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What is the relevance of option pricing for forest valuation in New Zealand?
Forest Policy and Economics 12 (2010) 299–307
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Forest Policy and Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / f o r p o l
What is the relevance of option pricing for forest valuation in New Zealand? Bruce Manley a,⁎, Kurt Niquidet b a b
New Zealand School of Forestry, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand Industry, Trade and Economics, Natural Resources Canada, 506 Burnside Rd W, Victoria, British Columbia, Canada
a r t i c l e
i n f o
Article history: Received 7 November 2008 Received in revised form 13 October 2009 Accepted 22 November 2009 Keywords: Forest valuation Option value Stochastic prices Rotation age
a b s t r a c t Three different option value approaches are used to estimate the value of a typical New Zealand plantation stand, under the assumption that log prices follow a random walk. Crop values are compared with the Faustmann value, the benchmark for forest valuation in New Zealand. The increase in forest value can be substantial when log prices are low and close to the exercise cost. Gains quickly diminish and become small, both as an absolute difference and as a percentage of forest value, as price increases. However, results are very sensitive to the log price model adopted. Assuming log prices are mean reverting gives higher values than Faustmann for all log prices. Stochastic Dynamic Programming (SDP) and Binomial Option Pricing (BOP) give very similar results. They evaluate the same harvest/defer harvest/never harvest options. A new Abandonment Adjusted Price approach gives results that have a similar pattern but are consistently lower than SDP and BOP. This approach only considers whether to harvest or not in the optimal year and does not allow the option of deferring harvest. At the present time, option valuation approaches have limited relevance for the practice of forest valuation in New Zealand. Practical issues (determination of the log price model, estimation of volatility, allowing for multiple log grades and modelling at the estate-level) need to be addressed before option value approaches can be routinely used for forest valuation. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In 1999 the New Zealand Institute of Forestry (NZIF) released a set of Forest Valuation Standards “for the physical and financial description and the valuation of commercial plantation forests in New Zealand”. In developing the standards, five different methods were considered: • Transaction based approach; i.e. use of comparable sales. • Cost based approaches including the historic cost, current cost and current replacement cost methods. • Immediate liquidation approach; i.e. stock value or current realisation value. • Expectation value approach; i.e. Faustmann approach. • Option pricing approach. The NZIF Forest Valuation Standards (NZIF, 1999) require that: “The market value of a crop of trees shall be derived from transaction evidence where this is available and suitable for the purpose in terms of reliability, comparability and volume of transactions. Where transaction evidence is not available, market value shall be established using the Expectation Value Approach. Where the ⁎ Corresponding author. Tel.: + 64 3 364 2122; fax: +64 3 364 2124. E-mail address: [email protected] (B. Manley). 1389-9341/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.forpol.2009.11.002
Expectation Value approach is used, the discount rate shall be determined with reference to transaction information.” Because of the limited transaction information available, the standard approach for forest valuation has been the expectation value approach calibrated by available market evidence. NZIF (1999) noted that “although attempts have been made to value forests using option pricing theory, the Working Party is not aware of any routine practical application. However there are ongoing developments in this area that have potential application in the future.” Subsequently, there has been ongoing debate about whether the focus on the Faustmann approach under-estimates the value of plantations, in particular whether it adequately reflects the option value of a forest. For example, Hughes (2000) used an approach based on the Black–Scholes option pricing formula to value the forest assets of Forestry Corporation of New Zealand. He calculated the option pricing value to be NZ$2.218 billion compared with the realised value of NZ$1.853 billion. The suggestion from this and other work1 was that the real options associated with harvesting provide an additional option value.
1 For example, Susaeta, A. 2005. Comparative analysis of the discounted cashflow approach and option pricing theory for forest valuation: a Chilean case study. MForSc thesis, University of Canterbury.
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So the question is: Why is it that, despite the apparent potential of the method, the option value (or pricing) approach has not been widely adopted for forest valuation? This paper attempts to answer this question by dealing with three key issues: • Under what circumstances will an option value approach give a different value to the expectation value approach? • Are these differences material enough to warrant the use of an option value approach? • What are the challenges and practical issues associated with applying an option value approach to forest valuation? The first and second issues deal with whether there is a need to use an option value approach. The third issue deals with whether it can be applied in practice. A useful review on the use of option value approaches was done by Plantinga (1998). He differentiates studies with stationary prices (e.g., Norstrom, 1975; Lohmander, 1988; Brazee and Mendelsohn, 1988; Haight and Holmes, 1991) from those with non-stationary prices (e.g., Clarke and Reed, 1989; Morck et al., 1989; Thomson, 1992; Reed, 1993). In the studies with stationary prices, the expected value of a stand is found to be higher when stochastic variation in price is exploited, compared to the Faustmann value. However, in studies using non-stationary prices, “there are no gains except when there are fixed costs (e.g., management costs, alternative land uses)”; i.e., the Faustmann value is sufficient. Plantinga (1988) found that “as in previous studies with stationary prices …, expected timber values are higher with a reservation price policy compared to the Faustmann model with expected prices. However, the expected values with non-stationary prices are identical to the Faustmann values, in contrast to the findings of Thomson (1992)”. Insley (2002) also showed the importance of the underlying stochastic price process in applying a real options approach to forest valuation. She found that “option value and optimal cutting time are significantly different under the mean reversion assumption compared to geometric Brownian motion”. For a numerical example, mean reversion gave higher option values when prices were below the mean. Analysis of historical log prices in New Zealand (Niquidet and Manley, 2007) indicates a non-stationary price process for virtually all log grades and regions. However most log price series available in New Zealand run for little more than 10 years. The domestic market was dominated by the government-owned NZ Forest Service and a few large companies until 1990. As a consequence, log prices were not set in an open, competitive market and limited log price information was publicly available. In fact, tests conducted by Niquidet and Manley (2007) on a longer time series of A-grade export logs for 1973–2001 were inconclusive. Guthrie and Kumareswaran (2003) used this A-grade series to develop a real options model with uncertain future timber prices to estimate the impact of carbon subsidies and taxes on the timing of harvest, the replanting decision and forest value. They modelled log prices using a stationary AR(1) process. We note the observation of Dixit and Pindyck (1994) that “using 30 or so years of data, it is difficult to statistically distinguish between a random walk and a mean-reverting process. As a result, one must often rely on theoretical considerations (for example, intuition concerning the operation of equilibrating mechanisms) more than statistical tests when deciding whether or not to model a price or other variable as a mean reverting process.” The difficulty is that theoretical considerations and intuition can lead to a range of assumptions. For example, Haight and Holmes (1991) note that “a random walk model is consistent with an informationally efficient market, which is generally accepted for most assets”.2 Gjolberg and Guttormsen (2002) make a case for prices being mean reverting — 2 McGough et al. (2004) present results that show that “market efficiency provides little justification for random walk prices”.
“there are theoretical justifications for commodity prices not to follow a random walk”. The approach taken in this paper is to: • Calculate the value of a representative New Zealand plantation stand using option value approaches under a non-stationary price model; • Compare the value from alternative option value approaches; • Compare these values with Faustmann values; • Understand the sensitivity of value to the log price model by assuming stationary price processes; • Identify the conditions under which option value exists; • Review practical issues in implementing option valuation for forest valuation in New Zealand. The over-arching question is — What is the relevance of option valuation for forest valuation in New Zealand? 2. Approach 2.1. Log price model The assumed log price model initially adopted is a non-stationary random walk with Geometric Brownian Motion (GBM). The discretetime version of this is that the price at time t (Pt) has a lognormal distribution (i.e., Pt is lognormal with ln(Pt) normally distributed). The model implies that: ln Pt
+ T ∼L½ln Pt
2
+ ðc−σ = 2ÞT; σ√T
ð1Þ
where Pt and Pt + T are the prices at time t and T years later µ is expected annual change in log price (expressed as a proportion) σ is the standard deviation, i.e., the volatility (expressed on an annual basis). We focus here on the case of annual prices; i.e., T = 1 ln Pt
+ 1 ∼L½ln Pt
2
+ ðc−σ = 2Þ; σ:
ð2Þ
The price model takes the form: ln Pt
2
+ 1
= ln Pt + ðc−σ = 2Þ + ε
ð3Þ
where ε is a random normal variable with mean 0 and variance = σ2 Given the properties of a lognormal distribution: EðPt
+ 1Þ
c
= Pt e :
2.2. Methods Three different option value approaches are considered. Both Binomial Option Pricing and Stochastic Dynamic Programming have been used in previous studies. We also evaluate a new approach based on the Black–Scholes option pricing model (Black and Scholes, 1973). 2.2.1. Binomial Option Pricing (BOP) model We use the two-state option pricing model developed by Cox et al. (1979) as applied by Thomson (1992). Suppose the current log price is Pt. Over the next year it will either rise to α · Pt (up state) or fall to β · Pt (down state) where σ
α=e
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β = 1 = α:
301
Table 1 Silvicultural regime.
The probability of an up move = (eµ − β) / (α − β). A binomial decision tree is constructed (starting from P0) showing prices at any time and their probabilities. This is used in conjunction with a yield table giving the volume at each time. Using a backwards recursive approach, the value at each node on the tree is calculated as the maximum of the value from immediate harvest and the expected value of growing the stand on for another year. At the final stage there is the option of never harvesting. 2.2.2. Stochastic Dynamic Programming (SDP) model We use the approach applied by Norstrom (1975). A price transition matrix is developed using Eq. (3) to give the probability of Pt + 1 taking on different values given a value of Pt. In general terms the approach is similar to the BOP model — the difference is that at each year a full range of prices have to be evaluated rather than just those prices on the binomial decision tree. Again a backwards recursive approach is used — the value at each price in each year is calculated as the maximum of the value from immediate harvest and the expected value of growing the stand on for another year. 2.2.3. Abandonment Adjusted Price (AAP) model This approach is based on the theory that Hull (Options, Futures and Other derivatives, edition 5: 2003 — pp 262–264) uses in the proof of the Black–Scholes formula. We note the concerns of Thomson (1991) that Black–Scholes cannot be used to solve the forest rotation problem. However our approach stops one step short of equating the value of a forest to the value of a call option as was done by Hughes (2000). The key question is what the log price will be at the time of harvest (T years from now). If the price (PT) is greater than the harvest (i.e. exercise) cost (K) then we will harvest and get a revenue of PT − K. If the price (PT) is less than the exercise cost (K) then we will not harvest and get a revenue of 0. We can calculate the expected revenue we will receive using equation 12A.1 of Hull. If PT is lognormally distributed and the standard deviation of ln PT is s then: E½maxðPT −K; 0Þ = EðPT ÞNðd1 Þ–KNðd2 Þ
ð4Þ
where d1 = ½ln½EðPT Þ = K + s2 = 2 = s d2 = d1 –s N(x) is the cumulative probability distribution function for a standardised normal variable; i.e., it is the probability that a standard normal variable is less than the value x. But EðPT Þ = P0 eμT and s = σ√T: So, μT
E½maxðPT −K; 0Þ = P0 e Nðd1 Þ–KNðd2 Þ
ð5Þ
Age
Operation
0 5 7 8 9
Plant 850 stems/ha Prune to 2.4 m Prune to 4.6 m Prune to 6.0 m Thin to 350 stems/ha
The implementation of the AAP approach is similar to the use of the Faustmann approach but with the stumpage price calculated using Eq. (5). The probability of harvesting, N(d2), is taken into account in determining whether there are ongoing costs associated with use of land. Alternative harvest ages are evaluated for a stand to find the optimum rotation. The BOP and SDP models explicitly evaluate the options of immediate harvest versus harvest in future years versus don't harvest at all. In contrast the AAP approach, having determined the optimum rotation age, only evaluates whether or not to harvest in that specific year; i.e., it considers only the options of harvest in the optimal year versus don't harvest at all. It might be expected to estimate a lower value than the other approaches. 2.3. Species, site and silvicultural regime Option value approaches are compared on a radiata pine stand grown on a standard silvicultural regime (Table 1). The stand is on an average New Zealand forest site (site index3 30 m, 300 Index4 25 m3/ ha/year). 2.4. Log grades and prices Specifications for the eight log grades used are given in Table 2. They represent the different types of log grades produced in New Zealand: • • • • •
Pruned logs — P1 and P2 Structural logs — S1 and S2 Utility logs — L1L2 and K Industrial logs — S3L3 Pulp logs — Pulp.
K grade is an export log which is sold to Korea — there are logs of similar specification sold to China and Japan. The time series of log prices collected by MAF (Ministry of Agriculture and Forestry) was used. Log prices were extracted for the December quarter for 1996 to 20075 and converted to real $2007 using the Reserve Bank of NZ CPI. We wish to evaluate option valuation in the multi-product situation that is typical in New Zealand. For this paper we have developed a weighted average log price, using as weights the relative proportion of the volume6 in each grade at age 30. Weightings are given in Table 3.
where d1 = fln½ðP0 eμT Þ = K + σ 2 T = 2g = ½σ√T rearranging: 3
2
d1 = fln½ðP0 Þ = K + ½μ + σ = 2Tg = ½σ√T d2 = d1 –σ√T: The expression E[max(PT − K, 0)] gives the expected stumpage price ($/m3) that will be received. This is then multiplied by the volume (m3/ha) to give the expected revenue.
Mean top height of 100 largest stems/ha at age 20 years. 300 Index is an index of volume productivity. It is the stem volume mean annual increment at age 30 years for a defined silvicultural regime of 300 stems/ha (Kimberley et al. 2005). 5 MAF have collected prices for all grades since September 1994. Only data since 1996 has been used here to get away from the “spotted owl” price spike of 1993/94. Although quarterly data is available, we only use December prices because we consider the case where the decision on whether or not to harvest is made annually. 6 Volumes were predicted using the Radiata Pine Calculator Pro Version 3 - New Zealand Tree Grower 24(2), May 2003, p.26. Calculators for radiata pine and Douglasfir. 4
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Table 2 Log grade specifications.
Table 3 Age 30 volume by log grade.
Log grade
Maximum branch size (cm)
Minimum small end diameter (cm)
Grade
Volume (m3/ha)
Proportion
P1 P2 S1 S2 L1L2 K S3L3 Pulp
0 (pruned) 0 (pruned) 6 6 12 10 12
40 35 40 30 30 22 20 10
P1 P2 S1 S2 L1L2 K S3L3 Pulp Total
84 51 28 176 24 176 29 102 669
0.13 0.08 0.04 0.26 0.04 0.26 0.04 0.15
The weighted average log price series is shown in Fig. 1.7 We calculate the volatility via the Hull approach. The volatility is calculated as the standard deviation of ui where ui = ln (Pt + 1 / Pt).8 Using the 1996 to 2007 data the volatility is estimated as 0.08. (The average drift of −0.03 is not significantly different from 0). 2.5. Weighting of volumes The log price series is based on log grade proportions at age 30. Potential harvest ages are 20 to 40 years. Consequently we need to make allowance for the value of the log grade mix at these ages relative to the log grade mix at age 30. We have incorporated the gradient of improving size and quality and hence log grade mix with increasing age by: • Calculating the harvest revenue at each age by multiplying log grade volume at each age by the average real 1996–2007 December price for the grade. • Calculating the ratio of the average log price at each age to the average log price at age 30. • Multiplying the volume at each age by this ratio. Fig. 2 shows the raw volumes and also the volumes adjusted for the relative size and quality (log grade mix) at each age. The approach is broadly equivalent to the age-dependent conversion factor used by Gong and Löfgren (2007). 2.6. Alternative scenarios The base case analysis assumes: • Log prices follow GBM with µ = 0 and σ = 0.08. • Current stand age = 10 years. • Exercise cost = $45/m3. This covers the cost of harvesting and transportation of logs to mill or wharf. It represents an average New Zealand harvesting operation on easy terrain with logs transported 90 km.9 • Discount rate = 6% real. • Annual cost (including land rental) = $100/ha/year.
Different valuation approaches are compared for initial log prices between $20 and $140/m3. The harvesting decision is based on maximum NPV (=discounted harvest revenues − discounted costs). Even after allowing for ongoing annual costs this may mean no harvesting. The Faustmann value was calculated using the same assumptions but with log prices constant at the initial value. Alternative scenarios were evaluated with independent changes made: • • • • •
Annual cost reduced from $100/ha/year to 0. Volatility is changed from 0.08 to 0.04 and 0.12. Discount rate is increased from 6% to 8%. Exercise cost is changed from $45/m3 to $30/m3 and $60/m3. Current stand age is increased from 10 to 20 years.
These values were chosen to illustrate the relative sensitivity of option value to different factors. The lower and higher exercise costs are indicative of the range of harvesting and transportation costs in New Zealand. 3. Results 3.1. Base case The Faustmann method provides the benchmark for the analysis. The options approaches give higher values only at lower prices (Figs. 3 and 4). The SDP and BOP methods give virtually identical results — differences from Faustmann exceed $200/ha at log prices of $65/m3 and less and are over $1000/ha at $40/m3. The AAP approach consistently provides lower values than these two approaches — less than 80% of the increase over Faustmann. Harvesting is 0/1 under the Faustmann approach. No harvesting occurs when the initial price is less than $40/m3. At prices of $40/m3 and above harvesting occurs – even at $40/m3 which is less than the
The annual cost is incurred for as long as the stand occupies the site. It includes the land rental. A common situation in New Zealand is for the tree crop and land to be owned by separate entities with the crop owner occupying the land under a lease or licence and paying an annual land rental.10
7 See Niquidet and Manley (2007) for a detailed analysis on New Zealand log price dynamics. The December 2008 nominal log price was $91/m3–$88/m3 in $2007. 8 Tests of the weighted log price data indicated that the variable is normal and independent. 9 NZX AgriFax Logging and Cartage Costs, December 17, 2007. 10 The NZIF Forest Valuation Working Party has adopted the approach of applying a notional land rental in the valuation of a tree crop on all land including freehold land (i.e., land owned by the tree crop owner). This has been adopted because leases and licences are a common form of land tenure — and also because of practical difficulties in basing land rental on Land Expectation Value (LEV).
Fig. 1. Weighted average log price for example stand at age 30 years ($/m3 delivered to mill or port). Real prices are expressed in $2007.
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Fig. 4. Increase in value of 10 year old stand (compared to Faustmann value) for each method with different initial log prices. Fig. 2. Total volume at each potential harvest age: (a) estimated volume; and (b) volume adjusted for size and quality relative to age 30.
exercise cost – this provides a higher NPV than the alternative of not harvesting which incurs ongoing annual costs. In contrast, under the options approaches, the probability of harvesting is less than 1 until prices are relatively high — the probability does not exceed 95% until initial log price reaches $85/m3 (Fig. 5). All methods give similar average rotation ages (for the area that is harvested) of age 27 to 28 years when the initial price is $80/m3 or more. Beneath this price the average rotation age for the Faustmann approach is less than that of the options approaches (Fig. 6).
approaches give virtually the Faustmann value. At a volatility of 0.12, these approaches give values that are greater than the Faustmann value even with an initial log price of $140/m3 (Fig 8). 3.4. Discount rate increased from 6% to 8% Increasing the discount rate causes the values and optimum rotation ages to reduce for all approaches. In terms of absolute values, the value increase given by the options approaches is lower with an 8% discount rate than a 6% discount rate. However, expressed as a ratio, the relativity between values under the options approaches and the Faustmann value are similar for either a 6% or 8% discount rate.
3.2. Annual costs reduced from $100/ha/year to 0
3.5. Exercise cost changed from $45/m3 to $30/m3 and $60/m3
The options approaches give a greater increase in value in the absence of annual costs including land rental (Fig. 7). In this case there is no opportunity cost associated with delaying harvesting, consequently the average harvest age for the options approaches increases by up to 2.8 years. Assuming zero annual costs will be unrealistic for most forestry situations and only applicable for a forest that is left unmanaged on land which has no cost.
Increasing the exercise cost increases the potential increase in value from the options approaches. The absolute level of the exercise cost has an impact over and above the relativity between exercise cost and log price (Fig. 9). 3.6. Current stand age increased from 10 to 20 years
The increase in value given by options approaches is very sensitive to the volatility of log prices. At a volatility of 0.04, option value
The option value approaches have a greater impact at age 20 compared to age 10 (Fig. 10 vs Fig. 4). This may appear to be counterintuitive — at age 10 there is a greater period of time until harvest to allow log prices to vary from the initial value. However this effect is less than the impact of discounting revenues over a longer period.
Fig. 3. Value of 10 year old stand (compared to Faustmann value) for each method with different initial log prices.
Fig. 5. Probability of 10 year old stand being harvested for each method with different initial log prices.
3.3. Volatility changed from 0.08 to 0.04 and 0.12
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Fig. 6. Average harvest age of 10 year old stand for each method with different initial log prices.
4. Discussion
Fig. 8. Impact of log price volatility on the increase in value of 10 year old stand (compared to Faustmann value) for BOP method with different initial log prices.
At the upper end, an option value approach would provide a higher value than Faustmann.
4.1. Values compared to Faustmann value 4.2. Option value approaches The increase in forest value over the Faustmann value can be substantial — but only when log prices are low and close to the exercise cost. Gains quickly diminish and become small both as an absolute difference and as a percentage of forest value as price increases. The biggest gains are typically when forest value is calculated to be negative. These gains assume that the initial (i.e., current) log price is the price used for forest valuation; i.e., that the current price is the starting point for use in the options approaches and, for the Faustmann approach, that the current price holds in perpetuity. In practice, negative values are rarely calculated. Valuers applying the Faustmann approach would generally not apply the current price as the price assumed for future harvest if it led to a negative value. Typically they would apply a moving average price or a forecast in this situation. The calculated value of the options associated with harvesting are greater than might have been anticipated from previous literature. Differences occur even when annual costs are not included. The December 2007 log price was $80/m3. At this level and with an exercise cost of $45/m3 there is little increase in value using an option pricing approach. However, harvesting and transport costs in New Zealand vary from $30 to $60/m3 depending on terrain and location.
Stochastic Dynamic Programming and Binomial Option Pricing give very similar results. This is to be expected given that the underpinning log price model (GBM) is common. Although they have different levels of detail about future price states, they do evaluate the same harvest/defer harvest/never harvest options. The AAP approach gives results that have a similar pattern but consistently produces lower values than SDP and BOP. The approach only considers whether to harvest or not in the optimal year and does not allow the option of harvesting in subsequent years as do the other two models. Essentially it values the option associated with walking away from a forestry investment and never harvesting. SDP and BOP also give the value of deferring harvesting to later ages. The difference between the AAP and SDP/BOP curves gives the option value for deferring the harvest.11 The AAP approach, having a closed form, has the advantage of being easy to apply — unlike the SDP and BOP approaches which require considerable customisation in order to be solved numerically for each situation. The AAP is only slightly more difficult to implement than using Faustmann. 4.3. Practical issues One reason that the Faustmann approach is widely used for forest valuation is that it is well understood and can be applied in practice. The option value approaches face some practical issues. 4.3.1. Reservation price In order to achieve the option values it is necessary to implement a reservation price strategy; i.e., only harvest if the price exceeds a minimum value. These can be determined for each method (e.g., Fig. 11). 4.3.2. Harvesting costs vary with age The assumption that exercise cost is constant for all ages is unrealistic. Harvesting costs do vary with age. All three options approaches evaluated readily allow harvest cost to vary deterministically with age. However, harvest cost fluctuations across time
Fig. 7. Increase in value of 10 year old stand (compared to Faustmann value) for each method with different initial log prices — annual costs 0.
11 Thomson (1992) provided a similar breakdown of the increase in NPV between the forestry abandonment option and the harvest age timing option.
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305
price models. We used the SDP approach with a random draw price model: ln Pt
+ 1
¯
2
= lnP −σ = 2 + ε
ð6Þ ¯
where ε is a random normal variable with mean 0 and σ = 0.08. lnP is mean(ln P). We also developed a stationary AR(1) model: ln Pt
Fig. 9. Impact of exercise cost on the increase in value of 10 year old stand (compared to Faustmann value) for BOP method with different initial log prices.
attributed to things like fuel prices potentially mean that the exercise price is also stochastic. 4.3.3. Log price model The underlying price model is a crucial assumption in implementing an option value approach. We also evaluated mean reverting log
Fig. 10. Increase in value of 20 year old stand (compared to Faustmann value) for each method with different initial log prices.
+ 1
¯
2
= 0:0678 lnP + 0:932 ln Pt −σ = 2 + ε:
ð7Þ
The mean reverting log price models lead to distinctly different option values (Fig. 12). The random draw and the AR(1) price models indicate that option value exists at all log prices, not just log prices that approach the exercise cost. The difference between the value increase for the random draw and AR(1) price models shows the impact of the assumption about the rate of mean reversion. The sensitivity of results to the underlying price model is one reason why forest valuers (and their clients) have not adopted option valuation techniques. 4.3.4. Volatility The increase in value from an option value approach is very sensitive to the estimate of future log price volatility. Again we have limited historical data to estimate this. In addition, volatility is assumed to be constant, where in many situations it too is stochastic. Option valuation methods have developed to incorporate GARCH (Generalised Autoregressive Conditional Heteroscedasticity) and other volatility processes (Lewis, 2000). Incorporating such situations into option valuation adds even more complexity and additional assumptions. 4.3.5. Multiple log grades In this analysis we have simplified things by using fixed (age 30) log grade proportions and adjusted volumes at other ages to allow for different size/quality. For practical application of option pricing it would be desirable to explicitly model the price for each log grade taking into account the volatility of each grade together with correlations between the prices of different grades. This would introduce complexity that we have not seen dealt with in either theory or practice. Forboseh et al. (1996) provide a strategy for a multi-product situation — but for only two log products with prices assumed to follow a bivariate normal distribution. Yin (2001) also
Fig. 11. Reservation price calculated using the SDP method for 10 year old stand with an initial price of $55/m3 and an exercise cost of $45/m3. Harvesting should occur when the price is above the specified value for each age.
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$55/m3 have a materially higher value estimated using an option value approach. Consequently, the circumstances under which the use of an option value approach for forest valuation is warranted appear limited under the assumption that log prices follow a random walk. Any conclusion about the relevance of option valuation is very sensitive to the assumption about the underlying price model. Fig. 12 shows that if log prices are mean reverting, option valuation approaches give values that are materially higher than the Faustmann approach for all log prices. Unfortunately we do not have the information to conclusively determine the model followed by log prices in New Zealand. The third question posed was: • What are the challenges and practical issues in applying an option value approach to forest valuation?
Fig. 12. Impact of alternative log pricing models on the increase in value of 10 year old stand using the SDP method (compared to Faustmann value) with different initial log prices. For random draw and AR(1) the initial price is assumed to be the average price. For comparative purposes, all models are assumed to have the same volatility (0.08).
uses a two product example with log price processes for sawtimber and pulpwood assumed to be independently normally distributed. Gong and Yin (2004) use SDP in another two product example — both sawtimber and pulpwood have AR(1) price models that are correlated. 4.3.6. Estate-level versus stand-level analysis The NZIF Forest Valuation Standards (NZIF, 1999) state that the valuation process “can be either estate-based or stand-based. However, in both cases, there needs to be an underlying management and harvesting strategy which is realistic for the forest being valued. This strategy should reflect what an economically rational owner would do taking into account wood supply commitments and logistical, marketing, social, political and environmental factors”. What that means, in practice, is that valuation is done within a forest estate modelling system that incorporates yield regulation and other constraints. An initial unconstrained model run in which each stand is harvested at the Faustmann optimum rotation provides an initial benchmark. The final valuation model typically has many stands being harvested at sub-optimal rotation ages in order to meet forest-level constraints. Estate-level modelling is relatively straightforward to do under deterministic log price assumptions. We see currently unresolved challenges in incorporating stochastic log prices into estate-level forest valuation. 5. Conclusion We can now review two of the questions posed in the Introduction: • Under what circumstances will an option value approach give a different value to the expectation value approach? • Are these differences material enough to warrant the use of an option value approach? Fig. 9 shows that when log prices are within $40/m3 of the exercise price, an option value approach can give different values to the Faustmann approach for a typical 10 year old stand. If $200/ha is used as a measure of materiality, then differences are material when log prices are within $10, $20 and $30 of exercise prices of $30, $45 and $60/m3 respectively. For a 20 year old stand (Fig. 10), the difference is material when log prices are within $15 of an exercise price of $45. These situations do arise in New Zealand plantation forestry — but are not the norm. With the December 2007 log price was $80/m3, only young stands with a harvesting and transportation cost over about
An advantage of the Faustmann approach is that it is well understood and relatively straightforward to apply to a wide range of forestry applications. For example, it can be applied to estate-level as well as stand-level valuation. It can readily deal with the situation of multiple log grades. It is not clear whether and how option valuation approaches can be applied at the estate-level and in situations where there are eight or more log grades. Consequently we conclude that, at the present time, option valuation approaches have limited relevance for the practice of forest valuation in New Zealand. Acknowledgement Andres Susaeta is thanked for discussion that contributed towards the ideas developed in this paper. References Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659. Brazee, R., Mendelsohn, R., 1988. Timber harvesting with fluctuating prices. Forest Science 34 (2), 359–372. Clarke, H.R., Reed, W.J., 1989. The tree-cutting problem in a stochastic environment: the case of age-dependent growth. Journal of Economics Dynamics and Control 13 (4), 569–595. Cox, J., Ross, S., Rubinstein, M., 1979. Option pricing: a simplified approach. Journal of Financial Economics 8, 229–263. Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press, Princeton, NJ. Forboseh, P.F., Brazee, R.J., Pickens, J.B., 1996. A strategy for multiproduct stand management with uncertain future prices. Forest Science 42 (1), 58–66. Gjolberg, O., Guttormsen, A.G., 2002. Real options in the forest: what if prices are meanreverting? Forest Policy and Economics 4, 13–20. Gong, P., Löfgren, K.G., 2007. Market and welfare implications of the reservation price strategy for forest harvest decisions. Journal of Forest Economics 13, 217–243. Gong, P., Yin, R., 2004. Optimal harvest strategy for slash pine plantations: the impact of autocorrelated prices for multiple products. Forest Science 50 (1), 10–19. Guthrie, G.A., Kumareswaran, D.K., 2003. Carbon subsidies and optimal forest management. New Zealand Institute for the Study of Competition and Regulation Inc. Haight, R.G., Holmes, T.P., 1991. Stochastic price models and optimal tree cutting: results for Loblolly pine. Natural Resource Modeling 5 (4), 424–443. Hughes, W.R., 2000. Valuing a forest as a call option: the sale of Forestry Corporation of New Zealand. Forest Science 46 (1), 32–39. Hull, J.C., 2003. Options, Futures and Other Derivatives, Fifth Edition. Prentice Hall, Upper Saddle River, NJ. Insley, M., 2002. A real options approach to the valuation of a forestry investment. Journal of Environmental Economics and Management 44, 471–492. Kimberley, M., West, G., Dean, M., Knowles, L., 2005. The 300 Index — a volume productivity index for radiata pine. New Zealand Journal of Forestry 50 (2), 13–18. Lewis, A.L., 2000. Option Valuation Under Stochastic Volatility: With Mathematica Code. Finance Press, Newport Beach, CA. McGough, B., Plantinga, A.J., Provencher, B., 2004. The dynamic behaviour of efficient timber prices. Land Economics 80 (1), 95–108. Morck, R., Schwartz, E., Stangeland, D., 1989. The valuation of forestry resources under stochastic prices and inventories. Journal of Financial and Quantitative Analysis 24 (4), 473–487. NZIF, 1999. Forest Valuation Standards. New Zealand Institute of Forestry, Christchurch, NZ. Niquidet, K., Manley, B., 2007. Price dynamics in the New Zealand log market. New Zealand Journal of Forestry 52 (3), 4–9. Norstrom, C.J., 1975. A stochastic model for the growth period decision in forestry. Swedish Journal of Economics 77, 329–337.
B. Manley, K. Niquidet / Forest Policy and Economics 12 (2010) 299–307 Plantinga, A.J., 1998. The optimal timber rotation: an option value approach. Forest Science 44 (2), 192–202. Reed, W.J., 1993. The decision to conserve or harvest old-growth forest. Ecological Economics 8, 45–69. Thomson, T.A., 1991. Financial maturity: a binomial option pricing approach. Proceedings of the Symposium on Systems Analysis in Forest Resources.
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Charleston, SC, 3–7 March. General Technical Report SE-74. USDA Forest Service, Southeastern Forest Experiment Station, pp. 369–376. Thomson, T.A., 1992. Optimal forest rotation when stumpage prices follow a diffusion process. Land Economics 68 (3), 329–342. Yin, R., 2001. Combining forest-level analysis with options valuation approach — a new framework for assessing forestry investment. Forest Science 47 (4), 475–483.
Contents lists available at ScienceDirect
Forest Policy and Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / f o r p o l
What is the relevance of option pricing for forest valuation in New Zealand? Bruce Manley a,⁎, Kurt Niquidet b a b
New Zealand School of Forestry, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand Industry, Trade and Economics, Natural Resources Canada, 506 Burnside Rd W, Victoria, British Columbia, Canada
a r t i c l e
i n f o
Article history: Received 7 November 2008 Received in revised form 13 October 2009 Accepted 22 November 2009 Keywords: Forest valuation Option value Stochastic prices Rotation age
a b s t r a c t Three different option value approaches are used to estimate the value of a typical New Zealand plantation stand, under the assumption that log prices follow a random walk. Crop values are compared with the Faustmann value, the benchmark for forest valuation in New Zealand. The increase in forest value can be substantial when log prices are low and close to the exercise cost. Gains quickly diminish and become small, both as an absolute difference and as a percentage of forest value, as price increases. However, results are very sensitive to the log price model adopted. Assuming log prices are mean reverting gives higher values than Faustmann for all log prices. Stochastic Dynamic Programming (SDP) and Binomial Option Pricing (BOP) give very similar results. They evaluate the same harvest/defer harvest/never harvest options. A new Abandonment Adjusted Price approach gives results that have a similar pattern but are consistently lower than SDP and BOP. This approach only considers whether to harvest or not in the optimal year and does not allow the option of deferring harvest. At the present time, option valuation approaches have limited relevance for the practice of forest valuation in New Zealand. Practical issues (determination of the log price model, estimation of volatility, allowing for multiple log grades and modelling at the estate-level) need to be addressed before option value approaches can be routinely used for forest valuation. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In 1999 the New Zealand Institute of Forestry (NZIF) released a set of Forest Valuation Standards “for the physical and financial description and the valuation of commercial plantation forests in New Zealand”. In developing the standards, five different methods were considered: • Transaction based approach; i.e. use of comparable sales. • Cost based approaches including the historic cost, current cost and current replacement cost methods. • Immediate liquidation approach; i.e. stock value or current realisation value. • Expectation value approach; i.e. Faustmann approach. • Option pricing approach. The NZIF Forest Valuation Standards (NZIF, 1999) require that: “The market value of a crop of trees shall be derived from transaction evidence where this is available and suitable for the purpose in terms of reliability, comparability and volume of transactions. Where transaction evidence is not available, market value shall be established using the Expectation Value Approach. Where the ⁎ Corresponding author. Tel.: + 64 3 364 2122; fax: +64 3 364 2124. E-mail address: [email protected] (B. Manley). 1389-9341/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.forpol.2009.11.002
Expectation Value approach is used, the discount rate shall be determined with reference to transaction information.” Because of the limited transaction information available, the standard approach for forest valuation has been the expectation value approach calibrated by available market evidence. NZIF (1999) noted that “although attempts have been made to value forests using option pricing theory, the Working Party is not aware of any routine practical application. However there are ongoing developments in this area that have potential application in the future.” Subsequently, there has been ongoing debate about whether the focus on the Faustmann approach under-estimates the value of plantations, in particular whether it adequately reflects the option value of a forest. For example, Hughes (2000) used an approach based on the Black–Scholes option pricing formula to value the forest assets of Forestry Corporation of New Zealand. He calculated the option pricing value to be NZ$2.218 billion compared with the realised value of NZ$1.853 billion. The suggestion from this and other work1 was that the real options associated with harvesting provide an additional option value.
1 For example, Susaeta, A. 2005. Comparative analysis of the discounted cashflow approach and option pricing theory for forest valuation: a Chilean case study. MForSc thesis, University of Canterbury.
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So the question is: Why is it that, despite the apparent potential of the method, the option value (or pricing) approach has not been widely adopted for forest valuation? This paper attempts to answer this question by dealing with three key issues: • Under what circumstances will an option value approach give a different value to the expectation value approach? • Are these differences material enough to warrant the use of an option value approach? • What are the challenges and practical issues associated with applying an option value approach to forest valuation? The first and second issues deal with whether there is a need to use an option value approach. The third issue deals with whether it can be applied in practice. A useful review on the use of option value approaches was done by Plantinga (1998). He differentiates studies with stationary prices (e.g., Norstrom, 1975; Lohmander, 1988; Brazee and Mendelsohn, 1988; Haight and Holmes, 1991) from those with non-stationary prices (e.g., Clarke and Reed, 1989; Morck et al., 1989; Thomson, 1992; Reed, 1993). In the studies with stationary prices, the expected value of a stand is found to be higher when stochastic variation in price is exploited, compared to the Faustmann value. However, in studies using non-stationary prices, “there are no gains except when there are fixed costs (e.g., management costs, alternative land uses)”; i.e., the Faustmann value is sufficient. Plantinga (1988) found that “as in previous studies with stationary prices …, expected timber values are higher with a reservation price policy compared to the Faustmann model with expected prices. However, the expected values with non-stationary prices are identical to the Faustmann values, in contrast to the findings of Thomson (1992)”. Insley (2002) also showed the importance of the underlying stochastic price process in applying a real options approach to forest valuation. She found that “option value and optimal cutting time are significantly different under the mean reversion assumption compared to geometric Brownian motion”. For a numerical example, mean reversion gave higher option values when prices were below the mean. Analysis of historical log prices in New Zealand (Niquidet and Manley, 2007) indicates a non-stationary price process for virtually all log grades and regions. However most log price series available in New Zealand run for little more than 10 years. The domestic market was dominated by the government-owned NZ Forest Service and a few large companies until 1990. As a consequence, log prices were not set in an open, competitive market and limited log price information was publicly available. In fact, tests conducted by Niquidet and Manley (2007) on a longer time series of A-grade export logs for 1973–2001 were inconclusive. Guthrie and Kumareswaran (2003) used this A-grade series to develop a real options model with uncertain future timber prices to estimate the impact of carbon subsidies and taxes on the timing of harvest, the replanting decision and forest value. They modelled log prices using a stationary AR(1) process. We note the observation of Dixit and Pindyck (1994) that “using 30 or so years of data, it is difficult to statistically distinguish between a random walk and a mean-reverting process. As a result, one must often rely on theoretical considerations (for example, intuition concerning the operation of equilibrating mechanisms) more than statistical tests when deciding whether or not to model a price or other variable as a mean reverting process.” The difficulty is that theoretical considerations and intuition can lead to a range of assumptions. For example, Haight and Holmes (1991) note that “a random walk model is consistent with an informationally efficient market, which is generally accepted for most assets”.2 Gjolberg and Guttormsen (2002) make a case for prices being mean reverting — 2 McGough et al. (2004) present results that show that “market efficiency provides little justification for random walk prices”.
“there are theoretical justifications for commodity prices not to follow a random walk”. The approach taken in this paper is to: • Calculate the value of a representative New Zealand plantation stand using option value approaches under a non-stationary price model; • Compare the value from alternative option value approaches; • Compare these values with Faustmann values; • Understand the sensitivity of value to the log price model by assuming stationary price processes; • Identify the conditions under which option value exists; • Review practical issues in implementing option valuation for forest valuation in New Zealand. The over-arching question is — What is the relevance of option valuation for forest valuation in New Zealand? 2. Approach 2.1. Log price model The assumed log price model initially adopted is a non-stationary random walk with Geometric Brownian Motion (GBM). The discretetime version of this is that the price at time t (Pt) has a lognormal distribution (i.e., Pt is lognormal with ln(Pt) normally distributed). The model implies that: ln Pt
+ T ∼L½ln Pt
2
+ ðc−σ = 2ÞT; σ√T
ð1Þ
where Pt and Pt + T are the prices at time t and T years later µ is expected annual change in log price (expressed as a proportion) σ is the standard deviation, i.e., the volatility (expressed on an annual basis). We focus here on the case of annual prices; i.e., T = 1 ln Pt
+ 1 ∼L½ln Pt
2
+ ðc−σ = 2Þ; σ:
ð2Þ
The price model takes the form: ln Pt
2
+ 1
= ln Pt + ðc−σ = 2Þ + ε
ð3Þ
where ε is a random normal variable with mean 0 and variance = σ2 Given the properties of a lognormal distribution: EðPt
+ 1Þ
c
= Pt e :
2.2. Methods Three different option value approaches are considered. Both Binomial Option Pricing and Stochastic Dynamic Programming have been used in previous studies. We also evaluate a new approach based on the Black–Scholes option pricing model (Black and Scholes, 1973). 2.2.1. Binomial Option Pricing (BOP) model We use the two-state option pricing model developed by Cox et al. (1979) as applied by Thomson (1992). Suppose the current log price is Pt. Over the next year it will either rise to α · Pt (up state) or fall to β · Pt (down state) where σ
α=e
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β = 1 = α:
301
Table 1 Silvicultural regime.
The probability of an up move = (eµ − β) / (α − β). A binomial decision tree is constructed (starting from P0) showing prices at any time and their probabilities. This is used in conjunction with a yield table giving the volume at each time. Using a backwards recursive approach, the value at each node on the tree is calculated as the maximum of the value from immediate harvest and the expected value of growing the stand on for another year. At the final stage there is the option of never harvesting. 2.2.2. Stochastic Dynamic Programming (SDP) model We use the approach applied by Norstrom (1975). A price transition matrix is developed using Eq. (3) to give the probability of Pt + 1 taking on different values given a value of Pt. In general terms the approach is similar to the BOP model — the difference is that at each year a full range of prices have to be evaluated rather than just those prices on the binomial decision tree. Again a backwards recursive approach is used — the value at each price in each year is calculated as the maximum of the value from immediate harvest and the expected value of growing the stand on for another year. 2.2.3. Abandonment Adjusted Price (AAP) model This approach is based on the theory that Hull (Options, Futures and Other derivatives, edition 5: 2003 — pp 262–264) uses in the proof of the Black–Scholes formula. We note the concerns of Thomson (1991) that Black–Scholes cannot be used to solve the forest rotation problem. However our approach stops one step short of equating the value of a forest to the value of a call option as was done by Hughes (2000). The key question is what the log price will be at the time of harvest (T years from now). If the price (PT) is greater than the harvest (i.e. exercise) cost (K) then we will harvest and get a revenue of PT − K. If the price (PT) is less than the exercise cost (K) then we will not harvest and get a revenue of 0. We can calculate the expected revenue we will receive using equation 12A.1 of Hull. If PT is lognormally distributed and the standard deviation of ln PT is s then: E½maxðPT −K; 0Þ = EðPT ÞNðd1 Þ–KNðd2 Þ
ð4Þ
where d1 = ½ln½EðPT Þ = K + s2 = 2 = s d2 = d1 –s N(x) is the cumulative probability distribution function for a standardised normal variable; i.e., it is the probability that a standard normal variable is less than the value x. But EðPT Þ = P0 eμT and s = σ√T: So, μT
E½maxðPT −K; 0Þ = P0 e Nðd1 Þ–KNðd2 Þ
ð5Þ
Age
Operation
0 5 7 8 9
Plant 850 stems/ha Prune to 2.4 m Prune to 4.6 m Prune to 6.0 m Thin to 350 stems/ha
The implementation of the AAP approach is similar to the use of the Faustmann approach but with the stumpage price calculated using Eq. (5). The probability of harvesting, N(d2), is taken into account in determining whether there are ongoing costs associated with use of land. Alternative harvest ages are evaluated for a stand to find the optimum rotation. The BOP and SDP models explicitly evaluate the options of immediate harvest versus harvest in future years versus don't harvest at all. In contrast the AAP approach, having determined the optimum rotation age, only evaluates whether or not to harvest in that specific year; i.e., it considers only the options of harvest in the optimal year versus don't harvest at all. It might be expected to estimate a lower value than the other approaches. 2.3. Species, site and silvicultural regime Option value approaches are compared on a radiata pine stand grown on a standard silvicultural regime (Table 1). The stand is on an average New Zealand forest site (site index3 30 m, 300 Index4 25 m3/ ha/year). 2.4. Log grades and prices Specifications for the eight log grades used are given in Table 2. They represent the different types of log grades produced in New Zealand: • • • • •
Pruned logs — P1 and P2 Structural logs — S1 and S2 Utility logs — L1L2 and K Industrial logs — S3L3 Pulp logs — Pulp.
K grade is an export log which is sold to Korea — there are logs of similar specification sold to China and Japan. The time series of log prices collected by MAF (Ministry of Agriculture and Forestry) was used. Log prices were extracted for the December quarter for 1996 to 20075 and converted to real $2007 using the Reserve Bank of NZ CPI. We wish to evaluate option valuation in the multi-product situation that is typical in New Zealand. For this paper we have developed a weighted average log price, using as weights the relative proportion of the volume6 in each grade at age 30. Weightings are given in Table 3.
where d1 = fln½ðP0 eμT Þ = K + σ 2 T = 2g = ½σ√T rearranging: 3
2
d1 = fln½ðP0 Þ = K + ½μ + σ = 2Tg = ½σ√T d2 = d1 –σ√T: The expression E[max(PT − K, 0)] gives the expected stumpage price ($/m3) that will be received. This is then multiplied by the volume (m3/ha) to give the expected revenue.
Mean top height of 100 largest stems/ha at age 20 years. 300 Index is an index of volume productivity. It is the stem volume mean annual increment at age 30 years for a defined silvicultural regime of 300 stems/ha (Kimberley et al. 2005). 5 MAF have collected prices for all grades since September 1994. Only data since 1996 has been used here to get away from the “spotted owl” price spike of 1993/94. Although quarterly data is available, we only use December prices because we consider the case where the decision on whether or not to harvest is made annually. 6 Volumes were predicted using the Radiata Pine Calculator Pro Version 3 - New Zealand Tree Grower 24(2), May 2003, p.26. Calculators for radiata pine and Douglasfir. 4
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Table 2 Log grade specifications.
Table 3 Age 30 volume by log grade.
Log grade
Maximum branch size (cm)
Minimum small end diameter (cm)
Grade
Volume (m3/ha)
Proportion
P1 P2 S1 S2 L1L2 K S3L3 Pulp
0 (pruned) 0 (pruned) 6 6 12 10 12
40 35 40 30 30 22 20 10
P1 P2 S1 S2 L1L2 K S3L3 Pulp Total
84 51 28 176 24 176 29 102 669
0.13 0.08 0.04 0.26 0.04 0.26 0.04 0.15
The weighted average log price series is shown in Fig. 1.7 We calculate the volatility via the Hull approach. The volatility is calculated as the standard deviation of ui where ui = ln (Pt + 1 / Pt).8 Using the 1996 to 2007 data the volatility is estimated as 0.08. (The average drift of −0.03 is not significantly different from 0). 2.5. Weighting of volumes The log price series is based on log grade proportions at age 30. Potential harvest ages are 20 to 40 years. Consequently we need to make allowance for the value of the log grade mix at these ages relative to the log grade mix at age 30. We have incorporated the gradient of improving size and quality and hence log grade mix with increasing age by: • Calculating the harvest revenue at each age by multiplying log grade volume at each age by the average real 1996–2007 December price for the grade. • Calculating the ratio of the average log price at each age to the average log price at age 30. • Multiplying the volume at each age by this ratio. Fig. 2 shows the raw volumes and also the volumes adjusted for the relative size and quality (log grade mix) at each age. The approach is broadly equivalent to the age-dependent conversion factor used by Gong and Löfgren (2007). 2.6. Alternative scenarios The base case analysis assumes: • Log prices follow GBM with µ = 0 and σ = 0.08. • Current stand age = 10 years. • Exercise cost = $45/m3. This covers the cost of harvesting and transportation of logs to mill or wharf. It represents an average New Zealand harvesting operation on easy terrain with logs transported 90 km.9 • Discount rate = 6% real. • Annual cost (including land rental) = $100/ha/year.
Different valuation approaches are compared for initial log prices between $20 and $140/m3. The harvesting decision is based on maximum NPV (=discounted harvest revenues − discounted costs). Even after allowing for ongoing annual costs this may mean no harvesting. The Faustmann value was calculated using the same assumptions but with log prices constant at the initial value. Alternative scenarios were evaluated with independent changes made: • • • • •
Annual cost reduced from $100/ha/year to 0. Volatility is changed from 0.08 to 0.04 and 0.12. Discount rate is increased from 6% to 8%. Exercise cost is changed from $45/m3 to $30/m3 and $60/m3. Current stand age is increased from 10 to 20 years.
These values were chosen to illustrate the relative sensitivity of option value to different factors. The lower and higher exercise costs are indicative of the range of harvesting and transportation costs in New Zealand. 3. Results 3.1. Base case The Faustmann method provides the benchmark for the analysis. The options approaches give higher values only at lower prices (Figs. 3 and 4). The SDP and BOP methods give virtually identical results — differences from Faustmann exceed $200/ha at log prices of $65/m3 and less and are over $1000/ha at $40/m3. The AAP approach consistently provides lower values than these two approaches — less than 80% of the increase over Faustmann. Harvesting is 0/1 under the Faustmann approach. No harvesting occurs when the initial price is less than $40/m3. At prices of $40/m3 and above harvesting occurs – even at $40/m3 which is less than the
The annual cost is incurred for as long as the stand occupies the site. It includes the land rental. A common situation in New Zealand is for the tree crop and land to be owned by separate entities with the crop owner occupying the land under a lease or licence and paying an annual land rental.10
7 See Niquidet and Manley (2007) for a detailed analysis on New Zealand log price dynamics. The December 2008 nominal log price was $91/m3–$88/m3 in $2007. 8 Tests of the weighted log price data indicated that the variable is normal and independent. 9 NZX AgriFax Logging and Cartage Costs, December 17, 2007. 10 The NZIF Forest Valuation Working Party has adopted the approach of applying a notional land rental in the valuation of a tree crop on all land including freehold land (i.e., land owned by the tree crop owner). This has been adopted because leases and licences are a common form of land tenure — and also because of practical difficulties in basing land rental on Land Expectation Value (LEV).
Fig. 1. Weighted average log price for example stand at age 30 years ($/m3 delivered to mill or port). Real prices are expressed in $2007.
B. Manley, K. Niquidet / Forest Policy and Economics 12 (2010) 299–307
303
Fig. 4. Increase in value of 10 year old stand (compared to Faustmann value) for each method with different initial log prices. Fig. 2. Total volume at each potential harvest age: (a) estimated volume; and (b) volume adjusted for size and quality relative to age 30.
exercise cost – this provides a higher NPV than the alternative of not harvesting which incurs ongoing annual costs. In contrast, under the options approaches, the probability of harvesting is less than 1 until prices are relatively high — the probability does not exceed 95% until initial log price reaches $85/m3 (Fig. 5). All methods give similar average rotation ages (for the area that is harvested) of age 27 to 28 years when the initial price is $80/m3 or more. Beneath this price the average rotation age for the Faustmann approach is less than that of the options approaches (Fig. 6).
approaches give virtually the Faustmann value. At a volatility of 0.12, these approaches give values that are greater than the Faustmann value even with an initial log price of $140/m3 (Fig 8). 3.4. Discount rate increased from 6% to 8% Increasing the discount rate causes the values and optimum rotation ages to reduce for all approaches. In terms of absolute values, the value increase given by the options approaches is lower with an 8% discount rate than a 6% discount rate. However, expressed as a ratio, the relativity between values under the options approaches and the Faustmann value are similar for either a 6% or 8% discount rate.
3.2. Annual costs reduced from $100/ha/year to 0
3.5. Exercise cost changed from $45/m3 to $30/m3 and $60/m3
The options approaches give a greater increase in value in the absence of annual costs including land rental (Fig. 7). In this case there is no opportunity cost associated with delaying harvesting, consequently the average harvest age for the options approaches increases by up to 2.8 years. Assuming zero annual costs will be unrealistic for most forestry situations and only applicable for a forest that is left unmanaged on land which has no cost.
Increasing the exercise cost increases the potential increase in value from the options approaches. The absolute level of the exercise cost has an impact over and above the relativity between exercise cost and log price (Fig. 9). 3.6. Current stand age increased from 10 to 20 years
The increase in value given by options approaches is very sensitive to the volatility of log prices. At a volatility of 0.04, option value
The option value approaches have a greater impact at age 20 compared to age 10 (Fig. 10 vs Fig. 4). This may appear to be counterintuitive — at age 10 there is a greater period of time until harvest to allow log prices to vary from the initial value. However this effect is less than the impact of discounting revenues over a longer period.
Fig. 3. Value of 10 year old stand (compared to Faustmann value) for each method with different initial log prices.
Fig. 5. Probability of 10 year old stand being harvested for each method with different initial log prices.
3.3. Volatility changed from 0.08 to 0.04 and 0.12
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Fig. 6. Average harvest age of 10 year old stand for each method with different initial log prices.
4. Discussion
Fig. 8. Impact of log price volatility on the increase in value of 10 year old stand (compared to Faustmann value) for BOP method with different initial log prices.
At the upper end, an option value approach would provide a higher value than Faustmann.
4.1. Values compared to Faustmann value 4.2. Option value approaches The increase in forest value over the Faustmann value can be substantial — but only when log prices are low and close to the exercise cost. Gains quickly diminish and become small both as an absolute difference and as a percentage of forest value as price increases. The biggest gains are typically when forest value is calculated to be negative. These gains assume that the initial (i.e., current) log price is the price used for forest valuation; i.e., that the current price is the starting point for use in the options approaches and, for the Faustmann approach, that the current price holds in perpetuity. In practice, negative values are rarely calculated. Valuers applying the Faustmann approach would generally not apply the current price as the price assumed for future harvest if it led to a negative value. Typically they would apply a moving average price or a forecast in this situation. The calculated value of the options associated with harvesting are greater than might have been anticipated from previous literature. Differences occur even when annual costs are not included. The December 2007 log price was $80/m3. At this level and with an exercise cost of $45/m3 there is little increase in value using an option pricing approach. However, harvesting and transport costs in New Zealand vary from $30 to $60/m3 depending on terrain and location.
Stochastic Dynamic Programming and Binomial Option Pricing give very similar results. This is to be expected given that the underpinning log price model (GBM) is common. Although they have different levels of detail about future price states, they do evaluate the same harvest/defer harvest/never harvest options. The AAP approach gives results that have a similar pattern but consistently produces lower values than SDP and BOP. The approach only considers whether to harvest or not in the optimal year and does not allow the option of harvesting in subsequent years as do the other two models. Essentially it values the option associated with walking away from a forestry investment and never harvesting. SDP and BOP also give the value of deferring harvesting to later ages. The difference between the AAP and SDP/BOP curves gives the option value for deferring the harvest.11 The AAP approach, having a closed form, has the advantage of being easy to apply — unlike the SDP and BOP approaches which require considerable customisation in order to be solved numerically for each situation. The AAP is only slightly more difficult to implement than using Faustmann. 4.3. Practical issues One reason that the Faustmann approach is widely used for forest valuation is that it is well understood and can be applied in practice. The option value approaches face some practical issues. 4.3.1. Reservation price In order to achieve the option values it is necessary to implement a reservation price strategy; i.e., only harvest if the price exceeds a minimum value. These can be determined for each method (e.g., Fig. 11). 4.3.2. Harvesting costs vary with age The assumption that exercise cost is constant for all ages is unrealistic. Harvesting costs do vary with age. All three options approaches evaluated readily allow harvest cost to vary deterministically with age. However, harvest cost fluctuations across time
Fig. 7. Increase in value of 10 year old stand (compared to Faustmann value) for each method with different initial log prices — annual costs 0.
11 Thomson (1992) provided a similar breakdown of the increase in NPV between the forestry abandonment option and the harvest age timing option.
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price models. We used the SDP approach with a random draw price model: ln Pt
+ 1
¯
2
= lnP −σ = 2 + ε
ð6Þ ¯
where ε is a random normal variable with mean 0 and σ = 0.08. lnP is mean(ln P). We also developed a stationary AR(1) model: ln Pt
Fig. 9. Impact of exercise cost on the increase in value of 10 year old stand (compared to Faustmann value) for BOP method with different initial log prices.
attributed to things like fuel prices potentially mean that the exercise price is also stochastic. 4.3.3. Log price model The underlying price model is a crucial assumption in implementing an option value approach. We also evaluated mean reverting log
Fig. 10. Increase in value of 20 year old stand (compared to Faustmann value) for each method with different initial log prices.
+ 1
¯
2
= 0:0678 lnP + 0:932 ln Pt −σ = 2 + ε:
ð7Þ
The mean reverting log price models lead to distinctly different option values (Fig. 12). The random draw and the AR(1) price models indicate that option value exists at all log prices, not just log prices that approach the exercise cost. The difference between the value increase for the random draw and AR(1) price models shows the impact of the assumption about the rate of mean reversion. The sensitivity of results to the underlying price model is one reason why forest valuers (and their clients) have not adopted option valuation techniques. 4.3.4. Volatility The increase in value from an option value approach is very sensitive to the estimate of future log price volatility. Again we have limited historical data to estimate this. In addition, volatility is assumed to be constant, where in many situations it too is stochastic. Option valuation methods have developed to incorporate GARCH (Generalised Autoregressive Conditional Heteroscedasticity) and other volatility processes (Lewis, 2000). Incorporating such situations into option valuation adds even more complexity and additional assumptions. 4.3.5. Multiple log grades In this analysis we have simplified things by using fixed (age 30) log grade proportions and adjusted volumes at other ages to allow for different size/quality. For practical application of option pricing it would be desirable to explicitly model the price for each log grade taking into account the volatility of each grade together with correlations between the prices of different grades. This would introduce complexity that we have not seen dealt with in either theory or practice. Forboseh et al. (1996) provide a strategy for a multi-product situation — but for only two log products with prices assumed to follow a bivariate normal distribution. Yin (2001) also
Fig. 11. Reservation price calculated using the SDP method for 10 year old stand with an initial price of $55/m3 and an exercise cost of $45/m3. Harvesting should occur when the price is above the specified value for each age.
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$55/m3 have a materially higher value estimated using an option value approach. Consequently, the circumstances under which the use of an option value approach for forest valuation is warranted appear limited under the assumption that log prices follow a random walk. Any conclusion about the relevance of option valuation is very sensitive to the assumption about the underlying price model. Fig. 12 shows that if log prices are mean reverting, option valuation approaches give values that are materially higher than the Faustmann approach for all log prices. Unfortunately we do not have the information to conclusively determine the model followed by log prices in New Zealand. The third question posed was: • What are the challenges and practical issues in applying an option value approach to forest valuation?
Fig. 12. Impact of alternative log pricing models on the increase in value of 10 year old stand using the SDP method (compared to Faustmann value) with different initial log prices. For random draw and AR(1) the initial price is assumed to be the average price. For comparative purposes, all models are assumed to have the same volatility (0.08).
uses a two product example with log price processes for sawtimber and pulpwood assumed to be independently normally distributed. Gong and Yin (2004) use SDP in another two product example — both sawtimber and pulpwood have AR(1) price models that are correlated. 4.3.6. Estate-level versus stand-level analysis The NZIF Forest Valuation Standards (NZIF, 1999) state that the valuation process “can be either estate-based or stand-based. However, in both cases, there needs to be an underlying management and harvesting strategy which is realistic for the forest being valued. This strategy should reflect what an economically rational owner would do taking into account wood supply commitments and logistical, marketing, social, political and environmental factors”. What that means, in practice, is that valuation is done within a forest estate modelling system that incorporates yield regulation and other constraints. An initial unconstrained model run in which each stand is harvested at the Faustmann optimum rotation provides an initial benchmark. The final valuation model typically has many stands being harvested at sub-optimal rotation ages in order to meet forest-level constraints. Estate-level modelling is relatively straightforward to do under deterministic log price assumptions. We see currently unresolved challenges in incorporating stochastic log prices into estate-level forest valuation. 5. Conclusion We can now review two of the questions posed in the Introduction: • Under what circumstances will an option value approach give a different value to the expectation value approach? • Are these differences material enough to warrant the use of an option value approach? Fig. 9 shows that when log prices are within $40/m3 of the exercise price, an option value approach can give different values to the Faustmann approach for a typical 10 year old stand. If $200/ha is used as a measure of materiality, then differences are material when log prices are within $10, $20 and $30 of exercise prices of $30, $45 and $60/m3 respectively. For a 20 year old stand (Fig. 10), the difference is material when log prices are within $15 of an exercise price of $45. These situations do arise in New Zealand plantation forestry — but are not the norm. With the December 2007 log price was $80/m3, only young stands with a harvesting and transportation cost over about
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