## Qdb

j = Decision to contract t=4

1 Actual p^youM are " ilhr rM -iiv 11: in> ' I would l.r.:' already been ■ n: r.; i tri j = Decision to contract t=4

At any point in time management has the option to decrease the scale and the value ol the factory by 25%, generating savings of $25 million

Underlying asset values

Management decisions (t - 5)

116 = Max (116, 116x 0.75 + 25) 90 = Max (36,86x 0.75 + 25)

Portfolio replication

N =(116-90)/(116-66) B =[116-N(116)]/(1 +8%) N =0.07, B = 14.4

Option value : 101

1 Actual p^youM are " ilhr rM -iiv 11: in> ' I would l.r.:' already been ■ n: r.; i tri

A replicating portfolio would consist of .8667 units of the present value of the project without flexibility, $100, plus 15.46 bonds worth $1. The value with flexibility is $101 as shown in Exhibit 20.7. We work backward, node by node, to arrive at a present value of $102 with the option to contract. The option increases the net present value of the project from - $5 to $2. So the option is worth $7.

Exhibit 20.8 shows the effect of an option to abandon the project. We assume that if abandoned, the salvage value of the project is $100. We don't abandon immediately because the value of the project and its salvage value are identical, and we do undertake the project because, given the flexibility to abandon, it is worth $106.32 (more than the initial investment of $105). If the project falls in value after the first period, we would abandon it.

To conclude this section, Exhibit 20.9 combines the various sources of flexibility into a single decision tree. If all three types of flexibility were available at once, the value of the project would be $113.49, rather than $100, which is its value without flexibility, and the correct decision would be to accept the project. Note that the value of the combined options, $13.49, is not the simple sum of their individual values, but it is greater than any of them individually. The values of the simple options are not additive because they interact in complex ways.

Exhibit 20.8 Option to Abandon

Evaluating Options

The overall approach for option valuation usually is a four-step process as illustrated in Exhibit 20.10. Step one is to calculate the base-case present value without flexibility using a traditional discounted cash flow model. The second step is to expand the DCF model into an event tree, mapping how the value of the project evolves, using explicit values, objective probabilities, and the weighted average cost of capital.6 It is also necessary at this stage to chose a multiplicative or additive stochastic process and to decide whether to model mean reversion.7 Since there is still no flexibility in the model, the present value of the project, based on the event tree, should equal the DCF value from the first step.

One of the important considerations in step 2 is that the uncertainty of a project within a company is not the same as the uncertainty of the variable or variables that drive that uncertainty. In one case, we found that the

6 Objective probabilities are estimates of the actual probability that an event will happen. Often experienced managers or scientists supply them. Sometimes they are extracted from historical databases. Occasionally, a forecasting model supplies them.

7 Mean reversion is a natural property of cyclical businesses. It implies that if prices are currently high, they are more likely to go down toward their long-term trend than to go up even further. When prices are currently low, the opposite is true.

Exhibit 20.10 Overall Approach—Four-Step Process

Steps 1. Compute base ease 2. Model the present value without uncertainty using flexibility using DCF event trees

3. Identify and incorporate managerial flexibilities creating a decision tree

### 4. Calculate option value

Objectives Compute base case Understand how the present value without present value develops flexibility at t = 0 with respect to the changing uncertainty

Analyzing the event tree to identify and incorporate managerial flexibility to respond to new information

Value the total project usiny a simple algebraic methodology and spreadsheet

Choose multiplicative or additive stochastic process

Comments Traditional present Still no flexibility; this value without flexibility value should equal the value from Step 1

Flexibility is incorporated into event trees, which transforms them into decision trees

The flexibility has altered the risk characteristics of the project, therefore the cost of capital has changed

Option value method will include the base case presenl value

Explicitly estimate uncertainty without flexibility plus the option (flexibility] value

Under high uncertainty and managerial flexibility option value will be substantial annualized standard deviation of a world commodity mineral was roughly 6 percent, but the annualized standard deviation of the percentage changes in the value of a mine that produced the commodity was about 35 percent. The mine had higher volatility than the commodity because of the fixed costs of operations that induced operating leverage. In most cases, the annualized volatility of the value of the project is impossible to observe directly. Exogenous effects such as uncertainty in prices, quantity sold, and input costs affect the uncertainty of projects.

We recommend using a Monte Carlo analysis of the variance of the DCF value of the project without flexibility as the best way of combining risks and taking into account the relationships among them. In another case, the price of the final product was an important source of uncertainty, but so too was the price of the major input. Furthermore, they were correlated. We used historical data on the spread between the two to drive the uncertainty in the Monte Carlo model. The spread reduced the two sources of uncertainty to one, and simultaneously took into account the correlation between the input and output prices.

Step 3 turns the event tree into a decision tree by identifying the types of managerial flexibility that are available and building them into the nodes of the tree. As we illustrated earlier, multiple sources of flexibility are possible at a single decision node, but it is important to have clear priority rules among them. Care must be taken on the sequence of decisions regarding flexibility, especially when the decision tree has compound options.

The fourth step is to recognize that the exercise of flexibility alters the risk characteristics of the project. That means the risk-adjusted discount rate is no longer the weighted average cost of capital that was used in step 1.

Instead we must use the replicating portfolio concept of option pricing to value the project with flexibility.

### Options in Practice

Drawing from our experience, we describe three case histories that illustrate increasingly complex uses of options. Names of corporations and data are disguised to ensure confidentiality. We make explicit comparisons among different valuation approaches to show the usefulness of the option-pricing approach. Finally, we discuss some conclusions and lessons learned from each application.

Most asset option-pricing applications are limited to those situations where the option value depends on the market price of a world commodity, such as oil, coal, copper, nickel, gold, or zinc. By using the marketed asset disclaimer and a lattice approach, however, it is possible to solve a much wider set of problems than ever before, in a way that is fairly easy to understand from a manager's point of view. Exhibit 20.11 shows the progress that has been made applying option-pricing models in a realistic setting.

Kryptonite Mining (Switching Option)

Kryptonite (not the real name of the mineral) is a globally traded commodity product. Kryptonite Mining Limited was the world's leading producer of Kryptonite, supplying over one-third of the Western world's demand. It had four production sites, each with a different layout of operating mines and extraction technology. The random movement of spot Kryptonite prices had been volatile in the past four years. Our study focused on developing a valuation method for each site as well as providing some guidance regarding the shut-down/reopen decision—a switching option. Initial estimates of Kryptonite Mining's NPV based on analysts' forecasts of Kryptonite prices measured only up to 45 percent of Kryptonite Mining's current market value of equity (see Exhibit 20.12). A scenario-based NPV analysis allowing for no explicit operational flexibility increased this estimate to 71

Exhibit 2G.11 Recent Advances Bring Wider Applicability

From

Uncertainty driven by world commodity product Higher mathematics necessary for application Single source of uncertainty Simple options Limited application

Source of uncertainty not necessarily market priced Algebra and Excel spreadsheets Multiple sources of uncertainty (rainbow options) Options on options (compound options, learning options) Many applications

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