Risk-adjusted valuation for real option decisions

Figure 1 - Calculation of expected values using the risk-adjusted discount rateAs a first step, we need to determine the expected payout of the option at the end of the tree. However, these can be calculated without creating an asset price tree. If the asset price at the beginning of the tree is S, then the value of the asset at the very top node of the six-step tree is Su.6 In our example it is 45(1.13883)6 = $98.17 million. The next bottom node is Su5d = $75.69 million and so on. With the terminal asset prices in hand, we can calculate the value of the option in each terminal node. Symbolically, let j denote terminal nodes, where S(T)j is the asset price and V(T)j is the option value for that node. Then V(T)j = max[0, K - S(T)j], where K is the strike price of the option ($50 million in our example).Now calculate the (true) probability of arriving at each of the terminal nodes, starting from the beginning of the tree. For example, six upward moves are required to reach the topmost terminal node (asset value = $98.17 million). At each node the probability of moving up is q, which is constant and does not depend on previous moves. Therefore, the probability of reaching the topmost node is q6 = (.515)6 = .01866. It takes five moves up and one move down to reach the next lowest node (asset value $75.69 million). This can happen in six ways (six possible paths along the tree to this node), each with probability q5 (1 - q). Thus, the total probability of reaching this node is 6q5 (1 - q) = .10542. This is the probability of exactly five successes in six binomial trials, where the probability of success in each trial is q. The function performing this calculation is standard in spreadsheets and can be used to calculate the probabilities for each of the terminal nodes.Using the values of terminal options and the probability of reaching each terminal node, we can calculate the expected value of the option at maturity. Let E denote the expectation taken with q(T)j, the probability of reaching node j from the beginning of the tree. Then E V(T) = Σ q(T)j V(T)j. For our example, this expectation is $7.2367 million.Once we have the expected payout at the terminal node, we need to find the appropriate risk-adjusted discount rate to calculate the present value of the option. This discount rate is usually different (often very different) from the corresponding discount rate for the valuation of the underlying project (calculated above as 14.85% for our example). Moreover, the riskiness of the option depends on the current price of the asset and the time remaining to maturity of the option. In the context of our tree, this risk is different at each node. Consequently, a constant risk-adjusted discount rate is inherently inconsistent with the nature of option pricing. We can find a constant discount rate that will provide the correct initial price, but the process is complicated.To find the right discount rate, we need to know the correct initial value of the option. Consider that this value is $8.20 million ($8.2045 million to be exact). Knowing the correct option value ($8.2045 million) and the true expected payout ($7.2367 million), we can easily derive the appropriate risk-adjusted discount rate. We will do this shortly, but first we note that the expected expected payout on the option using true probabilities is less than the original option value. This is because the option is a guarantee or, in other words, a form of policy insurance. Thus, it is worth more at the beginning of the tree. tree than its expected future payout. The premium is paid (over and above the expected payout) to insure against loss. To have a guarantee value today higher than its expected future value, the risk-adjusted discount rate must be negative. For our example, it is -24.84%. with monthly compounding. That is, the discount rate at which $7.2367 million has a present value of $8.2045 million six months into the future is -24.84%.For many people, this is a rather unexpected result. We are not used to evaluating risky investments using negative discount rates. What happens is that the option in our example has a negative beta in CAPM terminology. This happens because when the value of the underlying project (which has a positive beta) increases, the value of the option (the guarantee) decreases. Suppose someone failed to correctly recognize the different risk of the option and simply estimated it using the discount rate of the underlying project of 14.85%. The mispriced option value would be only $6.7219 million - an error of almost $1.5 million compared to the correct option value of $8.2045 million. This emphasizes the need to understand correct option valuation (Hodder et al, 2001).ConclusionThe method of real options allows us to present a risk-balanced economic model and show its effectiveness. In any case, at the initial point in time there is a single, most probable from the point of view of management, set of decisions, which is taken as the basis for the evaluation of net discounted income. Any changes related to risk factors that are used to evaluate using the real options method represent a mathematical expectation that has a large variability depending on the sensitivity of management decisions to the degree of risk.In evaluating the appropriateness of different valuation approaches, it is important to remember that many projects have more than one option but involve multiple decisions. For example, a company may decide to enter the market in stages and expand only after learning more about the product. At each stage, the company receives additional information and may reconsider its penetration strategy. Estimating the cost of such a project requires accurate option prices at all decision points (but not necessarily at all nodes). If the decision time is also flexible, then we need accurate prices at all intermediate nodes. The latter CEV methodology, where the risk premium varies across nodes, can be used as a basis for pricing such an option. The same can be said for a methodology in which risk-adjusted discount rates vary by node. However, procedures such as a single risk-adjusted discount rate that force the risk adjustment to be the same at each node would be inappropriate and potentially very costly. By making consistent decisions, we would be making choices based on inaccurate pricing and could make mistakes that would seriously impair the future potential of the project.For effective analysis of investment decisions under uncertainty, it is important to consider risk management options. Revising a decision and adjusting project parameters in accordance with changing situations is often necessary to achieve project goals. Opportunities to respond to positive and negative risks can be evaluated as real options because they are a right but not an obligation. Various options for changing the decision can be considered as options, including switching to the production of related products, temporarily stopping the project, and changing its scope.The search for real options is possible on the basis of the identification and analysis of the risks of the project, in particular to find them possible by conducting a SWOT analysis and scenario analysis.The value of real options creates such factors as the ability to manage risk and the degree of sensitivity of the project to the risk in question. The more severe the risk for the project, the more valuable the real option will be, which allows you to hedge it. A detailed analysis of project risks and countermeasures is needed to identify the most important options. It makes sense to build a probability and consequence matrix, conduct a sensitivity and predictability analysis.The model for evaluating a real option is determined by the possibility of changing the decision, as well as the quality and nature of the data for quantitative risk analysis. The source of input data for option valuation can be a business plan or strategic development plan of the enterprise. To evaluate a real option, it is necessary to identify those items of income and expenses that the investor is willing to change and quantify their adjustments in the event of various investment decisions.The option approach is implicitly encountered in the analysis of investment projects characterized by a high level of risk and assuming the possibility of maneuvering. In particular, it is used in the analysis of the financial condition of the company implementing the investment project, as well as in plans for strategic analysis of companies.Unsteady conditions lead to increasing risks, and the possibility of risk management becomes of great importance. Nevertheless, the assumptions of many models evaluating real options do not hold under nonstationary conditions. Therefore, many valuation models require modification and the results obtained may be ambiguous. For correct conclusions, a sensitivity analysis of option valuation on initial parameters is necessary.References Alesii, G. (2002). A CVP Analysis with Irreversible and Recurrent Real Options.Cassimon B., Baecker D.E., Engelen P.J., Van Wouwe M., Yurdanow V. Incorporating Technical risk in compound real option models to value a pharmaceutical R&D licensing opportunity // Research Policy. 2011. Vol. 40. Iss. 9. Р. 1200-1216.Cortazar, G., Schwartz, E. S., & Salinas, M. (1998). 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