BUS 552 Module 6 Binomial Trees Example

Up 42% or Down 30% Each Year: A Binomial Tree Valuation of an Indoor Shrimp Farm's Option to Expand

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MBA Program, Aspen University

BUS 552: Innovative Finance and Venture Capital

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Month Day, Year

Up 42% or Down 30% Each Year: A Binomial Tree Valuation of an Indoor Shrimp Farm's Option to Expand

Gulf Coast Aquafarms, a composite startup near Houston, Texas, raises Pacific white shrimp indoors in recirculating saltwater tanks, selling fresh, never-frozen shrimp to restaurants and grocers in the region. Its pilot facility has run for a year. If the pilot continues to perform, the company plans to build a full-scale farm in two years at a cost of about $45 million. Investors in its current round want to know what the right to build that farm is worth. This paper values it with a binomial tree.

The Option

The right to build the full farm is a real option: Gulf Coast can build if conditions are favorable and decline if not. Trigeorgis (1996) describes such expansion options as analogous to call options on the present value of the project's cash flows, with the investment cost as the exercise price. Here, the underlying asset is the present value of the full farm's future cash flows, estimated today at $40 million based on the pilot's production, expected prices and costs. The exercise price is the $45 million construction cost, and the decision will be made in two years. Because $40 million is below $45 million, the farm would not be worth building today, but its value could rise as the pilot proves itself and the market develops.

The Inputs

The tree requires the volatility of the underlying asset's value and the risk-free rate. Copeland and Antikarov (2001) recommend estimating the volatility of a real asset by simulating the project's own cash flows, combining uncertainties in price, cost and volume, and measuring the resulting variation in its value, rather than borrowing a stock's volatility. Gulf Coast's analyst did this, combining uncertainty in shrimp prices, survival rates and feed costs, and estimated annual volatility of about 35%. The risk-free rate is 4%, approximating a two-year Treasury yield.

Building the Tree

Cox et al. (1979) showed that an option can be valued by modeling the underlying asset as moving up or down by fixed factors each period and finding the value of a portfolio that replicates the option's payoffs. With annual steps, the up factor equals e raised to the power of the volatility, about 1.419, and the down factor is its reciprocal, about 0.705. The risk-neutral probability of an up move is the difference between one plus the risk-free rate and the down factor, divided by the difference between the up and down factors: 1.04 minus 0.705, divided by 1.419 minus 0.705, or about 0.469.

Working Backward

At year two, the option pays the farm's value minus $45 million if positive, and zero otherwise. Only the up-up node, at $80.6 million, justifies building, with a payoff of about $35.6 million. At the year one up node, the option's value is the risk-neutral expected payoff, 0.469 times $35.6 million plus 0.531 times zero, discounted one year at 4%, or about $16.0 million. At the year one down node, both possible payoffs are zero, so the option is worthless there. Today, the option is worth 0.469 times $16.0 million plus 0.531 times zero, discounted at 4%, or about $7.2 million.

Interpreting the Result

The option to build the full farm is worth about $7.2 million today, even though building it now would destroy $5 million of value. That value comes entirely from the chance that the farm's value rises substantially over two years, and from Gulf Coast's ability to walk away if it does not. For investors, the option is a meaningful part of what they are buying in the current round.

Sensitivity

Higher volatility raises the option's value because the company benefits from high outcomes while being protected from low ones. The result is therefore quite sensitive to the volatility estimate, the most uncertain input.

Checking With a Replicating Portfolio

The risk-neutral method can be checked by building a portfolio that copies the option's payoffs. At the year one up node, the option pays $35.6 million if the farm's value rises to $80.6 million and nothing if it falls to $40.0 million. A position equal to about 0.88 of the farm's value, partly financed by borrowing at the risk-free rate, produces exactly those payoffs, and its cost today equals the $16.0 million option value calculated above. Cox, Ross and Rubinstein's insight was that because such a portfolio can be built, the option's value does not depend on investors' attitudes toward risk, which is why risk-neutral probabilities give the right answer.

What the Value Means for the Round

Investors in the current round are buying a share of a company whose value includes the pilot's operations and the option to expand. If the pilot alone is worth little, much of the round's price rests on the option. An investor who believed volatility was closer to 25% would value the company lower than one who believed 45%, a difference of about $4 million in the option alone. Making that assumption explicit helps both sides understand why they disagree on price.

Limits

A two-period tree is a coarse approximation; a tree with monthly steps would give a more precise value, typically somewhat different. The model assumes the decision is made only at year two, while in practice Gulf Coast could build earlier if the pilot exceeds expectations. And the underlying value of $40 million is itself an estimate.

Conclusion

A binomial tree turns an uncertain future into a set of up and down paths, values the option at the end of each and works back to today. For Gulf Coast, the right to build a full farm is worth about $7.2 million, value that comes from the ability to expand only if the pilot and the market justify it. The method is simple enough to show every step, which makes its assumptions, especially volatility, easy to examine.

References

Copeland, T., & Antikarov, V. (2001). Real options: A practitioner's guide. Texere.

Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263. https://doi.org/10.1016/0304-405X(79)90015-1

Trigeorgis, L. (1996). Real options: Managerial flexibility and strategy in resource allocation. MIT Press.