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Tree Values

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FINA 461 Section 1 |
“Tree Values” Case Report
|
Huong Nguyen |

10/7/2010
|

Section 1 – Executive Summary
This report analyzes the case “Tree Values” to find an optimal way for Mr. Smith to manage his forestland and harvest the crop trees. The fundamental idea in this report is based on the concept of present value. A number of options are analyzed and the one with highest present value of pay off is considered.
Questions 9, 10, and 11 give Mr. Smith 3 options: * Option 1: Harvest all crop trees now and receive $8,160 * Option 2: Let the forest grow without thinning, then harvest all crop trees 60 years from now and receive $537,962.01 at harvesting, equivalent to $28,800.08 now * Option 3: Thin and manage the forest, then harvest all crop trees 50 years from now and receive $670,033.56 at harvesting, equivalent to $58,429.42 now
Based on the present value of the money received at harvesting, it is highly recommended that Mr. Smith should choose option 3. Furthermore, provided Mr. Smith decides to thin and manage his forest, in case he needs money soon to use for other purposes, he can harvest all of his crop trees at the 40th year to receive $410,608.68 at harvesting, equivalent to $58,325.19 at present, instead of waiting for 10 more years. This is because the present value of money received at the 40th year is just a little ($104.23) less than that of money received at the 50th year.

Section 2 – Analysis
Question 1
In order to choose the best offer, the amount of payment in each offer is discounted to present at the discount rate of 10%. This is just a basic discounting problem, using the formula
PV=FV(1+r)t
In this formula, FV is the amount of payment, r is the discount rate of 10%, t is the number of years from now in each offer. Table 1 in the appendix summarizes the present value of payment in each offer. It also shows that offer 8 has the highest payment’s present value of $140.61. Therefore, offer 8, which pays $274 seven years from now, will be chosen.
Question 2
The price of a tree at increments of 10 years between now and 60 years from now will be calculated first. This is done by using the following formula:
Price per Tree=$MBF1000×Board Feet per Tree
A tree has certain diameter at breast height (DBH) at certain time (e.g., 10” now, 11” at 10th year, 12” at 20th year, etc.). With different DBH, the tree would yield different board feet of lumber, which are found in Exhibit 1 of the case. Because there is no change in tree grade, the tree will be in grade 4 at any time during the period of 60 years. Therefore, the stumpage price (with no increase) is always equal to $40 per thousand board feet (MBF) as shown in Exhibit 2 of the case.
After the prices are calculated, they are discounted at the discount rate of 5% to present. Table 2 shows the price of the tree at 0th, 10th, 20th, 30th, 40th, 50th, and 60th years and the corresponding present value. The optimal time to harvest the tree will be chosen based on the year in which the tree has highest present value of its price. Therefore, (1) The optimal time to harvest the tree is 10 years from now (2) The tree would be sold for $1.60 at the 10th year, i.e., the present value of its price is $0.98.
Question 3
The price of a tree at increments of 10 years between now and 60 years from now will be calculated using the same formula as in question 2. However, because the stumpage price ($MBF) increases at the rate of 2% each year, it is calculated using the formula
$MBF=$40×(1+0.02)t, where t is the number of years from now.
As shown in Table 3, at the 30th year, the price of the tree has the highest present value of $1.42. Therefore, (1) The optimal time to harvest the tree is 30 years from now (2) The tree would be sold for $6.16 at the 30th year.
Question 4
This question is similar to question 2, except the tree grows 2” in DBH every 10 years. The DBH and corresponding board feet of the tree will be different at increments of 10 years during the 60-year period. This difference is shown in Table 4.
Using the same formula as in question 2, the price of the tree has highest present value of $1.67 at the 30th year. Therefore, (1) The optimal time to harvest the tree is 30 years from now (2) The tree would be sold for $7.20 at the 30th year.
Question 5
The DBH and board feet of the tree remain the same as in question 2. Only the tree grade changes at the 20th (from 4 to 3), 40th (from 3 to 2), and 60th (from 2 to 1) years. These changes result in different stumpage prices as shown in Table 5.
Using the same formula as in question 2, the price of the tree has highest present value of $4.29 at the 60th year. Therefore, (1) The optimal time to harvest the tree is 60 years from now (2) The tree would be sold for $80.10 at the 60th year.
Question 6
As indicated in the case, it takes approximately 10 years for a tree to grow 2" in diameter on a managed site and 20 years (half the rate on managed site) on an unmanaged site.
On an unmanaged forestland, a 10” DBH tree now will grow 1” in diameter every 10 years and its grade will change every 20 years. At present, 100% of the trees are in Grade 4 and 0% for other grades. This proportion remains unchanged 10 years from now. The proportion of five tree grades starts changing at the 20th year. The tree grade change is calculated based on the probabilities given in Exhibit 3 of the case and the following formulas:
Grade 420 = Grade 410 x (1 – 60%)
Grade 320 = Grade 410 x 60% + Grade 310 x (1 – 50%)
Grade 220 = Grade 310 x 50% + Grade 210 x (1 – 40%)
Grade 120 = Grade 210 x 40% + Grade 110 x (1 – 10%)
Veneer20 = Grade 110 x 10% + Veneer10
This gives proportion of 40% Grade 4, 60% Grade 3, and 0% for other grades at the 20th year. The proportion remains the same at the 30th year as at the 20th year. It will change again at the 40th year and is calculated in similar way using the formulas above, based on the proportion at the 30th year. Continuing this process for the next 40 years results in Table 6, which shows the proportion of five grades at each increment of 10 years in the period of 80 years.
On a managed forestland, a 10” DBH tree now will grow 2” in diameter every 10 years and its grade will change every 10 years. At present, 100% of the trees are in Grade 4 and 0% for other grades. The proportion of Grade 4, Grade 3, Grade 2, Grade 1, and Veneer trees at the nth year will be calculated based on the proportion at the (n-10)th year using the probabilities in Exhibit 3 of the case and the following formulas:
Grade 4n = Grade 4n-10 x (1 – 80%)
Grade 3n = Grade 4n-10 x 80% + Grade 3n-10 x (1 – 70%)
Grade 2n = Grade 3n-10 x 70% + Grade 2n-10 x (1 – 60%)
Grade 1n = Grade 2n-10 x 60% + Grade 1n-10 x (1 – 20%)
Veneern = Grade 1n-10 x 20% + Veneern-10
The results are shown in Table 7 in the appendix.
Questions 7-9
As the case indicates, there are a total of 40 acres in Mr. Smith’s forestland and about 60 crop trees per acre. This gives approximately 2,400 (40x60) crop trees in Mr. Smith’s forestland. In addition, because the crop trees are about evenly divided between 12” DBH and 14” DBH trees, there are approximately 1,200 (2,400/2) trees in each of two types of crop trees.
Because all of the crop trees are in Grade 4 now, their stumpage price is $40 per thousand board feet (Exhibit 2 of the case). From Exhibit 1 of the case, 12” DBH crop trees have 60 board feet per tree and 14” DBH crop trees have 110 board feet per tree. Therefore, the money received from harvesting each of two types of crop trees is:
12” DBH crop trees: $40/1000×60×1,200=$2,880
14” DBH crop trees: $40/1000×110×1,200=$5,280
So if Mr. Smith harvests all of his crop trees immediately, he will receive $8,160 ($2,880+$5,280).
Question 10
First, the price of 1,200 crop trees with 12” in diameter is calculated at each increment of 10 years during the period of 80 years as follows. At each 10-year increment, the number of trees in each tree grade is calculated using the formulas in question 6 for unmanaged forestland. The results are shown in Table 8. For each tree grade, the price of all the trees is calculated using the following formula:
Price=Number of Trees×$MBF×(1+0.02)t1000×Board Feet per Tree where $MBF is the stumpage price and t is the years from now. Then the price of trees in all tree grades are added up to get the price of 1,200 crop trees with 12” DBH at each 10-year increment.
Second, the price of 1,200 crop trees with 14” in diameter at each increment of 10 years is calculated in the same way as for the 12” DBH trees above. Then, at each 10-year increment, the price of 14” DBH trees is added to that of 12” DBH trees to find the total price of 2,400 crop trees. This total price is discounted at the discount rate of 5% to find its present value. The results are shown in Table 9.
The year in which the total price has the highest present value will be optimal to harvest. Therefore, if Mr. Smith lets his forest grow without thinning, he should harvest the crop trees at the 60th year to receive $537,962.01 at harvesting, or equivalently $28,800.08 at present.
Question 11
All the steps in this question are similar to those in question 10. However, there are some differences in the calculation. First, there are only 600 12” DBH trees and 600 14” DBH trees at present because half of the crop trees are cut in thinning process. Second, the number of trees in each tree grade at each 10-year increment is calculated based on the probabilities in Exhibit 3 of the case for trees on thinned, managed forestland. Third, the growth in DBH every 10 years is different: 2” instead of 1”. This difference leads to changes in DBH of the trees at each 10-year increment, hence different board feet per tree and price.
Table 10 shows the price of 1,200 crop trees with 12” DBH during the 80-year period. Table 11 shows the price of 1,200 crop trees with 14” DBH, the total price of 2,400 crop trees and its present value during the 80-year period. Based on this information, optimal time to harvest can be chosen.
If Mr. Smith decides to thin and manage his forest, 50 years from now will be optimal to harvest. He should expect to receive $670,033.56 at harvesting, or equivalently $58,429.42 at present.
Appendix
Table 1
Offer | Payment
($) | Years from now | Present
Value ($) | 1 | 100 | 0 | 100.00 |
2 | 120 | 1 | 109.09 |
3 | 142 | 2 | 117.36 |
4 | 166 | 3 | 124.72 |
5 | 192 | 4 | 131.14 |
6 | 219 | 5 | 135.98 |
7 | 246 | 6 | 138.86 |
8 | 274 | 7 | 140.61 |
9 | 300 | 8 | 139.95 |

Table 2
Years
from now | DBH
(inches) | Tree
Grade | Board
Feet/Tree | $MBF | Price/Tree($) | Present
Value ($) |
0 | 10 | 4 | 20 | 40.00 | 0.80 | 0.80 |
10 | 11 | 4 | 40 | 40.00 | 1.60 | 0.98 |
20 | 12 | 4 | 60 | 40.00 | 2.40 | 0.90 |
30 | 13 | 4 | 85 | 40.00 | 3.40 | 0.79 |
40 | 14 | 4 | 110 | 40.00 | 4.40 | 0.63 |
50 | 15 | 4 | 145 | 40.00 | 5.80 | 0.51 |
60 | 16 | 4 | 180 | 40.00 | 7.20 | 0.39 |

Years from now | DBH
(inches) | Tree
Grade | Board
Feet/Tree | $MBF | Price/Tree($) | Present
Value ($) |
0 | 10 | 4 | 20 | 40.00 | 0.80 | 0.80 |
10 | 11 | 4 | 40 | 48.76 | 1.95 | 1.20 |
20 | 12 | 4 | 60 | 59.44 | 3.57 | 1.34 |
30 | 13 | 4 | 85 | 72.45 | 6.16 | 1.42 |
40 | 14 | 4 | 110 | 88.32 | 9.72 | 1.38 |
50 | 15 | 4 | 145 | 107.66 | 15.61 | 1.36 |
60 | 16 | 4 | 180 | 131.24 | 23.62 | 1.26 |
Table 3

Years from now | DBH
(inches) | Tree
Grade | Board
Feet/Tree | $MBF | Price/Tree($) | Present
Value (S) |
0 | 10 | 4 | 20 | 40.00 | 0.80 | 0.80 |
10 | 12 | 4 | 60 | 40.00 | 2.40 | 1.47 |
20 | 14 | 4 | 110 | 40.00 | 4.40 | 1.66 |
30 | 16 | 4 | 180 | 40.00 | 7.20 | 1.67 |
40 | 18 | 4 | 280 | 40.00 | 11.20 | 1.59 |
50 | 20 | 4 | 350 | 40.00 | 14.00 | 1.22 |
60 | 22 | 4 | 430 | 40.00 | 17.20 | 0.92 |
Table 4

Years from now | DBH
(inches) | Tree
Grade | Board
Feet/Tree | $MBF | Price/Tree($) | Present
Value (S) |
0 | 10 | 4 | 20 | 40.00 | 0.80 | 0.80 |
10 | 11 | 4 | 40 | 40.00 | 1.60 | 0.98 |
20 | 12 | 3 | 60 | 120.00 | 7.20 | 2.71 |
30 | 13 | 3 | 85 | 120.00 | 10.20 | 2.36 |
40 | 14 | 2 | 110 | 260.00 | 28.60 | 4.06 |
50 | 15 | 2 | 145 | 260.00 | 37.70 | 3.29 |
60 | 16 | 1 | 180 | 445.00 | 80.10 | 4.29 |
Table 5

Table 6
Years
from now | DBH
(inches) | Grade 4 | Grade 3 | Grade 2 | Grade 1 | Veneer |
0 | 10 | 100.00% | 0.00% | 0.00% | 0.00% | 0.00% |
10 | 11 | 100.00% | 0.00% | 0.00% | 0.00% | 0.00% |
20 | 12 | 40.00% | 60.00% | 0.00% | 0.00% | 0.00% |
30 | 13 | 40.00% | 60.00% | 0.00% | 0.00% | 0.00% |
40 | 14 | 16.00% | 54.00% | 30.00% | 0.00% | 0.00% |
50 | 15 | 16.00% | 54.00% | 30.00% | 0.00% | 0.00% |
60 | 16 | 6.40% | 36.60% | 45.00% | 12.00% | 0.00% |
70 | 17 | 6.40% | 36.60% | 45.00% | 12.00% | 0.00% |
80 | 18 | 2.56% | 22.14% | 45.30% | 28.80% | 1.20% |

Table 7
Years
from now | DBH
(inches) | Grade 4 | Grade 3 | Grade 2 | Grade 1 | Veneer |
0 | 10 | 100.0000% | 0.0000% | 0.0000% | 0.0000% | 0.0000% |
10 | 12 | 20.0000% | 80.0000% | 0.0000% | 0.0000% | 0.0000% |
20 | 14 | 4.0000% | 40.0000% | 56.0000% | 0.0000% | 0.0000% |
30 | 16 | 0.8000% | 15.2000% | 50.4000% | 33.6000% | 0.0000% |
40 | 18 | 0.1600% | 5.2000% | 30.8000% | 57.1200% | 6.7200% |
50 | 20 | 0.0320% | 1.6880% | 15.9600% | 64.1760% | 18.1440% |
60 | 22 | 0.0064% | 0.5320% | 7.5656% | 60.9168% | 30.9792% |
70 | 24 | 0.0013% | 0.1647% | 3.3986% | 53.2728% | 43.1626% |
80 | 26 | 0.0003% | 0.0504% | 1.4748% | 44.6574% | 53.8171% |

Table 8
Years | DBH | Number of Trees in Grade | Board | | Present | from now | (inches) | 4 | 3 | 2 | 1 | Veneer | Feet/Tree | Price | Value |
0 | 12 | 1,200 | 0 | 0 | 0 | 0 | 60 | $2,880.00 | $2,880.00 |
10 | 13 | 1,200 | 0 | 0 | 0 | 0 | 85 | $4,973.50 | $3,053.30 |
20 | 14 | 480 | 720 | 0 | 0 | 0 | 110 | $17,260.76 | $6,505.40 |
30 | 15 | 480 | 720 | 0 | 0 | 0 | 145 | $27,735.57 | $6,417.39 |
40 | 16 | 192 | 648 | 360 | 0 | 0 | 180 | $71,158.94 | $10,107.82 |
50 | 17 | 192 | 648 | 360 | 0 | 0 | 230 | $110,837.44 | $9,665.44 |
60 | 18 | 77 | 439 | 540 | 144 | 0 | 280 | $239,094.23 | $12,800.03 |
70 | 19 | 77 | 439 | 540 | 144 | 0 | 315 | $327,886.34 | $10,776.37 |
80 | 20 | 31 | 266 | 544 | 346 | 14 | 350 | $580,870.74 | $11,720.21 |

Table 9
Years | DBH | Number of Trees in Grade | Board | | Present | Total | Total | from now | (inches) | 4 | 3 | 2 | 1 | Veneer | Feet/Tree | Price | Value | Price | Present Value |
0 | 14 | 1,200 | 0 | 0 | 0 | 0 | 110 | $5,280.00 | $5,280.00 | $8,160.00 | $8,160.00 |
10 | 15 | 1,200 | 0 | 0 | 0 | 0 | 145 | $8,484.20 | $5,208.56 | $13,457.70 | $8,261.86 |
20 | 16 | 480 | 720 | 0 | 0 | 0 | 180 | $28,244.89 | $10,645.20 | $45,505.65 | $17,150.60 |
30 | 17 | 480 | 720 | 0 | 0 | 0 | 230 | $43,994.35 | $10,179.30 | $71,729.92 | $16,596.69 |
40 | 18 | 192 | 648 | 360 | 0 | 0 | 280 | $110,691.68 | $15,723.27 | $181,850.61 | $25,831.09 |
50 | 19 | 192 | 648 | 360 | 0 | 0 | 315 | $151,799.11 | $13,237.45 | $262,636.55 | $22,902.89 |
60 | 20 | 77 | 439 | 540 | 144 | 0 | 350 | $298,867.78 | $16,000.04 | $537,962.01 | $28,800.08 |
70 | 21 | 77 | 439 | 540 | 144 | 0 | 385 | $400,749.97 | $13,171.12 | $728,636.32 | $23,947.48 |
80 | 22 | 31 | 266 | 544 | 346 | 14 | 430 | $713,641.20 | $14,399.12 | $1,294,511.94 | $26,119.34 |

Table 10
Years | DBH | Number of Trees in Grade | Board | | Present | from now | (inches) | 4 | 3 | 2 | 1 | Veneer | Feet/Tree | Price | Value |
0 | 12 | 600 | 0 | 0 | 0 | 0 | 60 | $1,440.00 | $1,440.00 |
10 | 14 | 120 | 480 | 0 | 0 | 0 | 110 | $8,367.18 | $5,136.72 |
20 | 16 | 24 | 240 | 336 | 0 | 0 | 180 | $31,326.15 | $11,806.50 |
30 | 18 | 5 | 91 | 302 | 202 | 0 | 280 | $91,024.83 | $21,061.09 |
40 | 20 | 1 | 31 | 185 | 343 | 40 | 350 | $184,247.49 | $26,171.56 |
50 | 22 | 0 | 10 | 96 | 385 | 109 | 430 | $335,016.78 | $29,214.71 |
60 | 24 | 0 | 3 | 45 | 366 | 186 | 430 | $468,257.90 | $25,068.43 |
70 | 26 | 0 | 1 | 20 | 320 | 259 | 430 | $630,298.92 | $20,715.51 |
80 | 28 | 0 | 0 | 9 | 268 | 323 | 430 | $826,888.15 | $16,684.10 |

Table 11
Years | DBH | Number of Trees in Grade | Board | | Present | Total | Total | from now | (inches) | 4 | 3 | 2 | 1 | Veneer | Feet/Tree | Price | Value | Price | Present Value |
0 | 14 | 600 | 0 | 0 | 0 | 0 | 110 | $2,640.00 | $2,640.00 | $4,080.00 | $4,080.00 |
10 | 16 | 120 | 480 | 0 | 0 | 0 | 180 | $13,691.75 | $8,405.54 | $22,058.92 | $13,542.27 |
20 | 18 | 24 | 240 | 336 | 0 | 0 | 280 | $48,729.56 | $18,365.66 | $80,055.71 | $30,172.16 |
30 | 20 | 5 | 91 | 302 | 202 | 0 | 350 | $113,781.04 | $26,326.37 | $204,805.87 | $47,387.46 |
40 | 22 | 1 | 31 | 185 | 343 | 40 | 430 | $226,361.20 | $32,153.63 | $410,608.68 | $58,325.19 |
50 | 24 | 0 | 10 | 96 | 385 | 109 | 430 | $335,016.78 | $29,214.71 | $670,033.56 | $58,429.42 |
60 | 26 | 0 | 3 | 45 | 366 | 186 | 430 | $468,257.90 | $25,068.43 | $936,515.80 | $50,136.86 |
70 | 28 | 0 | 1 | 20 | 320 | 259 | 430 | $630,298.92 | $20,715.51 | $1,260,597.85 | $41,431.02 |
80 | 30 | 0 | 0 | 9 | 268 | 323 | 430 | $826,888.15 | $16,684.10 | $1,653,776.30 | $33,368.20 |

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