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):

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.

P.S:

For negative growth (decrease), use negative g.

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

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)?

Answer:

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.

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