Four Steps to Valuing Flexibility  783 in the downward branch, so the payoffs in the decision tree are $116.20 in the upward branch and $100 in the downward branch. Using risk-neutral valu- ation this time, the abandonment option can be valued in the node at t = 4 at $104.90, as shown in Exhibit 39.13 (the same result a replicating portfolio would have generated). Working backward through time, the value for a fac- tory with the ability to abandon is $106.40, so that the abandonment option is worth $6.40. Now the value-maximizing decision strategy is to abandon the factory immediately in any year in which its value drops below $100. Multiple sources of flexibility can be combined within a single decision tree, as illustrated in Exhibit 39.14, using risk-neutral valuation. The value of the project, including the options to abandon and expand, would be $113.50 rather than $100, its stand-alone value without flexibility. With these options, the correct decision would be to accept the project. Note that the value of the combined expansion- abandonment flexibility, $13.50, is less than the sum of the individual flexibility values ($8.40 + $6.40 = $14.80) but greater than either of them individually. The val- ues of both options are not additive, because they interact in complex ways (for ex- ample, you cannot expand the factory once you have abandoned it). As indicated in Exhibit 39.14, the best decision strategy is to abandon the factory whenever its value25 drops below $100 and to expand only in year 5 if its value exceeds $75. EXHIBIT 39.13  Decision Tree: Option to Abandon Factory $ t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 106 Underlying asset values PV+ = 116 PV– = 86 PV = 100 212 182 157 157 136 135 119 118 116 106 105 100 100 100 NE NE NE NE NE NE Management decisions (t = 5) 116 = Max (116, 100) 100 = Max (86, 100) Risk-neutral valuation p* = (1 + rf – d ) / (u – d ) = (1.05 – 0.861) / (1.162 – 0.861) = 0.629 Value of option (t = 4) Option = Max ([p* × 116 + (1 – p*) 100] / 1.05, 100) = Max (105, 100) = 105 Decision to abandon Note: t = time, in years     NE = nonexisting state     PV = present value     p* = binomial (risk-neutral) probability     rf = risk-free rate     d = downward movement of value     u = upward movement of value       Liquidation value: $100 25 Note that this is the value of the factory including the option to expand. Therefore, abandonment occurs only in more unfavorable states of the world than in Exhibit 39.13. 784  Flexibility Real-Option Valuation and Decision Tree Analysis: A Numerical Example Our next example applies both the DTA and the ROV approaches in the valu- ation of a research and development project. Assume a company needs to decide whether to develop a new pharmaceutical drug. In our simplified ex- ample,26 the first step in development is a research phase of three years, in which the most promising chemical compounds are selected. The probability of success in the research phase is estimated at 15 percent. This is followed by a three-year testing phase, during which the compounds are tested in labora- tory and clinical settings. The chance of successfully completing the testing phase is 40 percent. If there are successful results, the drug can be released in the market. On failure in any phase, the company terminates development, and the product dies worthless. EXHIBIT 39.14  Decision Tree: Option to Expand or Abandon Factory $ t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 114 Underlying asset values PV+ = 116 PV– = 86 PV = 100 239 204 175 173 150 148 129 127 124 112 110 102 101 100 100 100 NE NE NE NE Decision to expand Decision to abandon Management decisions (t = 5) 124 = Max (116, 100, 116 × 1.2 – 15) 100 = Max (86, 100, 86 × 1.2 – 15) Risk-neutral valuation p* = (1 + rf – d ) / (u – d ) = (1.05 – 0.861) / (1.162 – 0.861) = 0.629 Value of option (t = 4) Option = Max ([p* × 124 + (1 – p*)100] / 1.05, 100, 100 × 1.2 – 15) = Max (110, 100, 105) = 110 Note: t = time, in years     NE = nonexisting state     PV = present value     p* = binomial (risk-neutral) probability     rf = risk-free rate     d = downward movement of value     u = upward movement of value       Liquidation value: $100       Incremental investment: $15       Incremental payoff: 20% 26 Pharmaceutical R&D is much more complex and consists of more phases than shown in this example. For a more extensive example of valuing flexibility in pharmaceutical research and development, see Kellogg and Charnes, “Real-Options Valuation for a Biotechnology Company.”