Pricing Commercial Timberland Returns in the United States

Abstract

Commercial timberland assets have attracted more attention in recent decades. One unique feature of this asset class roots in the biological growth, which is independent of traditional financial markets. Using both parametric and nonparametric approaches, we evaluate private- and public-equity timberland investments in the United States. Private-equity timberland returns are proxied by the NCREIF Timberland Index, whereas public-equity timberland returns are proxied by the value-weighted returns on a dynamic portfolio of the US publicly traded forestry firms that had or have been managing timberlands. The results from parametric analysis reveal that private-equity timberland investments outperform the market and have low systematic risk, whereas public-equity timberland investments fare similarly as the market. The nonparametric stochastic discount factor analyses reveal that both private- and public-equity timberland assets have higher excess returns.

Static estimations of the capital asset pricing model and Fama-French three-factor model are obtained by ordinary least squares, whereas dynamic estimations are by state space specifications with the Kalman filter. In estimating the stochastic discount factors, linear programming is used.

Appendices

Appendix 1: State Space Model with the Kalman Filter

The multivariate time-series model can be represented by the following state space form:

$$ {y}_t={Z}_t{\alpha}_t+{\varepsilon}_t,\kern0.5em {\varepsilon}_t\sim NID\left(0,{H}_t\right) $$

(34.10)

$$ {\alpha}_{t+1}={T}_t{\alpha}_t+{R}_t{\eta}_t,\kern1em {\eta}_t\sim NID\left(0,{Q}_t\right) $$

(34.11)

for t = 1, ⋯, N, where y _t is p × 1 vector of observed values at time t, Z _t is a p × m matrix of variables, α _t is m × 1 state vector, T _t is called the transition matrix of order m × m, and R _t is an m × r selection matrix with m ≥ r. The first equation is called the observation or measurement equation, and the second is called state equation. The parameters α _t , H _t , and Q _t in the system of equations can be estimated jointly by the maximum likelihood method with the recursive algorithm Kalman filter. The intention of filtering is to update the information of the system each time a new observation y _t is available, and the filtering equations are

$$ \begin{array}{l}{v}_t={y}_t-{Z}_t{a}_t,\\ {}{F}_t={Z}_t{P}_t{Z}_t^{\prime }+{H}_t,\\ {}{K}_t={T}_t{P}_t{Z}_t^{\prime }{F}_t^{-1},\\ {}{L}_t={T}_t-{K}_t{Z}_t,\\ {}{a}_{t+1}={T}_t{a}_t+{K}_t{v}_t,\\ {}{P}_{t+1}={T}_t{P}_t{L}_t^{\prime }+{R}_t{Q}_t{R}_t^{\prime },\end{array} $$

(34.12)

For t = 1, ⋯, N. The mean vector a ₁ and the variance matrix P ₁ are known for the initial state vector α ₁ (Durbin and Koopman 2001; Harvey 1989).

Appendix 2: Heuristic Proof of Equation 34.6

In a pure exchange economy with identical consumers, a typical consumer wishes to maximize the expected sum of time-separable utilities

$$ \begin{array}{l}\underset{C_t}{\mathrm{Max}}{E}_t\left[{\displaystyle \sum_{i=0}^{\infty }{\beta}^iU\left({C}_{t+i}\right)}\right]\\ {}s.t.{\displaystyle \sum_{j=1}^N{x}_t^j{p}_t^j+{C}_t}={W}_t+{\displaystyle \sum_{j=1}^N{x}_{t-1}^j\left({p}_t^j+{d}_t^j\right)}\end{array} $$

(34.13)

where x ^j _t is the amount of security j purchased at time t, p ^j _t is the price of security j at time t, W _t is the individual’s endowed wealth at time t, C _t is the individual’s consumption at time t, d ^j _t is the dividend paid by security j at time t, and β is time discount. Express C _t in terms of x ^j _t , and differentiate the objective function with respect to x ^j _t , then we can get the following first-order condition:

$$ {E}_t\left[U\prime \left({C}_t\right){p}_t^j\right]={E}_t\left[\beta U\prime \left({C}_{t+1}\right)\left({p}_{t+1}^j+{d}_{t+1}^j\right)\right] $$

(34.14)

for all j. After rearranging the terms, we can reach Eq. 34.6, where

$$ \begin{array}{l}{M}_t=\frac{\beta U\prime \left({C}_{t+1}\right)}{U\prime \left({C}_t\right)}\\ {}{R}_{t+1}=\frac{p_{t+1}^j+{d}_{t+1}^j}{p_t^j}-1.\end{array} $$

(34.15)

Appendix 3: NCREIF Timberland Index

The NCREIF Timberland Index has two components, the income return and the capital return. The income return is also known as EBITDDA return, which represents earnings before interest expenses, income taxes, depreciation, depletion, and amortization. The capital return is derived from land appreciation. The formulas to calculate these returns are

$$ I{R}_t=\frac{ EBITDD{A}_t}{M{V}_{t-1}+0.5\left(C{I}_t-P{S}_t+P{P}_{\mathrm{t}}- EBITDD{A}_t\right)} $$

(34.16)

$$ C{R}_t=\frac{M{V}_t-M{V}_{t-1}-C{I}_t+P{S}_t-P{P}_t}{M{V}_{t-1}+0.5\left(C{I}_t-P{S}_t+P{P}_t- EBITDD{A}_t\right)} $$

(34.17)

where IR _t and CR _t are the income return and capital return, respectively; EBITDDA _t equals the net operating revenue obtained from the tree farm (primarily from timber sales); CI _t equals the capitalized expenditures on the tree farm (e.g., forest regeneration and road construction); PS _t equals the net proceeds from sales of land from the tree farm; PP _t equals the gross costs of adding land to the tree farm; and MV _t equals the market value of the tree farm (Binkley et al. 2003).

Appendix 4: EViews Code for Estimating the CAPM and the Fama-French Three-Factor Model

Texts after the single quotation marks are notations.

' Specify the file location

CD "C:\Mei Bin\Publication\Handbook of FES"

' Create a workfile in EViews and read in quarterly data 1987Q1-2010Q4

Workfile Timberland_Jul2011 q 1987 2010

' 7 is the total number of series to be read in

Read(t = dat, s) quarterly.csv 7

' Group the 7 series

group quarterly NCREIF MktRf SMB HML RF port

' Estimate the CAPM for NCREIF, adjusted for autocorrelation

' LS means OLS. Equation given by dependent variable followed by a list of independent variables

' In excess returns

equation CAPM1.LS (NCREIF-RF) C MktRf AR(4)

' Estimate the Fama-French three-factor model for the NCREIF Timberland Index

equation FF31.LS (NCREIF-RF) C MktRf SMB HML AR(4)

' Estimate the CAPM for the portfolio, AR(4) term dropped due to its insignificance.

equation CAPM2.LS (port-RF) C MktRf

' Estimate the Fama-French three-factor model for the portfolio

equation FF32.LS (port-RF) C MktRf SMB HML

' State space estimation of time-varying parameters for the CAPM

' Define a state space model for the NCREIF Timberland Index

sspace KFcapm1

' Signal equation, the CAPM

' Define alpha to be time-varying (state variable)

KFcapm1.append @signal (NCREIF-RF) = sv1 + c(1)*MktRf + [var = exp(c(2))]

' State equation as a random walk

KFcapm1.append @state sv1 = sv1(-1) + [var = exp(c(3))]

' Starting values for the state space model. Values come the OLS estimation

KFcapm1.append @param c(1) 0.01 c(2) 0 c(3) 0

' Maximum likelihood estimation

KFcapm1.ml(showopts, m = 500, c = 0.0001, m)

Show KFcapm1.output

' Save the time-varying alphas and its RMSEs

KFcapm1.makestates(t = filt) CAPM1filt*

KFcapm1.makestates(t = filtse) CAPM1filtse*

' Generate the graph of time-varying alphas with the 95 % confidence intervals

series CAPM1_a_bandplus = CAPM1filtsv1 + 2*CAPM1filtsesv1

series CAPM1_a_bandminus = CAPM1filtsv1 - 2*CAPM1filtsesv1

' Group the series to be shown in a graph

Group CAPM1_a_curves CAPM1filtsv1 CAPM1_a_bandplus CAPM1_a_bandminus

' State space model for the portfolio of public forest products firms

sspace KFcapm2

' Define beta to be time-varying (state variable)

KFcapm2.append @signal (port-RF) = c(1) + sv1*MktRf + [var = exp(c(2))]

KFcapm2.append @state sv1 = sv1(-1) + [var = exp(c(3))]

KFcapm2.append @param c(1) 0.59 c(2) 4.3 c(3) -30

KFcapm2.ml(showopts, m = 500, c = 0.0001, m)

Show KFcapm2.output

KFcapm2.makestates(t = filt) CAPM2filt*

KFcapm2.makestates(t = filtse) CAPM2filtse*

series CAPM2_b_bandplus = CAPM2filtsv1 + 2*CAPM2filtsesv1

series CAPM2_b_bandminus = CAPM2filtsv1 - 2*CAPM2filtsesv1

Group CAPM2_b_curves CAPM2filtsv1 CAPM2_b_bandplus CAPM2_b_bandminus

sspace KFff1

' Time-varying alpha in the Fama-French three-factor model

KFff1.append @signal (NCREIF-RF) = sv1 + c(1)*MktRf + c(2)*SMB + c(3)*HML + [var = exp(c(4))]

KFff1.append @state sv1 = sv1(-1) + [var = exp(c(5))]

KFff1.append @param c(1) 0.02 c(2) -0.04 c(3) -0.05 c(4) 0 c(5) 0

KFff1.ml(showopts, m = 500, c = 0.0001, m)

Show KFff1.output

sspace KFff2

' Time-varying beta in the Fama-French three-factor model

KFff2.append @signal (port-RF) = c(1) + sv1*MktRf + c(2)*SMB + c(3)*HML + [var = exp(c(4))]

KFff2.append @state sv1 = sv1(-1) + [var = exp(c(5))]

KFff2.append @param c(1) 0 c(2) 0 c(3) 3 c(4) 3 c(5) 0

KFff2.ml(showopts, m = 500, c = 0.0001, m)

Show KFff2.output

' Save the workfile

wfsave Timberland_Jul2011

Appendix 5: Steps for the SDF Approach Using Excel Solver

First, choose minimizing the standard deviation of the SDFs as the objective function.

Second, set the mean of the SDFs equal to a predetermined value. This is constraint No.1.

Third, for each basis asset (industry group) in the industry portfolio, add one constraint according to $\frac{1}{T}{\displaystyle \sum_{t=1}^T{M}_t}\left(1+{R}_{i,t}\right)=1$. That is, add five more constraints when using the five-industry portfolio, whereas add ten more constraints when using the ten-industry portfolio. Fourth, specify the solutions to be nonnegative and solve for the SDFs. Fifth, use the SDFs to price timberland returns according to Eq. 34.9. Repeat steps 1–5 with a different value as the given mean of the SDFs.