Eur J Forest Res (2006) 125: 335–343 DOI 10.1007/s10342-006-0117-7

O R I GI N A L P A P E R

Markku J. Penttinen

Impact of stochastic price and growth processes on optimal rotation age

Received: 25 April 2005 / Accepted: 12 December 2005 / Published online: 12 April 2006 Springer-Verlag 2006

Abstract This paper analyses timber harvesting in the Finnish economic and wood production environment. Empirical evidence including stumpage prices, silvicultural costs, etc., since 1949 covers all non-industrial private forestry. Stumpage price and volume growth processes are proxied by GBMs. Optimal harvesting age solutions and numerical results recognising price drift, price and growth volatility, volume growth, value growth and stand establishment costs, as well as thinning beneﬁts, are provided for Scots pine. Moreover, comparative static and sensitivity solutions, including numerical results, show the impact of the discount rate, price drift, and price and growth volatilities on optimal harvesting age. Price volatility prolonged harvesting age by some 5–9 years, and growth volatility by about 1–2, but negative price drift for discount rates from 5 to 2% fell by roughly 6–10 years. Ignoring the future thinning beneﬁts prolonged the harvesting age only by 1–2 years, but ignoring future stand establishment costs reduced it by 2–4 years. Including the price drift and volatility violated the 70 year age limit in the Forest Act for discount rates exceeding 3.5%. The recommended harvesting age of 80 years could be established only by ignoring the price drift. In all, this study produces solutions and programs that can be incorporated into a forest management planning software product widely used in Finland (Hynynen et al. in For Ecol Manage 207(1–2):5–18, 2005). Keywords Stumpage price drift and volatility Æ Volume growth volatility Æ Variable value growth and stand establishment cost Æ Optimal rotation age Æ Pressler’s indicating percentage Communicated by Martin Moog M. J. Penttinen Finnish Forest Research Institute, Vantaa Research Centre, Unioninkatu 40 A, 00170 Helsinki, Finland E-mail: mar[email protected].ﬁ Tel.: +358-10-2112244 Fax: +358-10-2112104

Introduction On timber-harvesting decisions Ecological sustainability has motivated both forest protection and the requirements for postponing ﬁnal felling. However, felling criteria based on Finnish legal provisions and norms such as those enshrined in legislation (The Amendment to the Forest Act 1997) and silvicultural recommendations (Tapio 2001) have been compared with optimal rotation results based on economic principles (see e.g. Samuelson 1976), showing that the norms require harvesting too late (Hyytia¨inen and Tahvonen 2001). This has encouraged the present rewriting of the recommendations (cf. the German struggle, Mo¨hring 2001). The aim of this study is to produce explicit optimal timber harvesting age solutions, and to construct comparative static and sensitivity results which focus on the impact of price drift and volatility, growth volatility, thinning beneﬁts, silvicultural costs and discount rate on optimal timber harvesting age. All solutions have been programmed using MATLAB (2002) and tested against local economic and growth data. This study has been stimulated by the availability of the systematic stumpage prices 1949–2004 and reforestation costs. They allow estimation of both stochastic stumpage price process parameters and the reforestation cost functions. Empirical evidence covers the economic environment of non-industrial private forest (NIPF) owners. The deterministic volume growth models both for logs and pulpwood are based on individual tree models for Finnish pine forests (Hynynen et al. 2005). Previous work on age-based timber-harvesting with continuous time processes Timber-harvesting models incorporating price and/or growth volatilities have been analysed using various approaches, from discrete time dynamic programming

336

to option theory models. Only age-based timber-harvesting models employing continuous time have been considered here. Clarke and Reed (1989), hereafter abbreviated to CR, have built up the modelling and numerical volatility impact analysis using geometric Brownian motion (GBM) to proxy both stumpage prices and volume growth. However, they ignored silvicultural costs, value growth and Faustmannian inﬁnite rotations. Morck et al. (1989) assumed that the inventory of timber and aggregate wealth, not merely stumpage prices, followed GBM, and determined the optimal cutting rate depending on time remaining to end of lease for a logging company. Yin and Newman (1995) extended the CR contribution to include additional rental and management costs and provided comparative static results for the optimal rotation age, except for price drift. Yin and Newman (1997) applied these results with a numerical example for loblolly pine. Reed and Haight (1996) assumed GBM stumpage prices and volume growth, as CR, and used the Richards curve to proxy the deterministic volume growth. They predicted the present value distribution of forest plantation investment as a function of rotation age and even estimated reservation prices for harvesting. Yin (2001) introduced options valuation approach to forestry, contributed harvesting reservation prices under diﬀerent stand ages and compared the results with the static Faustmann rotation ones. Insley (2002) laid the foundation of the real options approach to forestry using GBM and mean reverting (MR) prices, and assumed a simple deterministic volume growth function. She contributed critical harvesting prices for diﬀerent stand ages and compared those for GBM and MR processes in a single rotation analysis. Insley and Wirjanto (2004) extended the results of Insley (2002), which applies dynamic programming (DP), to elaborate both DP and contingent claims (CC) analyses and compared empirical results based on GBM and MR prices both with DP and CC solutions. Insley’s model (2002) was recently extended by Insley and Rollins (2005) to a multirotation framework, producing critical harvesting prices at diﬀerent age levels as well as evaluating the impact of harvesting age restrictions on critical prices. Thomson (1992) used stochastic prices, like CR, in calculating harvest age under diﬀerent stumpage prices and in comparison with Faustmann steady state results. Klemperer et al. (1994) chose a similar price process, but analysed risk premiums as appropriate for long-term

(1998) used a similar model to Thomson (1992), as well as providing harvest age as a function of stumpage prices, and compared these results with those of the deterministic Faustmann approach. Yoshimoto and Shoji (2002) tested 13 stochastic models to capture the uncertainty of price and growth dynamics. Sødal (2003) suggested a simpliﬁed solution to the optimal rotation problem of Willassen (1998) using the mark-up approach to investment. The aim of this study The purpose of this study is to develop new solutions for (1) optimal harvesting age, (2) sensitivity analyses and (3) comparative statics in analytic form. The ﬁnal goal is (4) implementation of the solutions in the form of programs and their numerical testing using local empirical evidence. The aim of the work is to develop models and programs to be used in forest management planning software products (Hynynen et al. 2005; Salminen et al. 2003; also in Redsven et al. 2005; Hynynen et al. 2002).

Materials and methods Study site This study employs a Vaccinium type (VT) site with a dominant height of 24 m at 100 years of age, H100=24. The stochastic price and growth processes start when the last actual observations were obtained, for example, at 70 years, the lowest harvesting age in the Amendment to the Forest Act (1997). However, in this case the starting age of the processes is at least 55 years, when the last thinning occurs. The growth model applied covers the state of the art as provided by a local software product (Salminen et al. 2003; Hynynen et al. 2005). The soil expectation value deﬁnition Consider the forest owner’s harvesting decision after the last thinning at age sTH,n. At this time, sn>sTH,n, the present value Sn of the forest stand is the value of the ﬁnal harvest at age sn plus the soil expectation value of future rotations both discounted by sn years

fpn ðsn Þqðsn ; dn Þ cnþ1 ðsn ; dnþ1 Þþbnþ1 ðsn ; sTH;nþ1 ; dnþ1 Þ expðrsTH;nþ1 Þ ; Sn ðsn ; dnþ1 ; sTH;nþ1 Þ ¼ ½expðrsn Þ 1 forestry investment. Willassen (1998) applied the theory of stochastic impulse control and derived explicit solutions to multirotational optimal harvesting problems with revenue as the state variable. Yoshimoto and Shoji

ð1Þ

where r is the discount rate. All subsequent rotations n+1, n+2,... repeat the functions for stumpage price pn, the stand volume q, the regeneration costs cn+1 and the thinning beneﬁts bn+1.

337

Note that the ex post regeneration costs cn(0,dn) and the beneﬁts bn(sTH,n,dn) from all historical ex post thinnings have no impact on the harvesting decision; i.e., cn(0,dn)+bn(sTH,n,dn)exp(rsTH,n) is ignored from Eq. 1. Notational burden is avoided by including the beneﬁts bn+1(sn,sTH,n+1,dn+1), which actually stand for several future ex ante thinnings in the ex ante regeneration costs cn+1(sn,dn+1), although they are present. Moreover, in order to single out the impact of volatility, the planting intensity dn+1 is assumed to be determined in advance based on the site properties of the stand.

where the properties of the volatility generating Wiener processes for stumpage price wpt and volume growth wqt volatility have been used (see Øksendal 1985). The average price pa(t) depicts the average value of a cubic metre of wood and the volume qi(t) stands for the interpolated growth, which is based on the volumes obtained every ﬁfth year from the growth and yield models (Salminen et al. 2003; Hynynen et al. 2005). Thus the soil expectation value S(t) for t>02 discounted to age 0 is of the form: SðtÞ ¼

Stumpage price and volume growth processes Clarke and Reed (1989) deﬁned the ‘biological asset’s aggregate intrinsic value‘ as pt*qt, where pt stands for the price process and qt for the volume growth process, and the bold letters indicate random variables. The diﬀerentials of both ln pt and ln qt are proxied by Geometric Brownian motions (GBMs).1 This alternative leads to modiﬁcations of the GBM of price and volume which avoid the ’virtual certainty of (relative) ruin’ (Samuelson 1965, p. 17), in which the probability mass tends to zero. The GBMs imply that the distributions of the stumpage prices pt and the volume growth qt are log-normal (see Klemperer et al. 1994). The price pt has by Ito’s lemma a mean of p0 exp[D(t)+t r2p/2] (Klemperer et al. 1994), where D(t) is the integral D(t)=t0 a(s) ds, a(s) is the stumpage price process drift and rp is the stumpage price process volatility (Reed and Haight 1996). Similarly, the volume growth qt is log-normally distributed with a mean of q0exp[C(t)+tr2q/2], where C(t) is the integral C(t)=t0 g(s)ds, g(s) is the growth process drift and rq is the growth process volatility (CR; Reed and Haight 1996). The soil expectation value of a stand

fvðtÞ cðtÞg ; fexpðrtÞ 1g

ð3Þ

where the expectation v(t) of the felling value of the stand is Eq. 2 above and c(t) consists of the stand establishment costs and thinning beneﬁts of the next rotation. Note that, the roundwood prices for future thinnings may be given as average stumpage prices (Hyytia¨inen and Tahvonen 2001, p. 445). Here, the expectations for both log and pulpwood price processes separately proxy log and pulpwood prices at each future thinning, which depend on the optimal rotation age t only in the form of stumpage prices. The price trend means that there is no unique Faustmannian optimal rotation age for all rotations. Solutions for only the ﬁrst rotation are provided, as is the case using Pressler’s indicating percentage (Pressler 1860). However, the costs and beneﬁts are discounted to the initial t=0 in the context of net present value (NPV) according to Samuelson (1976). Optimal timber harvesting age If the stumpage price drift a(t) is constant a and the volume growth q(t) is deterministic, then the optimal rotation age can be deﬁned by q¢(T)/q(T)=rm/ [1exp(rmT)], where the modiﬁed discount rate rm=rar2p/2 (CR). They ignore growth volatility rq and reforestation costs c(t), which causes some discrepancy. However, their ignoring the value growth pa(t) and Faustmannian inﬁnite rotations impose limitations on the model. If the forest owner is risk-neutral,3 and her/his utility function is linear, then the expected NPV is considered as the objective function. The optimal rotation age T is deﬁned by the solution (Appendix 1):

The felling value of the stand vt also includes the potential gross sales revenue from the ﬁnal felling, vt=pt qt. Ito’s formula for the function vt(pt, qt) = pt qt yields dvt = dpt qt + pt dqt + dpt dqt, where dpt dqt= rprq qpq dt (Øksendal 1985). The expected value v(t)=E{vt} of the stand at the harvesting age t is then (CR; Yin and Newman 1995, 1997) r½1 cðT Þ=vðT Þ 0 2 vðtÞ ¼ p0 q0 exp DðtÞ þ tr2p =2 þ CðtÞ þ tr2q =2 þ tqpq rp rq ; aðT Þ þ gðT Þ þ rR =2 c ðT Þ=vðT Þ ¼ ½1 expðrT Þ ; ð4Þ ¼ p a ðtÞqi ðtÞ exp t a þ r2p =2 þ r2q =2 þ qpq rp rq ; ð2Þ 1

The stochastic diﬀerential equations are D ln pt = a(t) dt + rp wpt and D ln qt = g(t) dt+rq wqt, where wpt and wqt are normally distributed so-called Wiener processes, and have already been used by CR. Note that D stands or the diﬀerential ¶/¶t and ln for the natural logarithm.

2 The denominator in Eq. 3 tends to zero as t ﬁ 0 caused by series expansion of the future rotations. However, the length of future rotations approaching zero is beyond the domain of the model. 3 Actual local cuttings have been on average 3.9 m3/ha per year, and the average forest holding size is roughly 37 ha (Karppinen et al. 2002, p. 29, 47). Thus the annual cutting is of the order of 140 m3, corresponding to some 3,000€/average woodlot. This low average yearly sales speaks for risk neutrality.

338

where a(t)=a+b(t), a is the stumpage price drift rate, b(t) the value growth drift, b(t)=p¢(t)/p(t), g(t) the volume growth drift, g(t)=q¢(t)/q(t), and rR2 stands for rR2=r2p+r2q+2qpqrprq. Here, the reforestation costs c(t) after harvesting actually also consist of the thinning beneﬁts of the next generation. While this solution resembles that of Yin and Newman (1995, 1997), even their solution lacks silvicultural costs c(t), value growth pa(t) and inﬁnite rotations. On the numerical solution of the optimal harvesting age The average stumpage price pa(t) at age t is deﬁned in order to cope with the deterministic value growth (Gong 1998): pa(t)=qlog(t) plog+qpulpw(t) ppulpwch, in which plog and ppulpw are log and pulpwood stumpage prices, qlog(t) and qpulpw(t) are the sawn timber and pulpwood yields from harvesting 1 m3 of standing timber at age t, and ch is the harvesting cost per m3.4 The average stumpage price function pa(t) is based on the value growth drift b(t), b(t)=D ln pa(t)=pa¢(t)/pa(t). In all, the stumpage price p0 exp(D(t)), where D(t)=t0 a(s)ds, and a(t)=a+b(t). Similarly, the deterministic volume growth q(t)=q0exp(C(t)), where C(t)=t0 g(s)ds, is based on the volume growth drift g(t), g(t)=D ln q(t)=q¢(t)/ q(t). The volume growth q(t) has been calculated using growth and yield simulation software (Salminen et al. 2003; Hynynen et al. 2005), which incorporates the state-of-the-art growth models (Hynynen et al. 2002). Moreover, the optimality of the management trajectory from 0 to the last thinning relies on the optimising software (Hynynen et al. 2005). Unfortunately, since the models and the simulator provide log, pulpwood and total volumes, etc. only at 5-year intervals (Matala et al. 2003), an interpolation function is needed. Although a three-parameter Richards Curve has been proposed for volume growth by Reed and Haight (1996),5 one of its two-parameter reduction, namely, the logistic growth as in Kuuluvainen and Tahvonen (1999) provides the ﬂexibility and accuracy required for the interpolation of the volumes provided in 5-year steps. However, removals caused by the thinnings violate the ordinary logistic growth model, so that an additional parameter, a negative bottom quantity qb, must be included in the model. In the same way, the logistic and polynomial value growths have been applied as the hypotheses in estimating average stumpage price pa(t) at age t.

4 The ﬁxed logging costs lead to non-convexities (Boscolo and Vincent 2001), but are not applied here. 5 The three-parameter Richards curve is of the form q(t)=K exp(td)/{1+(h1) exp[K(td)]}. It can be reduced to the monomolecular (h=0), and the Gompertz (h ﬁ 1) as well as the logistic volume growth model (h=2) as special cases. For interpolation purposes, it is not important which model alternative is chosen.

Results Empirical estimation of the stumpage price and silvicultural cost parameters All price and cost series were converted to 2004 currency using the consumer price index before the ratios py/py1 were calculated. Here y means year, half-year, month or week. The systematic price statistics consisted of annual observations 1949–2004, semi-annual observations 1981–2004 and monthly observations 1985–2004 as well as weekly observations 1997–2004, all covering the whole of non-industrial private forestry (NIPF). The annual reforestation cost series covering NIPF was available for 1977–2004 for the reforestation cost function estimation. The maximum likelihood estimates of stumpage price drift a and the volatility rate rp of the GBM stumpage price process, pt, will be a* and rp*, where a* is the estimated mean and rp* is the estimated standard deviation of the series ln [py/py1] (Reed and Haight 1996; Insley 2002). According to Statistical Yearbook (2005) the cost for reforestation with nursery seedlings was an average of €750 ha1 and its compound annual growth rate was 1.54%. Statistical properties and parameter estimates of the stumpage price processes The normality of the annual softwood log and pine pulpwood net return series {[py/py1]1} 1949–2004 could be rejected (P0 and even convex by the positiveness of its second derivative of [exp(rt)1]. The numerator, v(t)c(t), is positive within the relevant age interval, i.e. from age 30. Consider the concavity of the numerator; i.e., whether v¢¢(t)c¢¢(t)

O R I GI N A L P A P E R

Markku J. Penttinen

Impact of stochastic price and growth processes on optimal rotation age

Received: 25 April 2005 / Accepted: 12 December 2005 / Published online: 12 April 2006 Springer-Verlag 2006

Abstract This paper analyses timber harvesting in the Finnish economic and wood production environment. Empirical evidence including stumpage prices, silvicultural costs, etc., since 1949 covers all non-industrial private forestry. Stumpage price and volume growth processes are proxied by GBMs. Optimal harvesting age solutions and numerical results recognising price drift, price and growth volatility, volume growth, value growth and stand establishment costs, as well as thinning beneﬁts, are provided for Scots pine. Moreover, comparative static and sensitivity solutions, including numerical results, show the impact of the discount rate, price drift, and price and growth volatilities on optimal harvesting age. Price volatility prolonged harvesting age by some 5–9 years, and growth volatility by about 1–2, but negative price drift for discount rates from 5 to 2% fell by roughly 6–10 years. Ignoring the future thinning beneﬁts prolonged the harvesting age only by 1–2 years, but ignoring future stand establishment costs reduced it by 2–4 years. Including the price drift and volatility violated the 70 year age limit in the Forest Act for discount rates exceeding 3.5%. The recommended harvesting age of 80 years could be established only by ignoring the price drift. In all, this study produces solutions and programs that can be incorporated into a forest management planning software product widely used in Finland (Hynynen et al. in For Ecol Manage 207(1–2):5–18, 2005). Keywords Stumpage price drift and volatility Æ Volume growth volatility Æ Variable value growth and stand establishment cost Æ Optimal rotation age Æ Pressler’s indicating percentage Communicated by Martin Moog M. J. Penttinen Finnish Forest Research Institute, Vantaa Research Centre, Unioninkatu 40 A, 00170 Helsinki, Finland E-mail: mar[email protected].ﬁ Tel.: +358-10-2112244 Fax: +358-10-2112104

Introduction On timber-harvesting decisions Ecological sustainability has motivated both forest protection and the requirements for postponing ﬁnal felling. However, felling criteria based on Finnish legal provisions and norms such as those enshrined in legislation (The Amendment to the Forest Act 1997) and silvicultural recommendations (Tapio 2001) have been compared with optimal rotation results based on economic principles (see e.g. Samuelson 1976), showing that the norms require harvesting too late (Hyytia¨inen and Tahvonen 2001). This has encouraged the present rewriting of the recommendations (cf. the German struggle, Mo¨hring 2001). The aim of this study is to produce explicit optimal timber harvesting age solutions, and to construct comparative static and sensitivity results which focus on the impact of price drift and volatility, growth volatility, thinning beneﬁts, silvicultural costs and discount rate on optimal timber harvesting age. All solutions have been programmed using MATLAB (2002) and tested against local economic and growth data. This study has been stimulated by the availability of the systematic stumpage prices 1949–2004 and reforestation costs. They allow estimation of both stochastic stumpage price process parameters and the reforestation cost functions. Empirical evidence covers the economic environment of non-industrial private forest (NIPF) owners. The deterministic volume growth models both for logs and pulpwood are based on individual tree models for Finnish pine forests (Hynynen et al. 2005). Previous work on age-based timber-harvesting with continuous time processes Timber-harvesting models incorporating price and/or growth volatilities have been analysed using various approaches, from discrete time dynamic programming

336

to option theory models. Only age-based timber-harvesting models employing continuous time have been considered here. Clarke and Reed (1989), hereafter abbreviated to CR, have built up the modelling and numerical volatility impact analysis using geometric Brownian motion (GBM) to proxy both stumpage prices and volume growth. However, they ignored silvicultural costs, value growth and Faustmannian inﬁnite rotations. Morck et al. (1989) assumed that the inventory of timber and aggregate wealth, not merely stumpage prices, followed GBM, and determined the optimal cutting rate depending on time remaining to end of lease for a logging company. Yin and Newman (1995) extended the CR contribution to include additional rental and management costs and provided comparative static results for the optimal rotation age, except for price drift. Yin and Newman (1997) applied these results with a numerical example for loblolly pine. Reed and Haight (1996) assumed GBM stumpage prices and volume growth, as CR, and used the Richards curve to proxy the deterministic volume growth. They predicted the present value distribution of forest plantation investment as a function of rotation age and even estimated reservation prices for harvesting. Yin (2001) introduced options valuation approach to forestry, contributed harvesting reservation prices under diﬀerent stand ages and compared the results with the static Faustmann rotation ones. Insley (2002) laid the foundation of the real options approach to forestry using GBM and mean reverting (MR) prices, and assumed a simple deterministic volume growth function. She contributed critical harvesting prices for diﬀerent stand ages and compared those for GBM and MR processes in a single rotation analysis. Insley and Wirjanto (2004) extended the results of Insley (2002), which applies dynamic programming (DP), to elaborate both DP and contingent claims (CC) analyses and compared empirical results based on GBM and MR prices both with DP and CC solutions. Insley’s model (2002) was recently extended by Insley and Rollins (2005) to a multirotation framework, producing critical harvesting prices at diﬀerent age levels as well as evaluating the impact of harvesting age restrictions on critical prices. Thomson (1992) used stochastic prices, like CR, in calculating harvest age under diﬀerent stumpage prices and in comparison with Faustmann steady state results. Klemperer et al. (1994) chose a similar price process, but analysed risk premiums as appropriate for long-term

(1998) used a similar model to Thomson (1992), as well as providing harvest age as a function of stumpage prices, and compared these results with those of the deterministic Faustmann approach. Yoshimoto and Shoji (2002) tested 13 stochastic models to capture the uncertainty of price and growth dynamics. Sødal (2003) suggested a simpliﬁed solution to the optimal rotation problem of Willassen (1998) using the mark-up approach to investment. The aim of this study The purpose of this study is to develop new solutions for (1) optimal harvesting age, (2) sensitivity analyses and (3) comparative statics in analytic form. The ﬁnal goal is (4) implementation of the solutions in the form of programs and their numerical testing using local empirical evidence. The aim of the work is to develop models and programs to be used in forest management planning software products (Hynynen et al. 2005; Salminen et al. 2003; also in Redsven et al. 2005; Hynynen et al. 2002).

Materials and methods Study site This study employs a Vaccinium type (VT) site with a dominant height of 24 m at 100 years of age, H100=24. The stochastic price and growth processes start when the last actual observations were obtained, for example, at 70 years, the lowest harvesting age in the Amendment to the Forest Act (1997). However, in this case the starting age of the processes is at least 55 years, when the last thinning occurs. The growth model applied covers the state of the art as provided by a local software product (Salminen et al. 2003; Hynynen et al. 2005). The soil expectation value deﬁnition Consider the forest owner’s harvesting decision after the last thinning at age sTH,n. At this time, sn>sTH,n, the present value Sn of the forest stand is the value of the ﬁnal harvest at age sn plus the soil expectation value of future rotations both discounted by sn years

fpn ðsn Þqðsn ; dn Þ cnþ1 ðsn ; dnþ1 Þþbnþ1 ðsn ; sTH;nþ1 ; dnþ1 Þ expðrsTH;nþ1 Þ ; Sn ðsn ; dnþ1 ; sTH;nþ1 Þ ¼ ½expðrsn Þ 1 forestry investment. Willassen (1998) applied the theory of stochastic impulse control and derived explicit solutions to multirotational optimal harvesting problems with revenue as the state variable. Yoshimoto and Shoji

ð1Þ

where r is the discount rate. All subsequent rotations n+1, n+2,... repeat the functions for stumpage price pn, the stand volume q, the regeneration costs cn+1 and the thinning beneﬁts bn+1.

337

Note that the ex post regeneration costs cn(0,dn) and the beneﬁts bn(sTH,n,dn) from all historical ex post thinnings have no impact on the harvesting decision; i.e., cn(0,dn)+bn(sTH,n,dn)exp(rsTH,n) is ignored from Eq. 1. Notational burden is avoided by including the beneﬁts bn+1(sn,sTH,n+1,dn+1), which actually stand for several future ex ante thinnings in the ex ante regeneration costs cn+1(sn,dn+1), although they are present. Moreover, in order to single out the impact of volatility, the planting intensity dn+1 is assumed to be determined in advance based on the site properties of the stand.

where the properties of the volatility generating Wiener processes for stumpage price wpt and volume growth wqt volatility have been used (see Øksendal 1985). The average price pa(t) depicts the average value of a cubic metre of wood and the volume qi(t) stands for the interpolated growth, which is based on the volumes obtained every ﬁfth year from the growth and yield models (Salminen et al. 2003; Hynynen et al. 2005). Thus the soil expectation value S(t) for t>02 discounted to age 0 is of the form: SðtÞ ¼

Stumpage price and volume growth processes Clarke and Reed (1989) deﬁned the ‘biological asset’s aggregate intrinsic value‘ as pt*qt, where pt stands for the price process and qt for the volume growth process, and the bold letters indicate random variables. The diﬀerentials of both ln pt and ln qt are proxied by Geometric Brownian motions (GBMs).1 This alternative leads to modiﬁcations of the GBM of price and volume which avoid the ’virtual certainty of (relative) ruin’ (Samuelson 1965, p. 17), in which the probability mass tends to zero. The GBMs imply that the distributions of the stumpage prices pt and the volume growth qt are log-normal (see Klemperer et al. 1994). The price pt has by Ito’s lemma a mean of p0 exp[D(t)+t r2p/2] (Klemperer et al. 1994), where D(t) is the integral D(t)=t0 a(s) ds, a(s) is the stumpage price process drift and rp is the stumpage price process volatility (Reed and Haight 1996). Similarly, the volume growth qt is log-normally distributed with a mean of q0exp[C(t)+tr2q/2], where C(t) is the integral C(t)=t0 g(s)ds, g(s) is the growth process drift and rq is the growth process volatility (CR; Reed and Haight 1996). The soil expectation value of a stand

fvðtÞ cðtÞg ; fexpðrtÞ 1g

ð3Þ

where the expectation v(t) of the felling value of the stand is Eq. 2 above and c(t) consists of the stand establishment costs and thinning beneﬁts of the next rotation. Note that, the roundwood prices for future thinnings may be given as average stumpage prices (Hyytia¨inen and Tahvonen 2001, p. 445). Here, the expectations for both log and pulpwood price processes separately proxy log and pulpwood prices at each future thinning, which depend on the optimal rotation age t only in the form of stumpage prices. The price trend means that there is no unique Faustmannian optimal rotation age for all rotations. Solutions for only the ﬁrst rotation are provided, as is the case using Pressler’s indicating percentage (Pressler 1860). However, the costs and beneﬁts are discounted to the initial t=0 in the context of net present value (NPV) according to Samuelson (1976). Optimal timber harvesting age If the stumpage price drift a(t) is constant a and the volume growth q(t) is deterministic, then the optimal rotation age can be deﬁned by q¢(T)/q(T)=rm/ [1exp(rmT)], where the modiﬁed discount rate rm=rar2p/2 (CR). They ignore growth volatility rq and reforestation costs c(t), which causes some discrepancy. However, their ignoring the value growth pa(t) and Faustmannian inﬁnite rotations impose limitations on the model. If the forest owner is risk-neutral,3 and her/his utility function is linear, then the expected NPV is considered as the objective function. The optimal rotation age T is deﬁned by the solution (Appendix 1):

The felling value of the stand vt also includes the potential gross sales revenue from the ﬁnal felling, vt=pt qt. Ito’s formula for the function vt(pt, qt) = pt qt yields dvt = dpt qt + pt dqt + dpt dqt, where dpt dqt= rprq qpq dt (Øksendal 1985). The expected value v(t)=E{vt} of the stand at the harvesting age t is then (CR; Yin and Newman 1995, 1997) r½1 cðT Þ=vðT Þ 0 2 vðtÞ ¼ p0 q0 exp DðtÞ þ tr2p =2 þ CðtÞ þ tr2q =2 þ tqpq rp rq ; aðT Þ þ gðT Þ þ rR =2 c ðT Þ=vðT Þ ¼ ½1 expðrT Þ ; ð4Þ ¼ p a ðtÞqi ðtÞ exp t a þ r2p =2 þ r2q =2 þ qpq rp rq ; ð2Þ 1

The stochastic diﬀerential equations are D ln pt = a(t) dt + rp wpt and D ln qt = g(t) dt+rq wqt, where wpt and wqt are normally distributed so-called Wiener processes, and have already been used by CR. Note that D stands or the diﬀerential ¶/¶t and ln for the natural logarithm.

2 The denominator in Eq. 3 tends to zero as t ﬁ 0 caused by series expansion of the future rotations. However, the length of future rotations approaching zero is beyond the domain of the model. 3 Actual local cuttings have been on average 3.9 m3/ha per year, and the average forest holding size is roughly 37 ha (Karppinen et al. 2002, p. 29, 47). Thus the annual cutting is of the order of 140 m3, corresponding to some 3,000€/average woodlot. This low average yearly sales speaks for risk neutrality.

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where a(t)=a+b(t), a is the stumpage price drift rate, b(t) the value growth drift, b(t)=p¢(t)/p(t), g(t) the volume growth drift, g(t)=q¢(t)/q(t), and rR2 stands for rR2=r2p+r2q+2qpqrprq. Here, the reforestation costs c(t) after harvesting actually also consist of the thinning beneﬁts of the next generation. While this solution resembles that of Yin and Newman (1995, 1997), even their solution lacks silvicultural costs c(t), value growth pa(t) and inﬁnite rotations. On the numerical solution of the optimal harvesting age The average stumpage price pa(t) at age t is deﬁned in order to cope with the deterministic value growth (Gong 1998): pa(t)=qlog(t) plog+qpulpw(t) ppulpwch, in which plog and ppulpw are log and pulpwood stumpage prices, qlog(t) and qpulpw(t) are the sawn timber and pulpwood yields from harvesting 1 m3 of standing timber at age t, and ch is the harvesting cost per m3.4 The average stumpage price function pa(t) is based on the value growth drift b(t), b(t)=D ln pa(t)=pa¢(t)/pa(t). In all, the stumpage price p0 exp(D(t)), where D(t)=t0 a(s)ds, and a(t)=a+b(t). Similarly, the deterministic volume growth q(t)=q0exp(C(t)), where C(t)=t0 g(s)ds, is based on the volume growth drift g(t), g(t)=D ln q(t)=q¢(t)/ q(t). The volume growth q(t) has been calculated using growth and yield simulation software (Salminen et al. 2003; Hynynen et al. 2005), which incorporates the state-of-the-art growth models (Hynynen et al. 2002). Moreover, the optimality of the management trajectory from 0 to the last thinning relies on the optimising software (Hynynen et al. 2005). Unfortunately, since the models and the simulator provide log, pulpwood and total volumes, etc. only at 5-year intervals (Matala et al. 2003), an interpolation function is needed. Although a three-parameter Richards Curve has been proposed for volume growth by Reed and Haight (1996),5 one of its two-parameter reduction, namely, the logistic growth as in Kuuluvainen and Tahvonen (1999) provides the ﬂexibility and accuracy required for the interpolation of the volumes provided in 5-year steps. However, removals caused by the thinnings violate the ordinary logistic growth model, so that an additional parameter, a negative bottom quantity qb, must be included in the model. In the same way, the logistic and polynomial value growths have been applied as the hypotheses in estimating average stumpage price pa(t) at age t.

4 The ﬁxed logging costs lead to non-convexities (Boscolo and Vincent 2001), but are not applied here. 5 The three-parameter Richards curve is of the form q(t)=K exp(td)/{1+(h1) exp[K(td)]}. It can be reduced to the monomolecular (h=0), and the Gompertz (h ﬁ 1) as well as the logistic volume growth model (h=2) as special cases. For interpolation purposes, it is not important which model alternative is chosen.

Results Empirical estimation of the stumpage price and silvicultural cost parameters All price and cost series were converted to 2004 currency using the consumer price index before the ratios py/py1 were calculated. Here y means year, half-year, month or week. The systematic price statistics consisted of annual observations 1949–2004, semi-annual observations 1981–2004 and monthly observations 1985–2004 as well as weekly observations 1997–2004, all covering the whole of non-industrial private forestry (NIPF). The annual reforestation cost series covering NIPF was available for 1977–2004 for the reforestation cost function estimation. The maximum likelihood estimates of stumpage price drift a and the volatility rate rp of the GBM stumpage price process, pt, will be a* and rp*, where a* is the estimated mean and rp* is the estimated standard deviation of the series ln [py/py1] (Reed and Haight 1996; Insley 2002). According to Statistical Yearbook (2005) the cost for reforestation with nursery seedlings was an average of €750 ha1 and its compound annual growth rate was 1.54%. Statistical properties and parameter estimates of the stumpage price processes The normality of the annual softwood log and pine pulpwood net return series {[py/py1]1} 1949–2004 could be rejected (P0 and even convex by the positiveness of its second derivative of [exp(rt)1]. The numerator, v(t)c(t), is positive within the relevant age interval, i.e. from age 30. Consider the concavity of the numerator; i.e., whether v¢¢(t)c¢¢(t)