the area on the graph, which, if all 500 simulated time paths were drawn, would encompass 476
paths.
4
When the depreciation approach is used, the path of net income displayed is highly uncertain.
Fig. 1 reveals that the confidence interval for the 500 draw values for a given year is extremely large
over 1 billion in some years. This variability occurs because large fires periodically cause the
timber stock to decline to near zero. Since the depreciation approach adjusts current income
flows by the value of the change in stock for that year, the variability in stock levels is reflected
directly in the net income measure.
The path of mean or expected net income that results when the wealth-approach is used is much
smoother. The confidence intervals increase over time but they do not fluctuate greatly. In addi-
tion, the path of net income is higher than the one that results from the depreciation approach.
Visual representations of the simulated net in- come values are also provided for selected years.
Year 10 was selected to illustrate the distribution of the 500 draws for a given year. The distribu-
tions for other years are very similar. Fig. 2 shows a histogram and cumulative frequency curve for
the net income values for year 10, for both the depreciation and the wealth approach. For the
depreciation approach, the first interval in the chart B 90 million includes some very small
values, since the net income predictions for a given year include values as low as about 1
billion. However, most values are grouped tightly around the upper end of the distribution in the
220 – 230 million interval. The corresponding chart for the wealth approach shows a distribu-
tion with a very similar shape but much smaller range. Most values lie between 180 and 190
million. There are fewer values at the extreme left of the distribution and the absolute minimum is
about 10 million.
8. Price risk simulations
As with the fire risk simulations, the simulation of price risk requires the distribution of future
prices to be explicitly defined. In this case, the distributions of pulp and lumber prices were esti-
mated using historical time series data. Lumber prices for western spruce – pine – fir f.o.b. mill net
or factory gate were obtained from Natural Re- sources Canada 1996 and Warren 1997 which
report annual average prices. Northern bleached hardwood and softwood pulp prices were ob-
tained from the Pulp and Paper North American Fact Book 1996.
5
Price movements over time can be modeled in many different ways. Here the characterization of
future price movements is relatively simple. The following regression equations were estimated for
each price series:
Autoregressive 1 AR1: P
t
= a
1
P
t − 1
+ g + e
t
5 Autoregressive 2 AR2:
P
t
= a
1
P
t − 1
+ a
2
P
t − 2
+ g + e
t
6 Logged Autoregressive 1 lnAR1:
ln P
t
= a
1
ln P
t − 1
+ g + e
t
7 Logged Autoregressive 2 lnAR2:
ln P
t
= a
1
ln P
t − 1
+ a
2
ln P
t − 2
+ g + e
t
8 The above equations were estimated using his-
torical annual average prices that were converted to 1996 Canadian dollars. A combination of fac-
tors including overall model significance P- value, coefficient significance, examination of
residuals was used to select the equations that would be used in the simulations. For hardwood
and softwood pulp the AR2 relationship appeared to explain price movements best, whereas for lum-
ber, lnAR1 was found to provide the best fit. A time variable was added to these specifications but
was not found to be significant.
5
Since, f.o.b. mill net pulp prices were not available, prices for pulp delivered to the US are used. This is accounted for in
the estimate of average variable cost AVC.
4
We remove 12 paths from the upper and lower ends of the distribution. Therefore, this actually results in a 95.2 confi-
dence interval.
Fig. 2. Distribution of net income values for year 10.
Table 3 Equations for future price movement
a
Equation type Coefficient
Number of years Intercept
Price series Standard error
g n
s
e
a
1
a
2
AR2 0.75598
HBKP −
0.54057 20
652.56185 151.87626
SBKP 20
AR2 0.86759
− 0.62316
699.86508 141.20234
lnAR1 NA
0.70015 1.75496
Lumber 0.201
22
a
HBKP, hardwood bleached kraft pulp; SBKP, softwood bleached kraft pulp.
The parameters of the regressions used to pre- dict future prices are listed in Table 3.
Note the relatively high standard error values for all three regressions. Even the selected models
leave much of the variation in prices unexplained. As noted above, the method for predicting future
prices is kept relatively simple in this analysis, however, a variety of more elaborate methods
might be applied. For example, the pulp and lumber market are sensitive to a number of fac-
tors including the interest rate, exchange rates and trade policy i.e. softwood lumber tax. Predic-
tions of the future path of these variables might also be used to forecast future pulp and lumber
prices. Similarly, the futures markets for pulp and lumber could be used to assist in predicting future
prices.
In order to isolate the influence of future price, the annual deciduous and coniferous harvests are
kept at a constant percentage of the stock equal to that which applied in 1996. This means that
opening and closing stocks and the equations that determine stock volumes over time in the fire
simulations are not required. The annual harvest level and change in the stock remain at the same
volumes as in 1996. The average variable cost estimates in Haener and Adamowicz 1999 is
used with the projected prices to calculate pulp and lumber rent. As with the fire simulations, 500
draws of 20 years each are made from the error term distributions.
9. Results
