Optimal Portfolio for N-Risky and N-Riskless Irreversible Investments

Abstract

When a firm decides to make irreversible investment expenditure, it neglects its right without obligation to invest but wait for new desirable information that might affect investment decision. Studies have shown irreversible investments with the underlying value process driven by stochastic differential equations. However, this study had no consideration for a portfolio of N-risky and Nriskless investments (N = 2; 3; ; T < ∞) in relation to the set of: Variable Production Costs (VPC), Periodic Loan Repayment (PLR) with penalty for default, Investment Reserve (IR) and Equipment Value (EV) equations. This study was designed to consider a portfolio of N-risky and N-riskless investments including a set of processes describing VPC for the portfolio, interval PLR for the same investments with penalty processes for default in PLR. A set of instantaneous Expected Revenue (ER) from the risky investments were formulated using Geometric Brownian Motion (GBM) with drift term restricted to zero and volatility strictly positive (Martingale condition). The resulting equation was solved using Ito’s formula. The IR and EV equations were formulated with predictive adaptive processes to examine liquidity of business plan and Equipment Depreciation (ED) value respectively. Maintenance Process (MP) was introduced into the EV equation. The equation describing portfolio option value was formulated using Bellman utility function and solved. Risks associated with the risky investments were measured using Arrow-Pratt measure. Linear Dynamic Optimization (LDO) was adopted to maximise the value of the N-risky and Nriskless investments; and also used for Capital Equipment (CE) value equation with MP to examine depreciation values. Application of this model to a Nigeria maritime industry showed that the set of ER obtained were fluctuating and met the Martingale condition. The equations obtained for the N-Risky and N-Riskless investments gave values as VPC ≅ #29:2M/month, PLR ≅ $37:9M/month, IR = #234:4M; ED ∈ 2 (0.3, 0.4). A set of threshold values for the N-irreversible investments were obtained. The equations obtained from Arrow Pratt yielded 18 to 20% as investors’ attitude towards risk management.