# Option-pricing and timberland's land-use conversion option.

I. INTRODUCTIONLand expectation value applications to forestry often assume that the land will be allocated to timber growing in perpetuity (Davis 1966, 358-66; Leuschner 1984, 58-65). However, the purchase of a timberland tract often provides the owner with the option to convert the land to some alternative use such as cropland or residential development. For example, 15 percent of timberland in the southeastern U.S. was judged to have at least a medium potential for conversion to cropland in 1980 (USDA Forest Service 1988, 216). [1] In relationship to residential development, Pope Resources (Poulsbo, WA), a timberland master limited partnership, made the following statement in its 1987 Annual Report that highlights the importance of considering the conversion option when valuing timberland:

[Pope] Resources currently has an inventory of 8,000 acres of land earmarked for sale, much of it in Jefferson and Kitsap counties on the Western side of Puget Sound. As population pressures make it uneconomic to continue growing timber on certain portions of the tree farm, these will be reclassified as development property. (Pope Resources 1987, 5)

In a survey of timberland investment advisory firms' managers, it was discovered that one timberland acquisition criterion used by some managers was a tract's perceived potential for conversion to a "higher use" (Zinkhan, Sizemore, Mason, and Ebner 1991). Better understanding of the valuation of land suitable for conversion to alternative uses might not only benefit buyers and sellers of timberland, but also might allow for improvements in the accuracy of land base models for forest resource supply analysis (see Parks and Alig 1988).

Ignoring the value of the option to convert to an alternative land use can result in an underestimation of the intrinsic value of timberland and other land forms. With its treatment as a right rather than an obligation, the value of this option can be positive, even if conversion is neither economically attractive at the present time nor expected to be so in the future. This explains why landowners might choose to maintain a property in a forested state even though its market price is greater than the estimated net present value from timber production.

After reviewing option pricing theory in the next section, an approach for applying it to the tasks of valuing timberland and evaluating a timberland bid is presented. Then, this valuation model is subjected to sensitivity analysis in order to explore the relative effects of changes in various economic and managerial assumptions.

II. REVIEW OF OPTION PRICING

THEORY

A call option on a common stock is a financial instrument which provides its owner with the right to buy the stock at a fixed exercise price. The nature of the maturity associated with this right depends upon whether the call option is of the European or American variety. European call options can be exercised only on a given future day, while American call options can be exercised on any day up to and including some maturity date. The derivation of a model by Black and Scholes (1973) for valuing European call options has been identified as one of the most significant contributions to financial theory (Jensen and Smith 1984; Martin, Cox, and MacMinn 1988, 5). A non-dividend-paying stock's call option value is dependent upon only five variables. There is a positive relationship between four of these variables and the call option's value, ceteris paribus: the underlying stock's price, the interest rate on riskless investments, the option's time to expiration (or maturity), and the volatility of the underlying stock's price. In contrast, the call option's value has an inverse relationship with the exercise price, ceteris paribus. Speculative investment texts provide logical explanations for these relationships (e.g., Chance 1989, 72-86; Marshall 1989, 478-503).

Discounted cash flow models cannot be used to value call options (Brealey and Myers 1988, 485). This is because the risk of the option changes (a) as the underlying stock price moves and (b) as time elapses even if the stock price remains constant. Such nonstationary risk prevents an analyst from determining an appropriate opportunity cost of capital for use in discounting. By creating hedged portfolios and adopting arbitrage arguments, Black and Scholes derived the following formula for valuing European call options on non-dividend-paying stocks:

C = [SN(d.sub.1]) - [Xe.sup.-rft] [N(d.sub.2])

where:

C = the value of the call option, S = the current stock price, [d.sup.1] = (log(S/X) + [r.sub.f]t + [[sigma].sup.2]t/2)/[sigma] [square root of t,] [d.sub.2] = (log(S/X) + [r.sub.f]t - [[sigma].sup.2]t/2)/[sigma] [square root of t,] N(d) + cumulative normal probability density function, X = exercise price associated with the option, t = time to expiration of the option, [[sigma].sup.2] = annualized variance of the continuously compounded returns on the stock, [r.sub.f] = continuously compounded return on a riskless investment with a maturity of t.

Despite the adoption of some rather restrictive assumptions, Chance (1989, 147) notes that "empirical tests have provided reasonable support for the option pricing models." With reliance on these assumptions, notice that neither a forecast of the future stock price nor a risk-adjusted discount rate enter the option valuation framework.

If the underlying stock is expected to pay any dividends during the life of the call option, then the current stock price (S) must be adjusted downward by an estimate of the value of such dividends. This adjustment is necessary since (a) the option owner is not entitled to any dividend payments during the life of the option and (b) such payments tend to reduce the stock price. A reduction in the stock price, in turn, tends to result in a decrease in the value of the associated call option, ceteris paribus.

There has been increased interest recently in the application of option pricing theory to the valuation of real options, such as capital budgeting projects (Martin et al. 1988, 497-512). Among its applications to natural resource problems are the valuation of oil reserves (Siegel, Smith, and Paddock 1987) and oil leases (Paddock, Siegel, and Smith 1988), the valuation of copper mines (Brennan and Schwartz 1985), the valuation of forest products company long-term cutting contracts (Shafter 1984), the valuation of timberland (Hughes 1987), and the valuation of young timber (Zinkhan 1990). In the process of applying option pricing to timberland valuation. Hughes treated timberland as a contingent claim whose value depends upon wood commodity prices. Hughes did not attempt to value the land-use conversion option. The timberland valuation methodology presented in the next section applies option pricing theory to the problem of valuing the land-use conversion option.

III. EVALUATING A TIMBERLAND BID

Suppose that a value-maximizing, income tax-exempt investor holds an acre of cutover timberland with some perceived potential for conversion to an alternative (alt) land use. Assume that it is not economically feasible to immediately convert to land-use alt. When offered for sale, the highest bid received for the tract is [POFF.sub.0]. [POFF.sub.0] would be unacceptable if less than the estimated value associated with the following alternative:

To hold the tract for at least the period from time 0 to time t', with the option of converting the tract from timber production to land-use alt at time t'.

The estimated value ([V.sub.0] associated with this alternative is:

[V.sub.0] = [LEV.sub.0] [+] [CONVERO.sub.0] [2]

where:

[LEV.sub.0] = land expectation value assuming the land will be utilized for timber production in perpetuity. [CONVERO.sub.0] = value of the option associated with converting from timber production to land-use alt at time t'.

Since the owner of the timberland must sacrifice an income source (i.e., timber production) in order to convert to land-use alt, this framework is somewhat similar in concept to convertible bond valuation approaches (see Smith 1984).

Options based on timberland are rather common. An interesting example in which a timberland put option was written by The Travelers Insurance Company and purchased by Diamond International Corporation has been described by Mason (1986). This timberland put option provided Diamond International with the right to sell its western U.S. timberland to The Travelers at a fixed price over a three-year period. The option was intended to establish a value floor for the timberland so that creditors would be willing to lend funds to Diamond International.

Because [CONVERO.sub.0] [is less than] 0 is not possible, [V.sub.0] [is greater than] [LEV.sub.o], always. Valuation approaches that rely exclusively on a tract's expected net cash flows from timber production, therefore, may underestimate the intrinsic value associated with a timberland tract which possesses some potential for conversion to an alternative land use.

Given the following assumptions, even [V.sub.0] may underestimate the intrinsic value of a cutover timberland tract with some potential for conversion to another land use:

1. In order to simplify the valuation problem, the land-use conversion option is treated as an European option rather than an American one. That is, the right to convert from timber production to land use alt can only be executed at time t'. The Black-Scholes option pricing model cannot be used to value American call options when the underlying asset pays a dividend of some sort prior to the time of expiration. This issue will be discussed in more detail.

2. Only one land-use conversion option is considered. In reality, a landowner has the option to convert to one of numerous land-use alternatives. (2)

3. A specific time t' is selected, where t' [is less than] [infinity]. The selected t' might provide a lower [CONVERO.sub.0] than when t' [right arrow] [infinity]. In reality, the time to expiration of a land-use conversion option associated with a given tract might approach [infinity]. Additional discussion of this issue is forthcoming.

4. Non-timber revenues to be generated in combination with timber-based activities are not explicitly considered. However, equation [2] could easily be adjusted to recognize the value of such non-timber ventures.

An estimation of [CONVERO.sub.0] can be undertaken by modifying four of the variable definitions in equation [1].

As a substitute for S,

S' = [PALT.sub.0] - [POFF.sub.0] [3]

where:

S' = the current spread between [PALT.sub.0] and [POFF.sub.0]. Based on the Black-Scholes option pricing model, [CONVERO.sub.0] [is greater than] 0 is a possible outcome only if S' > 0.

[PALT.sub.0] = the estimated price of a tract comparable to the acre of cutover timberland if immediately converted to land-use alt.

Assume that if [POFF.sub.0] < [LEV.sub.0], the bid would not be evaluated and [CONVERO.sub.0] would not be estimated. If [POFF.sub.0] [is greater than or equal to] [LEV.sub.0], a floor for the value of the tract to the owner is established at [POFF.sub.0]. For the purpose of estimating [CONVERO.sup.0] in a Black-Scholes option pricing framework, [POFF.sub.0] is treated as a proxy for the market-clearing price (or [PTIM.sub.0]) of the cutover timberland tract at time 0.

The dependence of [CONVERO.sub.0] on the spread between [PALT.sub.0] and [POFF.sub.0]--rather than between [PALT.sub.0] and [LEV.sub.0]--reflects the landowner's forfeiture of an option value as well as [LEV.sub.0] if the tract is sold or converted.

In order to apply the Black-Scholes option pricing model to the valuation of a call option, the continuously compounded rate of return on the underlying asset must be assumed to follow a normal distribution (see Marshall [1989, 495] for a summary of all the necessary assumptions associated with the model). For the application described herein, the price of the underlying asset is represented by S'. Another critical Black-Scholes option pricing assumption is that there are no transaction costs associated with buying or selling the underlying asset. This assumption is violated to varying degrees in all Black-Scholes option pricing model applications, especially if opportunity costs are considered. Although percentage-of-price transaction costs for land sales generally exceed those associated with the trading of financial assets, the relatively long holding period often associated with real estate reduces the severity of this imperfection.

If either land-use alt or the timber production activities are capable of generating cash income during the period from time 0 to time t', then an adjusted S' (SADJ') must be identified for incorporation into equation [1]. Similar to a dividend-paying stock, [PALT.sub.0] and [POFF.sub.0] must be adjusted downward in recognition of any anticipated positive net cash flows during the time period. Since the land-use conversion option is treated as a European option, it cannot be exercised until time t'. Once time t' is reached and the landowner has the right to convert, neither [PTIM.sub.t'] (the price of the bare timberland at time t') nor [PALT.sub.t'] (the price of a comparable tract allocated to land-use alt at time t') will reflect any net cash income received during the period. Since it might be optimal for the landowner to convert land uses prior both to the receipt of a given flow of net cash income from land-use alt and time t' (if possible), the value of a land-use conversion option specified as an American call option will be [is greater than or equal to] [CONVERO.sub.0], always.

To simplify the treatment of net cash income from timberland production during the period from time 0 to time t', assume that the landowner leases the cutover timberland at time 0. The contract life of the lease is t' years and the certain lease payments are to be paid continuously so that the annual net cash income yield (relative to [POFF.sub.0]) is maintained at [[delta].sub.tim]. In the lease contract, [[delta].sub.tim] is set at a level which will ensure that the present value of lease payments (at time 0) is equal to the present value of the expected net cash flows from timber production activities for one rotation of length t':

[POFF.sub.0] (1 - [e.sup.-[delta]timt']) = [NCF.sub.t'] / [(1 + r).sup.t']

where:

[NCF.sub.t'] = compounded value at time t' of real net cash flows from timber production activities during the period from time 0 to time t'.

r = risk-adjusted, real rate of return.

Solving for the contractual [delta.sub.tim]--under the assumption that the lessee pays all timber management costs and property taxes,

[delta.sub.tim] = -In(1 - ([NCF.sub.t'] / [(1 + r).sup.t']) / [POFF.sub.0])/t'.

Assumed equality between the present value of the leasing opportunity and the present value of net cash flows from timber production makes it unnecessary to revise the [LEV.sub.0] term in equation.

If the land-use alt is assumed to be capable of generating a constant annual net cash income yield [([delta]sub.alt t']), then:

SADJ' = [PALT.sub.0] [(e.sup.[delta]timt')]

Among the methods for estimating the annualized variance [([sigma].sup.2')] of the continuously compounded percentage change in S' is basing the projection on the historical volatility of the spread, PALT - PTIM.

The exercise price (X'), or critical spread, represents the minimum differential between [PALT.sub.t'] and [PTIM.sub.t'] in order for the explicit and implicit conversion costs to be met at time t':

X' = [CC.sub.t'] [+] [I.sub.t']

where:

[CC.sub.t'] = estimated explicit cost associated with converting from cutover timberland to land-use alt at time t'.

[I.sub.t'] = absolute value, compounded to time t' at r, of the negative impact on [LEV.sub.0] resulting from a deviation (if any) from the optimal rotation age. This deviation results when a harvest at time t' (immediately prior to conversion) does not coincide with the optimal harvest schedule reflected by [LEV.sub.0].

Although it is assumed that X' can be estimated with certainty, the dependence of X' on time complicates the application of American call option valuation models (e.g., Roll 1977; Whaley 1981) to the valuation of a land-use conversion option. The treatment of the land-use conversion option as a European option rather than an American option overcomes this complication.

Finally, an explanation of the selected time to expiration, t', is necessary. In theory, the time to expiration of a land-use conversion option generally approaches [infinity]. When using the Black-Scholes option pricing model in equation [1] to value call options on non-dividend-paying stocks, [lim.sub.t [right arrow] [infinity]] C = S, ceteris paribus. Two factors, however, complicate the relationship between t' and [CONVERO.sub.0]:

1. The influence of a deviation between [[delta].sub.alt] and [[delta].sub.tim] on SADJ'.

2. The sensitivity of the estimated explicit conversion costs ([CC.sub.t']), implicit conversion costs ([I.sub.t']), and thus X' to changes in t'.

Simulation of equations [3]-[7], [1] (after modifying the variables), and [2] with alternative levels of t' could be undertaken to estimate the t' that maximizes [V.sub.0].

After substituting SADJ', [[sigma].sup.2'], X', and t' for S, [[sigma].sup.2, X, and t in equation [1], the value of the land-use conversion option ([CONVERO.sub.0]) can be estimated.

[POFF.sub.0] should be rejected if:

[POFF.sub.0] < [V.sub.0]

As noted earlier, if [POFF.sub.0] > [V.sub.0], then acceptance/rejection of the bid is not obvious since the intrinsic value of the tract might exceed [V.sub.0].

[TABLE DATA OMITTED]

IV. TIMBERLAND VALUATION CASE

ILLUSTRATION

As noted in the lat section, the attractiveness of a timberland bid is dependent upon [CONVERO.sub.0]. The sensitivity of the level of [CONVERO.sub.0] (and its level relative to [LEV.sub.0]) to various economic and managerial inputs is investigated for a hypothetical tract. Required data are based on a variety of southern U.S. sites. This investigation assumes that an acre of cutover loblolly pine (Pinus taeda L.) timberland is being valued as of year-end 1988 from the perspective of an income tax-exempt institutional investor. The most promising land-use alt is assumed to be represented by farmland. [CONVERO.sub.0] is estimated under the assumption that t' is equal to the given rotation age. All other assumptions are listed in the Appendix.

As shown in Table 1, shortening the rotation age from 36 to 27 years is non-optimal whether value is measured according to [LEV.sub.0] or [V.sub.0]. A lower [CONVERO.sub.0] represents the outcome of conflicting forces. The decline in X' and the existence of a positive differential between [[delta].sub.alt] and [[delta].sub.tim] are forces which tend to increase [CONVERO.sub.0] as t' is reduced from 36 to 27 years. These forces are outweighed by the negative influence of the shorter time to expiration on the option's value. A call option's value is positively related to its time to expiration, ceteris paribus. Although not suggested by the results of this case, objective functions based upon alternative valuations ([V.sub.0] versus [LEV.sub.0]) could lead to divergent optimal rotation age estimations.

Since [CC.sub.t'] represents a component of X', its decrease from $314 to $220 results in an increase in [CONVERO.sub.0]. There is an inverse relationship between exercise price and a call option's value, ceteris paribus.

An increase in [PALT.sub.0] from $466 to $533 results in a gain in [CONVERO.sub.0]. In a call option pricing framework, there is a positive relationship between an underlying asset's price and option price, ceteris paribus. The [V.sub.0] associated with the increase in [PALT.sub.0] is the only one in Table 1 that is greater than the assumed highest bid of $250 (see assumption 3(c) in the Financial Assumptions and Data section of the Appendix). This reveals that the accept/reject decision of the tract owner is dependent upon the inputs incorporated into the option pricing model.

Because a call option's value is positively related to the underlying asset's return volatility, ceteris paribus, the reduction in [sigma]' from 41.5 percent to 20.8 percent results in a decrease in [CONVERO.sub.0].

As reflected in Table 1, the valuation of the land-use conversion option ranges from $16 per acre (or 7.6 percent of [LEV.sub.0]) to $52 per acre (or 24.6 percent of [LEV.sub.0]) for the parameter levels selected for this sensitivity analysis. Because no tests of whether or not the percentage changes in S' follow a normal distribution (as required for application of the Black-Scholes option pricing model) were undertaken, these simulation results should be interpreted with caution (see assumption 3(e) in the Financial Assumptions and Data section of the Appendix).

V. CONCLUSION

As timberland becomes a more popular portfolio asset--especially among institutional investors--it is likely that the managers of this asset will not hesitate to convert it to those land uses which are perceived as more economically attractive. Accurate models for pricing the conversion premium will therefore be increasingly demanded by investors and policymakers interested in land-use conversion. Timberland management simulation models which currently rely on net present value from timber growing as the criterion for decision making might eventually also reflect the value of the land-use conversion option.

Future research needs to focus on empirical testing of an option pricing-based timberland valuation model. Using land transaction databases maintained by private appraisers, value estimates computed with both the option pricing-based model and a conventional discounted cash flow model could be compared relative to market prices. The initial focus of such studies should be on those tracts which possess an obviously high potential for conversion to some alternative land use. Results associated with such empirical testing should provide hints concerning the selection of appropriate parameter levels for an option pricing-based timberland valuation model.

APPENDIX

Additional assumptions and clarifications used in the generation of the output in Table 1:

Biological and Management Assumptions

and Data

1. A site index of 80 (base age 50 years) (assumptions 1-4 are based on an analysis completed by Klemperer [1987]).

2. Loblolly pine pulpwood and sawtimber volumes (as of a particular age) are typical for the Florida-Georgia-Alabama region.

3. Even-aged management with no fertilizer applications or thinnings.

4. Optimal rotation age is treated as an exogenous variable. Two alternative rotation ages are considered: 27 years and 36 years. Projected yields per acre at these rotation ages are as follows:

age 27 age 36 cords of pulpwood 25.9 23.6 mbf (international 1/4-inch rule) of sawtimber 3.866 8.447

5. The best anticipated alternative land use is farmland, and immediate conversion to this land use will be considered immediately following harvest at time t'.

Financial Assumptions and Data

1. The Faustmann formula is used to estimate [LEV.sub.0].

2. Income taxes are ignored since the perspective of an income tax-exempt investor is taken.

3. Prices

a. Mean fourth quarter 1988 statewide prices (from Timber Mart-South) in the three-state area of Alabama, Florida, and Georgia after converting to international rule:

southern yellow pine pulpwood pine stumpage stumpage $114.40 per mbf $14.08 per cord

b. If immediately converted to farmland, the tract is assumed to have a value ([PALT.sub.0]) equal to either 70 percent ($466) or 80 percent ($533), respectively, of the mean farmland values in Alabama and Georgia (USDA Economic Research Service 1989). In order to eliminate the influence of citrus orchards, Florida values were not incorporated into the analysis.

c. The highest bid price ([POFF.sub.0]) for the tract is assumed to be $250. This figure is equal to the mean 1988 price per acre for bare land in Forest Investment Associates' (Atlanta, GA) Southern Timberland Index Fund.

d. The [sigma] levels, 41.5 percent and approximately 50 percent less-20.8 percent, are indicators of the volatility of the annual percentage changes in the spread between PALT and PTIM. The former level is based on the annual Keith Index of Cropland and Woodland Values for southeastern North Carolina (Tom Keith and Associates, Inc., Fayetteville, NC) for the period of 1971 to 1988. Before calculating the historical spreads, the cropland values were reduced by 30 percent.

e. Because of the limited historical data set available for S', tests of whether or not the percentage changes in S' follow a normal distribution were not undertaken. Therefore, it must be assumed here that percentage changes in S' over time do follow a normal distribution.

f. Incorporating the relevant data into equations [5]-[6], SADJ' was estimated. For the base assumptions in Table 1, [[delta].sub.tim] and [[delta]sub.alt] were found to be 2.5 percent and 3.2 percent, respectively.

4. Costs

a. Reforestation costs as of time 0 are based on a CPI-adjusted level of $140 in 1984.

b. Annual operating costs as of time 0 are based on a CPI-adjusted level of $3.00 in 1985 (Binkley and Washburn 1988).

c. Explicit conversion costs (as of time 0) of $220 and $314 per acre represent 140.1 and 200.0 percent, respectively, of the mean 1988 amounts reported for shear-rake-pile-bed site preparation treatments in the South (Straka, Watson, and DuBois 1989). [CC.sub.0] is assumed to increase to time t' at an annual inflation rate of 4.5 percent. [I.sub.t'] is assumed to be zero at the t' levels of 27 and 36 years.

5. Interest Rates

a. Risk-free rates of 8.7 percent (when t' = 27 years) and 8.6 percent (when t' = 36 years), based on long-term Treasury bond yields, are used in the Black-Scholes option pricing formula.

b. A real, risk-adjusted interest rate of 3.5 percent is incorporated into the Faustmann formula. This figure is based on a nominal risk-free rate of 8.2 percent, a capital asset pricing model beta of 0.0 (Thomson 1989; Zinkhan et al. 1991), and an expected inflation rate of 4.5 percent. For the purpose of estimating [[delta].sub.alt], a real, risk-adjusted interest rate of 3.3 percent was adopted. The latter reflects a capital asset pricing model beta for farmland of -0.04 (Arthur, Carter, and Abizadeh 1988).

Zinkhan is associate professor of business administration, Campbell University, Buies Creek, NC. This research was supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Economic Returns from Forestry Investments in the Southeast and Nation Research Unit, Research Triangle Park, NC. The author thanks Ralph J. Alig, Clark S. Binkley, Courtland L. Washburn, and an anonymous reviewer for their helpful comments.

(1) Based on the 1980 judgments of a multiagency group in each county as part of the National Resource Inventory. After considering such factors as the effort necessary for conversion, historical conversion rates of similar tracts, commodity prices, and costs, the group classified the timberland acreage into four categories (relative to their potential for conversion to cropland over the next 10 to 15 years): zero, low, medium, or high (USDA Forest Service 1988, 115).

(2) As noted by C. S. Binkley and C. L. Washburn in private correspondence, a methodology for estimating the value of multiple land-use alternatives might be developed by pursuing the approaches of Stulz (1982) and Johnson (1987).

References

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Publication: | Land Economics |

Date: | Aug 1, 1991 |

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