Optimal Investment Rules Research Articles

Hunting for superstars

The “superstar economy” is characterized by payoff functions that depend in a discontinuous way on the quality level of the corresponding products and services. Firm A might generate much higher returns than firm B, although A’s product is only marginally superior to B’s product. We look at an investor who considers to invest into start-ups that want to become active in one particular technological segment. Consequently only the very best few projects generate high returns. The investor is faced with a sequence of investment opportunities, observes the objective relative rankings of the corresponding projects seen so far, and must decide whether and how much to invest into the currently observed opportunity. Returns are realized at the end of the investment horizon. We derive the value functions and optimal investment rules for risk-neutral and risk averse investors. Under weak assumptions, the expected infinite horizon utility exceeds that of initial wealth. We show that for a risk-neutral investor “invest all or nothing”, depending on the project’s ranking and time of occurrence, is an optimal strategy. For a risk-averse investor the optimal rule is non-linear and path dependent. A simulation study is performed for risk-neutral and log-utility investors.

RISK-AVERSION: MATHEMATICAL AND ECONOMIC PERSPECTIVES

A necessary condition for a dominance of prospect [Formula: see text] over prospect [Formula: see text] by all commonly employed investment criteria is [Formula: see text]. Because of this requirement, optimal investment rules fail to distinguish between two alternative prospects where in practice one prospect is preferred over the other by all investors — a paradoxical result. We suggest the general almost second-degree stochastic dominance rule (GASSD) that relaxes this necessary condition, resolves the existing paradoxes and distinguishes between mathematically and economically relevant preferences. In the comparison of stocks and bonds for various horizons, the GASSD rule does not support the idiom “stocks for the long run”.

Debt recapitalization and value in waiting to finance a project

This paper shows that an option to recapitalize with debt accelerates the exercise of a real option to start an irreversible project investment. The debt-recapitalization add-on, while not directly affecting project cash flows, influences the timing of the project start by offering an extra gain from tax savings less debt costs. This finding demonstrates that capital structure decisions are closely linked to project investment decisions, a deviation from the standard optimal investment rule of the real option theory.

Paradoxes with Optimal Investment Rules

Assuming risk aversion, the Second degree Stochastic Dominance (SSD) criterion is employed for prospect ranking. However, in some cases the optimal SSD rule cannot distinguish between two alternative prospects when it is obvious that one prospect is preferred by practically all risk averters-a paradoxical result. The optimal Almost SSD (ASSD) rule which aims to avoid such paradoxical results also fails to do so. We suggest a new rule, the General ASSD rule (GASSD) to replace the ASSD rule, which in turn, solves the SSD as well the ASSD existing and potential paradoxes. The GASSD rule, unlike the SSD and ASSD rules, takes into account the magnitude of the mean return difference of the two prospects under consideration. The ASSD and the GASSD rules coincide only when these two means are equal.

Consumption, investment and life insurance strategies with heterogeneous discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent’s preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.

Irreversible Investment in Stochastically Cyclical Markets

This paper studies entry and exit decisions in markets whose demand alternates between growth and decline phases at uncertain times. We introduce a stochastic process that captures these features of random market evolution, and we provide key mathematical results related to first passage times which make the characterization of entry and exit behavior quite simple and straightforward (even when the process is subject to an endogenously determined upper or lower barrier). We characterize entry and exit patterns in a dynamic competitive equilibrium, and we show why our results differ from those obtained if demand follows a diffusion process (e.g., a Geometric Brownian Motion). Despite the stochastic process of the underlying variable has a continuous sample path in both cases, we demonstrate in our setting that positive rates of entry and exit discontinuously fall to zero owing to informational overshooting. Another advantage of our framework is that it can explain discontinuities in firm values even if sample paths are continuous. Our framework is also amenable to empirical implementations (as we show using Corts’ 2008 offshore oil drilling application), and to an intuitive interpretation of optimal (dis) investment rules based on Bernanke’s (1983) “bad news principle of irreversible investment.”

Consumption, Investment and Life Insurance Strategies with Heterogeneous Discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent's preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.

Modeling of timber harvesting options using timber prices as a mean reverting process with stochastic trend

Proper characterization of the timber price process plays a vital role in forest management decisions. The process of long-run timber prices and its implications for harvesting decisions are analyzed for a forest in Ontario, Canada. Timber prices are modeled as a mean reverting process with stochastic trend. The Kalman filter is used to estimate the state–space model. The forecasted prices from the model are used in real options analysis to determine the optimal investment time and optimal investment rule. The results provide insight different from that of other specifications used in earlier literature.

Portfolio Selection with Estimation Risk: A Test-Based Approach

An important challenge of portfolio allocation arises when the (true) characteristics of returns' distribution are replaced by sample estimates. Such substitutions introduce estimation risk, which adds to traditional financial risk. I develop a new framework to provide a feasible optimal investment rule that accounts for estimation risk. In borrowing from practitioners, I evaluate funds' allocations through their probability of defeating a chosen benchmark. More precisely, the P-value investment rule maximizes the p-value of a one-sided test, ensuring that the portfolio performance is above the given threshold. When the portfolio performance is measured by the Markowitz mean–variance criterion and when the estimation risk of the variance is ignored, the optimal investment rule is known in closed form. The P-value investment rule is a two-fund rule when the benchmark is fixed and a three-fund rule when the benchmark is estimated. In addition, ten investment strategies are compared on simulated and empirical data.

Optimal consumption and investment under time-varying relative risk aversion

We consider the continuous time consumption–investment problem originally formalized and solved by Merton in case of constant relative risk aversion. We present a complete solution for the case where relative risk aversion with respect to consumption varies with time, having in mind an investor with age-dependent risk aversion. This provides a new motivation for life-cycle investment rules. We study the optimal consumption and investment rules, in particular in the case where the relative risk aversion with respect to consumption is increasing with age.

Building an Optimal Portfolio in Discrete Time in the Presence of Transaction Costs

Portfolio theory covers different approaches to the construction of a portfolio offering maximum expected returns for a given level of risk tolerance where the goal is to find the optimal investment rule. Each investor has a certain utility for money which is reflected by the choice of a utility function. In this article, a risk averse power utility function is studied in discrete time for a large class of underlying probability distribution of the returns of the asset prices. Each investor chooses, at the beginning of an investment period, the feasible portfolio allocation which maximizes the expected value of the utility function for terminal wealth. Effects of both large and small proportional transaction costs on the choice of an optimal portfolio are taken into account. The transaction regions are approximated by using asymptotic methods when the proportional transaction costs are small and by using expansions about critical points for large transaction costs.

Optimal Consumption and Investment under Time-Varying Relative Risk Aversion

We consider the continuous time consumption-investment problem originally formalized and solved by Merton in case of constant relative risk aversion. We present a complete solution for the case where relative risk aversion varies with time, having in mind an investor with age-dependent risk aversion. This provides a new motivation for life-cycle investment rules. We study the optimal consumption and investment rules, in particular in the case where the relative risk aversion is increasing with age.

European CO 2 prices and carbon capture investments

We assess the option to install a carbon capture and storage (CCS) unit in a coal-fired power plant operating in a carbon-constrained environment. We consider two sources of risk, namely the price of emission allowance and the price of the electricity output. First we analyse the performance of the EU market for CO 2 emission allowances. Specifically, we focus on the contracts maturing in the Kyoto Protocol's first commitment period (2008 to 2012) and calibrate the underlying parameters of the allowance price process. Then we refer to the Spanish wholesale electricity market and calibrate the parameters of the electricity price process. We use a two-dimensional binomial lattice to derive the optimal investment rule. In particular, we obtain the trigger allowance prices above which it is optimal to install the capture unit immediately. We further analyse the effect of changes in several variables on these critical prices, among them allowance price volatility and a hypothetical government subsidy. We conclude that, at current permit prices, immediate installation does not seem justified from a financial point of view. This need not be the case, though, if carbon market parameters change dramatically, carbon capture technology undergoes significant improvements, and/or a specific governmental policy to promote these units is adopted.

The optimal economic lifetime of vintage capital in the presence of operating costs, technological progress, and learning

Abstract This paper studies the optimal vintage specific investment and disinvestment decision of a firm. The decision is formulated as a distributed optimal control problem with a constraint on the state variable. The economic model takes into accounts maintenance and operating costs, technological progress and learning. It allows firms to purchase not only the latest technology but also previous year vintages and completely disinvest old machines. It does not assume that the firms keep machines of all vintages as other similar models do. An analytical expression for the optimal economic lifetime of capital is derived and the qualitative dynamics of the optimal investment and disinvestment rules is analyzed.

Investment decision in oil and gas projects using real option and risk tolerance models

This paper presents a model for valuation and decision making, integrating discounted cash flow, real-options pricing and preference theory, aiming to cover the following questions: i) what is the current value of a project?; ii) what is the optimal investment rule?; iii) what is the optimal working interest? The traditional model suggests that, when the project value is above its investment cost, the corporation should invest immediately and incur in 100% working interest. The real option pricing suggests that the corporation should only invest if the project's current value is at least 1.85 times investment cost. The preference theory suggests funding only 44.38% working interest, and partners must acquire the remaining 55.62%. These tools must be integrated in order to allow a more realistic treatment of risk. In general, when the uncertainty (volatility) of cash flow components increases, the two models give more divergent results.

Environmental policy implications of extreme variations in pollutant stock levels and socioeconomic costs

This paper uses a real options approach to examine the impact of abrupt increases in carbon dioxide emissions and pollutant-related socio-economic costs. It derives optimal investment rules in the form of critical values for both pollutant stock levels and social costs, above which environmental policies should be adopted. Moreover, it determines the optimal emissions abatement level. Our analysis extends the methodology of Pindyck (2000) using jump diffusion processes. We show that if the stock of pollutant is subject to extreme variations and the emissions abatement level is chosen exogenously by the policymaker, then lower levels of the pollutant stock are required to trigger policy adoption. A similar, yet more prominent, effect is observed under the assumption that pollutant-related socio-economic costs and benefits are expected to exhibit abrupt changes. However, different results are obtained when we examine simultaneously the two interrelated decisions, namely, the optimal threshold of emissions abatement and the optimal abatement level. In this case, an increase in the size and/or probability of a jump increases the critical values of both pollutant stock levels and socio-economic costs but leads to higher optimal abatement.

Irreversible Investment in Stochastically Cyclical Markets

This paper studies entry and exit decisions in markets whose demand alternates between growth and decline phases at uncertain times. We introduce a stochastic process that captures these features of random market evolution, and we provide key mathematical results related to first passage times which make the characterization of entry and exit behavior quite simple and straightforward (even when the process is subject to an endogenously determined upper or lower barrier). We characterize entry and exit patterns in a dynamic competitive equilibrium, and we show why our results differ from those obtained if demand follows a diffusion process (e.g., a Geometric Brownian Motion). Despite the stochastic process of the underlying variable has a continuous sample path in both cases, we demonstrate in our setting that positive rates of entry and exit discontinuously fall to zero owing to informational overshooting. Another advantage of our framework is that it can explain discontinuities in firm values even if sample paths are continuous. Our framework is also amenable to empirical implementations (as we show using Corts’ 2008 offshore oil drilling application), and to an intuitive interpretation of optimal (dis) investment rules based on Bernanke’s (1983) “bad news principle of irreversible investment.”

Valuing flexibility: The case of an Integrated Gasification Combined Cycle power plant

In this paper we analyze the choice between two technologies for producing electricity. In particular, the firm has to decide whether and when to invest either in a Natural Gas Combined Cycle (NGCC) power plant or in an Integrated Gasification Combined Cycle (IGCC) power plant, which may burn either coal or natural gas. Instead of assuming that fuel prices follow standard geometric Brownian motions, here they are assumed to show mean reversion, specifically to follow an inhomogeneous geometric Brownian motion. First we consider the opportunity to invest in a NGCC power plant. We derive the optimal investment rule as a function of natural gas price and the remaining life of the right to invest. In addition, the analytical solution for a perpetual option to invest is obtained. Then we turn to the IGCC power plant. We analyse the valuation of an operating plant when there are switching costs between modes of operation, and the choice of the best operation mode. This serves as an input to evaluate the option to invest in this plant. Finally we derive the value of an opportunity to invest either in a NGCC or IGCC power plant, i.e. to choose between an inflexible and a flexible technology, respectively. Depending on the opportunity's time to maturity, we derive the pairs of coal and gas prices for which it is optimal to invest in NGCC, in IGCC, or simply not to invest. Numerical computations involve the use of one- and two-dimensional binomial lattices that support a mean-reverting process for coal and gas prices. Basic parameter values are taken from an actual IGCC power plant currently in operation. Sensitivity of some results with respect to the underlying stochastic process for fuel price is also checked.

R & D Strategic Investment in an Asymmetrical Case

This article analyzes R & D investment decisions in an asymmetrical case. The investment decisions share three important characteristics. First, the investment is completely irreversible. Second, there are two kinds of uncertainties over the future returns from the investment and over technology in R & D process, respectively. Third, there is strategic competition in the asymmetrical case. This article presents the optimal investment threshold values and the optimal investment rule of high-efficient firm (leader), and shows that the investment threshold values are reduced by competition of two firms. Finally, the mixed investment strategies for two firms, the probability that each firm separately exercises the option to invest, and the probability that two firms simultaneously exercise the option are given in the paper.

A theory of health investment under competing mortality risks

In this paper, we present a theory of health investment when there are multiple causes of death. We analyze the optimal investment rules and the comparative statics. In particular, we examine the conditions under which a cause-specific intervention has a spillover effect. By spillover effect, we mean a price reduction in one cause-specific health investment would increase all other investments. We also study the strength of the spillover effect, which is measured in terms of cross price elasticities. We find that, while a cause-specific intervention might not be wasteful, the spillover effect would not be large either.