Tuesday, December 31, 2013

Value with a constant growth factor

Many times we hear in the news, either financial news or economy in general, about growths with constant factor.  For example: "The appreciated value of residential property in townville grows at 4% per year', or "The stock price grows constantly at 5% per year", or "the population of villageville decreases by constant factor of 5%".

What does it mean?

Well, it is actually simple.  The value increases by factor of 4%.  If the current value is A0, next year its value is A0 + 4%*A0.  Next 2 years the value is A1 + 4%*A1 = A0*(1+4%)^2 and so on.  From this, we can deduct a general formula for a growth (which is a form of geometric series):

A(t) = A(0) * (1+g/100)^t

Where A(0) is the value at the initial evaluation (t = 0)
t = unit time for the growth
g = percentage of growth (in %), so we need to divide it by 100 there
A(t) = the value at t

From this formula, we also can find "Doubling Time", or the time needed for a value to be double.

Tdouble = ln(2) / ln( 1+g/100)

For example:
Michael bought his house in 2002 for \$315,000.  The average appreciation rate of properties in his area is 2%/year.  How long he has to keep his house to make the house value double (assuming the appreciation rate stays the same)?

Tdouble = ln(2) / ln( 1 + 2/100) = 0.693/0.0198 = 35 years.

P.S:
For negative growth (decrease), use negative g.

More complex calculation is if the growth factor is not constant.