# Average percent growth per year

1. Feb 4, 2015

### gamow99

1. The problem statement, all variables and given/known data

If a stock has increased 400% over the last 6 years, then how much does it grow on average per year.

2. Relevant equations

400 = 100(1+r/6)^6

3. The attempt at a solution

Divide both sides by 100

4 = (1+(r/6)^6

Take the sixth root of each side.

1.26 = 1 + (r/6)

Subtract each side by 1

.26 = r/6

Multiply each side by 6

1.5 = r

which can't be right because I doubt the stock has grown 50% per year over the last 6 years.

2. Feb 4, 2015

### Simon Bridge

Did you check the answer ... say by working the forward problem?
i.e. if the stock grows by 50% per year, then how long for it to double?
This stock doubled twice in 6 years.

6 iterations is not all that many - you could just brute-force the numbers.

Last edited: Feb 4, 2015
3. Feb 4, 2015

### gamow99

Well, I'd like to figure out why my equation is outputting the wrong answer.

4. Feb 4, 2015

### gamow99

I think I see what the problem is. It should be

400 = 100(1+r)^6

which after crunching the numbers is 26% per year which is still pretty high but after all it is FDN which is an ETF of the top 50 internet companies and it went from 14.87 on 2/1/2009 to 60.39 now

5. Feb 4, 2015

### gamow99

I cross-checked it with an interest calculator website and it appears that 26% is correct.

6. Feb 4, 2015

### mdrice12

You've actually calculated the correct answer in this process, but you didn't realize it.

When something grows by x percent each year, it is calculated as follows:

Final Value = (Initial Value) * (1 + x)# years

7. Feb 4, 2015

### Staff: Mentor

I'm not sure that the equation above is at all relevant.
Unrounded, it is 1.259921, to six decimal places. It's not a good idea to round intermediate results. If I were working the problem, I would use all of the precision that the calculator gives.
.259921 = r/6
I get r = 1.559526, rounded to six decimal places.
The average annual growth rate would be $\frac{\text{change in value}}{\text{length of time}}$. Since the stock has increased by 400%, its average annual growth rate over 6 years would be 400/6 ≈ 67%.

8. Feb 4, 2015

### Simon Bridge

Depends what is meant by "annual growth" doesn't it?

In this case the question is probably asking, "if the rate of growth were a constant over the 6 years, what would it have to be to quadruple the initial value?" The OPs attempts to answer the question would seem to bear out this reading of the problem statement.

This is commonly how people talk in financial courses around here.

... following the suggestions in post #2 would allow you to figure out just that, as well as show me where you needed help if you still got stuck. But you seem to have got there by means untold anyway. Perhaps you looked up the equation again?

You should always check your answer by putting it into an easier calculation and seeing if it makes sense.
i.e. 26%pa gives a doubling time of (70/26=) just under 3 years ... so time to quadruple would be 6 years, which is consistent with the problem statement.
So the calculation pans out.

Thinking in terms of how long to double is an effective way to get a feel for the numbers you are using.

9. Feb 5, 2015

### haruspex

If a stock bought at x increased in value by 400%, what is it worth now?

10. Feb 5, 2015

### Staff: Mentor

Simon, you and the OP are treating this problem as if were a compound interest problem, in which the interest accrued in one period becomes part of the principal for the next period. That's different from what's happening here. I could be wrong, but my interpretation of this problem is much more straightforward, with linear growth.

11. Feb 5, 2015

### Simon Bridge

I agree.

Strictly OP, and many others, are abusing the language when they talk like that.
People posting here do not always use strict definitions when they write, but it is sometimes possible to figure out what they are trying to say.

note: OPs figures - 14.97 to 60.39 ... call it 15 to 60, in 6 years is a percentage increase over that time of (60-15)(100)/15 = 300%, not 400%.
This is a clue that OP may not be adhering to the strict meanings of the words.
Technically, it was confirmation ... the way OP was using the figures in post #1 was already suggestive of this.
(Had he followed my suggestions in post #2, I'd have had the confirmation earlier.)

The calculation would more usually go like:
http://www.wikihow.com/Calculate-an-Annual-Percentage-Growth-Rate
... see "method 2".

100(f/s)^(1/y) = (100)(60/15)^(1/6) = 26%
... and this is typical.

However, strictly this should be "compound annual growth rate".
http://www.investopedia.com/terms/a/aagr.asp
http://www.investopedia.com/terms/c/cagr.asp

But you could be right... @gamow99, you want to clear this up for us?

Last edited: Feb 5, 2015
12. Feb 5, 2015

### haruspex

Whether it's simple interest or compound, to get the right answer you need to use the right growth factor. See my post #9.

13. Feb 5, 2015

### Ray Vickson

I think the question is ambiguous, so I will give four possible interpretations, each giving a different answer. (Some of the following may be what haruspex was referring to in Post #9; I am not absolutely sure). To fix the scenario, assume that at time t = 0 (the start of year 1) our stock is worth $100. What is it worth after 6 years (time t = 6, the end of year 6)? You wrote "... has increased 400% ...". This is not well-worded, but MY interpretation of it would be "increased by 400%", so that the increase was$400; added to the initial $100 that makes the worth equal to$500 at time t = 6. Then, in this interpretation we can either have arithmetic or geometric growth, from 100 to 500. Another interpretation would be " increase to 400% ..." so the worth at t = 6 would be \$400. Now, growth would be from 100 to 400 over 6 years, so the arithmetic and geometric growth models would, again, give two different answers.

14. Feb 5, 2015

### haruspex

Yes, that was my point, but I don't see it as ambiguous.