Abstract
This paper demonstrates the use of real option analysis to value the flexibility available to management decision making in introducing an express liner service using a new technology. The ship operator's decision is framed as an option to exchange the risky income stream of one strategy (asset) for the maximum of several alternatives as uncertainty is resolved over time. Copeland's and Antikarov's Marketable Asset Disclaimer is invoked; the intrinsic values of the underlying assets and their volatilities and correlations are modelled using traditional discounted cash flow techniques. The option is valued numerically in a multinomial tree. The result is the value of flexibility and is added to the present value of the original strategy to derive the present value of the flexible strategy. A sensitivity analysis is performed to examine the value of the flexible strategy as both the levels of uncertain factors and their volatilities change.
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Notes
Parameters used in the paper were adapted from the earlier study (Bendall and Stent, 2001) and validated by industry professionals. A sensitivity analysis is performed.
Correlated standard normal variates are first generated; the resulting observations on their cumulative densities become the inputs for the Inverse Transformation Method, which is then applied in the usual manner to generate observations on the individual marginal distributions. While the originating normal variates have the specified correlation coefficients, when measured for generated marginal distributions their values are perturbed slightly. A reference to copula functions is Dall'Aglio et al (1991).
Other stochastic variables were initialised randomly. Since all stochastic variables are dependent, the initial freight rates are in fact random, too. In almost all runs, their initial values were in the left side of their triangular distribution (Table 2). Their average initial value was near their 10th percentiles.
Thirty independent batches of 500 iterations each, to facilitate calculation of standard errors of both estimated PVs and the later option value. All estimates reported are the means of the 30 batch estimates. The model was written and simulations performed using the Matlab program (The MathWorks, Inc., Natick, MA, USA).
Servicing Penang subtracts value in PV terms. Penang's negative contribution would manifest directly if a fourth strategy, servicing Penang alone, were included. This strategy has not been included as it is an extremely unlikely outcome. Also, going beyond three strategies (one ship Klang/Penang, two ships Klang/Penang, one ship Klang) makes the later numerical calculation of option values prohibitively expensive.
The model was coded in C++ and tested against known analytical results. It was also tested against an alternative numerical procedure developed by Boyle et al (1989), which was coded in C++ as well for the purpose. The method is iterative with both the storage required for intermediate calculations and the total number of calculations growing exponentially as the number of states and iterations is increased. This is the prohibiting factor referred to in Endnote 5. With three states, n=79 iterations were performed to obtain the option values in the study. Varying n slightly about this value had little effect on the values obtained. The method also needs a value for a parameter λ. The value 1.25 was used. The option values obtained were insensitive to this parameter.
From Table 7, the ‘up’ factors over 3 years for a binomial model,
, are 1.18, 1.49 and 1.46 for the three strategies, respectively. Recalculating present values as was done in Table 6 with ‘high’ demands and freight rates for both Klang and Penang, scales them by the greater factors 1.21, 1.67 and 1.73, respectively. While the meaning of such values is imprecise they do support the validity of the market model. Capacity constraints are not a significant issue over a 3-year period.
The method follows Copeland and Antikarov (2000, Chapters 5 and 9). For the purpose of calculating 6-montly ratios, annual cash flows were assumed to fall evenly over the year. Further, since the ratios did not differ much over the three strategies, the same value, the median, was used for each, calculated from the full sample of 15,000 iterations. The ratios are expressed as percentage decreases in value for the first 6 months, the second 6 months, etc, in sequence: 5.2%, 5.4%, 6.0%, 6.4%, 6.3%, 6.7%.
The option value reported is the mean of the 30 independent estimates obtained from repeating the exercise for each of the 30 independent samples of 500 iterations, explained in Endnote 4. Another estimate of the option value is available from the full sample of 15,000 iterations. This value is the same as that reported, to within $0.01 million, suggesting that any statistical bias must be very small.
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Bendall, H., Stent, A. Ship Investment under Uncertainty: Valuing a Real Option on the Maximum of Several Strategies. Marit Econ Logist 7, 19–35 (2005). https://doi.org/10.1057/palgrave.mel.9100122
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DOI: https://doi.org/10.1057/palgrave.mel.9100122