Calculate annualized rate of return?

Question

I recently read that someone gained 3017% over 47 years (see: http://www.fool.com/investing/dividends-income/2007/01/09/how-to-earn-a-3017-return.aspx?source=iflfollnk0000003 )

I then went to figure out what the annualized rate, but can't remember the exact formula. I believe the problem would be written as x^47 = 301.7 to get the annual percent return. I was thinking the solution was log(47) / log(302) but when I did the math I got an obviously wrong result. I remember figuring out a really simple way when you have a calculator with the log function. Maybe it was the LN function because it sure isn't working how I am trying it.

My second guess was Log(10) / Log (302) * 47. That didn't work either. I remember having to do something to compensate for the fact that the windows calculator won't let you put in any base so you have to use base 10.

Answer 1

Oops. 3017% is 30.17 times, not 301.7 times.

x^47 = 30.17
47 log x = log 30.17
log x = (log 30.17)/47
x = 10^[(log 30.17)/47]
= 10^(1.4796/47)
= 10^0.03148
= 1.0752 [use "inv log" on windows calculator]
That is 7.52 percent per year compounded annually

But upon reading the Fool reference, this is not
what you are trying to calculate at all.

The claim is $10,500/year in current dividend income
on an investment of $350 in 1960.
Or 10 shares at $35 turning into 9100 shares today at $26.64
That's $350 to $242,000, a factor of 691 in 47 years,
which calculates to 15 percent annual return.

Sanity check with rule of 72, 15 percent will double
in 72/15 years or about 5 years.
47 years will double more than 9 times. 2^9 = 512.

Yes, a 15 percent return can make you rich if you have
patience, but 15% is hard to find. I'm lucky to get 8%.