# Incentive Compatible Decision Making: Real Options with Adverse Incentives

## Abstract

**:**

## 1. Introduction

## 2. The Model

_{t}is a standard Brownian motion process defined on probability space Ω and probability measure $P$ endowed with the usual Brownian filtration. We assume that the time t total operating cash flow C

_{t}is comprised of the systematic cash flow ${C}_{\mathrm{t}}^{\mathrm{s}}$ and an unsystematic component ${y}_{t}\left(\epsilon \right)$ where $\epsilon $ represents some unobservable action or effort on the part of the manager who undertakes the project and impacts on the distribution of the unsystematic cash flow ${y}_{t}\left(\epsilon \right)$. The effort$\epsilon $ is costly to the manager and we denote the expected value of ${y}_{t}\left(\epsilon \right)$ conditional on $\epsilon $, as $\overline{y}$ which is the same for t. For now, we will assume that the level of effort is known (induced through some wage mechanism) and suppress the effort argument. We will comment further about the effort level in a later section after address the implication of basing the wage mechanism on residual income. Based on the solution to the stochastic differential equation above, we get that the random operating cash flow generated by the investment project at time t is

**Assumption**

**1.**

**Assumption**

**2.**

#### 2.1. The Traditional NPV Approach

_{0}. We assume that once the investment project is operational, its cash flows continue on forever (unless the project is sold for some salvage value which is a special case of what is discussed in the next section). The traditional approach to assessing this project is then to discount the cash flows to infinity and subtract this cost. That is

#### 2.2. The Real Options Approach

_{B}.

_{B}equal to salvage value. Note, that I

_{B}can be easily generalized to be an increasing function of α or of time. The boundary B is initially assumed to be constant and we will later verify that this assumption is valid.

_{0}leads to the below closed form expression:

_{0}is the traditional NPV value. The third and fourth parts comprise the real options value of the investment and are analogous to a financial options framework where the option value is the discounted future cash flows net of exercise price conditional on the option being “in-the-money”. We can interpret $G\left(C,B\right)$ as the discounted probability of exercising the option and we point out that $\left|B-C\right|/\left(\sqrt{{\mu}^{2}+2r{\sigma}^{2}}\right)$ is the expected time (under certainty-equivalence) to exercising the real option.

#### 2.3. Optimal Exercise of Real Options

_{t}, then the smooth pasting condition is

^{NPV}, is

- -
- Initial cash flow, C = 100
- -
- Risk-free interest rate, r = 0.05
- -
- Expected periodic cash flow change, μ = 5
- -
- Volatility of cash flow changes, σ = 7.5
- -
- Price of risk of cash flows, θ = 0.25
- -
- Expansion factor, α = 1.5
- -
- Initial Capital Expenditure, I
_{0}= 2500 - -
- Add-on Capital Expenditure, I
_{B}= 2500

_{B}, μ, σ, θ, and α.

## 3. Multiple Objectives and the Agency Problem of Adverse Incentives

**Assumption**

**3.**

_{t}is the current book value of the capital assets used for the project. In terms of a depreciation rate δ it is clear that

**Definition**

**1.**

#### Project Market Values

_{0}= I

_{B}= 2500, μ = 5, σ = 5, T = 10, α = 1.7, δ = 0.25 and R* = 0.065, it shows that the real options boundary increases as hurdle rates increase and that at a hurdle rate of about 6%, the optimal boundary and the manager’s boundary coincide. Figure 1 also shows that the curves for different time horizons are almost identical implying that manager’s with different horizons may tend to behave in the same manner.

## 4. Incentive Compatible Cost-of-Capital

## 5. Conclusions

## Conflicts of Interest

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**Figure 5.**(

**A**) Iso-Incentive Curve (Dep. vs. Imp. Interest for different horizons); (

**B**) Iso-Incentive Curve (Dep. vs. Imp. Interest for different managerial hurdle rates).

**Figure 6.**(

**A**) Incentive Comp. Cost-of-capital vs. depreciation; (

**B**) Incentive Comp. Cost-of-capital vs. hurdle rate.

w.r.t. | Direction | |
---|---|---|

Investment Outlay | $\frac{\partial {B}^{*}}{\partial {I}_{0}}$ | >0 |

Expansion Factor | $\frac{\partial {B}^{*}}{\partial {I}_{B}}$ | <0 |

Volatility of Cash Flows | $\frac{\partial {B}^{*}}{\partial \sigma}$ | >,<0 |

Exp. Change in Cash Flows | $\frac{\partial {B}^{*}}{\partial \mu}$ | <0 |

Price of Risk | $\frac{\partial {B}^{*}}{\partial \theta}$ | >0 |

Risk-free rate | $\frac{\partial {B}^{*}}{\partial r}$ | >,<0 |

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**MDPI and ACS Style**

Osakwe, C.-J.U.
Incentive Compatible Decision Making: Real Options with Adverse Incentives. *Axioms* **2018**, *7*, 9.
https://doi.org/10.3390/axioms7010009

**AMA Style**

Osakwe C-JU.
Incentive Compatible Decision Making: Real Options with Adverse Incentives. *Axioms*. 2018; 7(1):9.
https://doi.org/10.3390/axioms7010009

**Chicago/Turabian Style**

Osakwe, Carlton-James U.
2018. "Incentive Compatible Decision Making: Real Options with Adverse Incentives" *Axioms* 7, no. 1: 9.
https://doi.org/10.3390/axioms7010009