A Modified 'Square Root' Process for Determining the Value of the Option to (Dis)invest

Abstract:   We determine optimal investment criteria for a capital project whose cash flows evolve in terms of a 'modified square root' process. The modified square root process has properties similar to the Cox, Ingersoll and Ross (1985) 'square root' process but in addition, encompasses the possibility of negative cash flows. Although closed form solutions for the valuation equations implied by the modified square root process are the exception rather than the rule we are able to show that analytic solutions, in the form of infinite power series expansions, will always exist. Furthermore, we also show that when a prescribed set of regularity conditions is satisfied, these power series expansions converge to closed form valuation functions. The optimality criteria for capital investment decisions are then simplified considerably.     

Capital Allocation in Financial Firms

This paper develops a theory of capital allocation in financial intermediaries where the cost of "risk capital" is a critical consideration. The implication for capital budgeting is that financial firms should use a modified NPV rule in which projects are valued by calculating the NPV of cash flows using marketdetermined discount rates and then subtracting a deadweight cost of capital that reflects the project's marginal contribution to firm-wide risk.
By taking account of deadweight costs—mainly monitoring and moral hazard costs associated with having too little equity capital as well as "free cash flow" agency costs and higher taxes associated with having too much—the capital allocation model predicts that financial firms will diversify across businesses with similar deadweight costs. Such diversification reduces the cost of risk capital for the individual businesses, thereby creating more profitable investment opportunities at the margin and enabling the businesses to operate on a larger scale. The authors note that their model has similarities to but also important differences from the standard applications of RAROC models.     

CAPITAL BUDGETING DECISIONS WHEN CASH FLOWS AND PROJECT LIVES ARE STOCHASTIC AND DEPENDENT

The purpose of this study is twofold: (1) to develop an operational economic state and simulation capital budgeting procedure for allowing cash flows and project lives to be dependent and (2) to provide empirical evidence of the impact of stochastic project lives on mean-variance and mean-semivariance capital budgeting decisions. The required number of input estimates for the proposed model is small. For individual projects, incorrectly assuming deterministic project lives when project lives are stochastic often results in large overestimates of expected net present values and large underestimates of the variance of the net present value. Similar results occur for the mean-variance and mean-semivariance portfolio models. The primary managerial implication of this study is that the inclusion of stochastic project lives in capital budgeting decisions is critical to obtain appropriate risk-return estimates.     

Time to wealth goals in capital accumulation

This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices, with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.     

Net working capital investment and capital budgeting analysis: Some pedagogical insights

Net working capital (NWC) investment, as a factor in discounted cash flow (DCF) analysis, receives little attention in the capital budgeting literature and accounting textbooks. The purpose of this paper is to explore the ways in which this important component of the analysis can be intregrated into the classroom and thus add to the student's overall understanding of capital budgeting. Four areas are discussed: (1) the significance of NWC investment in capital budgeting analysis, (2) the opportunity cost nature of the NWC investment, (3)measurement of the components of the NWC investment, and (4) use of the NWC investment to help restore the bottom line in DCF analysis to a pure cash flow basis. Integration of the fourth point into the topic of capital budgeting is found to be a convenient way to reinforce the student's understanding of the statement of cash flows.     

PROJECT HOLDING-PERIOD RATE OF RETURN AND THE MIRR

In this paper we examine the relationship between the Modified Internal Rate of Return (MIRR) and a project's expected holding-period rate of return assuming that cash flows are reinvested at the cost of capital. When cash flows are uncertain, the MIRR overstates the expected holding-period rate of return. The relationship between the MIRR and a project's expected holding-period rate of return is shown to be a simple function of conventional project statistics like the coefficient of variation of the present value of random cash flows, the profitability index, the cost of capital, and the project's life.     

Factors affecting biasing of capital budgeting cash flow forecasts: evidence from the hotel industry

This study contributes to a neglected aspect of the capital budgeting process, namely, the proposal development stage, which is primarily concerned with project cash flow estimation. Given that the deployment of sophisticated selection techniques is severely undermined when directed to input data suffering from bias, it is surprising that minimal empirical research has sought to explore for antecedent factors associated with biasing of capital budgeting cash flow forecasts. This paper reports the findings of a survey concerned with determining factors associated with biasing of capital budget cash flow forecasts in hotels that are mediated by a management contract. Statistically significant support is provided for the view that higher levels of biasing of capital budget cash flow forecasts occur in the presence of: high emphasis attached to the payback investment appraisal method; deficient reserve funds for furniture, fittings, and equipment (FF&E); low operator accessibility to reserve funds for FF&E; shorter periods of time to management contract expiry; and high emphasis attached to non-financial factors in capital budgeting appraisal.     

Integrating genetic algorithms and spreadsheets: a capital budgeting application

The role of the tax system in generating interactions between the post‐tax cash flows of different projects is discussed. When such interactions can occur, the capital budgeting process should be based around project combinations rather than individual projects. Evaluation of a project combination in net present value terms can easily be done using a spreadsheet. If the number of individual projects is large, then project combinations can be generated and an optimum combination of projects searched for using a genetic algorithm. The genetic algorithm approach has an advantage over alternative computational approaches, such as mixed integer programming, because of the more understandable representation of the problem it allows. Copyright © 2007 John Wiley & Sons, Ltd.     

Intertemporal capital budgeting

This paper analyzes the optimal capital budgeting mechanism when divisional managers are privately informed about the arrival of future investment projects. Consistent with field study evidence, an optimal allocation mechanism can include a stipulation that a capital request for discretionary investment will be declined with positive probability in the period after a significant investment was made even though this is ex post suboptimal. The model derives a number of empirical predictions regarding capital budgeting and the investment of financially constrained firms.     

THE ROLE OF THE PAYBACK PERIOD IN THE THEORY AND APPLICATION OF DURATION TO CAPITAL BUDGETING

The payback period has been a widely used capital budgeting tool in the analysis of capital projects. It has come under criticism for its inablility to consider all the project's flows in a present valued context. The purpose of this article is to present the duration measure in a capital budgeting framework and show how it is related to the payback period. The relationship is shown analytically and empirically. As a result, the payback period assumes a new identity which goes far to overcome the objections historically levied against it.     

Dynamic portfolio selection with process control

The risk inherent in the accumulation of investment capital depends on the true return distributions of the risky assets, the accuracy of estimated returns, and the investment strategy. This paper considers risk control with Value-at-Risk and Conditional Value-at-Risk, using control limits to determine times for portfolio rebalancing. Optimal strategies and control limits are determined for a geometric Brownian motion asset pricing model with random parameters. The approaches to risk control are applied to the fundamental problem of investment in stocks, bonds, and cash over time.     

The Role of Payback in the Investment Process

This study examines the use of the payback (PB) method as a means of evaluating a proposed asset's risk and its joint application with profit-oriented capital budgeting models. Previous research studies indicating a linkage between the PB method and risk analysis are reviewed. A certainty-equivalent model is used to demonstrate this relationship and the properties of the relationship exploited by PB when used as a heuristic. Results of the analysis indicate that using a hurdle PB as a filter for identifying proposals with acceptable risk and return attributes is consistent with more quantitatively oriented investment techniques under certain conditions. The study then examines the conceptual relationship between PB and profit-oriented capital budgeting models. Results suggest that PB and profit-oriented capital budgeting techniques measure different attributes of an investment and complement one another in describing and analysing its cash flows.     

Duration and Fisher's Rate of Return over Cost

This paper seeks to explain why the net present value profiles of investment projects may intersect with changes in the interest rate. Previous explanations have pointed to differences in the timing of the projects' cash flows. However, the timing of a project's cash flows has not been defined in the literature and the relative timing of projects' cash flows typically is illustrated by a simplistic example. We make use of the projects' durations to specify timing differences and provide a more explicit analysis of the role of timing differences in the intersection of present value profiles of projects.     

Internal versus External Financing: An Optimal Contracting Approach

We study optimal financial contracting for centralized and decentralized firms. Under centralized contracting, headquarters raises funds on behalf of multiple projects. Under decentralized contracting, each project raises funds separately on the external capital market. The benefit of centralization is that headquarters can use excess liquidity from high cash‐flow projects to buy continuation rights for low cash‐flow projects. The cost is that headquarters may pool cash flows from several projects and self‐finance follow‐up investments without having to return to the capital market. Absent any capital market discipline, it is more difficult to force headquarters to make repayments, which tightens financing constraints ex ante. Cross‐sectionally, our model implies that conglomerates should have a lower average productivity than stand‐alone firms.     

THE USER COST AND CAPITAL BUDGETING

This study describes the relationship between the user cost of capital and capital budgeting processes and presents a formula for computing the user cost of an investment, accommodating the existence of taxes, inflation, and a nonconstant level of production. A project acceptability criterion assigning positive net present value when cash operating income exceeds user cost is discussed and applied to several examples. The test is shown to be useful and easy to apply.     

An Empirical Study of the Adoption of Sophisticated Capital Budgeting Practices and Decision-Making Effectiveness

This paper examines the trend towards greater sophistication in investment selection techniques and control processes, and their impact on capital budgeting decision effectiveness. Based on a sample of 100 large UK firms, the study examines the capital budgeting practices employed over an 11-year period. Very significant increases in the uptake of sophisticated investment methods are reported, particularly in the analysis of project risk. These developments are partly explained by the rapid developments in computing within capital budgeting. Clear evidence is found to suggest that senior finance executives believe that the adoption of sophisticated investment practices gives rise to improved effectiveness in the evaluation and control of large capital projects.     

Valuación con opciones reales de proyectos con flujos correlacionados con fundamentales económicos y con saltos extremos Viabilidad del caso COMERCI UCB

This paper extends the method of discounted cash flows to value investment projects through incorporating real options. It is assumed the cash flows generated by the firm are correlated with macroeconomic fundamentals, particularly with the interest rate. It is also assumed that the cash flows have jumps whose size is given by an extreme value distribution. The flows are viewed as a portfolio of real options. The options arise from a stochastic dynamic optimization process where the investor (the entrepreneur) seeks to maximize his/her total profit discounted, subject to the wealth he/she possesses. This wealth includes the investment project, a risk-free bond, and a set of real options associated with the project.     

The odd notion of “reversible investment”

Irreversible investment and the attendant concept of real-option value have been well discussed. Complete reversibility has been frequently invoked but less studied, especially for the case of lumpy investment typically considered in capital budgeting. We examine a simple lumpy investment problem for the full range, from complete irreversibility to completely reversibility, with a focus on the latter. The optimal stopping rules under complete reversibility are to invest when the project generates enough net cash flow to cover Jorgenson’s opportunity cost of investment and to disinvest when it does not. Given the static nature of these rules, net present value as a timing criterion under reversibility is not pertinent. Investments that are partially reversible have much in common with completely irreversible investments but nothing in common with completely reversible investments. The case of reversible investment provides a foil for understanding that the distinguishing feature of investment as treated in corporate finance is that it entails at least some irreversibility.     

THE DETERMINATION AND USE OF INVESTMENT HURDLE RATES IN CAPITAL BUGETING: A SURVEY OF AUSTRALIAN PRACTICE*

This paper summarizes the results of a survey, conducted in 1979, which investigated Australian practice in the determination and use of investment hurdle rates, and in certain other areas of capital budgeting which impinge on hurdle rate practice. The study suggests a significant closure of the gap between theory and practice in capital budgeting in terms of the use of discounted cash flow techniques of capital project evaluation, and in terms of the use of some tools of finance such as the weighted average cost of capital. However, many developments in the determination and use of investment hurdle rates appear to have taken place at a slower rate, and it is possible that some “back-tracking” may be required in order to improve practice.     

Risk Management, Capital Budgeting, and Capital Structure Policy for Insurers and Reinsurers

This article builds on Froot and Stein in developing a framework for analyzing the risk allocation, capital budgeting, and capital structure decisions facing insurers and reinsurers. The model incorporates three key features: (i) value‐maximizing insurers and reinsurers face product‐market as well as capital‐market imperfections that give rise to well‐founded concerns with risk management and capital allocation; (ii) some, but not all, of the risks they face can be frictionlessly hedged in the capital market; and (iii) the distribution of their cash flows may be asymmetric, which alters the demand for underwriting and hedging. We show these features result in a three‐factor model that determines the optimal pricing and allocation of risk and capital structure of the firm. This approach allows us to integrate these features into: (i) the pricing of risky investment, underwriting, reinsurance, and hedging; and (ii) the allocation of risk across all of these opportunities, and the optimal amount of surplus capital held by the firm.