Present Value of a Perpetual Periodic Series
A.K.A.
The Soil Expectation Value
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Calculate it here!

Present Value of a Perpetual Periodic Series
where:
a = Periodic Payment amount
t = Interval in years between payments 
   (like the interval between timber harvests)
i = compound interest rate
V0 = Value at year 0
a (Periodic Payment) $
interest rate (0.00 format) 
t (interval between payments) 

V0 (Present Value) $


ForestTax.com Newsletter

The famous, often used, wonderful...
Soil Expectation Value
(also known as the Present Value of a Perpetual Periodic Series)

This one is very useful formulae because as we will see, it helps us define the value of bare land used for forestry; the Soil Expectation Value. You see, many forest properties yield income periodically at the end of a cutting cycle or at the end of a rotation, rather than annually (like agricultural land). It is often desirable to find the present value of a perpetual periodic series to determine the potential value of this land for forestry use. In fact, this formulae will help you to determine the value of bare land with no trees on it. It tells us the value of bare land with no higher and better use than to grow trees.

The present value of this series can be found by first assuming the following:

a = amount of periodic payment
t = interval between periodic payments
Then, the year "t" payment value in year 0 = 
The year 2t payment value in year 0 = 
The year 3t payment value in year 0 = 
The year infinity payment value in year 0 = 
Now we can summarize these payments by adding them together
 
V0+ + + … 
In this case we can see that each of the expressions in the series possess (1 + i)t in the denominator. This is a geometric series, like many we have seen before. Remember that the general formulae for a geometric series is:
So if we substitute
S = V0
b = a / (1+i)t
r = 1 / (1+i)t
Then the following will be true


Think about the substitution for (1-r)m and how that portion of the equation will react as m approaches infinity.

This portion will go to zero (try it on your calculator for "i = 9%" and "t = 90 years") put infinity at something like 100 and see how close to zero you are.

So if we multiply the numerator and denominator by (1 + i)t. we obtain a modified calculation.


Present Value of a Perpetual Periodic Series
Example (a)

At 9% interest, what is the present value of bare land for growing successive crops of Hybrid Poplar from a tract that yields $30,000 of net revenue every 10 years?
 

Therefore, we can interpret this number by saying that this land is worth $21,940 for the purpose of growing hybrid poplar. Is that the value of the land? Maybe. It is if there is no higher and more profitable use of the land. If the land has more profitable uses then that use will be the determining factor in its bare land value.

 
Example (b)

How would the result in ( a ) have been different if the stand was mature and ready for its first harvest tomorrow?

Keep the solution we calculated above and add in the current harvest value of $30,000 for a total asset value of $51,940.
 


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Kamiak Econometrics, a Division of Kamiak Ridge, LLC