How does real option value compare with Faustmann value when log prices follow fractional Brownian motion?

Forest Policy and Economics 85 (2017) 76–84

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How does real option value compare with Faustmann value when log prices follow fractional Brownian motion?

MARK

Bruce Manleya,⁎, Kurt Niquidetb a b

New Zealand School of Forestry, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand International Economic Analysis Department, Bank of Canada, Ottawa, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Forest valuation Stochastic prices Real option value Fractional Brownian motion Faustmann

Analysis of an extended price series from 1973 to 2016 for New Zealand A grade export logs confirms that the price series is not I(0) stationary. The rejection of the unit root test suggests that it is also not I(1). It is estimated that log prices are fractionally integrated with A grade prices being I(0.78) and the natural logarithm of A grade prices being I(0.83). The implication is that log prices should be modelled using fraction Brownian motion (FBM) rather than geometric Brownian motion (GBM) or as a stationary autoregressive process. The difference in the NPV calculated using FBM compared to the NPV of classic Faustmann or GBM depends on the fractional difference, log price and volatility. Analysis of the extended A grade price series indicates an H value of about 0.3. At this level the differences at stand age 0, from both Faustmann and GBM, are modest in terms of NPV. However the differences are marked in terms of reserve log price strategy, probability of harvest and rotation age. Differences in NPV between FBM and Faustmann increase and become material as volatility increases.

1. Introduction Forest valuation in New Zealand and many other countries is based on the Faustmann approach (e.g. NZIF, 1999). However, there is increasing interest in recognising the option value inherent in the flexibility that the forest owner has over the timing of harvest. Practitioners have been following research activity in this field and are seeking practical applications. Research on the use of real option approaches for forest valuation can be differentiated between studies with stationary prices (e.g., Norstrom, 1975; Lohmander, 1988; Brazee and Mendelsohn, 1988; Haight and Holmes, 1991) and 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 has been 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, 1998). Plantinga (1998) also 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”. 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”. Manley and Niquidet (2010) compared the real option value of a typical New Zealand plantation with the Faustmann value under different log price models. As with the earlier studies, real option value depended heavily on the log price model assumed. Under the assumption that log prices follow a non-stationary random walk with geometric Brownian motion (GBM), real option value was similar to Faustmann values estimated with constant log prices, except when log prices were very low and close to the harvesting cost. However, when log prices follow a mean–reverting random draw or AR(1) process, option value exceeds Faustmann value at all log prices. Niquidet and Manley (2007) analysed historical log prices in New Zealand and found that virtually all log prices followed a non-stationary process. The analysis indicated that log prices were likely non-stationary random walks (or at least contained a random walk component). A subsequent analysis (reported in Manley, 2013) with an updated time series confirmed this finding. However, a key limitation of this work is the length of the time series (from Q3 1994 to Q1 2011), which limits the size and power of the tests and perhaps more

Corresponding author at: New Zealand School of Forestry, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand. E-mail addresses: [email protected] (B. Manley), [email protected] (K. Niquidet).

http://dx.doi.org/10.1016/j.forpol.2017.08.017 Received 14 March 2017; Received in revised form 25 May 2017; Accepted 30 August 2017 1389-9341/ © 2017 Elsevier B.V. All rights reserved.

Forest Policy and Economics 85 (2017) 76–84

B. Manley, K. Niquidet

confirmed that there was close alignment of data from the two series over the common period of Q1 1992 to Q2 2001.

importantly the ability to pick up longer term cycles. Although the analysis of Niquidet and Manley (2007) indicated a non-stationary price process for virtually all log grades and regions, tests on a longer time series of A grade export logs for 1973–2001 were inconclusive. Niquidet and Sun (2012), in reviewing forest product prices, considered 32 publications of which 22 analysed log price or stumpage series. They found conflicting evidence about the price process followed. They noted that research in forest economics had focused on “the two extremes: (1) a nonstationary unit root process which is integrated of order one I(1), and (2) a stationary process integrated of order zero I(0)”. Their subsequent analysis of North American lumber and pulp prices led them to reject both stationary and non-stationary null hypotheses. They concluded that lumber and pulp prices are fractionally integrated (i.e. they are I(d) with 0 < d < 1) and display long memory. Estimates of the fractional difference parameter (d) ranged between 0.68 and 0.81 for lumber prices and was 0.64 for pulp. Kristoufek and Vosvrda (2014) analysed daily prices between 2000 and 2013 of front futures for 25 commodities. Their implied estimate of d for lumber was 0.86. For other commodities the range was between 0.73 (oats) and 1.2 (copper). Niquidet and Sun (2012) noted that “in the range − 0.5 < d < 0.5 time series are stationary and invertible, whereas nonstationary series are those with d ≥ 0.5. The data-generating process is mean-reverting if 0 ≤ d ≤ 1, although for d ≥ 0.5 the term mean-reversion may be misleading as it only applies to the property of shocks eventually dissipating but the expected mean of the series is undefined as the variance of the series is not finite”.

Nominal prices are converted to real using CPI. Tests are applied to both untransformed prices and also the natural logarithm of prices. The former are used to allow comparison with previous studies, while the latter are used in the price model adopted. Tests used are:

• DF-GLS test (Elliott et al., 1996) with null hypothesis of I(1); i.e., • • •

that there is a unit root and the series is non-stationary. The lag was selected by the modified Akaike information criterion (MAIC) (Ng and Perron, 2001). KPSS test (Kwiatkowski et al., 1992) with null hypothesis of I(0); i.e., that the series is stationary. GPH estimate (Geweke and Porter-Hudak, 1983) of fractional differencing parameter (d) for any series that has both hypotheses rejected. KPSS test on the demeaned fractionally differenced series (Shimotsu, 2006) to confirm whether it is stationary. If a series is I(d) then its dth difference follows a I(0) process.

2.2. Price models Manley and Niquidet (2010) and Manley (2013) used a log price model of a non-stationary random walk with GBM. The discrete-time 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:

1.1. Fractional Brownian motion

ln Pt + T = ln Pt + μT − σ 2T 2 +

Fractional integration is the discrete time counterpart of fractional Brownian motion (FBM). The above suggests that FBM rather than GBM may be more appropriate for modelling log prices over time. FBM was introduced by Mandelbrot and van Ness (1968) as a generalisation of Brownian motion in which, unlike Brownian motion or GBM, increments may not be independent. It has subsequently been used in a wide range of applications. For example, Baillie (1996) discussed applications of fractional integration in geophysical sciences, macroeconomics, asset pricing models, stock returns, exchange rates and interest rates. Other FBM examples are given by Elliott and Chan (2004) on options and Rostek and Schobel (2013) on financial modelling. The focus of this paper is to extend the previous work of Manley and Niquidet (2010) to include fractional Brownian motion. Initially we analyse New Zealand log prices including the extended A grade log price series. Having determined that fractional integration or long memory is indicated we then compare real option value under FBM with Faustmann value.

∑ εi

(1)

where Pt and Pt + T are 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) ε is a random normal variable with mean 0 and variance = σ2 In contrast, for FBM the log price model is:

ln Pt + T = ln Pt + μT − σ 2T 2H 2 +

∑ WiH

(2)

where WiH is fractional Gaussian noise (FGN) with mean 0, variance = σ2 and Hurst coefficient H. The autocorrelations are given by ρ(T) = ½ [(T + 1)2H − 2T2H + (T − 1)2H]. The summation extends from i = 1 to T.

2. Approach The Hurst coefficient (H) is related to the fractional difference or memory parameter (d) by the expression H = d + ½. Table 1 gives the autocorrelations for different values of T with H varying between 0 and 0.5. In the case where H = 0.5, the autocorrelations (for T > 0) all become 0 and Model (2) is equivalent to Model (1); i.e., GBM is a special case of FBM with H = 0.5. As a basis for comparison, we also developed a stationary AR(1) model:

2.1. Log price analysis Log prices analysed are:

• Ministry for Primary Industries (MPI) quarterly log prices from Q3 1994 to Q3 2016 for each of 9 log grades. • Weighted average log price (using relative volumes at age 30 for a specified stand as weights). • A grade price series from Q1 1973 to Q3 2016. This series is a 1

ln Pt + 1 = (1–θ) ln P + θ ln Pt –σ 2 2 + ε

with θ = 0.742. Examples are provided in Fig. 1 for FBM with H = 0.001, H = 0.3 and H = 0.49 as well as the AR model. Ten different price paths over 50 years, from an initial price of $100/m3, are shown for each price model. They illustrate that as H decreases, prices become more persistent. There is much less of a range when H = 0.001 compared to

composite series. Data was available for Q1 1973 to Q1 2001 from the now defunct company Fletcher Challenge Forests. We have extended this with MPI data through to Q3 2016. Before doing so we

1

(3)

Export A and export K price series start in Q1 1992.

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Table 1 Autocorrelations for different values of T (periods) and Hurst coefficient H.

Table 2 Log grade specifications for radiata pine.

H

0

0.1

0.2

0.3

0.4

0.5

Log grade

Maximum branch size (cm)

Minimum small end diameter (cm)

Proportion at age 30

T 1 2 3 4 5 6 7 8 9 10

−0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

− 0.43 − 0.03 − 0.01 − 0.01 0.00 0.00 0.00 0.00 0.00 0.00

−0.34 −0.04 −0.02 −0.01 −0.01 −0.01 −0.01 0.00 0.00 0.00

− 0.24 − 0.05 − 0.03 − 0.02 − 0.01 − 0.01 − 0.01 − 0.01 − 0.01 0.00

− 0.13 − 0.04 − 0.02 − 0.02 − 0.01 − 0.01 − 0.01 − 0.01 − 0.01 − 0.01

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

P1 P2 S1 S2 A K L1L2 S3L3 Pulp

0 (pruned) 0 (pruned) 6 6 10 10 12 12

40 30 40 30 30 20 30 20 10

0.13 0.08 0.04 0.13 0.13 0.26 0.04 0.04 0.15

Fig. 1. Examples of log price development for example log price models. Shown are (a) FBM with H = 0.001, (b) FBM with H = 0.3, (c) FBM with H = 0.49 and (d) AR(1). Note the different scales in the y-axis.

2.4. Log grades, volumes, prices and costs

H = 0.49. This is caused by the increasing level of negative correlation in the difference between successive values of ln Pt as H decreases (Table 1). It is also apparent that variation increases over time as H increases. This is as expected – variance for FBM is σ2T2H. Whereas the FBM series show no tendency to revert to the initial price the AR series tends to revert to the initial price (and mean value) of $100/m3.

Specifications for the nine log grades used are given in Table 2. They represent different grades produced in New Zealand.4 Volumes were estimated for each grade using the Radiata Pine Calculator (NZTG, 2003). Some simplification of the New Zealand multi-log grade situation is required in order to apply option valuation. Total recoverable volume is included in the model rather than log grade volume. However, total recoverable volume at each potential harvest age (20 to 50 years) is scaled to make allowance for the relative value (on a $/m3 basis) of the log grade mix at these ages compared to age 30. This is the same approach to incorporate size/quality/age gradients used by Manley and Niquidet (2010) and Manley (2013). It is broadly equivalent to the age-dependent conversion factor used by Gong and Lofgren (2007). In order to determine the level of volatility and drift for the model, a weighted average log price was developed using as weights the relative

2.3. Species, site and silvicultural regime The example used is a radiata pine stand grown on an average New Zealand forest site (site index2 30 m, 300 Index3 25 m3/ha/year) under a standard clearwood silvicultural regime (Plant 850 stems/ha, prune in 3 lifts to 6.0 m, thin at age 9 to 350 stems/ha).

2

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). 3

4 Pruned logs – P1 and P2. Structural logs – S1 and S2. Utility logs – export A, export K, L1L2. Industrial logs – S3L3. Pulp logs – Pulp.

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perpetuity. 2. The reserve price at age 50 is calculated. The value of not harvesting at age 50 is calculated by dividing the annual cost by the discount rate. The value of harvesting at age 50 (i.e., stumpage value) is set equal to the value of not harvesting.

V50∗ (P50 –H50) = −A i

(4)

where: V50 is the volume at age 50 years P50 is the price at age 50 H50 is the harvest cost at age 50 A is the annual cost (overhead cost and land rental) i is the discount rate Eq. (4) is solved for P50.

Fig. 2. Average log price using relative log grade volume at age 30 as weights ($/m3 delivered to mill or port). Real prices are expressed in Q3 2016 $.

3. The reserve price for age 49 is then calculated. A trial value for the reserve price at age 49 is estimated. The price model is used to generate a price at age 50 (given this price at age 49) for each of 5000 price paths. For each price path the maximum of (i) the stumpage revenue at age 50 and (ii) the value of not harvesting at age 50 is calculated. The average of the maximum values is calculated across all paths and discounted back for one year with annual costs deducted. This average value of not harvesting is compared with the stumpage value from harvesting at age 49. An iterative process is carried out:

proportion of the volume in each grade at age 30. The time series of log prices collected by MPI was used. December quarter log prices were extracted for 1994 to 2015 and converted to real Q3$2016 using the NZ CPI. The weighted average log price series is shown in Fig. 2. The December 2015 price is $98/m3 while the real average price for 1994 to 2015 is $114/m3. The volatility is calculated as the standard deviation of ui where ui = ln (Pt + 1/Pt) (Hull, 2006, p. 286). Using the 1994 to 2015 data the volatility (on an annual basis) is estimated as 0.12. The average annual drift of −0.025 is not significantly different from 0. For modelling purposes a volatility of 0.1 and a drift of 0 are assumed. Costs are included:

• If the continuation value is greater than the harvest value the price at age 49 is increased. • If the continuation value is less than the harvest value the price at age 49 is decreased. • The reserve price at age 49 is found when the continuation value

• Silvicultural costs for each operation are based on industry information. • Harvesting and transportation cost varies from $48/m at age 20 to $37/m at age 50. • An annual cost of $50/ha/year is applied to cover overhead costs. In

equals the harvest value.

3

3

4. The reserve prices for ages 48 years back to 20 years are then calculated using the same approach. Harvesting is not permitted at ages younger than 20 years. The key feature of the method is that the 5000 price paths are kept distinct; for example, for a trial value of the reserve price at age 48 each of the 5000 price paths has a price at each of ages 49 and 50 generated using Model (2). In the case of FBM the increments between ages 48 and 49 and between ages 49 and 50 are correlated (as shown in Table 1). The calculation of reserve prices explicitly keeps track of the unique paths with correlated increments. In contrast to the iterative approach required here, Petrasek and Perez-Garcia (2010) were able to calculate reserve prices at each age using a closed form equation because the increments in the price series that they assumed were independent. However, in the case of FBM and the associated FGN, increments in price are correlated and it is not possible to use a closed form approach.

addition a land rental of $70/ha/year is included. 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. The annual cost and land rental are incurred for as long as the stand occupies the site. A real discount rate of 8% is used.

2.5. Model structure Manley and Niquidet (2010) and Manley (2013) used the stochastic dynamic programming approach of Norstrom (1975). However, this approach will not work with FBM where, because of the autocorrelations associated with fractional Gaussian noise, specific price paths have to be modelled. Consequently the model used is based on the Monte Carlo simulation approach used by Ibanez and Zapatero (2004) to value American options. Their method “relies on the fact that the optimal exercise frontier is the locus of points where the exercise value matches the continuation value”. This approach was used by Petrasek and Perez-Garcia (2010) to solve the optimal timber harvest problem with timber and carbon prices that follow logarithmic mean-stochastic processes. Here we consider a single rotation. Initially the reserve price strategy is determined. There are a number of steps to generating reserve prices using the backward recursive approach.

Once the reserve prices for ages 20 to 50 years have been used the model is then used to carry out a simulation: 1. An initial price (P0) is specified and the price model is used to generate 100,000 price paths from age 1 to 50 years. 2. The model works back from age 50 years to age 20 years and determines for each path whether to harvest or continue depending on whether the price in that year (for that path) exceeds the reserve price. 3. The optimum rotation age for each path is the youngest age at which the preference is to harvest rather than continue. For this age stumpage revenue is calculated. 4. The value for each path at younger ages is calculated by discounting stumpage revenue and deducting annual costs and any silvicultural costs.

1. Yields from ages 20 to 50 are generated. The maximum harvest age is 50 years. If a stand is not harvested at age 50 it is assumed that it will never be harvested. The annual cost is assumed to apply in 79

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Table 3 H value for each model and corresponding d value (fractional differencing parameter) for the differenced series and the undifferenced series. Model

FBM 0.001 FBM 0.1 FBM 0.2 FBM 0.3 FBM 0.4 FBM 0.49 GBM

H value (relates to differenced series ln Pt + 1 − ln Pt)

Corresponding d value for differenced series

d value for undifferenced series (relates to series ln Pt)

0.001 0.1 0.2 0.3 0.4 0.49 0.5

−0.499 −0.4 −0.3 −0.2 −0.1 −0.01 0

0.501 0.6 0.7 0.8 0.9 0.99 1

Table 4 Results of DF-GLS and KPSS tests for quarterly MPI log price series. Test is on lnP. Export A and Export K price series are from Q1 1992 to Q3 2016. Other price series are from Q3 1994 to Q3 2016. Prices are Q3 2016 $/m3.

5. Finally values at each age are calculated by averaging the values across the 100,000 price paths.

Log grade

DF-GLS (MAIC)

KPSS

P1 P2 S1 S2 Export A Export K L1L2 S3L3 Pulp Weighted average

0.552 (1) −0.234 (1) 0.089 (2) −0.641 (1) −1.094 (3) −1.060 (5) −0.903 (3) −0.745 (2) −0.499 (2) −0.457 (2)

1.27 (6)*** 1.16 (6)*** 1.11 (6)*** 1.04 (6)*** 0.928 (6)*** 0.578 (6)** 0.458 (6)* 0.563 (6)** 0.719 (6)** 0.962 (6)**

Level of significance: * 10%, ** 5%, *** 1%. Lag order is given in parentheses.

The same general approach was also used for the GBM and AR(1) models. Reserve prices were generated using the iterative approach using increments generated using Model (1) and Model (3) respectively. Unlike FBM, the price increments for GBM and AR(1) are independent from year to year. Consequently reserve prices could be checked using the stochastic dynamic approach of Norstrom (1975) that had been used by Manley and Niquidet (2010). This uses a price transition matrix to give the probability of each value of Pt + 1 given the value of Pt. In the case of GBM and FBM, the same set of reserve prices is applicable to any initial price. However in the case of AR(1), a unique set of reserve prices has to be calculated for each initial price when, as was assumed here, the initial price is the long-term average price; i.e., Model (3) changes as the long-term average price changes. 2.6. Model sequence We ran a sequence of models in which the Hurst parameter varied from 0.001 to 0.49. FGN was generated in MatLab using the ffgn routine5 that uses Lowen's method for 0 < H < 0.5. For comparison purposes we also ran a GBM model and an AR(1) model. Models were run with initial prices varying between $40/m3 and $150/m3 in steps of $10/m3. The lower limit was chosen to be close to the exercise cost (i.e., harvesting and transportation cost) while the upper limit is close to the upper limit of historical log prices. For initial models a value of 0.1 is used for σ. Subsequent models have σ increased to 0.15 in order to understand the sensitivity of results to price volatility. Table 3 is provided to help interpret results. It shows the H value for each model and the fractional differencing parameter associated with the differenced series (d = H − ½) and the undifferenced series ln Pt. Measures used in the evaluation include:

Fig. 3. Extended A grade price series (Q3 2016 $/m3).

• The difference between the expected value at stand age 0 for the real • • •

options approach and the Faustmann NPV – essentially this is the option value differential. The reserve price strategy. The probability of harvesting occurring. The average rotation age of area harvested.

Fig. 4. Natural logarithm of extended A grade price series (Q3 2016 $/m3).

3. Results

findings were obtained when untransformed prices were analysed. Results for the MPI log price series would suggest that prices are I (1). However these price series only start in 1992 or 1994. In contrast the extended A grade series starts in 1973.6 The A grade series and the natural logarithm of A grade (lnA) series are shown in Figs. 3 and 4. Results in Table 5 show that both the test for stationarity and the test

3.1. Log price analysis All nine MPI quarterly log prices together with the weighted average price series have the hypothesis of stationarity rejected and the hypothesis of a unit root not rejected (Table 4). The test results reported were undertaken on the natural logarithm of prices. However, the same 5

6 The extended A grade series is the Fletcher Challenge Forests series from 1973 to 2001 extended to 2016 using the MPI Export A series.

Developed by Yingghun Zhou and Stilian Stoev.

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Table 5 Results of DF-GLS and KPSS tests for extended A grade price series. Test is on lnP. Price series are from Q1 1973 to Q3 2016. Prices are Q3 2016 $/m3. Log grade

DF-GLS (MAIC)

KPSS

A (1973–2016) Ln A (1973–2016)

− 2.250 (2)** − 1.745 (3)*

1.23 (9)*** 1.22 (10)***

Level of significance: * 10%, ** 5%, *** 1%. Lag order is given in parentheses. Table 6 GPH estimates of fractional differencing parameter. Log grade

d

t(Ho: d = 0)

KPSS

A (1973–2016) lnA

0.783 0.833

4.6752*** 4.7785***

0.118 (14) 0.112 (14)

Fig. 6. Difference in NPV (compared to Faustmann NPV) at stand age 0 for each price model with different initial log prices.

Level of significance: * 10%, ** 5%, *** 1%. KPSS test is on demeaned fractionally differenced series. Lag order is given in parentheses.

Fig. 7. Reserve price strategy for each log price model. The AR reserve prices are for an initial price of $110. Fig. 5. Faustmann NPV for stand age 0 at different log prices.

Faustmann NPV is shown in Fig. 6. GBM gives a higher NPV than Faustmann at all log prices. However the impact decreases with increasing log price. With FBM at low log prices, NPV decreases as H decreases. As log prices increase there is a reversal of this trend; NPV increases as H decreases. FBM with H = 0.49 is virtually identical to GBM. This is to be expected as GBM is equivalent to FBM with H = 0.50. FBM gives a lower NPV than GBM at low log prices but a higher NPV at high log prices. GBM provides an upper envelope for the NPV for FBM at log prices less than $80/m3 but a lower envelope at log prices greater than $100/m3. AR provides the greatest NPV at prices over $50/m3. The difference between AR NPV and Faustmann NPV increases as log price increases. The optimum reserve price strategy is different for each log price model (Fig. 7). The reserve price at age 49 is similar for all FBM models and for GBM. This is to be expected as the models have similar price increments over a single year. However at younger ages the impact of correlated price increments over multiple years becomes apparent. As the FBM value of H increases the reserve prices decrease until, with h = 0.49, they approach those of GBM. The reserve price, for ages less than the minimum Faustmann rotation age of 25, is infinite for all FBM series and for GBM. However AR reserve prices are finite down to age 20. The AR reserve price strategy varies with mean log price. The example shown is for a mean price of $110/m3. The strategy is quite different from those for the FBM and GBM models reflecting the meanreverting nature of the AR series. There are marked differences in the probability of a stand being harvested (Fig. 8). Under the Faustmann approach there is a 100% probability of the stand being harvested. The AR model is similar with 100% probability of harvest with log price over $50/m3. The probability of harvest is lowest with GBM. Even at an initial price of $90/m3

for a unit root are rejected. Rejection of the test for stationarity implies that prices are not I(0) while rejection of the test for a unit root implies that prices are not I(1). Together these findings suggest that log prices are I(d) with 0 < d < 1. The fractional difference parameter (d) is estimated as 0.783 for the extended A grade series and 0.833 for the logarithmic lnA series (Table 6). The lack of significance for the KPSS test on the demeaned fractionally difference series confirm that they are stationary. Consequently we can infer that both the A grade price and lnA exhibit true long memory. The values of the fractional difference parameter imply that the two series would be over-differenced if an I(1) model (i.e., GBM) is assumed. The d value for lnA suggests that the d value for the differenced lnA series would be about − 0.17 and the H coefficient about 0.33. It was subsequently estimated as − 0.225 with a corresponding H coefficient of 0.275. 3.2. Comparison of models As expected, Faustmann NPV7 increases with increasing log price8 (Fig. 5). Although Fig. 5 appears to show a linear relationship between NPV and log price the relationship is actually slightly curvilinear reflecting an increase in optimum rotation age at lower prices (Fig. 9). NPV is negative when log price is less than $92.70/m3. The difference in NPV for each price model relative to the 7 Although we are dealing with a stand at age 0 we refer to NPV rather than LEV because we only consider one rotation. 8 Faustmann NPV is calculated using a fixed log price; i.e., the initial price is used as an estimate of the price in all future years.

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Fig. 8. Probability of stand being harvested for each log price model with different initial log prices.

Fig. 10. Difference in NPV (compared to Faustmann NPV) for stand age 0 at different initial log prices for GBM and FBM 0.3 models with volatility at (i) 0.1 and (ii) 0.15.

there is still only a 90% probability of harvest. For the FBM price models, the probability of harvest increases as the H value decreases. There are also patterns in the mean rotation age (Fig. 9). The Faustmann rotation age is the youngest apart from when the initial price is $40/m3. It stays constant at 25 years once the initial log price reaches $80/m3. For the FBM models average rotation age increases as H decreases. Average rotation age for the AR price model is similar to that of GBM at low log prices but stays at a higher level at log prices above $80/m3.

used but was rejected for the extended A grade series. Prices for A grade and lnA are persistent in the sense that they exhibit high serial correlation. The negative autocorrelation pattern associated with the differenced series implies that a high (positive) value for the difference in the A grade price or lnA will be followed by a low (negative) value. Consequently prices will be more persistent under FBM (with d < 0.5) than GBM where autocorrelations of the differenced series are zero. The estimate of d, the fractional difference parameter, for the extended A grade series is 0.783. This is in the range (0.68 and 0.81) that Niquidet and Sun (2012) estimated for North American lumber prices. The fractional difference parameter is also referred to as a measure of long memory or long-range dependence or long-range persistence. It is a measure of whether shocks to price are transitory or whether they are long-lived. Niquidet and Sun (2011) concluded that shocks died out and prices were mean reverting for most of Canada's forest products. However shocks were persistent for softwood logs as well as lumber from coastal Douglas fir and western hemlock. A possible reason suggested for the permanence of shocks to softwood log price was the “time lags between logging, milling and lumber deliveries such that bids for logs are based on expected future prices for lumber rather than current spot prices”. However they concluded that a more plausible explanation was the effect of aggregation given that there are “so many distinct spatially separate softwood log markets, each with a considerable degree of species and grade variation”. Niquidet and Sun (2012) suggested the long-term nature of forest development as an explanation for long memory in forest products prices: “it is not surprising that the time series of prices for products derived from timber, where rotations are typically several decades in length, are very persistent”. However they also noted that the results could be driven by structural change associated with “significant technological advancements in virtually all stages of forest production (e.g., tree breeding, harvesting techniques, lumber recovery) have taken place and landowners' objectives have likely evolved”. A possible explanation for the export A grade price demonstrating FBM is the continuously evolving export log market. The export log market was initiated by exports to Japan. South Korea became important in the 1990s with India increasing in importance and China dominating since 2009 (Fig. 11). Market changes could well have resulted in a series of shocks to log prices, each of which has had an ongoing effect. The prices for export logs represent the average price that New Zealand forest growers have received each year for these grades. In a sense they are the weighted average of prices received for these grades in China, South Korea, India and Japan. The relative importance of each country has changed, in some cases dramatically. For example, the increase in log exports to China from 1 to 12 million m3 over 5 years

3.3. Sensitivity to volatility The difference in NPV relative to the Faustmann NPV increases as volatility increases. At stand age 0 and a volatility of 0.15, FBM 0.3 gives an NPV that is about $300/ha higher than the Faustmann NPV once log prices reach $60/m3 (Fig. 10). The difference is even higher at lower log prices. Differences are generally at least double those when volatility is 0.1. The relationship between the NPV of GBM and FBM 0.3 also depends on volatility. The crossover point between the two models increases from $80/m3 to $100/m3 as volatility increases from 0.1 to 0.15 (Fig. 10). 4. Discussion 4.1. I(d) rather than I(1) Pindyck (1999) pointed out the need for a long time series in order for the augmented Dickey-Fuller test to reject a unit root. Notwithstanding subsequent enhancements to the test, this would explain why the unit root was not rejected when the MPI A grade price series was

Fig. 9. Mean rotation age for each log price model with different initial log prices.

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Fig. 11. New Zealand log exports by country of destination. Volume is for all grades – radiata pine A grade has been an important component. (Source: MPI).

depend on the value of H, the initial log price and the volatility. There are some combinations at which differences are not material; for example when H = 0.3, log price = $100/m3 and volatility = 0.15 (Fig. 10). However there are other combinations at which differences will be material; for example, with the same H value and log price, but the lower volatility of 0.1, GBM only gives 60% of the increase in NPV shown by FBM over Faustmann (Fig. 10). Differences would be even greater at lower values of H. Differences in reserve price strategy, probability of harvest and rotation age also become material.

can be regarded as a shock to the market for New Zealand logs. 4.2. FBM NPV compared to Faustmann NPV Analysis of lnA suggests an H value of about 0.3. With H = 0.3, volatility = 0.1 and stand age 0, FBM gives an increase over the Faustmann NPV of $265/ha with a log price of $40/m3 decreasing to $83/ha when log price is $150/m3. When volatility increases to 0.15, the increase in FBM over Faustmann goes from $438/ha to $328/ha over the same price range. With H = 0.3, volatility = 0.1 and an initial price of $100/m3, FBM has an NPV that is higher than the Faustmann NPV by $155/ha at age 0. With volatility = 0.15 NPV is higher by $307/ha at age 0. These differences represent 30% and 60% respectively of the Faustmann NPV of $512/ha.

4.5. Limitations Using real options analysis, we have applied different price models to a weighted log grade mix. However, the indication of FBM is based on a single price series so findings must be considered tentative. There is a question-mark on the applicability of findings about an export log price series to domestic log prices. However there is strong linkage between domestic and export log prices. As the proportion of log exports has increased, the principle of export log parity has been increasingly applied in domestic log contract negotiations in New Zealand.

4.3. Probability of harvest/rotation age The patterns for FBM reflect the different reserve price strategies. The reserve price strategy is steeper for lower values of H; i.e., the lower the value of H the higher the level of reserve prices at age 25. They descend to similar levels at older ages. The higher reserve prices for lower values of H lead to older rotation ages. Although reserve prices become similar beyond age 48 years, the lower levels of persistence (and a greater spread of values) mean that a higher H value leads to a lower probability of harvest.

4.6. Practical issues Practical application of FBM will be difficult. Manley and Niquidet (2010) in considering GBM, comment that “For practical application of option pricing it would be desirable to explicitly model the price of each 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”. This comment also applies to the implementation of FBM into forest valuation. The underlying process may vary by each log grade and the aggregate series may also be different.

4.4. Are the differences with GBM material? Differences of NPV for FBM models compared to GBM reflect the persistence of prices. As H decreases prices become more persistent. Consequently when log prices are low, FBM leads to a lower NPV than GBM with NPV decreasing as H decreases. Conversely when log prices are high FBM leads to a higher NPV than GBM with NPV increasing as H decreases. The impact of GBM and AR price models are comparable to those found by Manley and Niquidet (2010). In reviewing the long-run behaviour of oil, coal and natural gas, Pindyck (1999) found that prices were mean-reverting but the rate of mean reversion was so slow “that for purposes of making investment decisions, one could just as well treat the price of oil as a geometric Brownian motion, or related random walk process”. Based on this Niquidet and Sun (2012) anticipated that “given that the rate of meanreversion from fractionally integrated series is very slow, for long-term investments it may well be that such an exercise leads to inconsequential differences in decision making and valuation compared to geometric Brownian motion”. Our findings are that differences in NPV between FBM and GBM

5. Conclusions Our analysis confirms that radiata pine log prices are not stationary. The extended A grade series indicates that FBM rather than GBM may be a more appropriate log price model. With GBM the increase in NPV (compared to Faustmann) can be substantial when log prices are low and near the exercise cost. The impact of FBM on NPV is less than that of GBM at low log prices with the NPV decreasing as H decreases to the extent that FBM can become less than the Faustmann NPV. However at higher log prices FBM gives a higher NPV than GBM with NPV increasing as H decreases. Analysis of the extended A grade price series indicates an H value of about 0.3. At this level the differences at stand age 0, from both 83

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Faustmann and GBM, are modest in terms of NPV. However the differences are marked in terms of reserve log price strategy, probability of harvest and rotation age. Differences in crop value between FBM 0.3 and Faustmann, although modest at the current log price of about $100/m3, are at the level of being material from a forest valuation perspective. Differences do become material as volatility increases. Given the limited long-term log price series to support it and the practical issues associated with implementation of option pricing with FBM, forest valuation based on Faustmann will continue to be the practical reality for forest valuation in New Zealand. It is easy to understand and apply. However it calculates values that are consistently lower than those calculated using an FBM price model because it ignores the option value associated with varying rotation age in response to changing log price. New Zealand forest growers, particularly smallscale owners, do bring forward harvest age when log prices are high and delay harvest when log prices are low. This is evident in the fluctuating national harvest volume (see NEFD, 2016). Consequently, the challenge is to make real option valuation models practical – in particular to develop applications that deal with multiple log grades. References Baillie, R.T., 1996. Long memory processes and fractional integration in econometrics. J. Econ. 73, 5–59. Brazee, R., Mendelsohn, R., 1988. Timber harvesting with fluctuating prices. `. For. Sci. 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. J. Econ. Dyn. Control. 13 (4), 569–595. Elliott, R.J., Chan, L., 2004. Perpetual American options with fractional Brownian motion. Quant. Finan. 4, 123–128. Elliott, G., Rothenberg, T., Stock, J.H., 1996. Efficient tests for an autoregressive unit root. Econometrica 64, 813–836. Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long-memory time series models. J. Time Ser. Anal. 4, 221–238. Gong, P., Lofgren, K.G., 2007. Market and welfare implications of the reservation price strategy for forest harvest decisions. J. For. Econ. 13, 217–243. Haight, R.G., Holmes, T.P., 1991. Stochastic price models and optimal tree cutting: results for loblolly pine. Nat. Resour. Model. 5 (4), 424–443. Hull, J.C., 2006. Options, Futures and Other Derivatives, sixth edition. Prentice Hall, Upper Saddle River, NJ. Ibanez, A., Zapatero, R., 2004. Monte Carlo valuation of American options through computation of the optimal exercise frontier. J. Financ. Quant. Anal. 39 (2), 253–275.

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