tau}\left(\frac{{x}^{\u2033}{X}_{\tau}^{1}}{{\delta}_{1}}+\frac{{y}^{\u2033}{Y}_{\tau}^{1}}{{\delta}_{2}}-I\right)\right]\hfill \\ \hfill \le & \lambda \underset{\tau}{sup}\mathbb{E}\left[{e}^{-r\tau}F({x}^{\prime},{y}^{\prime})\right]+(1-\lambda )\underset{\tau}{sup}\mathbb{E}\left[{e}^{-r\tau}F({x}^{\u2033},{y}^{\u2033})\right]\hfill \\ \hfill =& \lambda V({x}^{\prime},{y}^{\prime})+(1-\lambda )V({x}^{\u2033},{y}^{\u2033}).\hfill \end{array}$$
- (Continuity of V) This property follows from the general theory of stochastic processes: see, e.g., (Krylov 1980, Theorem 3.1.5).
- (Monotonicity of V) We proved that V is (strictly) increasing in x. Again, the result is trivial on S. Take $(x,y)\in D$ and let $\epsilon >0$ be such that $(x+\epsilon ,y)\in D$ (such $\epsilon $ exists since D is open; see below). Take any stopping time $\tau $. It then holds that:$$\mathbb{E}\left[{e}^{-r\tau}\left(\frac{(x+\epsilon ){X}_{\tau}^{1}}{{\delta}_{1}}+\frac{y{Y}_{\tau}^{1}}{{\delta}_{2}}-I\right)\right]\ge \mathbb{E}\left[{e}^{-r\tau}\left(\frac{x{X}_{\tau}^{1}}{{\delta}_{1}}+\frac{y{Y}_{\tau}^{1}}{{\delta}_{2}}-I\right)\right],$$
- (Closedness of D) Take a sequence ${({x}^{\left(n\right)},{y}^{\left(n\right)})}_{n\in \aleph}$ in S with limit $(x,y)$. Then, $V({x}^{\left(n\right)},{y}^{\left(n\right)})=F({x}^{\left(n\right)},{y}^{\left(n\right)})$ for all $n\in \aleph $. Since ${lim}_{n\to \infty}F({x}^{\left(n\right)},{y}^{\left(n\right)})=F(x,y)$ and V is continuous, it holds that $V(x,y)=F(x,y)$. This implies that $(x,y)\in S$.
- (Convexity of D) Suppose there exists $({x}^{\prime},{y}^{\prime}),({x}^{\u2033},{y}^{\u2033})\in S$ and $\lambda \in (0,1)$ such that $(x,y):=\lambda ({x}^{\prime},{y}^{\prime})+(1-\lambda )({x}^{\u2033},{y}^{\u2033})\in D$. It then holds that:$$V(x,y)>F(x,y)=\lambda F({x}^{\prime},{y}^{\prime})+(1-\lambda )F({x}^{\u2033},{y}^{\u2033})=\lambda V({x}^{\prime},{y}^{\prime})+(1-\lambda )V({x}^{\u2033},{y}^{\u2033}).$$This contradicts convexity of V.
- ($b\left(x\right)$ can be written as a sup) Take $(x,y)\in D$. There exists a stopping time ${\tau}^{*}$, such that $({X}_{{\tau}^{*}},{Y}_{{\tau}^{*}})\in D$, a.s. Hence,$$V(x,y)=\underset{\tau}{sup}\mathbb{E}\left[{e}^{-r\tau}F({X}_{\tau},{Y}_{\tau})\right]\ge \mathbb{E}\left[{e}^{-r{\tau}^{*}}F({X}_{{\tau}^{*}},{Y}_{{\tau}^{*}})\right]>F(x,y).$$Now take $\epsilon \in (0,y)$. Then:$$\begin{array}{cc}\hfill V(x,y-\epsilon )\ge & \mathbb{E}\left[{e}^{-r{\tau}^{*}}F({X}_{{\tau}^{*}},{Y}_{{\tau}^{*}})\right]\hfill \\ \hfill =& \mathbb{E}\left[{e}^{-r{\tau}^{*}}\left(\frac{x{X}_{{\tau}^{*}}^{1}}{{\delta}_{1}}+\frac{(y-\epsilon ){Y}_{{\tau}^{*}}^{1}}{{\delta}_{2}}-I\right)\right]\hfill \\ \hfill =& \mathbb{E}\left[{e}^{-r{\tau}^{*}}\left(\frac{x{X}_{{\tau}^{*}}^{1}}{{\delta}_{1}}+\frac{y{Y}_{{\tau}^{*}}^{1}}{{\delta}_{2}}-I\right)\right]-\mathbb{E}\left[{e}^{-r{\tau}^{*}}\frac{\epsilon {Y}_{{\tau}^{*}}^{1}}{{\delta}_{2}}\right]\hfill \\ \hfill \stackrel{(*)}{\ge}& \mathbb{E}\left[{e}^{-r{\tau}^{*}}\left(\frac{x{X}_{{\tau}^{*}}^{1}}{{\delta}_{1}}+\frac{y{Y}_{{\tau}^{*}}^{1}}{{\delta}_{2}}-I\right)\right]-\frac{\epsilon}{{\delta}_{2}}\hfill \\ \hfill >& \mathbb{E}\left[{e}^{-r{\tau}^{*}}\left(\frac{x{X}_{{\tau}^{*}}^{1}}{{\delta}_{1}}+\frac{y{Y}_{{\tau}^{*}}^{1}}{{\delta}_{2}}-I\right)\right]>F(x,y)>F(x,y-\epsilon ),\hfill \end{array}$$
- (b is non-increasing) This follows from the fact that for all $(x,y)\in D$ and all $\epsilon \in (0,x)$ it holds that $(x-\epsilon ,y)\in D$. This can be proved using a similar argument as above.
- (b is convex) Convexity of b follows from the fact that its epigraph is the convex set S.
- (b is continuous) Continuity of b on $(0,\infty )$ is immediate, because it is a convex function on an open convex set (see, for example, Berge 1963, Theorem 8.5.7). Continuity at $x=0$ follows from the fact that the stopping set is closed.
- (boundedness of b) The boundedness properties follow from continuity and ${x}^{*}$ and ${y}^{*}$ being the solutions of the optimal stopping problem on ${\Re}_{+}\times \left\{0\right\}$ and $\left\{0\right\}\times {\Re}_{+}$, respectively.

## Notes

1 | ${\mathbb{E}}_{(x,y)}$ denotes the expectation conditional on $({X}_{0},{Y}_{0})=(x,y).$ |

2 | Note that Adkins and Paxson (2011b) assume this is the case. |

3 | For simplicity, henceforth we assume that $\rho =0$. |

4 | The same holds for (Adkins and Paxson 2011a), see Table 2; Adkins and Paxson (2017b), see Table of Figure 1; Heydari et al. (2012), see Equation (19); Adkins and Paxson (2013a), see Equation (9); Adkins and Paxson (2013b), see Figure 2; Fleten et al. (2016), see Equation (17); Støre et al. (2018), see Equation (18); and Adkins and Paxson (2017a), see Table 3. |

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**Figure 1.**The threshold boundaries, $\widehat{b}$, for the following set of parameter values: ${\sigma}_{1}=0.2$, ${\alpha}_{1}=0.02$, ${\alpha}_{2}=0.02$, $r=0.1$, $\rho =0$, ${Q}_{1}=5$, ${Q}_{2}=10$, $I=2000$ and different values of ${\sigma}_{2}$.

**Figure 2.**The numerical threshold boundary for the following set of parameter values: ${\sigma}_{1}=0.2$, ${\alpha}_{1}=0.02$, ${\alpha}_{2}=0.02$, $r=0.1$, $\rho =0$, ${Q}_{1}=5$, ${Q}_{2}=10$, $I=2000$ and different values of ${\sigma}_{2}$.

**Figure 3.**The numerical threshold boundary for the following set of parameter values: $x=40.52$, ${\sigma}_{1}=0.2$, ${\sigma}_{2}=0.6$, ${\alpha}_{1}=0.02$, ${\alpha}_{2}=0.02$, $r=0.1$, $\rho =0$, ${Q}_{1}=5$, ${Q}_{2}=10$, $I=2000$ and different values of y.

**Figure 4.**The numerical value function and threshold boundary (solid black curve) for the following set of parameter values: ${\sigma}_{1}=0.2$, ${\sigma}_{2}=0.6$, ${\alpha}_{1}=0.02$, ${\alpha}_{2}=0.02$, $r=0.1$, $\rho =0$, ${Q}_{1}=5$, ${Q}_{2}=10$, $I=2000$ and different values of x and y.

**Table 1.**The value of the first two terms of Equation (29) for the following set of the parameter values: ${\sigma}_{1}=0.2$, ${\sigma}_{2}=0.6$, ${\alpha}_{1}=0.02$, ${\alpha}_{2}=0.02$, $r=0.1$, $\rho =0$, ${Q}_{1}=5$, ${Q}_{2}=10$, and $I=2000$.

$\widehat{\mathit{x}}$ | Contribution of Partial Derivatives |
---|---|

10 | −10,841.14 |

20 | −54,856.60 |

30 | −9040.40 |

**Table 2.**Percentage of cases when a firm undertakes an investment within the next 5 years for the set of parameter values: ${\sigma}_{1}=0.2$, ${\sigma}_{2}=0.6$, ${\alpha}_{1}=0.02$, ${\alpha}_{2}=0.02$, $r=0.1$, $\rho =0$, ${Q}_{1}=5$, ${Q}_{2}=10$, $I=2000$.

(a) Quasi-analytical boundary | |||

$({\mathit{x}}_{\mathbf{0}},{\mathit{y}}_{\mathbf{0}})$ | 5 | 10 | 15 |

10 | 10.06% | 23.97% | 39.03% |

15 | 21.69% | 42.87% | 61.69% |

20 | 40.32% | 68.34% | 90.21% |

(b) Numerical boundary | |||

$({\mathit{x}}_{\mathbf{0}},{\mathit{y}}_{\mathbf{0}})$ | 5 | 10 | 15 |

10 | 5.40% | 5.56% | 5.51% |

15 | 5.57% | 5.41% | 5.57% |

20 | 5.43% | 5.24% | 5.59% |

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**MDPI and ACS Style**

Compernolle, T.; Huisman, K.J.M.; Kort, P.M.; Lavrutich, M.; Nunes, C.; Thijssen, J.J.J.
Investment Decisions with Two-Factor Uncertainty. *J. Risk Financial Manag.* **2021**, *14*, 534.
https://doi.org/10.3390/jrfm14110534

**AMA Style**

Compernolle T, Huisman KJM, Kort PM, Lavrutich M, Nunes C, Thijssen JJJ.
Investment Decisions with Two-Factor Uncertainty. *Journal of Risk and Financial Management*. 2021; 14(11):534.
https://doi.org/10.3390/jrfm14110534

**Chicago/Turabian Style**

Compernolle, Tine, Kuno J. M. Huisman, Peter M. Kort, Maria Lavrutich, Cláudia Nunes, and Jacco J. J. Thijssen.
2021. "Investment Decisions with Two-Factor Uncertainty" *Journal of Risk and Financial Management* 14, no. 11: 534.
https://doi.org/10.3390/jrfm14110534